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## **Meet the editor**

Dr Hüseyin Canbolat was born in Adana in 1966. He received his BS and MS degrees from the Middle East Technical University, Ankara, Turkey, in 1989 and 1993 respectively, and a PhD degree in Electrical Engineering from Clemson University, Clemson, SC, USA, in 1997. As an Associate Professor he joined the Department of Electrical Engineering at Mersin University in 1998 and

at the beginning of 2012 he joined the Department of Electronics and Communication Engineering at Yildirim Beyazit University, Ankara, Turkey. His research interests include control systems with applications to robotic and mechatronic systems, microelectromechanical systems (MEMS) and renewable energy systems. He is a member of IEEE and the Chamber of Electrical Engineers of Turkey.

Contents

**Preface IX** 

Chapter 1 **Electrostatic Interactions in** 

Andrey G. Cherstvy

Chapter 3 **Air-Solids Flow Measurement** 

**Part 4 Mathematical Modelling 81** 

Chapter 5 **Nanowires: Promising Candidates for Electrostatic Control** 

Julien Dura, Sébastien Martinie,

Jianyong Zhang

Chapter 4 **Mathematical Models for** 

Toshko Boev

**Part 5 Nanoelectronics 111** 

**Part 2 Bioengineering 41** 

Chapter 2 **Electrostatics in Protein** 

**Dense DNA Phases and Protein-DNA Complexes 3** 

**Part 1 Electrostatics in Biological Sciences 1** 

**Engineering and Design 43**  I. John Khan, James A. Stapleton, Douglas Pike and Vikas Nanda

**Part 3 Measurement and Instrumentation 59** 

**Using Electrostatic Techniques 61** 

**Electrostatics of Heterogeneous Media 83** 

**in Future Nanoelectronic Devices 113** 

Yann-Michel Niquet and Autran Jean-Luc

Daniela Munteanu, François Triozon, Sylvain Barraud,

### Contents

#### **Preface XI**

	- **Part 2 Bioengineering 41**
	- **Part 3 Measurement and Instrumentation 59**
	- **Part 4 Mathematical Modelling 81**

#### **Part 5 Nanoelectronics 111**

Chapter 5 **Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 113**  Julien Dura, Sébastien Martinie, Daniela Munteanu, François Triozon, Sylvain Barraud, Yann-Michel Niquet and Autran Jean-Luc

#### X Contents

#### **Part 6 Electrostatic Actuation 137**

Chapter 6 **New Approach to Pull-In Limit and Position Control of Electrostatic Cantilever Within the Pull-In Limit 139**  Ali Yildiz, Cevher Ak and Hüseyin Canbolat

### Preface

Electrostatics is the branch of electricity dealing with the phenomena created by stationary charges. Electric charges exert force on each other according to the Coulomb's law.

Electrostatic phenomena include many examples. Some simple examples are everyday experiences of attracting the plastic wrap to a hand after it is removed from a package and small electroshocks during the first contact with a metal object, especially after working with an electrical appliance. However, there are more serious phenomena caused by electrostatic charges, such as the apparently spontaneous explosion of grain silos or damage to electronic components during manufacturing. It is also used in some modern day technology, such as operation of photocopiers.

Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces. Although charge exchange happens whenever any two surfaces contact and separate, the effects of charge exchange are usually only noticed when at least one of the surfaces has a high resistance to electrical flow. This is because the charges that transfer to or from the highly resistive surface are more or less trapped there for a long enough time for their effects to be observed. These charges then remain on the object until they either bleed off to the ground or are quickly neutralized by a discharge.

In this book, the authors provide state-of-the-art research studies on electrostatic principles or include the electrostatic phenomena as an important factor. The chapters cover diverse subjects, such as biotechnology, bioengineering, actuation of MEMS, measurement and nanoelectronics. Hopefully, the interested readers will benefit from the book in their studies. It is probable that the presented studies will lead the researchers to develop new ideas to conduct their research.

> **Dr Hüseyin Canbolat**  Yildirim Beyazit University, Ankara, Turkey

**Part 1** 

**Electrostatics in Biological Sciences** 

**Part 1** 

**Electrostatics in Biological Sciences** 

A. G. Cherstvy

 *Germany* 

*Charges.* Many constituents of living cells bear large charges on their surfaces. The list includes DNA/RNA nucleic acids [1], cellular lipid membranes [2], DNAbinding [3,4] and architectural [5,6] proteins, natural ion channels [7] and pores, elements of cytoskeleton networks [8], and molecular motors. ES interactions on the nanoscale often dominate the physical forces acting between these components in the last 13 nm prior to surfacesurface contact, often governing their spontaneous assembly and longrange spatial ordering. There has been a number of excellent reviews covering the general principles of ES effects in nucleic acids [9,10], proteins [11,12,13], lipid membranes [2,14,15], and some other biosoft matter systems [16]. Salt and pHsensitivity of ES forces provides cells with a useful handle to direct/tune the pathways of many biological processes. Among them are DNADNA, proteinDNA [17] and proteinprotein ES interactions [18], DNA compactification into higherorder structures [19,20], DNA spooling inside viral shells [21], actin aggregation, RNA folding [10,22], and ion translocation through membrane pores. ES forces modulate structure and control functioning of subcellular supramolecular assemblies [23,24] and can

Over the last years, the ES mechanisms of some DNArelated phenomena mentioned above have been developed in our group. General concepts of the PB theory often give a satisfactory physical description of ES properties of molecules in solution and macromolecular complexes. Below, we try to keep the presentation on illustrative level avoiding complicated algebra: all analycal expressions, the details of their derivation, and regimes of applicability can be found in the original papers cited. We rather focus on underlying physical mechanisms, comparing the system behavior under varying conditions. We often treat ES forces in dense, weakly fluctuating structures/complexes, where entropic effects are weak and can be neglected. Because of limited space, we focus on latest ES motivated developments from other groups, trying to position our research in this context.

1Abbreviations: ES, electrostatic; HB, hydrogen bond; PE, polyelectrolyte; PB, PoissonBoltzmann; PEG, polyethylene glycol; DH, DebyeHückel; EM, electron microscopy; AFM, atomic force microscopy; bp, base pair; kbp, kilo base pair; [DNA], DNA concentration; [salt], salt concentration; ds, double stranded;

ss, single stranded; CL, cationic lipids; LC, liquidcrystalline; GNP, gold nanoparticle; hom,

homologous; NCP, nucleosome core particle; PDB, Protein Data Bank.

affect cellcell interactions in tissues [25].

*Institute of Complex Systems, ICS2, Forschungszentrum Jülich, Jülich, Institute for Physics and Astronomy, University of Potsdam, PotsdamGolm,* 

A. G. Cherstvy

*Institute of Complex Systems, ICS2, Forschungszentrum Jülich, Jülich, Institute for Physics and Astronomy, University of Potsdam, PotsdamGolm, Germany* 

#### 

*Charges.* Many constituents of living cells bear large charges on their surfaces. The list includes DNA/RNA nucleic acids [1], cellular lipid membranes [2], DNAbinding [3,4] and architectural [5,6] proteins, natural ion channels [7] and pores, elements of cytoskeleton networks [8], and molecular motors. ES interactions on the nanoscale often dominate the physical forces acting between these components in the last 13 nm prior to surfacesurface contact, often governing their spontaneous assembly and longrange spatial ordering. There has been a number of excellent reviews covering the general principles of ES effects in nucleic acids [9,10], proteins [11,12,13], lipid membranes [2,14,15], and some other biosoft matter systems [16]. Salt and pHsensitivity of ES forces provides cells with a useful handle to direct/tune the pathways of many biological processes. Among them are DNADNA, proteinDNA [17] and proteinprotein ES interactions [18], DNA compactification into higherorder structures [19,20], DNA spooling inside viral shells [21], actin aggregation, RNA folding [10,22], and ion translocation through membrane pores. ES forces modulate structure and control functioning of subcellular supramolecular assemblies [23,24] and can affect cellcell interactions in tissues [25].

Over the last years, the ES mechanisms of some DNArelated phenomena mentioned above have been developed in our group. General concepts of the PB theory often give a satisfactory physical description of ES properties of molecules in solution and macromolecular complexes. Below, we try to keep the presentation on illustrative level avoiding complicated algebra: all analycal expressions, the details of their derivation, and regimes of applicability can be found in the original papers cited. We rather focus on underlying physical mechanisms, comparing the system behavior under varying conditions. We often treat ES forces in dense, weakly fluctuating structures/complexes, where entropic effects are weak and can be neglected. Because of limited space, we focus on latest ES motivated developments from other groups, trying to position our research in this context.

<sup>1</sup>Abbreviations: ES, electrostatic; HB, hydrogen bond; PE, polyelectrolyte; PB, PoissonBoltzmann; PEG, polyethylene glycol; DH, DebyeHückel; EM, electron microscopy; AFM, atomic force microscopy; bp, base pair; kbp, kilo base pair; [DNA], DNA concentration; [salt], salt concentration; ds, double stranded; ss, single stranded; CL, cationic lipids; LC, liquidcrystalline; GNP, gold nanoparticle; hom, homologous; NCP, nucleosome core particle; PDB, Protein Data Bank.

M=92% for trivalent (*z*=3) cations. In the DNA model as a thin long linear PE at vanishing

ξ

*<sup>M</sup>* = −1 1/( *z*

where ξ is the ratio of the Bjerrum length ( 7.1 *Bl* ≈ Å in water) to the axial PE intercharge separation, *b* ≈ 1.7 Å for the bare DNA. Recent experiments on DNA translocation through

*Cation binding.* DNA structure offers welldefined sites for counterion binding. Depending on chemical nature and valence, cations bind in DNA grooves, on DNA strands, or both. The distribution and binding equilibrium of adsorbed cations result in a distinct pattern of charges on DNA surface that, in turn, dictates the properties of DNADNA ES forces. It also affects intrinsic DNA helical structure and conformation DNA adopts in solution [31]. ES forces are believed to dominate the interaction of parallel DNAs in the last 20Å prior to surfacesurface contact, because of still relatively large residual DNA charge density after

DNADNA hydration force created by overlapping patterns of structured water molecules on DNA surfaces is another alternative [32]. Close similarities in the magnitude and decay length of repulsive forces in the last 12 nm prior to the contact measured by osmotic stress technique in simplesalt solutions for DNA, some netneutral polymers [33,34] and lipid membranes [14,35] favor the hydration force picture. Extreme sensitivity of DNADNA forces measured to the chemical nature and valence of cations added, not expected to affect strongly the closerange hydration forces, favors however the ES mechanism of DNADNA force generation. In particular, DNADNA attraction in the presence of multivalent cations

The pattern of condensed cations bears some/strong correlations to the helical symmetry of DNA phosphates, forming a "lattice" of alternating positivenegative charges along the DNA axis, Fig. 1. ES forces between these periodic arrays of charges might turn from repulsion to attraction for wellneutralized DNAs. Attractive DNADNA forces have been systematically measured by the osmotic stress technique in dense columnar hexagonal DNA assemblies in the presence of some di and many trivalent cations at ≈1 nm between the surfaces [36,37], Fig. 2, while purely repulsive forces have been detected with monovalent salts [38]. The list of DNA condensing agents includes multivalent cations (cohex3+, spermine4+, spermidine3+), some highly positively charged proteins and polypeptides (poly Lys and polyArg, protamines, H1 histones), as well as concentrated solutions of neutral PEG polymers. The latter are excluded from the DNA phase, exerting an external osmotic pressure onto the DNA lattice. Some ions from this list interact with DNA in natural environments, such as spermidine3+ present in many bacteria in 13 mM concentrations [39], protamines that are abundant in sperm heads, as well as putrescine2+ and spermidine3+ vital

*Duplexduplex ES forces.* A number of theoretical models have been developed in the last two decades to provide a physical rationale for DNADNA attraction, including some recent

M=76% of charge compensation for monovalent (*z*=1) and

) , (1)

θ[29,30],

θ

θ

nmsized solidstate nanopores enabled measuring the compensation fractions

[salt], the neutralization fraction is predicted to be

often in good agreement with the Manning theory.

can be rationalized by our ES models, see below.

for DNA compaction in some Teven bacteriophages [40].

Manning theory [28] predicts

condensation of counterions.

θ

*Outline.* The main aim of this chapter is to provide a review of recent advances in the theory of ES interactions in dense assemblies of DNAs and to discuss some ES aspects of protein DNA recognition and binding. These subjects have been the main area of my scientific activity in the last several years. ES effects on different levels of DNA organization *in vivo* and *in vitro* are considered below. We overview e.g. the biophysical principles behind DNA DNA ES interactions, DNA complexation with CLmembranes, DNA condensates, DNA cholesteric phases and touch on DNA spooling inside viruses. For DNAprotein complexes, the effects include ES recognition and binding. For these systems, we develop theoretical frameworks and computational approaches to describe physicalchemical mechanisms of structure formation that allow us later to anticipate some biological consequences.

First, we focus on theoretical concepts used in derivation of the ES interaction potential of two parallel doublehelical DNAs immersed in electrolyte solution [9]. The linear PB theory developed for this system [26] accounts for a lowdielectric DNA interior and spiral distribution of negative phosphate charges on DNA periphery. We discuss the regimes of applicability of this linear theory, in application to interaction of DNAs partly neutralized by adsorbed counterions. This theory and its modifications have allowed us to rationalize a number of experimental observations regarding the behavior of DNAs in columnar hexagonal phases (Sec. 2), dense cholesteric DNA assemblies (Sec. 5), the decay length of DNADNA ES repulsion in monovalent salts, the region of DNADNA attraction in the presence of multivalent cations (Sec. 2), as well as DNA condensation into toroids (Sec. 3). A separate domain of our research deals with interactioninduced adjustment of DNA helical structure, DNADNA sequence recognition and pairing (Sec. 7), as well as DNADNA friction (Sec. 8). We also overview DNA melting and hybridization in dense DNA lattices, Sec. 9.

In the second part, we focus on ES recognition between DNA and DNAbinding proteins in their complexes. We propose a model of DNA sequence recognition by relatively small proteins (e.g., transcription factors) based on complementarity of charge patterns on DNA target site and bound protein (Sec. 10). For relatively large proteins, we support the theoretical conclusions by a detailed bioinformatic statistical analysis of charge patterns along interfaces of various proteinDNA complexes, as extracted from their PDB entries. We decipher the reasons why large structural proteinDNA complexes of pro and eukaryotic organisms do involve a substantial ES component in their recognition (Sec. 10). On the contrary, DNA recognition by small DNAbinding proteins appears to be ESnonspecific, being likely governed by HB formation.

Every section below starts with a short introduction to the subject, followed by a presentation of basic theoretical concepts and discussion of main results, and it ends with some perspectives for future developments and possible model improvements. The content of this chapter is based on the recent perspective article [27].

#### 

*Counterion condensation.* BDNA is one of the most highly charged biohelices, with one elementary charge *e*0 per ≈nm2 on the surface at standard pH and physiological [salt]. These charges are the phosphate negative groups located on DNA periphery, forming a duplex with 1010.5 bp per helical turn of *H*≈34Å and nonhydrated DNA radius of *a* ≈ 9 Å. More that ∼75% of DNA charge is neutralized by counterions adsorbed onto it from solution. The

*Outline.* The main aim of this chapter is to provide a review of recent advances in the theory of ES interactions in dense assemblies of DNAs and to discuss some ES aspects of protein DNA recognition and binding. These subjects have been the main area of my scientific activity in the last several years. ES effects on different levels of DNA organization *in vivo* and *in vitro* are considered below. We overview e.g. the biophysical principles behind DNA DNA ES interactions, DNA complexation with CLmembranes, DNA condensates, DNA cholesteric phases and touch on DNA spooling inside viruses. For DNAprotein complexes, the effects include ES recognition and binding. For these systems, we develop theoretical frameworks and computational approaches to describe physicalchemical mechanisms of

First, we focus on theoretical concepts used in derivation of the ES interaction potential of two parallel doublehelical DNAs immersed in electrolyte solution [9]. The linear PB theory developed for this system [26] accounts for a lowdielectric DNA interior and spiral distribution of negative phosphate charges on DNA periphery. We discuss the regimes of applicability of this linear theory, in application to interaction of DNAs partly neutralized by adsorbed counterions. This theory and its modifications have allowed us to rationalize a number of experimental observations regarding the behavior of DNAs in columnar hexagonal phases (Sec. 2), dense cholesteric DNA assemblies (Sec. 5), the decay length of DNADNA ES repulsion in monovalent salts, the region of DNADNA attraction in the presence of multivalent cations (Sec. 2), as well as DNA condensation into toroids (Sec. 3). A separate domain of our research deals with interactioninduced adjustment of DNA helical structure, DNADNA sequence recognition and pairing (Sec. 7), as well as DNADNA friction (Sec. 8).

In the second part, we focus on ES recognition between DNA and DNAbinding proteins in their complexes. We propose a model of DNA sequence recognition by relatively small proteins (e.g., transcription factors) based on complementarity of charge patterns on DNA target site and bound protein (Sec. 10). For relatively large proteins, we support the theoretical conclusions by a detailed bioinformatic statistical analysis of charge patterns along interfaces of various proteinDNA complexes, as extracted from their PDB entries. We decipher the reasons why large structural proteinDNA complexes of pro and eukaryotic organisms do involve a substantial ES component in their recognition (Sec. 10). On the contrary, DNA recognition by small DNAbinding proteins appears to be ESnonspecific, being likely

Every section below starts with a short introduction to the subject, followed by a presentation of basic theoretical concepts and discussion of main results, and it ends with some perspectives for future developments and possible model improvements. The content

*Counterion condensation.* BDNA is one of the most highly charged biohelices, with one elementary charge *e*0 per ≈nm2 on the surface at standard pH and physiological [salt]. These charges are the phosphate negative groups located on DNA periphery, forming a duplex with 1010.5 bp per helical turn of *H*≈34Å and nonhydrated DNA radius of *a* ≈ 9 Å. More that ∼75% of DNA charge is neutralized by counterions adsorbed onto it from solution. The

structure formation that allow us later to anticipate some biological consequences.

We also overview DNA melting and hybridization in dense DNA lattices, Sec. 9.

of this chapter is based on the recent perspective article [27].

governed by HB formation.

Manning theory [28] predicts θM=76% of charge compensation for monovalent (*z*=1) and θM=92% for trivalent (*z*=3) cations. In the DNA model as a thin long linear PE at vanishing [salt], the neutralization fraction is predicted to be

$$\theta\_{\mathsf{M}} = 1 - 1 \;/\left(z\xi\right),\tag{1}$$

where ξ is the ratio of the Bjerrum length ( 7.1 *Bl* ≈ Å in water) to the axial PE intercharge separation, *b* ≈ 1.7 Å for the bare DNA. Recent experiments on DNA translocation through nmsized solidstate nanopores enabled measuring the compensation fractions θ [29,30], often in good agreement with the Manning theory.

*Cation binding.* DNA structure offers welldefined sites for counterion binding. Depending on chemical nature and valence, cations bind in DNA grooves, on DNA strands, or both. The distribution and binding equilibrium of adsorbed cations result in a distinct pattern of charges on DNA surface that, in turn, dictates the properties of DNADNA ES forces. It also affects intrinsic DNA helical structure and conformation DNA adopts in solution [31]. ES forces are believed to dominate the interaction of parallel DNAs in the last 20Å prior to surfacesurface contact, because of still relatively large residual DNA charge density after condensation of counterions.

DNADNA hydration force created by overlapping patterns of structured water molecules on DNA surfaces is another alternative [32]. Close similarities in the magnitude and decay length of repulsive forces in the last 12 nm prior to the contact measured by osmotic stress technique in simplesalt solutions for DNA, some netneutral polymers [33,34] and lipid membranes [14,35] favor the hydration force picture. Extreme sensitivity of DNADNA forces measured to the chemical nature and valence of cations added, not expected to affect strongly the closerange hydration forces, favors however the ES mechanism of DNADNA force generation. In particular, DNADNA attraction in the presence of multivalent cations can be rationalized by our ES models, see below.

The pattern of condensed cations bears some/strong correlations to the helical symmetry of DNA phosphates, forming a "lattice" of alternating positivenegative charges along the DNA axis, Fig. 1. ES forces between these periodic arrays of charges might turn from repulsion to attraction for wellneutralized DNAs. Attractive DNADNA forces have been systematically measured by the osmotic stress technique in dense columnar hexagonal DNA assemblies in the presence of some di and many trivalent cations at ≈1 nm between the surfaces [36,37], Fig. 2, while purely repulsive forces have been detected with monovalent salts [38]. The list of DNA condensing agents includes multivalent cations (cohex3+, spermine4+, spermidine3+), some highly positively charged proteins and polypeptides (poly Lys and polyArg, protamines, H1 histones), as well as concentrated solutions of neutral PEG polymers. The latter are excluded from the DNA phase, exerting an external osmotic pressure onto the DNA lattice. Some ions from this list interact with DNA in natural environments, such as spermidine3+ present in many bacteria in 13 mM concentrations [39], protamines that are abundant in sperm heads, as well as putrescine2+ and spermidine3+ vital for DNA compaction in some Teven bacteriophages [40].

*Duplexduplex ES forces.* A number of theoretical models have been developed in the last two decades to provide a physical rationale for DNADNA attraction, including some recent

*Attraction vs. repulsion.* The theory predicts ES attraction of wellneutralized DNAs with the majority of cations adsorbed in the major groove, as pioneered in Ref. [50]. This arrangement of charges facilitates a periodic positivenegative charge alternation along the DNA axis. In physical terms, a DNADNA attraction emerges from a *zipperlike ES matching* of phosphate groups of one DNA with the cations adsorbed in a regular fashion in the grooves of another DNA. Many large or extended multivalent DNAcondensing cations are indeed known to bind preferentially into the major DNA groove, preferred both from interaction and steric point of view. Correlated ES potential alternations thus generate charge interlocking along the DNADNA contact and give rise to DNADNA ES attraction, see below. The mathematical apparatus used for deriving DNADNA forces, ES and chemical features of counterion binding, as well as applicability regimes of this meanfiled

continuum DHBjerrum PB theory are discussed in details in excellent review [9].

the detailed statistical theory of dense DNA assemblies has been worked out [60].

recent perspective [67].

as follows [50]

θ

the first helical interaction harmonics *<sup>n</sup> a* [9]

π σ

> ε

( ) ( )( )

*m*

θ

κ

Further developments of this theory enabled us to incorporate fine realistic details of DNA structure, such as a discrete nature of adsorbed cations [51] and sequencespecific pattern of the twist angles [52] between the adjacent DNA bps [53]. The models for description of interaction and *T*mediated rearrangements of condensed cations on DNA surfaces [54], torsional flexibility of DNA backbone [55], some solitonlike DNA twist "defects" [56], and DNA helical "straightening" in dense phases [57] have also been developed. ES forces between nonparallel infinitely long [58] and finitelength [59] DNAs were computed and

*Basic equations.* A number of outcomes of this theory are in quantitative agreement with a number of exprimental observations available for DNA assemblies. These include the decay length of DNADNA repulsion in simple salt solutions and attraction at *R*=2832Å in the presence of multivalent cations, Fig. 2. Also, DNA azimuthal frustrations [55,61], DNA straightening [62], and a reduced positional order observed in dense DNA lattices [63] have been rationalized. Recent developments unraveled the effects of DNA thermal undulations [64,65] and have shown that duplexduplex ES forces might get amplified in DNA columnar phases at finite *T*, as compared to *T*=0 case. Recently, the implications of binding equilibrium of finitesize ions on DNADNA ES forces have been clarified [66]. A number of biological consequences of computed ES duplexduplex forces were analyzed in excellent

DNADNA ES interaction energy in electrolyte solution can be approximated as the sum of

These positive coefficients decay nearly exponentially with DNADNA separation *R*, Fig. 3, and their values depend on partitioning of cations on DNA and DNA charge compensation

δφ+ <sup>2</sup> ( )

δφ. (2)

( ) ( )( )

<sup>∑</sup> <sup>ɶ</sup> ,

*D nm n m n n m nn n m n*

κκ

 κ

*m*

( )

0

θ

<sup>∞</sup> <sup>−</sup>

*aK a aK a K a*

'

*aK a*

 κ

*mm m*

' '

() ()

<sup>ɶ</sup> . (3)

κκ

 κ

*ERL L a R a R a R* ( , ) ≈ − <sup>0</sup> ( ) <sup>1</sup> ( ) cos cos2

<sup>2</sup> <sup>2</sup> 2 22 <sup>2</sup>

*<sup>a</sup> K R f n f K RI a a R*

8 1 , , '

( ) ( ) ( )

1,2 2 16 , ,

*<sup>a</sup> fm f K R a R*

<sup>=</sup> <sup>=</sup>

<sup>2</sup> 2 22

κ

=−∞ <sup>−</sup> <sup>=</sup> <sup>−</sup>

θ

( )

 κ

0 0 2 2 , <sup>1</sup>

 κ

π σ

> ε

advances [41,42]. In one group of models, the spatiotemporal correlations of cations stem from the inherent DNA structure, which render DNADNA attraction possible via a "zipper effect". In other models, beyond the PB limit, the *correlated* fluctuations in the density profiles of condensed cations give rise to attraction [43,44,45], even for DNAs modeled as a uniformly charged PE rods. The period of oscillatory charge density waves on PE surfaces in these models is largely decoupled from intrinsic DNA charge periodicity. To save space, we address the reader to a comprehensive review [9] focused primarily on ES DNADNA forces. It provides a broad coverage, physical comparison, and analysis of applicability regimes for various models of PE likecharge attraction. In this chapter, we target primarily *new developments* in the theory of DNADNA and DNAprotein ES interactions. DNADNA attraction has also been extensively investigated by computer simulations [46,47,48,49], for diverse models for DNA structure, the shape and binding specificity of counterions, as well as for various solvent models implemented.

Fig. 1. Schematics of cationdecorated DNA duplexes (a) and interacting hom vs. nonhom sequences (b, c). Positivenegative charge zipper motif that ensures DNADNA ES attraction is shown in part (b). The image is reprinted from Ref. [53], subject to APS2001 Copyright.

The helicity of DNA charges renders the ES potential close to the double helix helically symmetric. When two DNAs approach one another in electrolyte, these helical potential profiles overlap. This affects nontrivially DNADNA ES forces, on top of ES repulsion of uniformly charged rods. The exact theory of ES forces between two long parallel double helical macromolecules was developed in 1997 by A. Kornyshev and S. Leikin [26]. This elegant linear PB theory explicitly accounts for the DNA charge helicity and its low dielectric hydrophobic core (permittivity of 2 *<sup>c</sup>* ε≈ ).

The model implies two distinct populations of cations around the DNA. The first one is the Manning's fraction of cations is strongly/irreversibly adsorbed in DNA grooves/strands, while the remaining DNA charge is shielded by electrolyte ions in DH linear manner. DNA ES potential renormalized in this fashion often does not exceed 25 mV, rendering the linear PB model applicable to description of interacting clouds of mobile ions around two partly neutralized DNAs. Both DNA phosphates and condensed cations in the middle of DNA grooves are modeled below as thin continuous helical lines of charges. Thermal smearing of charge pattern can be incorporated via the DebyeWaller factor [9] that reduces the magnitude of the helical harmonics 1,2 *a* , see below.

advances [41,42]. In one group of models, the spatiotemporal correlations of cations stem from the inherent DNA structure, which render DNADNA attraction possible via a "zipper effect". In other models, beyond the PB limit, the *correlated* fluctuations in the density profiles of condensed cations give rise to attraction [43,44,45], even for DNAs modeled as a uniformly charged PE rods. The period of oscillatory charge density waves on PE surfaces in these models is largely decoupled from intrinsic DNA charge periodicity. To save space, we address the reader to a comprehensive review [9] focused primarily on ES DNADNA forces. It provides a broad coverage, physical comparison, and analysis of applicability regimes for various models of PE likecharge attraction. In this chapter, we target primarily *new developments* in the theory of DNADNA and DNAprotein ES interactions. DNADNA attraction has also been extensively investigated by computer simulations [46,47,48,49], for diverse models for DNA structure, the shape and binding specificity of counterions, as well

Fig. 1. Schematics of cationdecorated DNA duplexes (a) and interacting hom vs. nonhom sequences (b, c). Positivenegative charge zipper motif that ensures DNADNA ES attraction is shown in part (b). The image is reprinted from Ref. [53], subject to APS2001 Copyright.

The helicity of DNA charges renders the ES potential close to the double helix helically symmetric. When two DNAs approach one another in electrolyte, these helical potential profiles overlap. This affects nontrivially DNADNA ES forces, on top of ES repulsion of uniformly charged rods. The exact theory of ES forces between two long parallel double helical macromolecules was developed in 1997 by A. Kornyshev and S. Leikin [26]. This elegant linear PB theory explicitly accounts for the DNA charge helicity and its low

> ε≈ ).

The model implies two distinct populations of cations around the DNA. The first one is the Manning's fraction of cations is strongly/irreversibly adsorbed in DNA grooves/strands, while the remaining DNA charge is shielded by electrolyte ions in DH linear manner. DNA ES potential renormalized in this fashion often does not exceed 25 mV, rendering the linear PB model applicable to description of interacting clouds of mobile ions around two partly neutralized DNAs. Both DNA phosphates and condensed cations in the middle of DNA grooves are modeled below as thin continuous helical lines of charges. Thermal smearing of charge pattern can be incorporated via the DebyeWaller factor [9] that reduces the

as for various solvent models implemented.

dielectric hydrophobic core (permittivity of 2 *<sup>c</sup>*

magnitude of the helical harmonics 1,2 *a* , see below.

*Attraction vs. repulsion.* The theory predicts ES attraction of wellneutralized DNAs with the majority of cations adsorbed in the major groove, as pioneered in Ref. [50]. This arrangement of charges facilitates a periodic positivenegative charge alternation along the DNA axis. In physical terms, a DNADNA attraction emerges from a *zipperlike ES matching* of phosphate groups of one DNA with the cations adsorbed in a regular fashion in the grooves of another DNA. Many large or extended multivalent DNAcondensing cations are indeed known to bind preferentially into the major DNA groove, preferred both from interaction and steric point of view. Correlated ES potential alternations thus generate charge interlocking along the DNADNA contact and give rise to DNADNA ES attraction, see below. The mathematical apparatus used for deriving DNADNA forces, ES and chemical features of counterion binding, as well as applicability regimes of this meanfiled continuum DHBjerrum PB theory are discussed in details in excellent review [9].

Further developments of this theory enabled us to incorporate fine realistic details of DNA structure, such as a discrete nature of adsorbed cations [51] and sequencespecific pattern of the twist angles [52] between the adjacent DNA bps [53]. The models for description of interaction and *T*mediated rearrangements of condensed cations on DNA surfaces [54], torsional flexibility of DNA backbone [55], some solitonlike DNA twist "defects" [56], and DNA helical "straightening" in dense phases [57] have also been developed. ES forces between nonparallel infinitely long [58] and finitelength [59] DNAs were computed and the detailed statistical theory of dense DNA assemblies has been worked out [60].

*Basic equations.* A number of outcomes of this theory are in quantitative agreement with a number of exprimental observations available for DNA assemblies. These include the decay length of DNADNA repulsion in simple salt solutions and attraction at *R*=2832Å in the presence of multivalent cations, Fig. 2. Also, DNA azimuthal frustrations [55,61], DNA straightening [62], and a reduced positional order observed in dense DNA lattices [63] have been rationalized. Recent developments unraveled the effects of DNA thermal undulations [64,65] and have shown that duplexduplex ES forces might get amplified in DNA columnar phases at finite *T*, as compared to *T*=0 case. Recently, the implications of binding equilibrium of finitesize ions on DNADNA ES forces have been clarified [66]. A number of biological consequences of computed ES duplexduplex forces were analyzed in excellent recent perspective [67].

DNADNA ES interaction energy in electrolyte solution can be approximated as the sum of the first helical interaction harmonics *<sup>n</sup> a* [9]

$$E\left(R, L\right) \approx L\left[a\_0\left(R\right) - a\_1\left(R\right)\cos\delta\phi + a\_2\left(R\right)\cos 2\delta\phi\right].\tag{2}$$

These positive coefficients decay nearly exponentially with DNADNA separation *R*, Fig. 3, and their values depend on partitioning of cations on DNA and DNA charge compensation θas follows [50]

$$a\_0(R) = \frac{8\pi^2 \overline{\sigma}^2 a^2}{\varepsilon} \left| \frac{\left(1 - \theta\right)^2 \mathcal{K}\_0(\kappa\_D R)}{\left[\kappa a \mathcal{K}\_1(\kappa a)\right]^2} - \sum\_{n,m=-\alpha}^{\alpha} \frac{\overline{f}\left(n, \theta, f\right)^2 \mathcal{K}\_{n-m}\left(\kappa\_n R\right) I\_m\left(\kappa\_n a\right)}{\left[\kappa\_n a \mathcal{K}\_n\left(\kappa\_n a\right)\right]^2 \mathcal{K}\_n\left(\kappa\_n a\right)} \right|,$$

$$a\_{m+1,2}\left(R\right) = \frac{16\pi^2 \overline{\sigma}^2 a^2}{\varepsilon} \frac{\overline{f}\left(m, \theta, f\right)^2 \mathcal{K}\_0\left(\kappa\_m R\right)}{\left[\kappa\_m a \mathcal{K}\_m\left(\kappa\_m a\right)\right]^2}.\tag{3}$$

*Future challenges.* Below, we overview some challenges for the current theory. One of them is water structuring in the hydration shells around the DNA. Namely, the most interesting features of intermolecular forces, including the attraction region, emerge at DNA densities when the shells of "structured waters" on interacting helices can overlap. Also, a distance dependent "effective" dielectric constant on the length scale of 12 water diameters [11], a modified decay of electric fields close to DNA, a finite diameter and precise geometrical form of DNAcondensing cations (e.g., linear flexible polyamines vs. compact cohex3+ ions), as well as a limited applicability of the linear PB model, all these points require more accurate theories to be developed close to DNA surface. The solvation of DNA also requires a microscopic treatment of dielectric environments and polarization states upon counterion binding to the DNA. Not only in the theory, these factors also complicate quantitative predictions of DNADNA forces by means of computer simulations. Similar complications in description of ES forces on the nanoscale emerge in modeling of DNAprotein, DNA

Fig. 2. Theoretically predicted (a) and experimentally measured (b) DNADNA forces in dense DNA assemblies at 50 mM MnCl2. The region of DNADNA attraction at *R*=2832Å detected in experiments corresponds to a spontaneous shrinkage/collapse of the DNA lattice. No azimuthal frustrations on DNA lattice were considered in the model, i.e. cos 0

DNA pairs. Note that, contrary to majority of divalent cations, Mn2+ and Cd2+ are capable of generating DNADNA attraction under the osmotic stress of PEG [36]. This technique allows to overcome the longrange DH repulsive branch of the potential and thus enhance the helix

= 0.85 , 0 *n* = 50 mM. The figure is reprinted from Ref. [54], subject to ACS2002 Copyright.

*Structure of toroids.* One biological manifestation of cationmediated DNADNA attraction is DNA condensation into compact toroidal structures observed in bacteria, viruses, and sperm cells *in vivo* and studied thoroughly *in vitro* [71]. For instance, some bacteria pack their DNAs into robust toroids to protect the genetic material and minimize the frequency of dsDNA breaks [72]. These radiationresistant bacteria retain strongly elevated [Mn2+] in their cells to regulate packaging of chromatin fibers, via likely attractive DNADNA forces [73]. In mammalian sperm cells, very long DNA is condensed with the help of highly basic Argrich

mediated DNADNA forces shielded with a shorter screening length, 1 1 /

θ

δφ

κ

≡ for all

. Parameters:

membrane, as well as proteinprotein complexes (discussed in Sec. 10).

Here, the first term in 0 *a* describes the ES repulsion between uniformly charged "DNA rods", that dominates at large *R*. The second term in 0 *a* is the imagecharge repulsion between the charges on one DNA from image charges (of the same sign) created in a low dielectric core of another DNA. The duplexspecific DNADNA forces are described by 1,2 *a* > 0 amplitudes. For ideally helical DNAs, the interaction energy scales linearly with the DNA length *L*, while for randomlysequenced nonideal DNA fragments a more intricate dependence arises, see Sec. 7. With the cations adsorbed prevalently in the major groove and at large θ values, the 1 *a* term responsible for ES helixhelix attraction grows. Many DNA condensing multivalent cations are indeed known to adsorb into the major DNA groove.

In these expressions, parameter *f* controls the partitioning of cations on DNA (at *f*=0 all cations occupy the major groove), ( ) , , ( ) 1 1 cos ( ) ( ) *<sup>n</sup> <sup>s</sup> fn f f* θ θ = +− − − *f n* θ φ ɶ ɶ , 0.4 φ π *<sup>s</sup>* ≈ ɶ is the azimuthal halfwidth of DNA minor groove, σ is the surface charge density of DNA phosphates, and *K*n(*x*), *I*n(*x*), *K*n'(*x*), *I*n'(*x*) are the modified Bessel functions of order *n* and their derivatives.

We note that the decay lengths of *<sup>n</sup>* 1,2 *<sup>a</sup>* <sup>=</sup> harmonics , ( )<sup>2</sup> 2 2 1/ 1/ 2 / κκπ *<sup>n</sup>* = + *n H* , is a non trivial function. Not only it contains the DH screening length in 1:1 solution with [salt]= *n*<sup>0</sup> , namely 0 1/ 1/ 8 *<sup>D</sup> <sup>B</sup>* λκ π = = *l n* , but also depends on the DNA helical repeat *H*. We remind here that at physiological conditions 7 10 λ*<sup>D</sup>* ≈ − Å, that is *n*<sup>0</sup> ∼0.150.1 M of simple salt.

The imageforce repulsion is screened with about half as short decay length, compared to the direct chargecharge repulsion. Effectively, the electric field travels a double distance to image charge. This gives rise to a shortrange branch of DNADNA ES repulsion at *R*<24Å or so (for typical parameters), see Fig. 2. For intermediate *R*=2832Å, the ES helixhelix attraction overwhelms the shortrange imageforce repulsion and the direct DH charge repulsion [68]. This renders the net DNADNA ES force *attractive* in this range. Typically, at about *R*>35Å the direct DH rodrod repulsion prevails and DNAs again repel each other.

Note however that the predicted in Fig. 2 shortrange repulsion domain is shifted by 35Å to smaller DNADNA distances, as compared to the measured DNA pressuredistance curves. A possible explanation is that the first, tight hydration shell of DNA, not included in the theory, might effectively increase DNA diameter in experiments and thus prevent direct DNA contacts at *R*≈20Å, shifting the energy curves measured towards larger *R* values.

Another effect is azimuthal frustrations of DNA molecules observed in dense hexagonal DNA lattices [61]. In the theory, they emerge from XYspinlike cos cos2 δφ δφ − dependence of the interaction potential on the mutual DNA rotation angle, δφ . Optimization of the interaction energy over all 6 neighboring DNAs on the lattice inevitably "frustrates" the azimuthal order [69]. Frustrated Pottslike states, reminiscent of those for magnetic spin systems, are often preferred for DNA hexagonal lattice in the model [61]. Namely, in the elementary triangle on a lattice, the two differences of the azimuthal DNA angles are

$$
\Delta\_1 = \pm \arccos \left[ 1 \;/ \; 4 + \sqrt{1 + 2a\_1 \;/\; a\_2} \;/ \; 4 \right], \tag{4}
$$

while the third one is 2 times larger [70].

Here, the first term in 0 *a* describes the ES repulsion between uniformly charged "DNA rods", that dominates at large *R*. The second term in 0 *a* is the imagecharge repulsion between the charges on one DNA from image charges (of the same sign) created in a low dielectric core of another DNA. The duplexspecific DNADNA forces are described by 1,2 *a* > 0 amplitudes. For ideally helical DNAs, the interaction energy scales linearly with the DNA length *L*, while for randomlysequenced nonideal DNA fragments a more intricate dependence arises, see Sec. 7. With the cations adsorbed prevalently in the major groove and

condensing multivalent cations are indeed known to adsorb into the major DNA groove.

cations occupy the major groove), ( ) , , ( ) 1 1 cos ( ) ( ) *<sup>n</sup>*

azimuthal halfwidth of DNA minor groove,

 π

here that at physiological conditions 7 10

θ

We note that the decay lengths of *<sup>n</sup>* 1,2 *<sup>a</sup>* <sup>=</sup> harmonics , ( )<sup>2</sup> 2 2 1/ 1/ 2 /

λ

In these expressions, parameter *f* controls the partitioning of cations on DNA (at *f*=0 all

phosphates, and *K*n(*x*), *I*n(*x*), *K*n'(*x*), *I*n'(*x*) are the modified Bessel functions of order *n* and

trivial function. Not only it contains the DH screening length in 1:1 solution with [salt]= *n*<sup>0</sup> ,

The imageforce repulsion is screened with about half as short decay length, compared to the direct chargecharge repulsion. Effectively, the electric field travels a double distance to image charge. This gives rise to a shortrange branch of DNADNA ES repulsion at *R*<24Å or so (for typical parameters), see Fig. 2. For intermediate *R*=2832Å, the ES helixhelix attraction overwhelms the shortrange imageforce repulsion and the direct DH charge repulsion [68]. This renders the net DNADNA ES force *attractive* in this range. Typically, at about *R*>35Å the direct DH rodrod repulsion prevails and DNAs again repel each other. Note however that the predicted in Fig. 2 shortrange repulsion domain is shifted by 35Å to smaller DNADNA distances, as compared to the measured DNA pressuredistance curves. A possible explanation is that the first, tight hydration shell of DNA, not included in the theory, might effectively increase DNA diameter in experiments and thus prevent direct DNA contacts at *R*≈20Å, shifting the energy curves measured towards larger *R* values.

Another effect is azimuthal frustrations of DNA molecules observed in dense hexagonal

interaction energy over all 6 neighboring DNAs on the lattice inevitably "frustrates" the azimuthal order [69]. Frustrated Pottslike states, reminiscent of those for magnetic spin systems, are often preferred for DNA hexagonal lattice in the model [61]. Namely, in the

elementary triangle on a lattice, the two differences of the azimuthal DNA angles are

DNA lattices [61]. In the theory, they emerge from XYspinlike cos cos2

of the interaction potential on the mutual DNA rotation angle,

while the third one is 2 times larger [70].

= = *l n* , but also depends on the DNA helical repeat *H*. We remind

 θ

σ

values, the 1 *a* term responsible for ES helixhelix attraction grows. Many DNA

*<sup>s</sup> fn f f*

= +− − − *f n*

ɶ ɶ , 0.4

θ

κκπ

δφ

δφ

<sup>1</sup> 1 2 =±arccos 1 / 4 1 2 / / 4 + + *a a* , (4)

 δφ− dependence

. Optimization of the

*<sup>D</sup>* ≈ − Å, that is *n*<sup>0</sup> ∼0.150.1 M of simple salt.

 φ

is the surface charge density of DNA

*<sup>n</sup>* = + *n H* , is a non

φ

 π*<sup>s</sup>* ≈ ɶ is the

at large

θ

their derivatives.

namely 0 1/ 1/ 8 *<sup>D</sup> <sup>B</sup>* λκ

*Future challenges.* Below, we overview some challenges for the current theory. One of them is water structuring in the hydration shells around the DNA. Namely, the most interesting features of intermolecular forces, including the attraction region, emerge at DNA densities when the shells of "structured waters" on interacting helices can overlap. Also, a distance dependent "effective" dielectric constant on the length scale of 12 water diameters [11], a modified decay of electric fields close to DNA, a finite diameter and precise geometrical form of DNAcondensing cations (e.g., linear flexible polyamines vs. compact cohex3+ ions), as well as a limited applicability of the linear PB model, all these points require more accurate theories to be developed close to DNA surface. The solvation of DNA also requires a microscopic treatment of dielectric environments and polarization states upon counterion binding to the DNA. Not only in the theory, these factors also complicate quantitative predictions of DNADNA forces by means of computer simulations. Similar complications in description of ES forces on the nanoscale emerge in modeling of DNAprotein, DNA membrane, as well as proteinprotein complexes (discussed in Sec. 10).

Fig. 2. Theoretically predicted (a) and experimentally measured (b) DNADNA forces in dense DNA assemblies at 50 mM MnCl2. The region of DNADNA attraction at *R*=2832Å detected in experiments corresponds to a spontaneous shrinkage/collapse of the DNA lattice. No azimuthal frustrations on DNA lattice were considered in the model, i.e. cos 0 δφ ≡ for all DNA pairs. Note that, contrary to majority of divalent cations, Mn2+ and Cd2+ are capable of generating DNADNA attraction under the osmotic stress of PEG [36]. This technique allows to overcome the longrange DH repulsive branch of the potential and thus enhance the helix mediated DNADNA forces shielded with a shorter screening length, 1 1 /κ . Parameters: θ= 0.85 , 0 *n* = 50 mM. The figure is reprinted from Ref. [54], subject to ACS2002 Copyright.

#### 

*Structure of toroids.* One biological manifestation of cationmediated DNADNA attraction is DNA condensation into compact toroidal structures observed in bacteria, viruses, and sperm cells *in vivo* and studied thoroughly *in vitro* [71]. For instance, some bacteria pack their DNAs into robust toroids to protect the genetic material and minimize the frequency of dsDNA breaks [72]. These radiationresistant bacteria retain strongly elevated [Mn2+] in their cells to regulate packaging of chromatin fibers, via likely attractive DNADNA forces [73]. In mammalian sperm cells, very long DNA is condensed with the help of highly basic Argrich

0 200 400 600

DNA Length, L, kbp

Let us mention one more example of dense DNA assembly, 3D DNA origami structures, where *extremely dense* DNA packing at *R*≈2225Å is realized [78,79]. A successful assembly necessitates ∼1020 mM of MgCl2: the divalent cations are likely to reduce the ES repulsion of DNA strands during the assembly process. The latter is driven by the chemical energy of

*Model and outcomes.* Utilizing these facts, we constructed a simple model of DNA toroid growth by generations [80]. Due to a finite value of the DNA bending persistence length *l*p [81,82], DNA toroids are often preferred over rodlike or (hollow) spherical condensates. During the first stage of compaction, initial DNA circular loop is thermally nucleated and stabilized, with the curvature radius of ∼*l*p. The growth of DNA toroids is controlled by DNADNA attractive ES contacts and by unfavorable energy of DNA elastic deformations.

As the toroidal crosssection increases, the fraction of "missing" DNADNA attractive contacts on the toroid periphery progressively decreases (the volumetosurface ratio grows). This improves the ES attractive energy gain per unit length of DNA compacted, approaching the value one gets for the DNA columnar hexagonal phase, where the pair DNADNA interaction is tripled due to six neighboring DNAs. Concurrently, however, DNA wrapping near the inner hole of DNA "donut" costs higher bending energies. The optimal toroidal radius *K* and thickness *M* obey the scaling relations 2/5 1/5 2/5 *K E Ll* | |0 *<sup>p</sup>*

<sup>−</sup> ∝ [80], as functions of DNADNA attraction strength at optimal DNA density *E ER* <sup>0</sup> = ( *opt* ) and DNA length *L*. According to Eqs. 2,3, in the presence of DNA condensing ions DNADNA cohesive energy can reach *E*<sup>0</sup> =–(0.01÷0.1) *<sup>B</sup> k T* per bp along the DNADNA contact. It plays the role of the surface tension controlling toroidal dimensions, see Fig. 5. The model reveals that DNA toroids become "fat" as the DNA persistence decreases and DNADNA attraction increases: torodial mean radius decreases and thickness

Several theoretical models of DNA toroidal condensation with nonhexagonal and non circular crosssections have been proposed in the literature [83,84]. We also want to mention that, although locally the lattice of the wrapped DNA preserves the hexagonal symmetry to make best use of attractive intermolecular contacts, the path taken by a *continuous* long

Fig. 5. Radii of DNA toroids of generation *n*, as obtained at relatively strong DNADNA attraction of 0 *E kT* = −0.05 / *<sup>B</sup>* Å [80]. The sawtooth variation of toroid dimensions is due to

association of complementary ssDNA fragments into dsDNA fragments.

0

5

Toroid Generation, n

<sup>−</sup> ∝ and

10

15

20

0

200

Toroid Radii,

the growthbygeneration model implemented.

1/5 2/5 1/5 *Th E L l* | |0 *<sup>p</sup>*

grows.

K and k, 

400

600

proteins protamines into the assembly of interconnected small toroids, as visualized by the AFM technique [74]. DNA compaction inside T5 bacteriophage in the presence of spermine4+ also exhibits some toroidallike arrangements for a part of DNA spool, that is likely to optimize the energetics of DNA packing/encapsidation inside viral shells [75,76].

Fig. 3. Dependence of the helical ES harmonics at typical DNA parameters: θ = 0.8 , *f*=0.3, *a*=9Å, 1/ 7 κ = Å. The solid curves are plotted for two DNAs in solution; the results for dense DNA lattices with the Donnan equilibrium are the dashed curves. In the region of DNADNA attraction, the first helical 1 *a* term dominates the interaction energy. The figure is reprinted from Ref. [27], subject to RCS2011 Copyright.

*In vitro*, DNA condensates formed in solutions of cohex3+, as visualized by cryoEM, often reveal a spoollike DNA organization into tori with ∼50 nm outer and ∼15 nm inner radii [77], with nearly hexagonal local DNA lattice order, Fig. 4. When several DNA chains comprise a torus, the most frequently encountered condensates contain an optimal number of DNA strands. Often, nearly hexagonal toroidal crosssections are observed, with a completely filled outer DNA shell, which give the most stable aggregates. Such structures maximize the number of attractive DNADNA contacts inside the toroid and minimize the number of (relatively unfavourable) DNA contacts with the solvent. It is important to note that DNADNA separations in toroids are often *R*≈28Å, being in the range of DNADNA attraction as measured by the osmotic stress technique and as predicted by the theory of DNADNA ES interactions, see Fig. 2.

Fig. 4. CryoEM images of DNA toroids constructed from 23 λphage 48.5 kbp long DNAs in 0.2 mM solution of cohex3+ (A, B). The mean *K* and inner *k* toroidal radii are indicated. One possible model of a defectfree DNA spooling into a torus of generation *n*=7 in shown in part (C). The image is reprinted from Ref. [80], with permission of IOP.

proteins protamines into the assembly of interconnected small toroids, as visualized by the AFM technique [74]. DNA compaction inside T5 bacteriophage in the presence of spermine4+ also exhibits some toroidallike arrangements for a part of DNA spool, that is likely to

*a*1

*a*0

*a*2 20 30 40 50

DNADNA Separation, R,

= Å. The solid curves are plotted for two DNAs in solution; the results for

dense DNA lattices with the Donnan equilibrium are the dashed curves. In the region of DNADNA attraction, the first helical 1 *a* term dominates the interaction energy. The figure

*In vitro*, DNA condensates formed in solutions of cohex3+, as visualized by cryoEM, often reveal a spoollike DNA organization into tori with ∼50 nm outer and ∼15 nm inner radii [77], with nearly hexagonal local DNA lattice order, Fig. 4. When several DNA chains comprise a torus, the most frequently encountered condensates contain an optimal number of DNA strands. Often, nearly hexagonal toroidal crosssections are observed, with a completely filled outer DNA shell, which give the most stable aggregates. Such structures maximize the number of attractive DNADNA contacts inside the toroid and minimize the number of (relatively unfavourable) DNA contacts with the solvent. It is important to note that DNADNA separations in toroids are often *R*≈28Å, being in the range of DNADNA attraction as measured by the osmotic stress technique and as predicted by the theory of

Fig. 4. CryoEM images of DNA toroids constructed from 23 λphage 48.5 kbp long DNAs in 0.2 mM solution of cohex3+ (A, B). The mean *K* and inner *k* toroidal radii are indicated. One possible model of a defectfree DNA spooling into a torus of generation *n*=7 in shown

*Th k* 

in part (C). The image is reprinted from Ref. [80], with permission of IOP.

θ

= 0.8 , *f*=0.3,

optimize the energetics of DNA packing/encapsidation inside viral shells [75,76].

10<sup>4</sup>

is reprinted from Ref. [27], subject to RCS2011 Copyright.

DNADNA ES interactions, see Fig. 2.

50 nm

*K*

Fig. 3. Dependence of the helical ES harmonics at typical DNA parameters:

0.001

*an*, *kB*T

*a*=9Å, 1/ 7 κ

0.01

0.1

Fig. 5. Radii of DNA toroids of generation *n*, as obtained at relatively strong DNADNA attraction of 0 *E kT* = −0.05 / *<sup>B</sup>* Å [80]. The sawtooth variation of toroid dimensions is due to the growthbygeneration model implemented.

Let us mention one more example of dense DNA assembly, 3D DNA origami structures, where *extremely dense* DNA packing at *R*≈2225Å is realized [78,79]. A successful assembly necessitates ∼1020 mM of MgCl2: the divalent cations are likely to reduce the ES repulsion of DNA strands during the assembly process. The latter is driven by the chemical energy of association of complementary ssDNA fragments into dsDNA fragments.

*Model and outcomes.* Utilizing these facts, we constructed a simple model of DNA toroid growth by generations [80]. Due to a finite value of the DNA bending persistence length *l*p [81,82], DNA toroids are often preferred over rodlike or (hollow) spherical condensates. During the first stage of compaction, initial DNA circular loop is thermally nucleated and stabilized, with the curvature radius of ∼*l*p. The growth of DNA toroids is controlled by DNADNA attractive ES contacts and by unfavorable energy of DNA elastic deformations.

As the toroidal crosssection increases, the fraction of "missing" DNADNA attractive contacts on the toroid periphery progressively decreases (the volumetosurface ratio grows). This improves the ES attractive energy gain per unit length of DNA compacted, approaching the value one gets for the DNA columnar hexagonal phase, where the pair DNADNA interaction is tripled due to six neighboring DNAs. Concurrently, however, DNA wrapping near the inner hole of DNA "donut" costs higher bending energies. The optimal toroidal radius *K* and thickness *M* obey the scaling relations 2/5 1/5 2/5 *K E Ll* | |0 *<sup>p</sup>* <sup>−</sup> ∝ and 1/5 2/5 1/5 *Th E L l* | |0 *<sup>p</sup>* <sup>−</sup> ∝ [80], as functions of DNADNA attraction strength at optimal DNA density *E ER* <sup>0</sup> = ( *opt* ) and DNA length *L*. According to Eqs. 2,3, in the presence of DNA condensing ions DNADNA cohesive energy can reach *E*<sup>0</sup> =–(0.01÷0.1) *<sup>B</sup> k T* per bp along the DNADNA contact. It plays the role of the surface tension controlling toroidal dimensions, see Fig. 5. The model reveals that DNA toroids become "fat" as the DNA persistence decreases and DNADNA attraction increases: torodial mean radius decreases and thickness grows.

Several theoretical models of DNA toroidal condensation with nonhexagonal and non circular crosssections have been proposed in the literature [83,84]. We also want to mention that, although locally the lattice of the wrapped DNA preserves the hexagonal symmetry to make best use of attractive intermolecular contacts, the path taken by a *continuous* long

membranes (one DNA layer per one membrane), compensating the charge. Note that for f actin, due to a mismatch in the charge densities, the unit cell of the lamellar stack consists of

For DNACL complexes, most stable assemblies often occur at the *isoelectric point* of exact charge matching between DNAs and CL membranes [100]. The assembly process is accompanied by almost complete release of condensed counterions from the DNA and membrane surface [101]. Concomitantly, the translational entropy of these "evaporated" counterions is maximized. The DNADNA separations measured in DNACL complexes are in the range 25Å<*R*<60Å and they

The ES stabilization mechanism of DNACL complexes based on numerical solutions of the *nonlinear* PB equation has been established a decade ago, in the model of nonfluctuating rodlike DNAs [102,103,104] and refined for more realistic setups in recent coarsegrained computer simulations [105,106,107]. For the lamellar complexes, a particular attention was paid to ESdriven DNAmediated adjustment of CL charge density profile [108] and membrane undulations [109] that might help to improve DNAmembrane charge matching. *Model and results.* Recently, we developed a similar ES model based on exact solutions of the *linear* PB theory, with the dielectric boundaries (DNAsolvent, membranewater) and DNA

planar membranes and for *<sup>c</sup> HII* phase with CL membranes wrapped around DNAs, the distribution of ES potential in electrolyte has been calculated. The variation of the complex ES energy was computed as a function of DNA lattice density and CLfraction on the membrane. Both appear to exhibit a *nonmonotonic* behavior, in good agreement with the numerical results of the nonlinear model [103]. The energy minimum found from the model roughly corresponds to electroneutral assemblies [110]. For the lamellar phase, the energy well near the minimum is attributable to ES compressibility of the DNA lattice. A scaling law for its modulus obtained, <sup>2</sup> *B R comp* ∝ 1 / , agrees well with the experimental data [111].

phase, the ES energy of DNAinduced undulatory membrane deformations can be

The laws of DNADNA ES forces along and across CL membranes were also examined [110]. For instance, for two thin PE rods on a "salty" interface with mobile charges [112] one can get a powerlaw decay of ES interactions, in contrast to a nearly exponential decay of rodrod ES screening known in 3D, Fig. 7. Confinement of electrolyte solution between the adjacent membrane layers also modifies the law of screening along the membranes. Namely, for a point charge, an exponential decay of ES potential at small distances *d* turns into <sup>3</sup> 1 /*d* powerlaw at large distances, when the electrolyte in intermembrane spaces is rendered quasi2D. The ES forces across a lowdielectric membrane are also renormalized non trivially [110]. All these features affect the properties of DNA transversal and longitudinal

The theory of duplexduplex ES interactions enables us also to rationalize [110] the DNA

Co2+, Ca2+). These are capable of triggering DNA condensation in 2D DNAmembrane system [113], but not in 3D solution. It was observed that at a critical concentration of

phases in the presence of some divalent cations (Mg2+,

correlations in DNACL lamellar phases, as measured in experiments [111].

α

α

phase with

helicity taken explicitly into account [110]. Within this approach, both for *<sup>c</sup> L*

*two* negatively charged factin layers on *both* sides of a CLmembrane.

are often consistent with the picture of counterionfree assemblies.

For *<sup>c</sup> L*α

included later on in more elaborate models.

DNA separations measured in *<sup>c</sup> L*

DNA upon wrapping into a toroid is still debatable [85,86]. Similar complications emerge for DNA packing inside viral capsids, see Sec. 11.

*Perspectives.* How stable are DNA toroids? Recent singlemolecule optical tweezers manipulation experiments have enabled the researchers to decipher the physical mechanisms behind toroidal stability and gain some insights into DNA condensation dynamics [87]. In particular, a stepwise DNA unwrapping from toroids by applied forces was detected, corresponding to multiple DNA loops released from the condensate. The number of turns released is a function of applied tension of 110 pN and of saltdependent DNADNA attraction. Theoretical statistical mechanics models of forceinduced DNA unwrapping from DNA "donuts" have also been developed in recent years [88,89].

One intriguing perspective to enrich the morphology of DNA toroids observed in 3D is DNA condensation on positively charged 2D interfaces. Some analysis of deformations of model toroids on wetting/nonwetting surfaces was performed e.g. in Ref. [90], without accounting however for ES DNADNA and DNAsurface effects. Indeed, DNA condensation on 2D attractive surfaces (for instance, on CL membranes) is expected to follow different pathways and result in different final morphologies, as compared to DNA aggregation in 3D solutions.

Recently, a coilglobule DNA transition on *unsupported* CL membranes has indeed been reported in Ref. [91]. DNA globules of ∼0.1÷0.4 m in size emerge on membranes in 1:1 salt solely due to the presense of mobile positive lipids. They act as counterions for DNA, neutralizing its charge along the DNAmembrane contact. Although a precise morphology of condensates could not be resolved, the hydrodynamic radii of DNA globules were dramatically reduced with the increasing fraction of positive lipids in the membrane. Physically, some patches of positive lipids get bound to DNA deposited on the CL membrane, progressively wrapping around and compacting the DNA coil into a dense globule. The membrane deformations accompanying this process are vital, because *supported* membranes with the same lipid composition do not exhibit a coilglobule DNA transition [92]. Mixing and rearrangement of membrane lipids is also expected to play a role, similarly to the adjustment of lipid charges in DNA complexes with CL membranes, reviewed in the next Section.

#### 

*Structure of complexes.* Selfassembly of CL membranes with oppositely charged biomacromolecules has been extensively studied experimentally for DNA [93], factin [94], microtubules [95], and some filamentous viruses [96]. Dense assemblies of DNA with CL membranes are promising nonviral transfection vectors for gene therapy applications [97], successfully targeting nowadays several types of cancer [98].

Surface charge density on the CL membranes, +0.3÷1 *e*0/nm2, is often comparable to that of DNA and thus ES forces dominate their complexation into different phases. Depending on the fraction of cationic lipids, membrane flexibility, and lipid composition, dense well ordered lamellar *<sup>c</sup> L*α [93] and invertedhexagonal *<sup>c</sup> HII* [99] phases are commonly observed in experiment, Fig. 6. The *<sup>c</sup> HII* phases are preferred for artificially soft or intrinsically pre curved membranes, when a tight cylindrical wrapping of membrane lipids around the DNA takes place. For the lamellar phase, ordered layers of parallel DNAs alternate with CL

DNA upon wrapping into a toroid is still debatable [85,86]. Similar complications emerge

*Perspectives.* How stable are DNA toroids? Recent singlemolecule optical tweezers manipulation experiments have enabled the researchers to decipher the physical mechanisms behind toroidal stability and gain some insights into DNA condensation dynamics [87]. In particular, a stepwise DNA unwrapping from toroids by applied forces was detected, corresponding to multiple DNA loops released from the condensate. The number of turns released is a function of applied tension of 110 pN and of saltdependent DNADNA attraction. Theoretical statistical mechanics models of forceinduced DNA

One intriguing perspective to enrich the morphology of DNA toroids observed in 3D is DNA condensation on positively charged 2D interfaces. Some analysis of deformations of model toroids on wetting/nonwetting surfaces was performed e.g. in Ref. [90], without accounting however for ES DNADNA and DNAsurface effects. Indeed, DNA condensation on 2D attractive surfaces (for instance, on CL membranes) is expected to follow different pathways and result in different final morphologies, as compared to DNA

Recently, a coilglobule DNA transition on *unsupported* CL membranes has indeed been reported in Ref. [91]. DNA globules of ∼0.1÷0.4 m in size emerge on membranes in 1:1 salt solely due to the presense of mobile positive lipids. They act as counterions for DNA, neutralizing its charge along the DNAmembrane contact. Although a precise morphology of condensates could not be resolved, the hydrodynamic radii of DNA globules were dramatically reduced with the increasing fraction of positive lipids in the membrane. Physically, some patches of positive lipids get bound to DNA deposited on the CL membrane, progressively wrapping around and compacting the DNA coil into a dense globule. The membrane deformations accompanying this process are vital, because *supported* membranes with the same lipid composition do not exhibit a coilglobule DNA transition [92]. Mixing and rearrangement of membrane lipids is also expected to play a role, similarly to the adjustment

of lipid charges in DNA complexes with CL membranes, reviewed in the next Section.

*Structure of complexes.* Selfassembly of CL membranes with oppositely charged biomacromolecules has been extensively studied experimentally for DNA [93], factin [94], microtubules [95], and some filamentous viruses [96]. Dense assemblies of DNA with CL membranes are promising nonviral transfection vectors for gene therapy applications [97],

Surface charge density on the CL membranes, +0.3÷1 *e*0/nm2, is often comparable to that of DNA and thus ES forces dominate their complexation into different phases. Depending on the fraction of cationic lipids, membrane flexibility, and lipid composition, dense well

in experiment, Fig. 6. The *<sup>c</sup> HII* phases are preferred for artificially soft or intrinsically pre curved membranes, when a tight cylindrical wrapping of membrane lipids around the DNA takes place. For the lamellar phase, ordered layers of parallel DNAs alternate with CL

[93] and invertedhexagonal *<sup>c</sup> HII* [99] phases are commonly observed

successfully targeting nowadays several types of cancer [98].

unwrapping from DNA "donuts" have also been developed in recent years [88,89].

for DNA packing inside viral capsids, see Sec. 11.

aggregation in 3D solutions.

ordered lamellar *<sup>c</sup> L*

α

membranes (one DNA layer per one membrane), compensating the charge. Note that for f actin, due to a mismatch in the charge densities, the unit cell of the lamellar stack consists of *two* negatively charged factin layers on *both* sides of a CLmembrane.

For DNACL complexes, most stable assemblies often occur at the *isoelectric point* of exact charge matching between DNAs and CL membranes [100]. The assembly process is accompanied by almost complete release of condensed counterions from the DNA and membrane surface [101]. Concomitantly, the translational entropy of these "evaporated" counterions is maximized. The DNADNA separations measured in DNACL complexes are in the range 25Å<*R*<60Å and they are often consistent with the picture of counterionfree assemblies.

The ES stabilization mechanism of DNACL complexes based on numerical solutions of the *nonlinear* PB equation has been established a decade ago, in the model of nonfluctuating rodlike DNAs [102,103,104] and refined for more realistic setups in recent coarsegrained computer simulations [105,106,107]. For the lamellar complexes, a particular attention was paid to ESdriven DNAmediated adjustment of CL charge density profile [108] and membrane undulations [109] that might help to improve DNAmembrane charge matching.

*Model and results.* Recently, we developed a similar ES model based on exact solutions of the *linear* PB theory, with the dielectric boundaries (DNAsolvent, membranewater) and DNA helicity taken explicitly into account [110]. Within this approach, both for *<sup>c</sup> L*α phase with planar membranes and for *<sup>c</sup> HII* phase with CL membranes wrapped around DNAs, the distribution of ES potential in electrolyte has been calculated. The variation of the complex ES energy was computed as a function of DNA lattice density and CLfraction on the membrane. Both appear to exhibit a *nonmonotonic* behavior, in good agreement with the numerical results of the nonlinear model [103]. The energy minimum found from the model roughly corresponds to electroneutral assemblies [110]. For the lamellar phase, the energy well near the minimum is attributable to ES compressibility of the DNA lattice. A scaling law for its modulus obtained, <sup>2</sup> *B R comp* ∝ 1 / , agrees well with the experimental data [111]. For *<sup>c</sup> L*α phase, the ES energy of DNAinduced undulatory membrane deformations can be included later on in more elaborate models.

The laws of DNADNA ES forces along and across CL membranes were also examined [110]. For instance, for two thin PE rods on a "salty" interface with mobile charges [112] one can get a powerlaw decay of ES interactions, in contrast to a nearly exponential decay of rodrod ES screening known in 3D, Fig. 7. Confinement of electrolyte solution between the adjacent membrane layers also modifies the law of screening along the membranes. Namely, for a point charge, an exponential decay of ES potential at small distances *d* turns into <sup>3</sup> 1 /*d* powerlaw at large distances, when the electrolyte in intermembrane spaces is rendered quasi2D. The ES forces across a lowdielectric membrane are also renormalized non trivially [110]. All these features affect the properties of DNA transversal and longitudinal correlations in DNACL lamellar phases, as measured in experiments [111].

The theory of duplexduplex ES interactions enables us also to rationalize [110] the DNA DNA separations measured in *<sup>c</sup> L*α phases in the presence of some divalent cations (Mg2+, Co2+, Ca2+). These are capable of triggering DNA condensation in 2D DNAmembrane system [113], but not in 3D solution. It was observed that at a critical concentration of

0 0.1 0.2 0.3 0.4

*E*min, *kB*T

�0.005 0 0.005

0 0.2 0.4

Κ, 1

) stays constant. The inset depicts the ES DNADNA

Reciprocal Screening Length, Κ, 1

θ

=80%, *f*=0.3.

spermine4+ experimentally [114] (dots) and predicted theoretically [110]. The theory curves with the Donnan saturation (dashed) are more realistic, while the solid curves are for external salt levels also inside the DNA lattice. Multivalent cations only affect the screening

*Properties of DNA twisted phases.* DNA chirality on the nanoscale manifests itself via DNA DNA interactions in a formation of twisted, ∼160350 mg/ml dense, LC DNA phases on the microscale [116,117]. Cholesteric phases also emerge upon assembly of other biohelices, e.g., collagen fibers, filamentous viruses [118] and guanosine [119]. Some polypeptides feature LC phases too. Nature uses the ability of DNA to form chiral phases for packing the genomes in

Typically, *lefthanded* DNA LC phases are detected, with the cholesteric pitch of *P*∼1÷4m for ∼150bp long nucleosomal DNA fragments [123]. The pitch dependences on the [salt], temperature *T*, DNA lattice density, and external osmotic stress are however nontrivial functions. For example, the pitch *P* decreases at higher [NaCl] in the range 0.21 M [123]. It reaches *P*∼20m for DNA phases with some multivalent cations added [124] and can be

A number of theories of DNA cholesteric ordering have been developed based on the helical nature of DNA charges [125] in order to rationalize these and many other observations. Some geometrical models imply that righthanded cholesteric pitch is favored by a steric hindrance of DNAs [126], while lefthanded phases should originate from ES interactions [127]. Other, purely ES models, treat the DNA charge helicity explicitly and predict a *right handed* twist direction to be favored for two righthanded DNA duplexes in a close contact [59]. Such twisting direction ensures a more parallel and more ESbeneficial arrangement of negative phosphate strands of one partially neutralized DNA with the "strands" of adsorbed counterions on the neighboring helix. This fact results however in a *righthanded*

*Model and its conclusions.* Based on the theory of ES interactions of two skewed DNAs [58], we examined the ES stability of DNA cholesteric phases calculating the strength of DNA

Λ�DNA, 48.5 kbp sodium�DNA, 146 bp spermine�DNA, 146 bp

Fig. 8. Optimal DNADNA separations measured in dense DNA precipitates with

θ

some bacteriophages [120], bacteria [121], and in sperm of many vertebrates [122].

reversed by addition of short basic polymers such as polyLysine and chitosan.

twist of DNA cholesterics, opposite to a number of experimental observations.

Energy Minimum,

in the model: DNA charge fraction (1

energy in the local energy minimum. Parameters:

 R*E*min, 

divalent salt, ∼2060 mM, the DNADNA separations drop abruptly from ≈45Å for nearly electroneutral complexes down to "universal" DNA compaction density with *R*≈29Å. Note again that DNADNA ES attraction is realized at these *R*, Fig. 2. Also, for multivalent spermine4+ and spermidine3+, the DNADNA distances in DNACL phase appear to be close to those measured in 3D lipidfree DNA condensates [114] and also agree with the predictions of our theory of DNADNA ES interactions, Fig. 8. One can conclude that 2D geometry of DNACL *<sup>c</sup> L*α phase facilitates interDNA counterionmediated attraction, rendering ES attraction possible for divalent cations and at DNA charge neutralization fractions measured to be θ ≈0.63 [113], lower than the Manning estimate of θM =0.80.85.

*Applications.* A perspective for future research is to enrich the physical understanding of DNA release from subm sized DNACL complexes and of their translocation into the cell cytoplasm across negatively charged cell membranes [98,115]. Both processes are necessary for efficient gene delivery, with the transfection efficiency of DNACL complexes still remaining low compared to viralbased gene carriers [98]. It is known e.g. that spermine and spermidine not only compactify DNAs in DNACL complexes, but can also trigger a DNA release from them via DNA condensation into dense globular/toroidal aggregates in solution. Another pathway for DNA release is designing lipid membrane being unstable in particular cellular cytoplasmatic environments (addition of special "helper" lipids).

Fig. 6. Schematics of 2D DNA condensation in the lamellar DNACLmembrane phase with divalent cations (a). Inverted hexagonal *<sup>c</sup> HII* phase (b). Part a) is reprinted from Ref. [113], copyright 2000NAS, USA.

Fig. 7. Energy density of rodrod ES repulsions along a "salty" lipid membrane with the inverse Debye length 1 /κ*<sup>s</sup>* and in 3D electrolyte solution.

Fig. 8. Optimal DNADNA separations measured in dense DNA precipitates with spermine4+ experimentally [114] (dots) and predicted theoretically [110]. The theory curves with the Donnan saturation (dashed) are more realistic, while the solid curves are for external salt levels also inside the DNA lattice. Multivalent cations only affect the screening in the model: DNA charge fraction (1θ) stays constant. The inset depicts the ES DNADNA energy in the local energy minimum. Parameters: θ=80%, *f*=0.3.

divalent salt, ∼2060 mM, the DNADNA separations drop abruptly from ≈45Å for nearly electroneutral complexes down to "universal" DNA compaction density with *R*≈29Å. Note again that DNADNA ES attraction is realized at these *R*, Fig. 2. Also, for multivalent spermine4+ and spermidine3+, the DNADNA distances in DNACL phase appear to be close to those measured in 3D lipidfree DNA condensates [114] and also agree with the predictions of our theory of DNADNA ES interactions, Fig. 8. One can conclude that 2D

rendering ES attraction possible for divalent cations and at DNA charge neutralization

*Applications.* A perspective for future research is to enrich the physical understanding of DNA release from subm sized DNACL complexes and of their translocation into the cell cytoplasm across negatively charged cell membranes [98,115]. Both processes are necessary for efficient gene delivery, with the transfection efficiency of DNACL complexes still remaining low compared to viralbased gene carriers [98]. It is known e.g. that spermine and spermidine not only compactify DNAs in DNACL complexes, but can also trigger a DNA release from them via DNA condensation into dense globular/toroidal aggregates in solution. Another pathway for DNA release is designing lipid membrane being unstable in

particular cellular cytoplasmatic environments (addition of special "helper" lipids).

≈0.63 [113], lower than the Manning estimate of

(a) (b)

<sup>=</sup> ( )

 <sup>=</sup>

κ

Fig. 6. Schematics of 2D DNA condensation in the lamellar DNACLmembrane phase with divalent cations (a). Inverted hexagonal *<sup>c</sup> HII* phase (b). Part a) is reprinted from Ref. [113],

Fig. 7. Energy density of rodrod ES repulsions along a "salty" lipid membrane with the

*<sup>s</sup>* and in 3D electrolyte solution.

phase facilitates interDNA counterionmediated attraction,

θ

M =0.80.85.

geometry of DNACL *<sup>c</sup> L*

fractions measured to be

copyright 2000NAS, USA.

inverse Debye length 1 /

κ

εε

<sup>ε</sup>

( )

 κ

α

θ

*Properties of DNA twisted phases.* DNA chirality on the nanoscale manifests itself via DNA DNA interactions in a formation of twisted, ∼160350 mg/ml dense, LC DNA phases on the microscale [116,117]. Cholesteric phases also emerge upon assembly of other biohelices, e.g., collagen fibers, filamentous viruses [118] and guanosine [119]. Some polypeptides feature LC phases too. Nature uses the ability of DNA to form chiral phases for packing the genomes in some bacteriophages [120], bacteria [121], and in sperm of many vertebrates [122].

Typically, *lefthanded* DNA LC phases are detected, with the cholesteric pitch of *P*∼1÷4m for ∼150bp long nucleosomal DNA fragments [123]. The pitch dependences on the [salt], temperature *T*, DNA lattice density, and external osmotic stress are however nontrivial functions. For example, the pitch *P* decreases at higher [NaCl] in the range 0.21 M [123]. It reaches *P*∼20m for DNA phases with some multivalent cations added [124] and can be reversed by addition of short basic polymers such as polyLysine and chitosan.

A number of theories of DNA cholesteric ordering have been developed based on the helical nature of DNA charges [125] in order to rationalize these and many other observations. Some geometrical models imply that righthanded cholesteric pitch is favored by a steric hindrance of DNAs [126], while lefthanded phases should originate from ES interactions [127]. Other, purely ES models, treat the DNA charge helicity explicitly and predict a *right handed* twist direction to be favored for two righthanded DNA duplexes in a close contact [59]. Such twisting direction ensures a more parallel and more ESbeneficial arrangement of negative phosphate strands of one partially neutralized DNA with the "strands" of adsorbed counterions on the neighboring helix. This fact results however in a *righthanded* twist of DNA cholesterics, opposite to a number of experimental observations.

*Model and its conclusions.* Based on the theory of ES interactions of two skewed DNAs [58], we examined the ES stability of DNA cholesteric phases calculating the strength of DNA

densities. In the first case, it is due to the decay of DNADNA ES interactions, Eq. 3, while in the small*R* region the inherent azimuthal *frustrations* of DNADNA potential (Eq. 4) destroy DNA orientational order. The existing ES theory of DNA cholesterics [59] has been modified to incorporate the Donnan electrochemical equilibrium of ions [129] in DNA lattices. This effect appears to be particularly important at low [electrolyte], as follows from the equation

> ( ) ( ( ) )

2 2 2 4 1

*lλ θ bR / ̟ a*

3 2

<sup>−</sup> ≈ + <sup>−</sup>

Thus, in dense nearly electroneutral DNA assemblies the DNA lattice density, rather than

Despite a good agreement for the pitch value, the direction of winding of DNA cholesteric layers and the shift of stability domains at different [salt], see Fig. 10, cannot be rationalized by this ES theory in its current form. Righthanded pitch is anticipated at relevant DNA densities of *R*=3545Å, with a possible change to lefthanded rotation for very dense DNA packing or atypical counterion patterns on DNA [125]. In the model of dense LC phases with thermally undulating, rather than straight DNAs, a righttoleft pitch inversion might originate from an enhanced contribution of ES image forces. The latter can favor the opposite sense of DNADNA crossing, compared to the direct duplexduplex ES forces. This

1/4 <sup>2</sup>

. (5)

−

Λ*D*�30

10 7

3

20 30 40 50

DNA�DNA Separation, R,

Fig. 11. Effective screening length in dense DNA assemblies for different [salt] in the bulk, as

*Right and lefthanded phases.* A major breakthough in understanding of LC DNA phases was accomplished recently in Ref. [130]. Namely, an *inversion* of cholesteric handedness in dense LC DNA phases build from ∼6÷20 bp short DNA fragments has been revealed. Subtle changes in DNA sequence and fragment length were shown to trigger this inversion. These very short DNA fragments stack on each other to form elongated DNAs with a sequence specific 3D structure [131,132]. For a stacking procedure that generates more regular helical DNA strands, predominantly lefthanded LC phases were observed, similarly to those for

for renormalized screening length inside a DNA phase [128]

*D D*

 λ

λ

conjecture requires a detailed future analysis.

Λ*D*Don, 

0

calculated in the cylindrical cell model according to Eq. 5.

Screening

 Length,

5

10

15

1

*Don B D*

the bulk [salt], dictates the ionic conditions of solvent between the DNAs, Fig. 11.

DNA azimuthal correlations [128], Fig. 9. The DNA "triad model" was implemented, in the groundstate, with no fluctuations, and with a perfect DNA azimuthal register on the lattice [59]. The theory predicts a *nonmonotonic* pitch dependence on the DNA density for ≈150bp DNA fragments, in agreement with experiments [123]. Also, the range of DNA densities of *R*=3545Å predicted is often close to the measured stability domains of DNA cholesterics, Fig. 10. This indicates that longrange ES forces, rather than shortrange steric hindrance of the grooved DNA surfaces, are likely to be responsible for DNA LC ordering.

Fig. 9. Stability domains of DNA cholesteric phases. Azimuthal correlations of 150 bp DNAs are strong inside the green domain, as predicted by the theory [59] with the Donnan equilibrium [129]. The energy of azimuthal DNA rotation on the hexagonal lattice exceeds *<sup>B</sup> k T* in the green region. The LC twist elastic constant is 22 *K* < 0 inside the red domain, the DNA azimuthal rigidity constant is *k* 0 φ < for magenta and blue domains (these regions are nonphysical). Parameters: 7 λ*<sup>D</sup>* = Å, *f*=0.3.

Fig. 10. The value of DNA cholesteric pitch, as calculated from the theory [128] with (dashed) and without (solid curve) the Donnan equilibrium in the DNA lattice. DNA assembly is pressurized to retain a proper [DNA]. Experimental points for DNA LC phases of 146 bp DNAs are taken from Fig. 5a of Ref. [123]. Parameters: *f*=0.3, θ=0.65, and 7 λ*<sup>D</sup>* = Å that corresponds to ∼0.2 M of NaCl.

Strong DNADNA azimuthal correlations are vital for formation of DNA cholesterics. Our ES model [128] predicts that these correlations vanish both at small and large DNA lattice

DNA azimuthal correlations [128], Fig. 9. The DNA "triad model" was implemented, in the groundstate, with no fluctuations, and with a perfect DNA azimuthal register on the lattice [59]. The theory predicts a *nonmonotonic* pitch dependence on the DNA density for ≈150bp DNA fragments, in agreement with experiments [123]. Also, the range of DNA densities of *R*=3545Å predicted is often close to the measured stability domains of DNA cholesterics, Fig. 10. This indicates that longrange ES forces, rather than shortrange steric hindrance of

0 0.2 0.4 0.6 0.8 1

DNA Charge Compensation, Θ

30 35 40 45 50

DNA�DNA Separation, R,

Strong DNADNA azimuthal correlations are vital for formation of DNA cholesterics. Our ES model [128] predicts that these correlations vanish both at small and large DNA lattice

Fig. 10. The value of DNA cholesteric pitch, as calculated from the theory [128] with (dashed) and without (solid curve) the Donnan equilibrium in the DNA lattice. DNA assembly is pressurized to retain a proper [DNA]. Experimental points for DNA LC phases

of 146 bp DNAs are taken from Fig. 5a of Ref. [123]. Parameters: *f*=0.3,

< for magenta and blue domains (these regions are

θ

=0.65, and 7 λ*<sup>D</sup>* = Å

1 M NaCl 0.5 M NaCl 0.2 M NaCl

Fig. 9. Stability domains of DNA cholesteric phases. Azimuthal correlations of 150 bp DNAs

are strong inside the green domain, as predicted by the theory [59] with the Donnan equilibrium [129]. The energy of azimuthal DNA rotation on the hexagonal lattice exceeds *<sup>B</sup> k T* in the green region. The LC twist elastic constant is 22 *K* < 0 inside the red domain, the

φ

*<sup>D</sup>* = Å, *f*=0.3.

the grooved DNA surfaces, are likely to be responsible for DNA LC ordering.

20

λ

6

0

2

Cholesteric

 Pitch, P,

Μ

m

4

30

40

Separation,

DNA azimuthal rigidity constant is *k* 0

nonphysical). Parameters: 7

that corresponds to ∼0.2 M of NaCl.

 R,

50

60

densities. In the first case, it is due to the decay of DNADNA ES interactions, Eq. 3, while in the small*R* region the inherent azimuthal *frustrations* of DNADNA potential (Eq. 4) destroy DNA orientational order. The existing ES theory of DNA cholesterics [59] has been modified to incorporate the Donnan electrochemical equilibrium of ions [129] in DNA lattices. This effect appears to be particularly important at low [electrolyte], as follows from the equation for renormalized screening length inside a DNA phase [128]

$$\mathcal{A}\_{\rm D}{}^{\rm Con} \approx \mathcal{A}\_{\rm D} \left[ 1 + \left( \frac{4l\_{\rm B} \lambda\_{\rm D}{}^{2} \left( 1 - \theta \right)}{b \left( R^{2} \sqrt{3} \right) \left( 2\pi \right) - a^{2}} \right)^{2} \right]^{-1/4} \cdot \tag{5}$$

Thus, in dense nearly electroneutral DNA assemblies the DNA lattice density, rather than the bulk [salt], dictates the ionic conditions of solvent between the DNAs, Fig. 11.

Despite a good agreement for the pitch value, the direction of winding of DNA cholesteric layers and the shift of stability domains at different [salt], see Fig. 10, cannot be rationalized by this ES theory in its current form. Righthanded pitch is anticipated at relevant DNA densities of *R*=3545Å, with a possible change to lefthanded rotation for very dense DNA packing or atypical counterion patterns on DNA [125]. In the model of dense LC phases with thermally undulating, rather than straight DNAs, a righttoleft pitch inversion might originate from an enhanced contribution of ES image forces. The latter can favor the opposite sense of DNADNA crossing, compared to the direct duplexduplex ES forces. This conjecture requires a detailed future analysis.

Fig. 11. Effective screening length in dense DNA assemblies for different [salt] in the bulk, as calculated in the cylindrical cell model according to Eq. 5.

*Right and lefthanded phases.* A major breakthough in understanding of LC DNA phases was accomplished recently in Ref. [130]. Namely, an *inversion* of cholesteric handedness in dense LC DNA phases build from ∼6÷20 bp short DNA fragments has been revealed. Subtle changes in DNA sequence and fragment length were shown to trigger this inversion. These very short DNA fragments stack on each other to form elongated DNAs with a sequence specific 3D structure [131,132]. For a stacking procedure that generates more regular helical DNA strands, predominantly lefthanded LC phases were observed, similarly to those for

Toptobottom NCPNCP stacking contacts are likely to be hydrophobic, governing formation of NCP columns in NCP crystals, semicrystalline NCP phases, NCP multi layered helices, etc. The sidetoside contacts, with the wrapped DNAs being often in close contact, on the contrary, are likely ES in nature, Fig. 12. Close similarities in condensation and resolubilization of DNAs and NCPs support this statement. These [salt]dependent ES

The stability of dense NCP phases is controlled by sidetoside DNAmediated NCPNCP contacts. In these phases, the DNAs on contacting NCPs are separated by only 515Å of

the NCPs are oriented in columns so that their dyad axes point on average perpendicular to the bilayer plane and the NCP sides with 2 DNA turns are buried inside the bilayer. The NCP azimuthal orientations are frustrated and distributed within ±≈35° from this preferred direction [145]. Stronger DNAmediated contacts of NCPs *inside* the bilayer govern its formation, being likely responsible for peculiar NCP azimuthal frustrations observed.

Fig. 12. Schematics of internucleosomal interactions, with the positive histone cores shown in blue and superhelically wrapped DNA depicted in red. This inplane orientation of

Fig. 13. Schematics of NCPNCP interactions modulated by charge periodicity of wrapped DNA. Zipperlike charge motif is viewed along the DNA superhelical axis, for a referential counterion adsorption into the DNA major groove. Data of NCP 1aoi.pdb structure was

*Model of interNCP ES forces.* Crystal structures of NCPs show ≈8 full DNA helical turns per one superhelical turn of the wrapped DNA. DNA turns in NCPs are separated by the same "magic" *R*≈25Å. At these distances parallel DNAs attract each other in dense assemblies with MnCl2, Fig. 2, and they are close to equilibrium separations in DNA toroids, Sec. 3. An important ingredient of the model is the structural fact that two turns of DNA wrapped in NCPs bear a strong positional *register* forming a "supergroove"[146]. As we have seen in Sec. 2, the DNA in electrolyte solution exhibits a distinct pattern of alternating charges along

used. The image is reproduced from [147] with permission of IOP.

λ

*<sup>D</sup>* . For the NCP bilayer phase [145],

contacts control the formation of mesophases of isolated NCPs *in vitro* [144].

electrolyte, that is typically within one Debye length

NCPs corresponds to a 2:2 DNADNA contact.

≈150 bp DNA fragments as well as for kbplong DNAs. Righthanded phases were observed for short sequences and more azimuthally flexible stacking connections of DNA nanorods.

The analysis of data (for a variety of stacked DNA oligomers, at different [DNA] controlled by sample dehydration, etc.) enabled the authors to reconcile the results in terms of a single physical parameter [130]. Namely, for DNA lattices with isotropicnematic phase transition below ≈620 mg/ml, the lefthanded DNA cholesterics are formed. DNA sequences that experience this transition at larger [DNA], give rise to righthanded LC phases. Thus, for DNADNA distances of *R* ≤ 32Å the righthanded DNADNA crossings seem to be favored [133], whereas LC twist is righthanded for DNA fragments shorter than 14 bp [130]. Some of these experimental trends have been supported in recent simulation study [134].

Also, a general trend was detected [130] that shorter DNA oligomers form cholesteric phases with a shorter, subm pitch. This fact is consistent with the ES DNA theory [59] being also akin to *P*(*L*) variation for longer DNAs [135]. For shortest DNAs, the pitch of ∼0.3 m was detected, much smaller than 24 m for ∼150 bp fragments. Systematic analysis of [salt] and *T*dependence revealed also several unexplained features. Some sequences exhibit e.g. a pitch reversal at [DNA] of 620 mg/ml. Also, for the majority of DNA oligomers, the pitch increases with *T* indicating an unwinding of cholesteric structures, regardless of their handedness. For many sequences, the pitch was almost insensitive to [salt], contrary to a strong *P*([salt]) dependence measured for ≈150 bp DNAs [123]. All in all, this detailed investigation of twisted phases of stacked nanoDNAs [130] has enriched enormously a widely accepted view of "lefthandedonly" DNA cholesterics, challenging future theoretical investigations of DNA twisted phases.

#### 

*DNA compaction.* NCPmediated DNA compaction in eukaryotic chromosomes ensures enormous compactification "power" often required to pack meterlong genomes in ∼m sized cell nuclei [19]. This process takes place on several hierarchical levels, with the initial step being the 30nm chromatin fiber, an organized array of NCPs connected by a continuous DNA. Structure of this fiber is sensitive to many biological factors such as ionic environment, DNA linker length, concentration of H1 linker histones [136], charge state of histone tails, etc. It is still debated what conditions favor solenoidal [137] vs. zigzag [138] arrangements of NCPs in chromatin fibers. *In vitro*, recent fiber reconstitution experiments provided a great deal of information about inherent stability, NCP linear density, and diameter of chromatin fibers for various lengths of DNA linker [139]. The energetics of DNAhistone [140], H1NCP, and NCPNCP [141] contacts is vital for physical understanding of chromatin fiber structure, stability and functioning.

Similarly to unwrapping of DNA toroids that enabled to rationalize their stability [87], recent measurements of forceinduced stretching of chromatin fibers revealed the NCPNCP cohesive energies of ≈3.4 [142] and later of ≈14 *<sup>B</sup> k T* [143], depending on concentrations of mono and divalent cations. Ionic conditions dramatically influence also the forces required to disrupt the NCPNCP contacts in chromatin fibers (typically ∼25 pN) and to induce a DNA unwrapping from NCPs (∼620 pN). These cohesive energies guarantee the fiber stability, but also allow for some "unwrapping plasticity" and "breathing dynamics" of complexed NCPs, both required for DNA transcription to take place.

≈150 bp DNA fragments as well as for kbplong DNAs. Righthanded phases were observed for short sequences and more azimuthally flexible stacking connections of DNA nanorods. The analysis of data (for a variety of stacked DNA oligomers, at different [DNA] controlled by sample dehydration, etc.) enabled the authors to reconcile the results in terms of a single physical parameter [130]. Namely, for DNA lattices with isotropicnematic phase transition below ≈620 mg/ml, the lefthanded DNA cholesterics are formed. DNA sequences that experience this transition at larger [DNA], give rise to righthanded LC phases. Thus, for DNADNA distances of *R* ≤ 32Å the righthanded DNADNA crossings seem to be favored [133], whereas LC twist is righthanded for DNA fragments shorter than 14 bp [130]. Some

of these experimental trends have been supported in recent simulation study [134].

theoretical investigations of DNA twisted phases.

understanding of chromatin fiber structure, stability and functioning.

complexed NCPs, both required for DNA transcription to take place.

Also, a general trend was detected [130] that shorter DNA oligomers form cholesteric phases with a shorter, subm pitch. This fact is consistent with the ES DNA theory [59] being also akin to *P*(*L*) variation for longer DNAs [135]. For shortest DNAs, the pitch of ∼0.3 m was detected, much smaller than 24 m for ∼150 bp fragments. Systematic analysis of [salt] and *T*dependence revealed also several unexplained features. Some sequences exhibit e.g. a pitch reversal at [DNA] of 620 mg/ml. Also, for the majority of DNA oligomers, the pitch increases with *T* indicating an unwinding of cholesteric structures, regardless of their handedness. For many sequences, the pitch was almost insensitive to [salt], contrary to a strong *P*([salt]) dependence measured for ≈150 bp DNAs [123]. All in all, this detailed investigation of twisted phases of stacked nanoDNAs [130] has enriched enormously a widely accepted view of "lefthandedonly" DNA cholesterics, challenging future

*DNA compaction.* NCPmediated DNA compaction in eukaryotic chromosomes ensures enormous compactification "power" often required to pack meterlong genomes in ∼m sized cell nuclei [19]. This process takes place on several hierarchical levels, with the initial step being the 30nm chromatin fiber, an organized array of NCPs connected by a continuous DNA. Structure of this fiber is sensitive to many biological factors such as ionic environment, DNA linker length, concentration of H1 linker histones [136], charge state of histone tails, etc. It is still debated what conditions favor solenoidal [137] vs. zigzag [138] arrangements of NCPs in chromatin fibers. *In vitro*, recent fiber reconstitution experiments provided a great deal of information about inherent stability, NCP linear density, and diameter of chromatin fibers for various lengths of DNA linker [139]. The energetics of DNAhistone [140], H1NCP, and NCPNCP [141] contacts is vital for physical

Similarly to unwrapping of DNA toroids that enabled to rationalize their stability [87], recent measurements of forceinduced stretching of chromatin fibers revealed the NCPNCP cohesive energies of ≈3.4 [142] and later of ≈14 *<sup>B</sup> k T* [143], depending on concentrations of mono and divalent cations. Ionic conditions dramatically influence also the forces required to disrupt the NCPNCP contacts in chromatin fibers (typically ∼25 pN) and to induce a DNA unwrapping from NCPs (∼620 pN). These cohesive energies guarantee the fiber stability, but also allow for some "unwrapping plasticity" and "breathing dynamics" of Toptobottom NCPNCP stacking contacts are likely to be hydrophobic, governing formation of NCP columns in NCP crystals, semicrystalline NCP phases, NCP multi layered helices, etc. The sidetoside contacts, with the wrapped DNAs being often in close contact, on the contrary, are likely ES in nature, Fig. 12. Close similarities in condensation and resolubilization of DNAs and NCPs support this statement. These [salt]dependent ES contacts control the formation of mesophases of isolated NCPs *in vitro* [144].

The stability of dense NCP phases is controlled by sidetoside DNAmediated NCPNCP contacts. In these phases, the DNAs on contacting NCPs are separated by only 515Å of electrolyte, that is typically within one Debye length λ*<sup>D</sup>* . For the NCP bilayer phase [145], the NCPs are oriented in columns so that their dyad axes point on average perpendicular to the bilayer plane and the NCP sides with 2 DNA turns are buried inside the bilayer. The NCP azimuthal orientations are frustrated and distributed within ±≈35° from this preferred direction [145]. Stronger DNAmediated contacts of NCPs *inside* the bilayer govern its formation, being likely responsible for peculiar NCP azimuthal frustrations observed.

Fig. 12. Schematics of internucleosomal interactions, with the positive histone cores shown in blue and superhelically wrapped DNA depicted in red. This inplane orientation of NCPs corresponds to a 2:2 DNADNA contact.

Fig. 13. Schematics of NCPNCP interactions modulated by charge periodicity of wrapped DNA. Zipperlike charge motif is viewed along the DNA superhelical axis, for a referential counterion adsorption into the DNA major groove. Data of NCP 1aoi.pdb structure was used. The image is reproduced from [147] with permission of IOP.

*Model of interNCP ES forces.* Crystal structures of NCPs show ≈8 full DNA helical turns per one superhelical turn of the wrapped DNA. DNA turns in NCPs are separated by the same "magic" *R*≈25Å. At these distances parallel DNAs attract each other in dense assemblies with MnCl2, Fig. 2, and they are close to equilibrium separations in DNA toroids, Sec. 3. An important ingredient of the model is the structural fact that two turns of DNA wrapped in NCPs bear a strong positional *register* forming a "supergroove"[146]. As we have seen in Sec. 2, the DNA in electrolyte solution exhibits a distinct pattern of alternating charges along

the range of DNAmediated and tailsmediated NCPNCP interactions might indeed overlap, being thus hard to distinguish. And, the histone tails also follow the symmetry of DNA in NCPs, protruding into solution through the aligned minor grooves of the DNA superhelix.

To reconcile numerous observations for semidense NCP phases, NCP crystals, and chromatin fibers, a rigorous theory of ES NCPNCP interactions is to be developed in the future. For arbitrary orientations of NCPs in space, one has to take into account the helicity of DNA charges, a heterogeneous distribution of histone charges on the side and top/bottom NCP surfaces, a lowdielectric core of histones and the DNA, DH ES screening by electrolyte, and the counterion separationdependent condensation. All these effects make such a theory a

*DNA structure.* So far, ES forces between ideallyhelical parallel or skewed DNAs were described. The locality of intermolecular potential gives rise to the same ES energies for DNA fragments hom in sequence, but not ideally helical. Below, the sequence effects on DNADNA ES forces are overviewed. To save space, we dwell here on several subjects only, addressing the reader to an excellent recent perspective [67] that covers all aspects of ES DNADNA recognition and also suggests biological phenomena where it is of potential importance. One immediate application of the theory is to provide a physical rationale [148] for recognition and

The physical mechanism of ES DNADNA sequence recognition was pioneered in Ref. [53] for parallel torsionally rigid DNAs and later extended for duplexes with a realistic value of torsional rigidity [55]. The ES recognition emerges in the model solely due to the inherent bpspecific nonidealities of DNA helix, as extracted from the analysis of structural data on DNADNA and DNAprotein crystals [151]. In particular, the DNA bp twist angles are known to exhibit a strong variation [52,152] fluctuating in a range of 2840° that gives rise to

In the theory, these variations for a randomlysequenced DNA form the *helical coherence* 

degree of DNA nonidealities that strongly affect DNADNA ES forces. A finite DNA twist persistence length 75 *twl* ≈ nm allows for some interactioninduced DNA torsional adjustments to take place. These restore to some extent the DNA helical register along such

*Results.* We define below the recognition energy *E L*( ) as the difference in ES interaction energy for hom and randomly sequenced DNA fragments of length *L*. For torsionally *rigid* DNA fragments, with azimuthally free ends, in the leading *a*1approximation we get [53,70]

 <sup>−</sup> ≈ + − . In the opposite limit of torsionally adaptable, *very soft* DNA duplexes, the recognition is also described by a simple formula [70]

torsional adaptation length. For a standard value of DNA twist modulus, 750 *C kT* = *<sup>B</sup>* Å, the exact analytical expressions for *E L*( ) are however quite cumbersome [55]. Roughly, at

λ

*<sup>c</sup>* = ≈ *H* nm at a typical value ≈ 5 o. This length controls the

*<sup>t</sup>* ( ) *R C aR* = / 2 <sup>1</sup> ( ) is the *R*dependent DNA

formidable problem of the mathematical physics, even within the linear PB theory.

pairing of hom genes on genomic dsDNAs [149,150] during cell division.

≈36±5° angle deviations.

*length* that is ( ) <sup>2</sup> / 10 45 λ

( ) / 2( ) <sup>1</sup> /2 2 *<sup>L</sup> <sup>c</sup> EL a L e c c*

 λ

( ) ( ) 2 / <sup>1</sup> 1 2 2

<sup>−</sup> ≈ −−

 λ

*c EL a L e* λ

λ

*t t L <sup>t</sup>*

λ

sequenceunrelated DNA fragments [55].

λ

λ

, where

its axis (negative phosphates vs. positive condensed cations). Thus, this positive/negative DNA *charge zipper* along the sidetoside contact of two inplane NCPs modulates their azimuthal interactions with the period of ≈ π/ 4 , see Fig. 13.

For inplane NCPs with parallel axes, we demonstrated that this azimuthal modulation gives rise to quantization of NCP orientations in nucleosomal bilayers with periodicity of ≈45° [147]. Azimuthal optimization of sidetoside contacts of NCPs in bilayers resembles the azimuthal adaptation of short DNA fragments in columns of nanoDNAs in the cholesteric phases, Sec. 5. Both effects likely originate from DNA helixhelix ES interactions.

Generally speaking, the DNA contribution to NCPNCP ES energy is 4 times stronger for the NCP sides with 2 DNA turns as compared to the NCP sides with only 1 DNA turn. For the inplane NCPs, we calculated the ES forces and attractionrepulsion phase diagram, see Fig. 14. We implemented a simple model of ES doublelayer repulsion for the histone cores, modeled as uniformly charged spheres with charge *Q*. For the DNA part, the Derjaguin approximation was used to compute the ES forces between the bent DNA duplexes, interacting locally according to Eqs. 2,3.

For a typical histone charge *Q*=+220*e*0, at physiological salt conditions, the model predicts for the 2:2 DNADNA contacts the maximal NCPNCP attractive forces of 2 pN for θ = 0.8 , *f*=0.3 (typical parameters used in the theory, as in Fig. 14). The NCPNCP attraction reaches 8 pN at θ = 0.9 , *f*=0.3 (better DNA charge neutralization), and even 60 pN at θ = 0.8 , *f*=0 (stronger DNADNA attraction due to binding of cations into the major groove [50]). As the histone positive charge grows, the DH repulsion of NCP cores overwhelms the DNADNA ES attraction. Thus, at larger *Q*/*e*0 values, the NCPNCP attraction region at 2535Å between DNA fragments along the NCPNCP contact disappears, see the inset in Fig. 14.

Fig. 14. ES force between two NCPs with parallel axes. Optimal NCP azimuthal alignment is assumed: DNADNA attraction. Here *R* is the distance between DNA axes on NCPs contacting sidetoside (*R*=2*a*=18Å is direct DNADNA contact). Thick and thin curves are for 2:2 and 1:1 DNADNA contacts, respectively. ES NCPNCP attractionrepulsion diagram is shown in the inset. Parameters: θ = = 0.8, 0.3, *f* 1/ 7 κ= Å.

DNAmediated ES attractions of 2:2 vs. 1:1 DNA sides of NCPs are likely to trigger the bilayer formation and NCP azimuthal frustrations. Another possibility is that histone tails bridge neighboring NCPs in bilayers in azimuthally dependent manner [145]. At typical conditions, the range of DNAmediated and tailsmediated NCPNCP interactions might indeed overlap, being thus hard to distinguish. And, the histone tails also follow the symmetry of DNA in NCPs, protruding into solution through the aligned minor grooves of the DNA superhelix.

To reconcile numerous observations for semidense NCP phases, NCP crystals, and chromatin fibers, a rigorous theory of ES NCPNCP interactions is to be developed in the future. For arbitrary orientations of NCPs in space, one has to take into account the helicity of DNA charges, a heterogeneous distribution of histone charges on the side and top/bottom NCP surfaces, a lowdielectric core of histones and the DNA, DH ES screening by electrolyte, and the counterion separationdependent condensation. All these effects make such a theory a formidable problem of the mathematical physics, even within the linear PB theory.

#### 

its axis (negative phosphates vs. positive condensed cations). Thus, this positive/negative DNA *charge zipper* along the sidetoside contact of two inplane NCPs modulates their

For inplane NCPs with parallel axes, we demonstrated that this azimuthal modulation gives rise to quantization of NCP orientations in nucleosomal bilayers with periodicity of ≈45° [147]. Azimuthal optimization of sidetoside contacts of NCPs in bilayers resembles the azimuthal adaptation of short DNA fragments in columns of nanoDNAs in the cholesteric phases, Sec. 5. Both effects likely originate from DNA helixhelix ES interactions. Generally speaking, the DNA contribution to NCPNCP ES energy is 4 times stronger for the NCP sides with 2 DNA turns as compared to the NCP sides with only 1 DNA turn. For the inplane NCPs, we calculated the ES forces and attractionrepulsion phase diagram, see Fig. 14. We implemented a simple model of ES doublelayer repulsion for the histone cores, modeled as uniformly charged spheres with charge *Q*. For the DNA part, the Derjaguin approximation was used to compute the ES forces between the bent DNA duplexes,

For a typical histone charge *Q*=+220*e*0, at physiological salt conditions, the model predicts

*f*=0.3 (typical parameters used in the theory, as in Fig. 14). The NCPNCP attraction reaches

(stronger DNADNA attraction due to binding of cations into the major groove [50]). As the histone positive charge grows, the DH repulsion of NCP cores overwhelms the DNADNA ES attraction. Thus, at larger *Q*/*e*0 values, the NCPNCP attraction region at 2535Å between

Q*e*0

20 30 40 50

0 150 300

DNADNA Spacing on NCPs, R,

Fig. 14. ES force between two NCPs with parallel axes. Optimal NCP azimuthal alignment is

= = 0.8, 0.3, *f* 1/ 7

DNAmediated ES attractions of 2:2 vs. 1:1 DNA sides of NCPs are likely to trigger the bilayer formation and NCP azimuthal frustrations. Another possibility is that histone tails bridge neighboring NCPs in bilayers in azimuthally dependent manner [145]. At typical conditions,

assumed: DNADNA attraction. Here *R* is the distance between DNA axes on NCPs contacting sidetoside (*R*=2*a*=18Å is direct DNADNA contact). Thick and thin curves are for 2:2 and 1:1 DNADNA contacts, respectively. ES NCPNCP attractionrepulsion diagram

θ

Q*e*<sup>0</sup>220

attr. 25 30 35 40

> κ= Å.

R,

= 0.9 , *f*=0.3 (better DNA charge neutralization), and even 60 pN at

θ= 0.8 ,

= 0.8 , *f*=0

θ

for the 2:2 DNADNA contacts the maximal NCPNCP attractive forces of 2 pN for

DNA fragments along the NCPNCP contact disappears, see the inset in Fig. 14.

2

NCP

is shown in the inset. Parameters:

NCP force, F, pN

0

2

4

6

/ 4 , see Fig. 13.

π

azimuthal interactions with the period of ≈

interacting locally according to Eqs. 2,3.

8 pN at

θ

*DNA structure.* So far, ES forces between ideallyhelical parallel or skewed DNAs were described. The locality of intermolecular potential gives rise to the same ES energies for DNA fragments hom in sequence, but not ideally helical. Below, the sequence effects on DNADNA ES forces are overviewed. To save space, we dwell here on several subjects only, addressing the reader to an excellent recent perspective [67] that covers all aspects of ES DNADNA recognition and also suggests biological phenomena where it is of potential importance. One immediate application of the theory is to provide a physical rationale [148] for recognition and pairing of hom genes on genomic dsDNAs [149,150] during cell division.

The physical mechanism of ES DNADNA sequence recognition was pioneered in Ref. [53] for parallel torsionally rigid DNAs and later extended for duplexes with a realistic value of torsional rigidity [55]. The ES recognition emerges in the model solely due to the inherent bpspecific nonidealities of DNA helix, as extracted from the analysis of structural data on DNADNA and DNAprotein crystals [151]. In particular, the DNA bp twist angles are known to exhibit a strong variation [52,152] fluctuating in a range of 2840° that gives rise to ≈36±5° angle deviations.

In the theory, these variations for a randomlysequenced DNA form the *helical coherence length* that is ( ) <sup>2</sup> / 10 45 λ*<sup>c</sup>* = ≈ *H* nm at a typical value ≈ 5 o. This length controls the degree of DNA nonidealities that strongly affect DNADNA ES forces. A finite DNA twist persistence length 75 *twl* ≈ nm allows for some interactioninduced DNA torsional adjustments to take place. These restore to some extent the DNA helical register along such sequenceunrelated DNA fragments [55].

*Results.* We define below the recognition energy *E L*( ) as the difference in ES interaction energy for hom and randomly sequenced DNA fragments of length *L*. For torsionally *rigid* DNA fragments, with azimuthally free ends, in the leading *a*1approximation we get [53,70] ( ) / 2( ) <sup>1</sup> /2 2 *<sup>L</sup> <sup>c</sup> EL a L e c c* λ λ λ <sup>−</sup> ≈ + − . In the opposite limit of torsionally adaptable, *very soft* DNA duplexes, the recognition is also described by a simple formula [70] ( ) ( ) 2 / <sup>1</sup> 1 2 2 *t t L <sup>t</sup> c EL a L e* λ λ λ λ <sup>−</sup> ≈ −− , where λ*<sup>t</sup>* ( ) *R C aR* = / 2 <sup>1</sup> ( ) is the *R*dependent DNA

torsional adaptation length. For a standard value of DNA twist modulus, 750 *C kT* = *<sup>B</sup>* Å, the exact analytical expressions for *E L*( ) are however quite cumbersome [55]. Roughly, at

In another study, singlemolecule magnetic tweezers measurements revealed an efficient sequencespecific pairing of λdsDNAs with hom regions longer than ∼ 5 kbp, at [salt] and [DNA] close to those *in vivo* [157]. The paired structures of hom DNAs were sheared by *F*≈1020 pN forces and pairing was more profound in the presence of MgCl2, indicative of ES nature of this effect. Some other properties, such as a strong enhancement of pairing efficiency with [simple salt] up to 1 M as well as a nonmonotonic *T*dependence favored however rather some nonCoulomb origin of pairing forces. Also, in experiments, the precision of hom DNADNA register along the DNA pair measured is often ∼25 m, much

Fig. 16. This width in experiments is also independent on the length of the paired hom DNA segments. Future developments of this highly promising technique might provide more information about the axial proximity of paired hom DNA fragments and thus enable us to

To summarize, we do believe that helixspecific ES forces can govern DNADNA sequence recognition in dense DNA phases [156], at high [DNA] and suppressed DNA fluctuations. DNADNA homology associations *in vivo* are however often maintained at much larger separations and take place between fluctuating DNAs [158]. The pairing remains efficient at DNADNA distances of 100300 nm, much longer than the Debye screening length that limits the action radius of ES forces. DNADNA hom pairing should thus also involve a recognition mechanism other than the direct ES forces [159], probably recruiting proteins for

λ

 

 

Å that give 1 0.015 *<sup>B</sup> a kT* ≈ /Å. The helical coherence

*<sup>c</sup>* = 1020 nm, is much shorter than in DNA

λ

*<sup>c</sup>* = Å,

*<sup>c</sup>* ≈ ÷ nm [148], see

larger than the width of the recognition energy well in the model, ∼ 10 50

proteinmediated homologyspecific DNADNA contacts.

*R*=30Å,

crystals,

θ

λ

= = = 0.8, 0.3, 1 / 7 *f*

length for DNAs in solutions and wet fibers

κ

estimate the effective "range of action" of these sequencespecific DNADNA forces.

 

≈

 

> λ

Fig. 16. Pictorial shape of the recognition energy well for sliding of two DNAs with a hom domain, as obtained from Eq. (1) of Ref. [148] for rigid DNAs. DNA hom segments are marked in green; nonhom sections are in red. Hom fragments are pinned near the well bottom by a stronger ES attraction, relative to the rest of DNA. Parameters: 100

λ

*<sup>c</sup>* = 50100 nm [57], where the helices are "straighten" by mutual interactions.

 

relevant parameters, the theory predicts that two DNA fragments with unrelated sequences attract each other nearly half as strong as two hom DNA sequences do, see Fig. 15a.

The ES recognition energy predicted grows linearly with the length of DNAs in contact [55], resembling thereby some properties of DNA hom recombination *in vitro* in the absence of specific DNApairing proteins, Fig. 15c. The recognition energy exceeds several *kBT* for closely aligned DNA fragments of ∼200500 bp in length. This energy is large enough to ensure a stable pairing of hom DNA segments at ambient temperature. It can also be sufficient to trigger unpairing of DNA single strands, required as initial step of hom recombination.

Fig. 15. a) Computed ES interaction energy in a pair of hom DNAs (thin), randomly sequenced torsionally rigid DNA fragments (dotdashed), and randomlysequenced DNA fragments with a realistic twist rigidity *C* (solid curve). b) The corresponding DNADNA ES recognition energy. Parameters: *R*=30Å, 7 λ*<sup>D</sup>* = Å, θ=0.8, *f*=0.3, 750 *twl* ≈ Å. c) Measured frequency of hom recombination events in T4 phage [153]. It shows a minimal length of DNA homology of ≈50 bp necessary for recombination to start and a linear growth of frequency with DNA homology length. This resembles a linear growth of the recognition energy for long DNA sequences in b). The images are reprinted from Ref. [55] with permission of the ACS and from Ref. [153] with permission of Elsevier.

For pulling two ds DNAs one over another, with the hom bp domains in them, the energy well for ES recognition has recently been theoretically computed [148]. For very closely juxtaposed DNAs at *R*=30Å, the recognition energies of up to 510 *<sup>B</sup> k T* and pinning forces near the bottom of the well of ∼2 pN were predicted for typical DNA parameters, Fig. 16.

Sequencespecific DNA recognition and pairing for intact duplexes was indeed observed for yeast hom DNA chromosomal loci, in the absence of any recAfamily proteins [154]. It was attributed to some sequencespecific DNADNA forces, capable of initiating and maintaining a proximity of hom DNA fragments. Recently, several experimental techniques have been utilized to elucidate the properties of DNADNA hom pairing in denser arrangements [155,156,157]. In one study on dense DNA cholesteric spherullites, the segregation of ∼300 bp DNA fragments with identical bp sequences into separate LC populations has been clearly identified [156] (telepathic DNAs). This offered a first proof of direct DNADNA recognition, based solely on DNA bp sequence information. The effect was attributed to more favorable ES interactions of hom DNA fragments as compared to nonrelated ones. ES DNADNA bp specific forces [148], are likely to be responsible for the observed segregation of hom sequences at these relatively high [DNA] corresponding to *R*=32÷40Å.

relevant parameters, the theory predicts that two DNA fragments with unrelated sequences

The ES recognition energy predicted grows linearly with the length of DNAs in contact [55], resembling thereby some properties of DNA hom recombination *in vitro* in the absence of specific DNApairing proteins, Fig. 15c. The recognition energy exceeds several *kBT* for closely aligned DNA fragments of ∼200500 bp in length. This energy is large enough to ensure a stable pairing of hom DNA segments at ambient temperature. It can also be sufficient to

Fig. 15. a) Computed ES interaction energy in a pair of hom DNAs (thin), randomly sequenced torsionally rigid DNA fragments (dotdashed), and randomlysequenced DNA fragments with a realistic twist rigidity *C* (solid curve). b) The corresponding DNADNA ES

frequency of hom recombination events in T4 phage [153]. It shows a minimal length of DNA homology of ≈50 bp necessary for recombination to start and a linear growth of frequency with DNA homology length. This resembles a linear growth of the recognition energy for long DNA sequences in b). The images are reprinted from Ref. [55] with

For pulling two ds DNAs one over another, with the hom bp domains in them, the energy well for ES recognition has recently been theoretically computed [148]. For very closely juxtaposed DNAs at *R*=30Å, the recognition energies of up to 510 *<sup>B</sup> k T* and pinning forces near the bottom of the well of ∼2 pN were predicted for typical DNA parameters, Fig. 16.

Sequencespecific DNA recognition and pairing for intact duplexes was indeed observed for yeast hom DNA chromosomal loci, in the absence of any recAfamily proteins [154]. It was attributed to some sequencespecific DNADNA forces, capable of initiating and maintaining a proximity of hom DNA fragments. Recently, several experimental techniques have been utilized to elucidate the properties of DNADNA hom pairing in denser arrangements [155,156,157]. In one study on dense DNA cholesteric spherullites, the segregation of ∼300 bp DNA fragments with identical bp sequences into separate LC populations has been clearly identified [156] (telepathic DNAs). This offered a first proof of direct DNADNA recognition, based solely on DNA bp sequence information. The effect was attributed to more favorable ES interactions of hom DNA fragments as compared to nonrelated ones. ES DNADNA bp specific forces [148], are likely to be responsible for the observed segregation of hom sequences

θ

=0.8, *f*=0.3, 750 *twl* ≈ Å. c) Measured

λ*<sup>D</sup>* = Å,

permission of the ACS and from Ref. [153] with permission of Elsevier.

at these relatively high [DNA] corresponding to *R*=32÷40Å.

recognition energy. Parameters: *R*=30Å, 7

attract each other nearly half as strong as two hom DNA sequences do, see Fig. 15a.

trigger unpairing of DNA single strands, required as initial step of hom recombination.

In another study, singlemolecule magnetic tweezers measurements revealed an efficient sequencespecific pairing of λdsDNAs with hom regions longer than ∼ 5 kbp, at [salt] and [DNA] close to those *in vivo* [157]. The paired structures of hom DNAs were sheared by *F*≈1020 pN forces and pairing was more profound in the presence of MgCl2, indicative of ES nature of this effect. Some other properties, such as a strong enhancement of pairing efficiency with [simple salt] up to 1 M as well as a nonmonotonic *T*dependence favored however rather some nonCoulomb origin of pairing forces. Also, in experiments, the precision of hom DNADNA register along the DNA pair measured is often ∼25 m, much larger than the width of the recognition energy well in the model, ∼ 10 50 λ*<sup>c</sup>* ≈ ÷ nm [148], see Fig. 16. This width in experiments is also independent on the length of the paired hom DNA segments. Future developments of this highly promising technique might provide more information about the axial proximity of paired hom DNA fragments and thus enable us to estimate the effective "range of action" of these sequencespecific DNADNA forces.

To summarize, we do believe that helixspecific ES forces can govern DNADNA sequence recognition in dense DNA phases [156], at high [DNA] and suppressed DNA fluctuations. DNADNA homology associations *in vivo* are however often maintained at much larger separations and take place between fluctuating DNAs [158]. The pairing remains efficient at DNADNA distances of 100300 nm, much longer than the Debye screening length that limits the action radius of ES forces. DNADNA hom pairing should thus also involve a recognition mechanism other than the direct ES forces [159], probably recruiting proteins for proteinmediated homologyspecific DNADNA contacts.

Fig. 16. Pictorial shape of the recognition energy well for sliding of two DNAs with a hom domain, as obtained from Eq. (1) of Ref. [148] for rigid DNAs. DNA hom segments are marked in green; nonhom sections are in red. Hom fragments are pinned near the well bottom by a stronger ES attraction, relative to the rest of DNA. Parameters: 100 λ*<sup>c</sup>* = Å, *R*=30Å, θ = = = 0.8, 0.3, 1 / 7 *f* κ Å that give 1 0.015 *<sup>B</sup> a kT* ≈ /Å. The helical coherence length for DNAs in solutions and wet fibers λ*<sup>c</sup>* = 1020 nm, is much shorter than in DNA crystals, λ*<sup>c</sup>* = 50100 nm [57], where the helices are "straighten" by mutual interactions.

another DNA and their action along the chain accumulates enhancing the magnitude of DNADNA friction (i.e., the force to remove the system from the favorable state of fully overlapping hom blocks increases). This renders the detection of such pinning mode amenable for the current experimental technique [164] with the resolution of several repeats *H*. We would like to encourage the exprimental groups to check whether such blocky homologous DNAs experience different frictional forces in tight DNA plies. On the contrary, the resolution of at least *H*/2 is necessary to probe the predicted above ES DNADNA

One potential aplication of DNADNA friction discussed is on DNA ejection from dsDNA phages, when densely packed DNA strands have to slide passing each other upon reorganization of DNA layers inside the capsid during DNA compactification and ejection.

Fig. 17. Schematics of ES potential barriers Φ(*z*) near the BDNA surface [166]. Negative DNA phosphate strands are shown in red, the counterions adsorbed in the DNA major

Upon heating up to ≈50100°C, depending on GCcontent and bp sequence, DNAs melt cooperatively in solution and their strands separate. Being thoroughly studied at low [DNA] [167,168,169], DNA melting in dense assemblies, when intermolecular forces become comparable to the internal DNA binding energies, remains not completely understood. The effects of DNA sequence on DNA melting in hexagonal assemblies are discussed below, within a simple thermodynamical model [170]. Namely, we predict that melting of hom DNAs is inhibited, while dstoss DNA transition for unrelated DNA sequences is facilitated. In particular, it is straightforward to show that under the conditions favoring duplexduplex ES attraction, ideally helical hom DNAs melt at higher *T* due to a stabilization of DNA helical regions by mutual interactions. For hom DNA fragments, the model predicts a rise of the melting temperature *Tm* (typically by 310° at *R*=2328Å between DNAs in the assembly) and more cooperative DNA melting transitions. The shift of *Tm* scales with the strength of attraction, namely 0 1 3 ∝ − *T aa <sup>m</sup>* . It can thus be controlled in future melting experiments in dense DNA phases via addition of attractionmediating cations, e.g., counterions with enhanced binding into the DNA major groove and propensity to induce DNA aggregation.

*<sup>c</sup>* , see Fig. 16. Therefore, every bpblock pins with its hom partner on

halfwidth of ∼

λ

groove are depicted as blue helices.

friction *Ffr* on the scale of DNA helicity of 3.4 nm.

#### 

Modern nanotribology applications necessitate a detailed understanding of frictional forces between biomolecules on the nanoscale [160,161]. For DNA, recent advances in single molecule manipulation techniques has allowed measuring the forces required to pull one DNA over another one in a tight superhelical DNA ply, the dual optical trap. Tight winding of two DNAs in the ply can facilitate their interactions [162]. Upon shearing the ply, in the presence of DNAassociated DNAbridging HNS proteins, the frictional forces up to ∼25 pN were detected [163]. They emerge from disruption of DNAproteinDNA bridges formed every several *H* along the ply. Also, when some small proteins bind to the dsDNA and sterically impede DNA pulling, the friction of ∼25 pN was detected. For the "bare" DNAs, one could expect that inherent DNA helicity on *H*=3.4 nm scale might itself generate some friction. With the same apparatus, no measurable friction was detected however [164]: surprizingly, the forces remained <1 pN, independently on the length of DNA ply, DNA pulling speed, and the presence of DNAcondensing (spermine4+) cations. Diameters of DNA plectonemes in experiments were estimated to be ∼510 nm.

In this and the next Section, we discuss two manifestations of sequencedependent DNA DNA interactions considered in Sec. 7, for DNADNA friction and DNA melting. Using the theory of DNADNA ES forces, Sec. 2, we examined different regimes for DNADNA nano friction depending on the character of DNA sequence [165]. For ideally helical non fluctuating and closely juxtaposed DNAs, the ES friction emerges due to spatial correlations of ES potential along DNA surfaces, Fig. 17. At relevant [salt], these correlations are only pronounced in the first ∼10Å from the DNA surface. ES frictional force in this regime oscillates with the period of *H*=3.4 nm, while its magnitude grows linearly with the length of DNA *L*. Namely, the force of static friction is 1 *F aL H fr* = 2 / π . For slow DNA pulling, this gives rise to a *stickslip motion* on the nanoscale [165]. The friction however remains rather low. Even for very tight DNA plies, with thickness of ∼40 Å and parameters favoring DNA DNA attraction (large 1 *a* ), the upper estimate for frictional force in a ply of *N LH* = = / 10 DNA turns long is as small as ≈4 pN.

Several effects are likely to reduce this upper limit. It is the case for pulling nonideally helical DNAs (random bp sequence). For such DNAs, the "corrugations" in DNA helical structure progressively accumulate with the length [53] and ES potential variations along the DNADNA contact become decorrelated, as discussed in Sec. 7. This, in turn, strongly impedes ES friction that attains in this limit an exponential decay with the pulling distance of one DNA with respect to another one. As ES forces decay exponentially with *R*, it is not surprising that for DNA plies that are typically much thicker than 40Å, being formed by fluctuating DNAs with quasirandom sequences, no measurable friction has been detected in DNApulling experiments [164].

The situation might however change, when tight DNA plies are realized (by larger static stretching forces applied to DNA ends) and for DNA sequences with some degree of bp homology. One important example is DNAs fragments artificially designed to contain *repetitive bp hom blocks* with the length of ∼50300 bp. Then, one could expect some homologymediated DNA pinning events upon mutual pulling of DNAs at the positions when these hom blocks on two DNAs overlap. The theory predicts [148] that these pinned states have a measurable

Modern nanotribology applications necessitate a detailed understanding of frictional forces between biomolecules on the nanoscale [160,161]. For DNA, recent advances in single molecule manipulation techniques has allowed measuring the forces required to pull one DNA over another one in a tight superhelical DNA ply, the dual optical trap. Tight winding of two DNAs in the ply can facilitate their interactions [162]. Upon shearing the ply, in the presence of DNAassociated DNAbridging HNS proteins, the frictional forces up to ∼25 pN were detected [163]. They emerge from disruption of DNAproteinDNA bridges formed every several *H* along the ply. Also, when some small proteins bind to the dsDNA and sterically impede DNA pulling, the friction of ∼25 pN was detected. For the "bare" DNAs, one could expect that inherent DNA helicity on *H*=3.4 nm scale might itself generate some friction. With the same apparatus, no measurable friction was detected however [164]: surprizingly, the forces remained <1 pN, independently on the length of DNA ply, DNA pulling speed, and the presence of DNAcondensing (spermine4+) cations. Diameters of

In this and the next Section, we discuss two manifestations of sequencedependent DNA DNA interactions considered in Sec. 7, for DNADNA friction and DNA melting. Using the theory of DNADNA ES forces, Sec. 2, we examined different regimes for DNADNA nano friction depending on the character of DNA sequence [165]. For ideally helical non fluctuating and closely juxtaposed DNAs, the ES friction emerges due to spatial correlations of ES potential along DNA surfaces, Fig. 17. At relevant [salt], these correlations are only pronounced in the first ∼10Å from the DNA surface. ES frictional force in this regime oscillates with the period of *H*=3.4 nm, while its magnitude grows linearly with the length of

gives rise to a *stickslip motion* on the nanoscale [165]. The friction however remains rather low. Even for very tight DNA plies, with thickness of ∼40 Å and parameters favoring DNA DNA attraction (large 1 *a* ), the upper estimate for frictional force in a ply of *N LH* = = / 10

Several effects are likely to reduce this upper limit. It is the case for pulling nonideally helical DNAs (random bp sequence). For such DNAs, the "corrugations" in DNA helical structure progressively accumulate with the length [53] and ES potential variations along the DNADNA contact become decorrelated, as discussed in Sec. 7. This, in turn, strongly impedes ES friction that attains in this limit an exponential decay with the pulling distance of one DNA with respect to another one. As ES forces decay exponentially with *R*, it is not surprising that for DNA plies that are typically much thicker than 40Å, being formed by fluctuating DNAs with quasirandom sequences, no measurable friction has been detected

The situation might however change, when tight DNA plies are realized (by larger static stretching forces applied to DNA ends) and for DNA sequences with some degree of bp homology. One important example is DNAs fragments artificially designed to contain *repetitive bp hom blocks* with the length of ∼50300 bp. Then, one could expect some homologymediated DNA pinning events upon mutual pulling of DNAs at the positions when these hom blocks on

two DNAs overlap. The theory predicts [148] that these pinned states have a measurable

π

. For slow DNA pulling, this

DNA plectonemes in experiments were estimated to be ∼510 nm.

DNA *L*. Namely, the force of static friction is 1 *F aL H fr* = 2 /

DNA turns long is as small as ≈4 pN.

in DNApulling experiments [164].

halfwidth of ∼ λ*<sup>c</sup>* , see Fig. 16. Therefore, every bpblock pins with its hom partner on another DNA and their action along the chain accumulates enhancing the magnitude of DNADNA friction (i.e., the force to remove the system from the favorable state of fully overlapping hom blocks increases). This renders the detection of such pinning mode amenable for the current experimental technique [164] with the resolution of several repeats *H*. We would like to encourage the exprimental groups to check whether such blocky homologous DNAs experience different frictional forces in tight DNA plies. On the contrary, the resolution of at least *H*/2 is necessary to probe the predicted above ES DNADNA friction *Ffr* on the scale of DNA helicity of 3.4 nm.

One potential aplication of DNADNA friction discussed is on DNA ejection from dsDNA phages, when densely packed DNA strands have to slide passing each other upon reorganization of DNA layers inside the capsid during DNA compactification and ejection.

Fig. 17. Schematics of ES potential barriers Φ(*z*) near the BDNA surface [166]. Negative DNA phosphate strands are shown in red, the counterions adsorbed in the DNA major groove are depicted as blue helices.

#### 

Upon heating up to ≈50100°C, depending on GCcontent and bp sequence, DNAs melt cooperatively in solution and their strands separate. Being thoroughly studied at low [DNA] [167,168,169], DNA melting in dense assemblies, when intermolecular forces become comparable to the internal DNA binding energies, remains not completely understood. The effects of DNA sequence on DNA melting in hexagonal assemblies are discussed below, within a simple thermodynamical model [170]. Namely, we predict that melting of hom DNAs is inhibited, while dstoss DNA transition for unrelated DNA sequences is facilitated.

In particular, it is straightforward to show that under the conditions favoring duplexduplex ES attraction, ideally helical hom DNAs melt at higher *T* due to a stabilization of DNA helical regions by mutual interactions. For hom DNA fragments, the model predicts a rise of the melting temperature *Tm* (typically by 310° at *R*=2328Å between DNAs in the assembly) and more cooperative DNA melting transitions. The shift of *Tm* scales with the strength of attraction, namely 0 1 3 ∝ − *T aa <sup>m</sup>* . It can thus be controlled in future melting experiments in dense DNA phases via addition of attractionmediating cations, e.g., counterions with enhanced binding into the DNA major groove and propensity to induce DNA aggregation.

than 12

λ

dependent interactions overviewed in Sec. 7. The physical mechanism here is more formidable. Namely, the melted ss DNA domains/bubbles are going to be created on ds DNAs to optimize *unfavorable ES repulsions* predicted for bprandom DNA stretches longer

cooperative in this case. An experimental support of some of these findings might come

Lastly, recently, I contributed to the invention of a method and apparatus to detect DNA melting and hybridization events with the help of bionanosensors functionalized with dense DNA lattices. The situations of direct DNA attachment to the sensor surface and DNA deposition via GNPs have been considered. For the former, the sensor signals recorded upon DNA hybridization were rationalized [177] based on the theoretical model of redistribution of mobile counterions in spaces between DNAs (Donnan equilibrium). For GNPDNA deposition, a more detailed ES screening model was developed [178]. This second detection setup allowed us to systematically monitor the change in the sensor response depending on mismatches in bp composition between probe and target ss DNA sequences. Such sensitivity is vital for a number of biological and biomedical applications involving detection of DNA sequences. This technique might also enable a detection of

*ProteinDNA recognition.* Despite enormous experimental and theoretical efforts, the recognition laws of DNAbinding proteins and their cognate sites on ds DNA remain quite obscure. Geometric shape complementarity of proteins with DNA grooves and protein DNA charge matching often drive the complex formation. Protein structures and their DNA recognition domains are extremely diverse that makes it hard to establish some universal rules of DNAprotein recognition. Several types of interactions can contribute to protein

The ES forces are known to dominate a nonspecific binding mode for a number of DNA protein complexes, e.g., weakly bound lac repressor [179]. DNAbinding domains of many relatively small proteins contain positively charged patches that ensure their ES attraction to the DNA. For large protein assemblies, the situation is often quite similar. For RNA Poly II, for instance, a strongly positively charged cleft is identified in the crystal structure along the path taken by DNARNA hybrid upon transcription [24]. For ribosomes, the basic residues are also located in protrusions of the structure expected to be involved in binding of

Indeed, Lys+ and Arg+ residues in DNAprotein complexes are often located only several Å away from DNA phosphates, Fig. 19. ES DNAprotein contacts are however often believed to bear little specificity to DNA sequence [180], merely providing a proximity of proteins to the DNA and allowing more sequencespecific and orientationdependent HB contacts to recognize HB donors and acceptors inside DNA bases [181]. Below, to confront this opinion, we propose an analytical model for proteinDNA ES recognition. Afterwards, via a systematic computational analysis of PBD structures of proteinDNA complexes, we justify

from *Tm* measurements in strongly confined wet DNA films [176].

sequencespecific effects in DNA melting predicted above.

tRNA/mRNA during translation [23].

DNA binding, with the ES and HB contacts being often the dominant ones.

this model for large architectural complexes of pro and eukaryotes.

*<sup>c</sup>* [55], see the dotdashed curve in Fig. 15a. The melting transition becomes less

Also, the model predicts a change in the character of the melting transition, from the second to the first order at a critical strength of DNADNA ES attraction [170], Fig. 18. Then, at the melting point the fraction of ds helical DNA regions exhibits a discontinuous change. At the isotropic cholesteric DNA transition, at moderate [DNA]≈100150 mg/ml, a clear indication of such *Tm* jumps by several degrees was observed and quantified already long ago [171].

Recently, for dense aggregates of 1050 nm GNPs linked by short DNA fragments [172] an extremely sharp DNA melting transition has been monitored, with a width of transition of 12°C, much smaller than ∼1020°C for melting of the same DNA in solution. This *enhanced cooperativity* of melting was attributed to shortrange DNADNA interactions that trigger an accumulation of ions from electrolyte in the overlapping double layers around DNAs [173], in a Donnanlike fashion. For dense DNA bundles connecting DNAfunctionalized GNPs [174], with DNADNA distances *R* = − 25 40 Å, a higher local [salt] near DNAs are realized, which in turn tends to stabilize the dsDNA state and increase *Tm* . The effective growth of *Tm* at these [DNA] was calculated to be 520° [173], based on a linear increase of *Tm* with log[[salt]] [172]. Once the melting of DNA bundle connecting GNPs starts, it progressively releases the excess counterions thus destabilizing the remaining ds DNA links. This might cause a sharp melting of the entire GNPsDNAs assembly, as detected in experiments [172].

Our DNA melting model for dense aggregates of identical DNA helices [170] does account for the Donnan equilibrium in the DNA lattice, that would give a corresponding [salt] induced rise of *Tm* [175]. The additional *Tm* shifts illustrated in Fig. 18 are however solely due to ES attraction of hom ds DNAs that appears also to be capable of inducing abrupt changes in the average DNA helicity.

Fig. 18. Melting profiles predicted for long hom DNAs in the hexagonal assembly at varying DNA density. Dots on the curves indicate the *Tm* at which an abrupt charge in DNA helicity occurs (between the stable branches of the melting curve, due to a Zshaped DNA melting isotherm realized at large enough DNADNA attraction). Parameters are the same as in Fig. 16. The figure is reprinted from [170], subject to ACS2005 Copyright.

For ds DNA random in sequence, pressurized externally to form dense assemblies, one can expect on the contrary a destabilization of the helical state [170] by DNADNA ES length

Also, the model predicts a change in the character of the melting transition, from the second to the first order at a critical strength of DNADNA ES attraction [170], Fig. 18. Then, at the melting point the fraction of ds helical DNA regions exhibits a discontinuous change. At the isotropic cholesteric DNA transition, at moderate [DNA]≈100150 mg/ml, a clear indication of such *Tm* jumps by several degrees was observed and quantified already long ago [171].

Recently, for dense aggregates of 1050 nm GNPs linked by short DNA fragments [172] an extremely sharp DNA melting transition has been monitored, with a width of transition of 12°C, much smaller than ∼1020°C for melting of the same DNA in solution. This *enhanced cooperativity* of melting was attributed to shortrange DNADNA interactions that trigger an accumulation of ions from electrolyte in the overlapping double layers around DNAs [173], in a Donnanlike fashion. For dense DNA bundles connecting DNAfunctionalized GNPs [174], with DNADNA distances *R* = − 25 40 Å, a higher local [salt] near DNAs are realized, which in turn tends to stabilize the dsDNA state and increase *Tm* . The effective growth of *Tm* at these [DNA] was calculated to be 520° [173], based on a linear increase of *Tm* with log[[salt]] [172]. Once the melting of DNA bundle connecting GNPs starts, it progressively releases the excess counterions thus destabilizing the remaining ds DNA links. This might cause a sharp melting of the entire GNPsDNAs assembly, as detected in experiments [172]. Our DNA melting model for dense aggregates of identical DNA helices [170] does account for the Donnan equilibrium in the DNA lattice, that would give a corresponding [salt] induced rise of *Tm* [175]. The additional *Tm* shifts illustrated in Fig. 18 are however solely due to ES attraction of hom ds DNAs that appears also to be capable of inducing abrupt

40 30 25 R23

0.01 0 0.01 0.02 0.03 0.04 0.05

*T Tm Tm*

Normalized T,

Fig. 18. Melting profiles predicted for long hom DNAs in the hexagonal assembly at varying DNA density. Dots on the curves indicate the *Tm* at which an abrupt charge in DNA helicity occurs (between the stable branches of the melting curve, due to a Zshaped DNA melting isotherm realized at large enough DNADNA attraction). Parameters are the same as in Fig.

For ds DNA random in sequence, pressurized externally to form dense assemblies, one can expect on the contrary a destabilization of the helical state [170] by DNADNA ES length

changes in the average DNA helicity.

0

16. The figure is reprinted from [170], subject to ACS2005 Copyright.

0.2

Fraction of ds

DNA

0.4

0.6

0.8

1

dependent interactions overviewed in Sec. 7. The physical mechanism here is more formidable. Namely, the melted ss DNA domains/bubbles are going to be created on ds DNAs to optimize *unfavorable ES repulsions* predicted for bprandom DNA stretches longer than 12 λ*<sup>c</sup>* [55], see the dotdashed curve in Fig. 15a. The melting transition becomes less cooperative in this case. An experimental support of some of these findings might come from *Tm* measurements in strongly confined wet DNA films [176].

Lastly, recently, I contributed to the invention of a method and apparatus to detect DNA melting and hybridization events with the help of bionanosensors functionalized with dense DNA lattices. The situations of direct DNA attachment to the sensor surface and DNA deposition via GNPs have been considered. For the former, the sensor signals recorded upon DNA hybridization were rationalized [177] based on the theoretical model of redistribution of mobile counterions in spaces between DNAs (Donnan equilibrium). For GNPDNA deposition, a more detailed ES screening model was developed [178]. This second detection setup allowed us to systematically monitor the change in the sensor response depending on mismatches in bp composition between probe and target ss DNA sequences. Such sensitivity is vital for a number of biological and biomedical applications involving detection of DNA sequences. This technique might also enable a detection of sequencespecific effects in DNA melting predicted above.

### 

*ProteinDNA recognition.* Despite enormous experimental and theoretical efforts, the recognition laws of DNAbinding proteins and their cognate sites on ds DNA remain quite obscure. Geometric shape complementarity of proteins with DNA grooves and protein DNA charge matching often drive the complex formation. Protein structures and their DNA recognition domains are extremely diverse that makes it hard to establish some universal rules of DNAprotein recognition. Several types of interactions can contribute to protein DNA binding, with the ES and HB contacts being often the dominant ones.

The ES forces are known to dominate a nonspecific binding mode for a number of DNA protein complexes, e.g., weakly bound lac repressor [179]. DNAbinding domains of many relatively small proteins contain positively charged patches that ensure their ES attraction to the DNA. For large protein assemblies, the situation is often quite similar. For RNA Poly II, for instance, a strongly positively charged cleft is identified in the crystal structure along the path taken by DNARNA hybrid upon transcription [24]. For ribosomes, the basic residues are also located in protrusions of the structure expected to be involved in binding of tRNA/mRNA during translation [23].

Indeed, Lys+ and Arg+ residues in DNAprotein complexes are often located only several Å away from DNA phosphates, Fig. 19. ES DNAprotein contacts are however often believed to bear little specificity to DNA sequence [180], merely providing a proximity of proteins to the DNA and allowing more sequencespecific and orientationdependent HB contacts to recognize HB donors and acceptors inside DNA bases [181]. Below, to confront this opinion, we propose an analytical model for proteinDNA ES recognition. Afterwards, via a systematic computational analysis of PBD structures of proteinDNA complexes, we justify this model for large architectural complexes of pro and eukaryotes.

, also make the well deeper. At zero [salt], the well depth drops as 3 <sup>∝</sup> 1 /*R* with proteinDNA separation *R*, while in electrolyte the decay is exponential,

2 *B B*

ε= .

− =−

*c k Tl M R z E z*

Here *z* is the mutual proteinDNA sliding distance with respect to the position of complete DNA and protein homology overlap at *z* =0, see Fig. 21. The charges are assumed to interact through a weaklypolarizable lowdielectric medium between DNA and

ε

ε

60 30 0 30 60

Sliding Distance, z,

κ= Å.

Fig. 21. ES recognition energy well upon sliding of a 1D "protein" over 1D DNA lattice. Parameters: *M*=11 charges in the hom region, *R*=10Å, <sup>2</sup> = 2 Å2. The dashed curve is zero

In this 1D model, the recognition well is accompanied by *energetic barriers*, both for the exact solution and simplistic expression in Eq. 6. The barriers disappear when charge displacements perpendicular to the DNAprotein plane are also taken into account. A generalization of this 1D model for protein and DNA charge displacements for 2D and 3D is more realistic, but only computationally feasible. Also note that some specific, not fully random displacement fields

 , can mimic charge patterns on a particular DNA sequence and for a given protein. ProteinDNA ES recognition well then resembles DNADNA barrierfree ES hom recognition well, sketched in Fig. 16. The calculation of DNAprotein ES recognition is methodologically

Let us consider one physical implication of this ES recognition. We have calculated [182] that this well is capable to slow down the protein diffusion, provoking protein trapping for ∼sms near this hom site on DNA. This time is long enough to allow some conformational changes in the protein structure (domain motions, rotation of sidechains, allosteric transition, etc.). Various protein conformation being sampled might trigger a stronger

salt limit, the solid curve describes a reduced ES recognition at 1/ 7

*<sup>D</sup>* ] . For weak charge fluctuations in nosalt limit the model returns an elegant

( )

+

*R z*

5/2 2 2 2

(6)

22 2

by 22 2 *n m* = +

λ

∝ − exp / [ *R*

and δ δ

expression for the average ES recognition energy well

2

0

2

4

Recognition

 Energy,

E,

*kB*T

6

similar to that for DNADNA recognition funnel, Fig. 16.

(chemical or HB) protein binding to this particular DNA fragment.

protein, with the dielectric constant of 2 *<sup>c</sup>*

( )

*Model of ES recognition.* A simple 1D model of DNAprotein recognition based on complementarity of their charge patterns was proposed in Ref. [182]. DNA and protein charge lattices were set commensurate for the cognate site and decorrelated for the rest of the DNA, Fig. 20. In the model, some random charge displacement fields along DNA *<sup>n</sup>*

and protein *<sup>m</sup>* δ mimic a sequence specificity of their charge patterns. This idealized protein is attracted stronger to a particular hom/matching segment on the idealized DNA.

Fig. 19. The distribution of ES potential on DNAprotein complexes of specifically bound lac repressor 1l1m.pdb, zinc finger ZIF268 1aay.pdb, leucine zipper GCN4 1ysa.pdb, and 146 bp NCP 1aoi.pdb (from left to right). The structures are visualized by MDL Chime and Protein Explorer programs, using the PDB files of the complexes. Images are not to scale.

Fig. 20. Schematic 1Dmodel of "proteinDNA" ES recognition. Charge positions on DNA and protein vary in a random fashion about quasiperiodic positions on 1D lattice.

The average ES recognition energy of a protein to this target site has been derived in the linear PB theory, for random realization of and δ fields. Both long and shortrange order situations for the charges on DNA were investigated (only longrange order results are shown here). For the parameters typical for lacrepressorlike proteins (∼10 positive charges at *R*=1nm from the DNA) the ES recognition well amounts to ≈310*k*B*T* in depth and a couple of nm in width, Fig. 21. Thus, this shortrange well cannot serve as "ES funnel" that would direct diffusing proteins from far away on the DNA to this chargehom binding site. This ES well is thus not expected to facilitate strongly the protein diffusion on DNA, the phenomenon known to take place e.g. for lac and galrepressors.

The well depth scales linearly with the number of charges in the hom domain, *M*. Larger magnitudes of charge deviations from their quasiperiodic positions on the lattice, described

*Model of ES recognition.* A simple 1D model of DNAprotein recognition based on complementarity of their charge patterns was proposed in Ref. [182]. DNA and protein charge lattices were set commensurate for the cognate site and decorrelated for the rest of the DNA, Fig. 20. In the model, some random charge displacement fields along DNA *<sup>n</sup>*

Fig. 19. The distribution of ES potential on DNAprotein complexes of specifically bound lac repressor 1l1m.pdb, zinc finger ZIF268 1aay.pdb, leucine zipper GCN4 1ysa.pdb, and 146 bp NCP 1aoi.pdb (from left to right). The structures are visualized by MDL Chime and Protein

δ

Fig. 20. Schematic 1Dmodel of "proteinDNA" ES recognition. Charge positions on DNA and protein vary in a random fashion about quasiperiodic positions on 1D lattice.

The average ES recognition energy of a protein to this target site has been derived in the

order situations for the charges on DNA were investigated (only longrange order results are shown here). For the parameters typical for lacrepressorlike proteins (∼10 positive charges at *R*=1nm from the DNA) the ES recognition well amounts to ≈310*k*B*T* in depth and a couple of nm in width, Fig. 21. Thus, this shortrange well cannot serve as "ES funnel" that would direct diffusing proteins from far away on the DNA to this chargehom binding site. This ES well is thus not expected to facilitate strongly the protein diffusion on DNA, the

The well depth scales linearly with the number of charges in the hom domain, *M*. Larger magnitudes of charge deviations from their quasiperiodic positions on the lattice, described

Explorer programs, using the PDB files of the complexes. Images are not to scale.

phenomenon known to take place e.g. for lac and galrepressors.

linear PB theory, for random realization of and

is attracted stronger to a particular hom/matching segment on the idealized DNA.

mimic a sequence specificity of their charge patterns. This idealized protein

fields. Both long and shortrange

δ

and protein *<sup>m</sup>*

δ

by 22 2 *n m* = + δ , also make the well deeper. At zero [salt], the well depth drops as 3 <sup>∝</sup> 1 /*R* with proteinDNA separation *R*, while in electrolyte the decay is exponential, ∝ − exp / [ *R* λ*<sup>D</sup>* ] . For weak charge fluctuations in nosalt limit the model returns an elegant expression for the average ES recognition energy well

$$
\Delta E \left( \Delta z \right) = -\frac{k\_\text{g} T l\_\text{g} M \Omega^2 \varepsilon}{2 \varepsilon\_\circ} \frac{R^2 - 2 \Delta z^2}{\left( R^2 + \Delta z^2 \right)^{5/2}} \tag{6}
$$

Here *z* is the mutual proteinDNA sliding distance with respect to the position of complete DNA and protein homology overlap at *z* =0, see Fig. 21. The charges are assumed to interact through a weaklypolarizable lowdielectric medium between DNA and protein, with the dielectric constant of 2 *<sup>c</sup>* ε= .

Fig. 21. ES recognition energy well upon sliding of a 1D "protein" over 1D DNA lattice. Parameters: *M*=11 charges in the hom region, *R*=10Å, <sup>2</sup> = 2 Å2. The dashed curve is zero salt limit, the solid curve describes a reduced ES recognition at 1/ 7 κ= Å.

In this 1D model, the recognition well is accompanied by *energetic barriers*, both for the exact solution and simplistic expression in Eq. 6. The barriers disappear when charge displacements perpendicular to the DNAprotein plane are also taken into account. A generalization of this 1D model for protein and DNA charge displacements for 2D and 3D is more realistic, but only computationally feasible. Also note that some specific, not fully random displacement fields and δ , can mimic charge patterns on a particular DNA sequence and for a given protein. ProteinDNA ES recognition well then resembles DNADNA barrierfree ES hom recognition well, sketched in Fig. 16. The calculation of DNAprotein ES recognition is methodologically similar to that for DNADNA recognition funnel, Fig. 16.

Let us consider one physical implication of this ES recognition. We have calculated [182] that this well is capable to slow down the protein diffusion, provoking protein trapping for ∼sms near this hom site on DNA. This time is long enough to allow some conformational changes in the protein structure (domain motions, rotation of sidechains, allosteric transition, etc.). Various protein conformation being sampled might trigger a stronger (chemical or HB) protein binding to this particular DNA fragment.

For the NCPs, the histone positive charges are mainly localized in the outer "ring" of the

the core, further away from the DNA. Our analysis demonstrates that N+ atoms on Arg+ and Lys+ track the positions of *individual* DNA phosphates, as visualized in the histogram 1 2 *s s* − for all N+ charges in DNA vicinity, Fig. 23. The *bimodal distributions* were detected both for individual NCPs (a good statistics can be achieved for a single particle) and for the entire family of 146 bp long complexes [184]. This indicates that N+ on Lys and Arg are encountered measurably more often close to one of the neighboring DNA phosphates than between the two, maximizing the ES attraction to DNA. As the structure and positions of DNA phosphates strongly correlate to DNA bp sequence [52], this detected "charge tracking" for DNA sequences wrapped in NCPs yields sequencespecific ES DNAprotein forces. This supports our model hypothesis above about commensurate charge lattices on DNA target sequence and on the protein. This fact can contribute to NCP positioning on genomic DNAs, interfering with the mechanism of sequencespecific DNA bendability believed to govern this process [185]. A similar bimodal distribution was obtained for prokaryotic NCP analogs (not shown) [184].

4 2 0 2 4

Fig. 23. A bimodal distribution of 1 2 *s s* − distances for 14 NCP complexes. It indicates the ES recognition of individual DNA phosphates by the closest N+ atoms on Arg and Lys basic

The specificity of ES binding of Arg and Lys of core histones into the DNA grooves on NCPs has recently been examined by other groups too [186]. The ESdirected localization of Arg+ in the minor grooves in ATrich DNA regions was confirmed for NCPs in Ref. [187]. These studies emphasize that for DNA sequences wrapped in NCPs the ATtracts have particularly narrow minor grooves that offers "attractive" sites for Arg+ and Lys+ binding every time the DNA minor groove faces the histone octamer. Arg was claimed to be preferred over Lys in the DNA minor grooves because of a lower selfenergy cost to remove a larger guanidinium group of Arg+ from its hydrated state in solution. The reason is the

Being valid for large structural complexes, the ES complementarity model fails for small DNAprotein complexes, with the standard simple motifs of DNA recognition (e.g., helix turnhelix, zinc finger, leucine zipper). For a large set of small proteins from these families, we could not detect any statistical preference in distribution of Lys and Arg close to DNA phosphates [184]. To make a more definite conclusion, some redundancy in protein

Born ES selfenergy that scales inversely proportional with the "ion" radius.

*s*<sup>1</sup>*s*2,

and Glu

acids being rather inside

octamer, close to the wrapped DNA, see Fig. 19d, with Asp

*N*, Sum

residues of the histone core proteins.

Our hypothesis is thus a *twostep* mechanism of recognition for some proteins. First, a DNA binding protein scans the ES surface of DNA for a chargecomplementary site. In this "searching" mode, the protein structure is flexible and adaptable to the pattern of interaction sites on DNA. When a commensurate DNA fragment is found, some interactioninduced folding solidifies the protein structure, switching it into the "binding" mode that enables stronger and more specific contacts with the DNA. Cumulative ES and HB contacts rigidify the protein structure and give rise to formation of specifically bound DNAprotein complexes.

This kind of hotspot twostep recognition mechanism is pretty common in structural molecular biology, both for proteinDNA and proteinprotein complexes. For the latter, the twostep docking directs the assembly pathway into the native structure by "anchoring" of a shape complementary relatively rigid "key" domain of one protein into a "lock" domain in the surface of another protein [183]. This process is accompanied by a large burial of solvent accessible surface area and this water release amplifies further docking of proteins into a tight complex.

*Analysis of PDB structures.* To justify the analytical predictions above, the detailed analysis of PDB structures of different classes of proteins in their complexes with the DNA has been performed. The distribution of NH2+ groups on Arginine+ and Lysine+ residues in DNA binding domains of DNAprotein complexes has been examined [184]. In particular, large structural complexes were studied: the NCPs of eukaryotes and architectural proteins of prokaryotes, both involving extensive DNA wrapping around the basic protein cores and featuring mainly ES mode of binding. A homewritten Mathematica 6 computer code was used for extracting the coordinates of DNA phosphates PO4 and protein's N+ and O atoms on the charged amino acids from the PDB files. We could analyse the distances from N+ atoms on Arg+ and Lys+ that are within ≈7Å from the closest ( <sup>1</sup> *s* ) and next closest ( <sup>2</sup> *s* ) DNA phosphates on the same DNA strand, see Fig. 22. The statistics of ES contacts and salt bridges in these DNAprotein complexes has thus been restored. Smaller cutoff distances of 35Å can also be used, to minimize the contributions of charges across the narrow DNA groove. Note that fluctuationinduced uncertainties in positions of protein charges in crystals of many DNA protein complexes are often ∼12Å. This is much smaller than the relevant periodicity in the system, the phosphatephosphate separation along the DNA helical strand, *ph s* ≈7Å.

Fig. 22. Definition of 1,2 *s* distances for the protein positive charges (in blue) which are closer than ~ *<sup>B</sup> r l* ≈7Å to the negative DNA phosphates (red helix).

Our hypothesis is thus a *twostep* mechanism of recognition for some proteins. First, a DNA binding protein scans the ES surface of DNA for a chargecomplementary site. In this "searching" mode, the protein structure is flexible and adaptable to the pattern of interaction sites on DNA. When a commensurate DNA fragment is found, some interactioninduced folding solidifies the protein structure, switching it into the "binding" mode that enables stronger and more specific contacts with the DNA. Cumulative ES and HB contacts rigidify the protein structure and give rise to formation of specifically bound DNAprotein complexes. This kind of hotspot twostep recognition mechanism is pretty common in structural molecular biology, both for proteinDNA and proteinprotein complexes. For the latter, the twostep docking directs the assembly pathway into the native structure by "anchoring" of a shape complementary relatively rigid "key" domain of one protein into a "lock" domain in the surface of another protein [183]. This process is accompanied by a large burial of solvent accessible surface area and this water release amplifies further docking of proteins into a tight complex. *Analysis of PDB structures.* To justify the analytical predictions above, the detailed analysis of PDB structures of different classes of proteins in their complexes with the DNA has been performed. The distribution of NH2+ groups on Arginine+ and Lysine+ residues in DNA binding domains of DNAprotein complexes has been examined [184]. In particular, large structural complexes were studied: the NCPs of eukaryotes and architectural proteins of prokaryotes, both involving extensive DNA wrapping around the basic protein cores and featuring mainly ES mode of binding. A homewritten Mathematica 6 computer code was

used for extracting the coordinates of DNA phosphates PO4 and protein's N+ and O

system, the phosphatephosphate separation along the DNA helical strand, *ph s* ≈7Å.

than ~ *<sup>B</sup> r l* ≈7Å to the negative DNA phosphates (red helix).

Fig. 22. Definition of 1,2 *s* distances for the protein positive charges (in blue) which are closer

the charged amino acids from the PDB files. We could analyse the distances from N+ atoms on Arg+ and Lys+ that are within ≈7Å from the closest ( <sup>1</sup> *s* ) and next closest ( <sup>2</sup> *s* ) DNA phosphates on the same DNA strand, see Fig. 22. The statistics of ES contacts and salt bridges in these DNAprotein complexes has thus been restored. Smaller cutoff distances of 35Å can also be used, to minimize the contributions of charges across the narrow DNA groove. Note that fluctuationinduced uncertainties in positions of protein charges in crystals of many DNA protein complexes are often ∼12Å. This is much smaller than the relevant periodicity in the

atoms on

For the NCPs, the histone positive charges are mainly localized in the outer "ring" of the octamer, close to the wrapped DNA, see Fig. 19d, with Asp and Glu acids being rather inside the core, further away from the DNA. Our analysis demonstrates that N+ atoms on Arg+ and Lys+ track the positions of *individual* DNA phosphates, as visualized in the histogram 1 2 *s s* − for all N+ charges in DNA vicinity, Fig. 23. The *bimodal distributions* were detected both for individual NCPs (a good statistics can be achieved for a single particle) and for the entire family of 146 bp long complexes [184]. This indicates that N+ on Lys and Arg are encountered measurably more often close to one of the neighboring DNA phosphates than between the two, maximizing the ES attraction to DNA. As the structure and positions of DNA phosphates strongly correlate to DNA bp sequence [52], this detected "charge tracking" for DNA sequences wrapped in NCPs yields sequencespecific ES DNAprotein forces. This supports our model hypothesis above about commensurate charge lattices on DNA target sequence and on the protein. This fact can contribute to NCP positioning on genomic DNAs, interfering with the mechanism of sequencespecific DNA bendability believed to govern this process [185]. A similar bimodal distribution was obtained for prokaryotic NCP analogs (not shown) [184].

Fig. 23. A bimodal distribution of 1 2 *s s* − distances for 14 NCP complexes. It indicates the ES recognition of individual DNA phosphates by the closest N+ atoms on Arg and Lys basic residues of the histone core proteins.

The specificity of ES binding of Arg and Lys of core histones into the DNA grooves on NCPs has recently been examined by other groups too [186]. The ESdirected localization of Arg+ in the minor grooves in ATrich DNA regions was confirmed for NCPs in Ref. [187]. These studies emphasize that for DNA sequences wrapped in NCPs the ATtracts have particularly narrow minor grooves that offers "attractive" sites for Arg+ and Lys+ binding every time the DNA minor groove faces the histone octamer. Arg was claimed to be preferred over Lys in the DNA minor grooves because of a lower selfenergy cost to remove a larger guanidinium group of Arg+ from its hydrated state in solution. The reason is the Born ES selfenergy that scales inversely proportional with the "ion" radius.

Being valid for large structural complexes, the ES complementarity model fails for small DNAprotein complexes, with the standard simple motifs of DNA recognition (e.g., helix turnhelix, zinc finger, leucine zipper). For a large set of small proteins from these families, we could not detect any statistical preference in distribution of Lys and Arg close to DNA phosphates [184]. To make a more definite conclusion, some redundancy in protein

DNAprotein complex formation [4,194]. And, it is possible that protein and DNA charged groups complexed together via ES attraction do not fully compensate for the energetic loses upon their "ES desolvation". The latter depends crucially on the εvalue assigned to the protein and its immediate vicinity. For DNAprotein complexes, the entropic effects of condensed cations released from the DNA, with the number defined by the slope of log[binding constant] on log [salt], are often presented as the main *driving force* for the complexation. Here, the situation is rather similar to counterion release from DNA(CL membrane) complexes, Sec. 4. In both cases, we however tend to think that the direct ES attraction between the oppositely charged components of the system governs/directs the complex formation, while the entropic free energy gain due to the release of condensed

In this chapter, we focused on recent developments and new viewpoints on ES effects for a number of biological DNArelated systems. Several experimental achievements and DNA related phenomena discovered in the last years have been overviewed, which challenge both theoretical and computational modeling. Some analytical insights from our recent studies are discussed, which uncover general principles behind chargemediated DNA DNA, NCPNCP, and DNAprotein interactions. We aimed at describing macroscopic effects having their possible origin in ES interactions as well as at trying to establish correlations between the structure of the system components and their function. The advanced theoretical and computational approaches developed in our studies on DNA DNA, DNAmembrane and DNAproteins interactions can find their applications in bio

The PE models for DNA and available structure information for the proteins have been applied to some nanotechnology applications, the principles of biomolecular DNAprotein recognition, and selfassembly. Despite inherent limitations of the meanfield PBlike theories applied to the DNA, the approaches developed often enabled us to rationalize the structural properties of the system as dictated by intermolecular forces. The conceptual framework proposed in the chapter allows us to anticipate the physical effects in these DNArelated systems that are still too large for modern *ab initio* computer simulations. Clearly, more work is to be done to achieve a quantitative understanding of these complex phenomena. In particular, the physical properties of interconnected NCPs in 30 nm chromatin fibers and DNA packaging inside bacteriophages feature a number of important biological details to be incorporated in future theoretical models. Another area is ES effects in proteinmediated loop formation in DNA [195], DNA plectonemes [196,197] and cyclization [198], as well as DNA wrapping in NCPs [199]. These interesting phenomena are

One hot and intriguing domain of our ESrelated biological research is DNA packaging inside viral capsids and selfassembly of viral shells from the capsid proteins [27]. Both processes are highly sensitive to salt conditions that control proteinprotein and DNADNA ES forces. We argue here that the accurate physical description of DNA compactification inside viral shells demands the application of all theories and models presented in the main

counterions accompanies this process.

technology and nanoengineering.

however beyond the scope of this contribution.

text. Let us list the effects one by one.

structures is to be excluded and a grouping into smaller, more specific protein subfamilies is to be performed.

A tentative explanation is however as follows. For large complexes, with ∼30100 ES DNA protein contacts, the ES energy gain being EScommensurate can reach ∼1030*k*B*T* and proteins appear to utilize it for sequencespecific binding to DNA. For small proteins, with only ∼310 ES contacts and much weaker ES binding, other interactions (such as HBs) are likely to direct the recognition of specific DNA sequences in complexes.

Note that DNAprotein ES commensurability for NCPs and their prokaryotic analogs resembles a *zipperlike* positioning of positive and negative amino acids along interfaces of many proteinprotein complexes [18,188]. For the latter, despite hydrophobic residues often dominate the overall binding affinity, these noncharged amino acids might be too abundant to ensure a proper degree of the binding specificity. The latter might stem form *charge patchiness* and propensity of HB formation between the residues along the contact surface of the bound proteins [189]. Analogously, for DNADNA interactions, overviewed in Sec. 2, we have seen that DNADNA attraction is only possible for entirely commensurate or hom sequences, for which *charge zipper motif* is realized. All these similarities for proteinDNA, proteinprotein, and DNADNA interactions reflect different aspects of the universal principle of ES complementarity in structural molecular biology.

*Challenges and Perspectives.* Several issues of description of biophysics of DNAprotein interactions challenge future theoretical developments. First, the computational analysis of PDB data presented above provides us only with a *statistical preference* of distribution of protein charges in DNA proximity. To evaluate the ES binding energies of DNAprotein complexes, one needs to speculate about the value of the dielectric constant *<sup>c</sup>* ε in spaces between DNA and protein. Because of dielectric saturation effects in confined/hydrated water molecules on the charged objects [190], its value can vary widely, *<sup>c</sup>* ε ∼2÷30 [11,12]. So does the ES interaction energy [191]. Another important ES issue is the charged state of ionizable protein residues in a particular neighborhood in DNAprotein complexes (e.g., Hist), with their pKa value being affected by local ES potential, geometrical shape of the protein surface, [salt], local dielectric permittivity, etc. [11,192].

NonES van der Waals and HB contacts, as well as the entropic terms associated with water release and counterion evaporation upon proteinDNA binding, are to be quantified in the future models as well. To make unambiguous conclusion about the mechanism of binding specificity for a given DNAprotein complex, the ES preference of Arg/Lys positioning with respect to DNA phosphates has to be supplemented by the analysis of HB formation propensity between the protein residues and DNA bases [181]. Also, one has to keep in mind that the protein (and DNA) structure visualized by xrays in crystals exposed to special crystallization buffers [193] might measurably differ from real structures stable at physiological conditions.

There exists an opinion in the literature that ES contacts of charged residues in proteinDNA complexes [4] and along proteinprotein interfaces [11] might (somewhat counter intuitively) *destabilize instead of stabilize* their binding. The argument goes as follows. The release of ESprofitable structured water shells around the constituents often accompanies DNAprotein complex formation [4,194]. And, it is possible that protein and DNA charged groups complexed together via ES attraction do not fully compensate for the energetic loses upon their "ES desolvation". The latter depends crucially on the εvalue assigned to the protein and its immediate vicinity. For DNAprotein complexes, the entropic effects of condensed cations released from the DNA, with the number defined by the slope of log[binding constant] on log [salt], are often presented as the main *driving force* for the complexation. Here, the situation is rather similar to counterion release from DNA(CL membrane) complexes, Sec. 4. In both cases, we however tend to think that the direct ES attraction between the oppositely charged components of the system governs/directs the complex formation, while the entropic free energy gain due to the release of condensed counterions accompanies this process.

#### 

structures is to be excluded and a grouping into smaller, more specific protein subfamilies

A tentative explanation is however as follows. For large complexes, with ∼30100 ES DNA protein contacts, the ES energy gain being EScommensurate can reach ∼1030*k*B*T* and proteins appear to utilize it for sequencespecific binding to DNA. For small proteins, with only ∼310 ES contacts and much weaker ES binding, other interactions (such as HBs) are

Note that DNAprotein ES commensurability for NCPs and their prokaryotic analogs resembles a *zipperlike* positioning of positive and negative amino acids along interfaces of many proteinprotein complexes [18,188]. For the latter, despite hydrophobic residues often dominate the overall binding affinity, these noncharged amino acids might be too abundant to ensure a proper degree of the binding specificity. The latter might stem form *charge patchiness* and propensity of HB formation between the residues along the contact surface of the bound proteins [189]. Analogously, for DNADNA interactions, overviewed in Sec. 2, we have seen that DNADNA attraction is only possible for entirely commensurate or hom sequences, for which *charge zipper motif* is realized. All these similarities for proteinDNA, proteinprotein, and DNADNA interactions reflect different aspects of the universal

*Challenges and Perspectives.* Several issues of description of biophysics of DNAprotein interactions challenge future theoretical developments. First, the computational analysis of PDB data presented above provides us only with a *statistical preference* of distribution of protein charges in DNA proximity. To evaluate the ES binding energies of DNAprotein

between DNA and protein. Because of dielectric saturation effects in confined/hydrated

does the ES interaction energy [191]. Another important ES issue is the charged state of ionizable protein residues in a particular neighborhood in DNAprotein complexes (e.g., Hist), with their pKa value being affected by local ES potential, geometrical shape of the

NonES van der Waals and HB contacts, as well as the entropic terms associated with water release and counterion evaporation upon proteinDNA binding, are to be quantified in the future models as well. To make unambiguous conclusion about the mechanism of binding specificity for a given DNAprotein complex, the ES preference of Arg/Lys positioning with respect to DNA phosphates has to be supplemented by the analysis of HB formation propensity between the protein residues and DNA bases [181]. Also, one has to keep in mind that the protein (and DNA) structure visualized by xrays in crystals exposed to special crystallization buffers [193] might measurably differ from real structures stable at

There exists an opinion in the literature that ES contacts of charged residues in proteinDNA complexes [4] and along proteinprotein interfaces [11] might (somewhat counter intuitively) *destabilize instead of stabilize* their binding. The argument goes as follows. The release of ESprofitable structured water shells around the constituents often accompanies

ε

∼2÷30 [11,12]. So

ε

in spaces

complexes, one needs to speculate about the value of the dielectric constant *<sup>c</sup>*

water molecules on the charged objects [190], its value can vary widely, *<sup>c</sup>*

likely to direct the recognition of specific DNA sequences in complexes.

principle of ES complementarity in structural molecular biology.

protein surface, [salt], local dielectric permittivity, etc. [11,192].

physiological conditions.

is to be performed.

In this chapter, we focused on recent developments and new viewpoints on ES effects for a number of biological DNArelated systems. Several experimental achievements and DNA related phenomena discovered in the last years have been overviewed, which challenge both theoretical and computational modeling. Some analytical insights from our recent studies are discussed, which uncover general principles behind chargemediated DNA DNA, NCPNCP, and DNAprotein interactions. We aimed at describing macroscopic effects having their possible origin in ES interactions as well as at trying to establish correlations between the structure of the system components and their function. The advanced theoretical and computational approaches developed in our studies on DNA DNA, DNAmembrane and DNAproteins interactions can find their applications in bio technology and nanoengineering.

The PE models for DNA and available structure information for the proteins have been applied to some nanotechnology applications, the principles of biomolecular DNAprotein recognition, and selfassembly. Despite inherent limitations of the meanfield PBlike theories applied to the DNA, the approaches developed often enabled us to rationalize the structural properties of the system as dictated by intermolecular forces. The conceptual framework proposed in the chapter allows us to anticipate the physical effects in these DNArelated systems that are still too large for modern *ab initio* computer simulations. Clearly, more work is to be done to achieve a quantitative understanding of these complex phenomena. In particular, the physical properties of interconnected NCPs in 30 nm chromatin fibers and DNA packaging inside bacteriophages feature a number of important biological details to be incorporated in future theoretical models. Another area is ES effects in proteinmediated loop formation in DNA [195], DNA plectonemes [196,197] and cyclization [198], as well as DNA wrapping in NCPs [199]. These interesting phenomena are however beyond the scope of this contribution.

One hot and intriguing domain of our ESrelated biological research is DNA packaging inside viral capsids and selfassembly of viral shells from the capsid proteins [27]. Both processes are highly sensitive to salt conditions that control proteinprotein and DNADNA ES forces. We argue here that the accurate physical description of DNA compactification inside viral shells demands the application of all theories and models presented in the main text. Let us list the effects one by one.

G. Wuite and G. Zanchetta for many stimulating discussions and scientific correspondence. Without a constant help and encouragement of my collaborators (see joint publications below) this chapter would not be possible. A part of this work was supported by the German Research Foundation, DFG Grants CH 707/51 and CH 707/22. Because of space restrictions, only a small fraction of relevant studies has been cited in the text (my apologies

to the authors which could not be mentioned).

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Many dsDNA bacteriophages pack their DNA in a very dense and wellorganized fashion [200,201,202]. DNA densities can reach *R*≈2328 Å between DNA axes, creating osmotic pressures of up to ∼50 atm inside the shells [203,204,205]. At such DNA densities, the effects of DNA helical structure onto DNADNA ES forces are going to be extremely pronounced, see

Fig. 24. These effects have been however largely neglected in the existing theories of DNA compaction in ds DNA viruses.

As the capsids of some dsDNA viruses are penetrable for small ions [40], the presence of di and trivalent cations in the solution can render DNADNA ES forces inside the capsids more attractive [212]. This will ease DNA packaging into and inhibit the DNA ejection from such capsids. Indeed, only 1 mM of spermine4+ in the buffer blocks nearly 90% of DNA inside the λphage capsids [206]. The counterionmediated DNADNA attraction inside viral capsids is a clear target for our ES interaction theory presented in Sec. 2.

Typical for dsDNA viruses are the concentric rings of DNA [207,208,209], with the DNA layers that are closer to the viral shell being resolved better by the cryoEM image reconstruction, see Fig. 24. This corresponds to a coaxial inversespool model of DNA packing, with the outer (more ordered) shells of the DNA spool being filled first. Recently, oriented DNA toroids condensed with spermine4+ inside T4 phages [75] and DNA "domainwall" transitions upon DNA ejection from T5 phages [210] have been clearly visualized by cryoEM. The interaction of toroidal DNA condensates overviewed in Sec. 3 with the confining protein shells of the capsid is a proper model to describe this "deformed toroid" conformation of spooling DNA [76]. Cholesteric ES effects, see Sec. 5, onto the DNA packing properties were also argued to be important for many ds DNA viruses. Lastly, the pauses in DNA packaging and ejection caused by necessary rearrangements of DNA spool [211] might stem from ES friction between the densely packed DNA layers inside the capsid, see Sec. 8.

Fig. 24. The result of 3D reconstruction of cryoEM images with DNA layers inside P22 virus (a), with DNA layers being more ordered near the portal region (b, bottom). Images are the courtesy of J. Johnson.

#### 

I am thankful to T. Bellini, R. Everaers, N. Kleckner, A. Kolomeisky, A. Kornyshev, D. Lee, S. Leikin, S. Malinin, W. Olson, E. Petrov, A. Poghossian, M. Prentiss, E. Starostin, R. Winkler, G. Wuite and G. Zanchetta for many stimulating discussions and scientific correspondence. Without a constant help and encouragement of my collaborators (see joint publications below) this chapter would not be possible. A part of this work was supported by the German Research Foundation, DFG Grants CH 707/51 and CH 707/22. Because of space restrictions, only a small fraction of relevant studies has been cited in the text (my apologies to the authors which could not be mentioned).

#### 

Many dsDNA bacteriophages pack their DNA in a very dense and wellorganized fashion [200,201,202]. DNA densities can reach *R*≈2328 Å between DNA axes, creating osmotic pressures of up to ∼50 atm inside the shells [203,204,205]. At such DNA densities, the effects of DNA helical structure onto DNADNA ES forces are going to be extremely pronounced, see Fig. 24. These effects have been however largely neglected in the existing theories of DNA

As the capsids of some dsDNA viruses are penetrable for small ions [40], the presence of di and trivalent cations in the solution can render DNADNA ES forces inside the capsids more attractive [212]. This will ease DNA packaging into and inhibit the DNA ejection from such capsids. Indeed, only 1 mM of spermine4+ in the buffer blocks nearly 90% of DNA inside the λphage capsids [206]. The counterionmediated DNADNA attraction inside viral

Typical for dsDNA viruses are the concentric rings of DNA [207,208,209], with the DNA layers that are closer to the viral shell being resolved better by the cryoEM image reconstruction, see Fig. 24. This corresponds to a coaxial inversespool model of DNA packing, with the outer (more ordered) shells of the DNA spool being filled first. Recently, oriented DNA toroids condensed with spermine4+ inside T4 phages [75] and DNA "domainwall" transitions upon DNA ejection from T5 phages [210] have been clearly visualized by cryoEM. The interaction of toroidal DNA condensates overviewed in Sec. 3 with the confining protein shells of the capsid is a proper model to describe this "deformed toroid" conformation of spooling DNA [76]. Cholesteric ES effects, see Sec. 5, onto the DNA packing properties were also argued to be important for many ds DNA viruses. Lastly, the pauses in DNA packaging and ejection caused by necessary rearrangements of DNA spool [211] might stem from ES

Fig. 24. The result of 3D reconstruction of cryoEM images with DNA layers inside P22 virus (a), with DNA layers being more ordered near the portal region (b, bottom). Images are the

I am thankful to T. Bellini, R. Everaers, N. Kleckner, A. Kolomeisky, A. Kornyshev, D. Lee, S. Leikin, S. Malinin, W. Olson, E. Petrov, A. Poghossian, M. Prentiss, E. Starostin, R. Winkler,

capsids is a clear target for our ES interaction theory presented in Sec. 2.

friction between the densely packed DNA layers inside the capsid, see Sec. 8.

compaction in ds DNA viruses.

courtesy of J. Johnson.


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**Part 2** 

**Bioengineering** 


## **Part 2**

**Bioengineering** 

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 *USA* 

**Electrostatics in** 

 **Protein Engineering and Design** 

I. John Khan, James A. Stapleton, Douglas Pike and Vikas Nanda

The electrostatic interactions between charged atoms in natural proteins play a central role in specifying protein topology, modulating stability of the molecule, and allowing for the impressive catalytic properties of enzymes. In this chapter, we discuss how protein engineers use the principles of electrostatics and computational protein modeling to develop new proteins for biomedical and biotechnological applications. First, a general introduction is given to familiarize the reader with the important factors to consider in protein electrostatics, and the nature of these electrostatic forces. The next section describes various levels of theory used for modeling electrostatics in proteins. The last sections focus on specific applications in two conceptual classes: the engineering of ionic interactions (1) on protein surfaces, and (2) within the hydrophobic protein core. In both cases, the aim is to

Interacting ionic species undergo rearrangement of their charge distributions under the influence of each other and their local environment. In electrostatics, we consider the static electrical field that is formed between these charged species once charge rearrangement has occurred. In the context of a protein, this amounts to looking at the many interactions among the polar and/or charged residues scattered throughout the three dimensional structure. Uncharged polar residues can form hydrogen bonding interactions with the hydroxyl (serine and threonine) and amide (asparagine and glutamine) hydrogen bond donors and acceptors on their side chains. Ionizable, or charged, residues have the following titratable side groups: carboxyl (asparate and glutamate), sulfhydryl (cysteine), hydroxyl (tyrosine), guanidino (arginine), amino (lysine), and imidazole (histidine). The ionization state of a titratable residue depends on its pKa value or proton affinity, which represents the pH at which there is equilibrium between the neutral and charged forms of their respective

Electrostatic interactions with the local environment influence the pKa values of titratable residues. These factors are manifested in the following relationship for the pKa of a buried

**1. Introduction** 

functional groups.

promote stability or to control molecular recognition.

residue (Bashford and Karplus 1990; Kaushik *et al.* 2006):

**2. Important factors influencing protein electrostatics** 

*University of Medicine and Dentistry of New Jersey, Piscataway, NJ,* 

### **Electrostatics in Protein Engineering and Design**

I. John Khan, James A. Stapleton, Douglas Pike and Vikas Nanda *University of Medicine and Dentistry of New Jersey, Piscataway, NJ, USA* 

### **1. Introduction**

The electrostatic interactions between charged atoms in natural proteins play a central role in specifying protein topology, modulating stability of the molecule, and allowing for the impressive catalytic properties of enzymes. In this chapter, we discuss how protein engineers use the principles of electrostatics and computational protein modeling to develop new proteins for biomedical and biotechnological applications. First, a general introduction is given to familiarize the reader with the important factors to consider in protein electrostatics, and the nature of these electrostatic forces. The next section describes various levels of theory used for modeling electrostatics in proteins. The last sections focus on specific applications in two conceptual classes: the engineering of ionic interactions (1) on protein surfaces, and (2) within the hydrophobic protein core. In both cases, the aim is to promote stability or to control molecular recognition.

#### **2. Important factors influencing protein electrostatics**

Interacting ionic species undergo rearrangement of their charge distributions under the influence of each other and their local environment. In electrostatics, we consider the static electrical field that is formed between these charged species once charge rearrangement has occurred. In the context of a protein, this amounts to looking at the many interactions among the polar and/or charged residues scattered throughout the three dimensional structure. Uncharged polar residues can form hydrogen bonding interactions with the hydroxyl (serine and threonine) and amide (asparagine and glutamine) hydrogen bond donors and acceptors on their side chains. Ionizable, or charged, residues have the following titratable side groups: carboxyl (asparate and glutamate), sulfhydryl (cysteine), hydroxyl (tyrosine), guanidino (arginine), amino (lysine), and imidazole (histidine). The ionization state of a titratable residue depends on its pKa value or proton affinity, which represents the pH at which there is equilibrium between the neutral and charged forms of their respective functional groups.

Electrostatic interactions with the local environment influence the pKa values of titratable residues. These factors are manifested in the following relationship for the pKa of a buried residue (Bashford and Karplus 1990; Kaushik *et al.* 2006):

Electrostatics in Protein Engineering and Design 45

environment, the dipoles of water molecules are free to reorient, hence the dielectric coefficient of water is relatively high with a value of ~78. In contrast, the dielectric coefficient of the protein is lower due to the limited mobility of the protein chain and the nonpolar nature of many amino acid residues. The dielectric coefficient within a protein varies with location, with values of 2-4 for regions having residues that are virtually inaccessible to water (i.e., the hydrophobic core), increasing to values of ~37 near the surface of the protein (Anslyn and Dougherty 2006). As a rule of thumb, we consider the range of electrostatic interactions to be dependent on the dielectric property of the medium according to the plots shown in Figure 1. For example, the energy between point charges in water (*ε* = 78) cannot be discriminated from baseline thermal energy at a separation of ~2 Å as a result of the charge screening by dipoles of water molecules. In the region below the protein surface (*ε* = 10) the effective separation increases to ~ 14Å, and within the hydrophobic core (*ε* = 4) the effective range can be greater than 30 Å. This difference implies that polar and charged residues have greater electrostatic potential when they are buried within the protein. The other factor influencing electrostatic interactions is the ionic strength which also has a screening effect of charge, particularly at the

Here we focus on the treatment of electrostatics in protein engineering design. For a more general discussion of electrostatics in proteins, we refer the reader to several excellent reviews (Neves-Petersen and Petersen 2003; Bosshard *et al.* 2004; Jelesarov and Karshikoff

**0 5 10 15 20 25 30**

**baseline thermal energy (kBT)**

**Distance (Å)**

Fig. 1. The change in free energy (ΔG) associated with separating two point charges of opposite sign when surrounded by a medium of dielectric coefficient ε in the absence of salt. The values are calculated using Coulomb's law based on point charges of +0.5 and -0.5. The dashed line represents the baseline thermal energy at 298 K, *kBT*, where *kB* is the Boltzmann

surface of the protein.

2009; Pace *et al.* 2009; Kukic and Nielsen 2010).

**-6**

**-5**

**-4**

**-3**

**ΔG (kcal/mol)**

**-2**

**-1**

ε = 4

**0**

constant and *T* is the temperature.

$$\mathbf{p} \mathbf{K}\_{\mathrm{a}} = \mathbf{p} \mathbf{K}\_{\mathrm{a,model}} + \Delta \mathbf{p} \mathbf{K}\_{\mathrm{desolv}} + \Delta \mathbf{p} \mathbf{K}\_{\mathrm{back}} + \Delta \mathbf{p} \mathbf{K}\_{\mathrm{coulomb}} \tag{1}$$

The pKa value for each model residue (pKa,model) has been experimentally determined. This value represents the pKa of the residue when it is completely surrounded by water. Adjustments to the pKa,model as the residue becomes buried are due to the following three factors. The first is the ΔpKdesolv, which is the change in pKa due to the unfavorable removal of an ionized residue from water to the hydrophobic protein core (i.e., a desolvation penalty). The second is the ΔpKback, which is the change in pKa due to interactions of the buried ionized residue with background charges present within the protein. Background charges are defined as the partial charges of atoms that are manifested as either permanent or induced dipoles in molecules. Examples of background charges are the permanent dipole of a water molecule or the permanent dipole that is formed by an –helical domain of the protein. The third is the ΔpKcoulomb, which is the change in pKa due to charge-charge interactions among buried ionized residues (or with metal ions, if present). The signs of these ΔpK values will depend on whether the ionized residue is positive or negative, and on the strength of the electrostatic interactions.

The dominant forces to consider in protein electrostatics are the ion-ion, hydrogen bonding, ion-permanent dipole, and permanent dipole-permanent dipole interactions. The strength of these interactions are distance-dependent, as shown in Table 1, with the force of ion-ion pairing being exerted over a significantly longer range compared to weak non-electrostatic forces. For example, the electrostatic force between two charged residues Lys+ and Gludecreases over a distance as 1/*r*, whereas the van der Waals attraction between uncharged atoms decreases over a distance as 1/*r*6, where *r* is on the order of atomic distance. The attraction between the oppositely charged residues, such as Lys+ and Glu-, forms a salt bridge, where by definition, both the centroids of their side groups and the charged atoms lie within a range of 4-8 Å (Kumar and Nussinov 2002). Salt bridges, hydrogen bonding, and background charges are commonplace in protein structures, yet proteins are stabilized not only by these electrostatic forces but also by non-electrostatic interactions as well. Examples of non-electrostatic interactions are hydrophobic interactions, van der Waals interactions, disulfide bridges, or covalent bonds.


Table 1. Relative range of electrostatic and non-electrostatic interactions in proteins

Two other important factors influencing electrostatics in proteins are (1) the dielectric properties of the protein and its surrounding aqueous environment, and (2) the ionic strength of the aqueous environment. The dielectric coefficient (*ε*) is an indication of polarizability – how readily dipoles can reorient within the medium. In an aqueous surface of the protein.

44 Electrostatics

The pKa value for each model residue (pKa,model) has been experimentally determined. This value represents the pKa of the residue when it is completely surrounded by water. Adjustments to the pKa,model as the residue becomes buried are due to the following three factors. The first is the ΔpKdesolv, which is the change in pKa due to the unfavorable removal of an ionized residue from water to the hydrophobic protein core (i.e., a desolvation penalty). The second is the ΔpKback, which is the change in pKa due to interactions of the buried ionized residue with background charges present within the protein. Background charges are defined as the partial charges of atoms that are manifested as either permanent or induced dipoles in molecules. Examples of background charges are the permanent dipole of a water molecule or the permanent dipole that is formed by an –helical domain of the protein. The third is the ΔpKcoulomb, which is the change in pKa due to charge-charge interactions among buried ionized residues (or with metal ions, if present). The signs of these ΔpK values will depend on whether the ionized residue is positive or negative, and on the strength of the electrostatic interactions. The dominant forces to consider in protein electrostatics are the ion-ion, hydrogen bonding, ion-permanent dipole, and permanent dipole-permanent dipole interactions. The strength of these interactions are distance-dependent, as shown in Table 1, with the force of ion-ion pairing being exerted over a significantly longer range compared to weak non-electrostatic forces. For example, the electrostatic force between two charged residues Lys+ and Gludecreases over a distance as 1/*r*, whereas the van der Waals attraction between uncharged atoms decreases over a distance as 1/*r*6, where *r* is on the order of atomic distance. The attraction between the oppositely charged residues, such as Lys+ and Glu-, forms a salt bridge, where by definition, both the centroids of their side groups and the charged atoms lie within a range of 4-8 Å (Kumar and Nussinov 2002). Salt bridges, hydrogen bonding, and background charges are commonplace in protein structures, yet proteins are stabilized not only by these electrostatic forces but also by non-electrostatic interactions as well. Examples of non-electrostatic interactions are hydrophobic interactions, van der Waals interactions,

**interaction Example Distance** 

Ionic Electrostatic Lys+ --- Glu- (salt bridge) 1/r

Ionic/dipole Electrostatic Asp- --- H2O 1/r2

peptide bond

electrostatic Dispersion forces 1/r6

dipole 1/r3

bonding Electrostatic Ser --- carboxyl of

(permanent) Electrostatic Helix dipole --- helix

Table 1. Relative range of electrostatic and non-electrostatic interactions in proteins

Two other important factors influencing electrostatics in proteins are (1) the dielectric properties of the protein and its surrounding aqueous environment, and (2) the ionic strength of the aqueous environment. The dielectric coefficient (*ε*) is an indication of polarizability – how readily dipoles can reorient within the medium. In an aqueous

Non-

**dependence** 

Bond length ~ 2.7 Å

disulfide bridges, or covalent bonds.

**Type of** 

Hydrogen

Dipole/dipole

Dipole/dipole (induced)

pK pK pK pK pK a a,model desolv back coulomb (1)

environment, the dipoles of water molecules are free to reorient, hence the dielectric coefficient of water is relatively high with a value of ~78. In contrast, the dielectric coefficient of the protein is lower due to the limited mobility of the protein chain and the nonpolar nature of many amino acid residues. The dielectric coefficient within a protein varies with location, with values of 2-4 for regions having residues that are virtually inaccessible to water (i.e., the hydrophobic core), increasing to values of ~37 near the surface of the protein (Anslyn and Dougherty 2006). As a rule of thumb, we consider the range of electrostatic interactions to be dependent on the dielectric property of the medium according to the plots shown in Figure 1. For example, the energy between point charges in water (*ε* = 78) cannot be discriminated from baseline thermal energy at a separation of ~2 Å as a result of the charge screening by dipoles of water molecules. In the region below the protein surface (*ε* = 10) the effective separation increases to ~ 14Å, and within the hydrophobic core (*ε* = 4) the effective range can be greater than 30 Å. This difference implies that polar and charged residues have greater electrostatic potential when they are buried within the protein. The other factor influencing electrostatic interactions is the ionic strength which also has a screening effect of charge, particularly at the

Here we focus on the treatment of electrostatics in protein engineering design. For a more general discussion of electrostatics in proteins, we refer the reader to several excellent reviews (Neves-Petersen and Petersen 2003; Bosshard *et al.* 2004; Jelesarov and Karshikoff 2009; Pace *et al.* 2009; Kukic and Nielsen 2010).

Fig. 1. The change in free energy (ΔG) associated with separating two point charges of opposite sign when surrounded by a medium of dielectric coefficient ε in the absence of salt. The values are calculated using Coulomb's law based on point charges of +0.5 and -0.5. The dashed line represents the baseline thermal energy at 298 K, *kBT*, where *kB* is the Boltzmann constant and *T* is the temperature.

Electrostatics in Protein Engineering and Design 47

physical chemical methods for measuring protein structure, thermodynamic stability and molecular motions improve, the corresponding force fields that model these processes are updated, improving their predictive power. In protein engineering, the modeling task becomes significantly more difficult relative to molecular dynamics, as we are not only concerned with determining the optimal molecular configuration, but also varying the amino acid sequence to modulate properties of the protein. Even for small proteins, the number of possible sequences to consider is immense: for example, a 100-residue protein has 20 possible amino acid choices at each position, resulting in a total space of 20100 ≈ 10130 sequences. Combine this with the configurational degrees of freedom of the protein chain and it is clear that an enumeration of all possible states is computationally impossible. To circumvent this search problem, a number of simplifications or coarse graining approaches are used, and no single level of chemical accuracy is universally applied to all protein engineering problems. The requirements of the design problem dictate the level of theory to use. We present several models of electrostatics with varying levels of chemical accuracy

The simplest electrostatics treatments do not incorporate atomic detail and assign discrete values to classes of interactions. This reduces a three-dimensional modeling problem to one dimension and is most useful in the design of molecules where positions that are adjacent in structure can be directly inferred from the amino acid sequence. This scenario is found in fibrous proteins such as -helical bundles and collagen fibrils (Spek *et al.* 1998). Due to the structural periodicity of the -helix and the collagen triple-helix, it is possible to anticipate which sequence positions are adjacent in structure (Figure 2). Using this information, a scoring function can be used to optimize these interactions. In many cases, the interactions are designed manually without computation(Berger *et al.* 1996; Lombardi *et al.* 1996; Bryson *et al.* 1998; Olson *et al.* 2001; Shi *et al.* 2001). Amino acids of opposite charge are introduced at adjacent positions such that the maximal number of charge pairs is satisfied. When the design goal is of sufficient complexity that computational intervention is required discrete scores are assigned to interactions (Nautiyal *et al.* 1995). One simple scoring function

> Arg /Arg 2 Glu /Glu 3 Arg /Glu 1

In this scenario, any structurally adjacent arginine (Arg) pairs are penalized by two kcals/mole. The penalty for adjacent glutamates (Glu) is greater in anticipation of their shorter side chains which bring repulsive charges in closer proximity. Only favorable Arg/Glu interactions are rewarded. The total energy for a given sequence is the sum of all residue-pair scores. If the number of sequences to sample is small, sequence-space can be fully searched. For larger design problems, Monte Carlo methods such as simulated evolution are often used (Hellinga and Richards 1994). Because they ignore molecular details, these scores are far from accurate, but they allow the rapid evaluation of large ensembles of sequences. The discrete scoring function in equation 3 has been used to design stable helical oligomers with specific composition – e.g. combining -helical chains A and B yielded an A2B2 tetramer, without forming A4 or B4 species (Summa *et al.* 2002). The same scoring function was recently extended to design of collagen heterotrimers where three

 

(3)

that are employed in protein engineering.

recently applied to both collagen and -helical proteins is:

#### **3. Theory and modeling of electrostatics in protein engineering**

Mechanical modeling of protein structure and dynamics requires a *force field*, a set of atomic and inter-atomic parameters that define how atoms interact in the context of the macromolecule. Among these parameters are the radius and partial charge of specific atom types, optimal bond lengths for atom pairs or optimal angles between groups of three atoms. These parameters are combined in an *objective energy function* which includes bonding (covalent) and nonbonding (electrostatics, van der Waals, hydrogen bonding) interactions to reflect the stability of a specific protein configuration:

$$\begin{aligned} \mathbf{E} &= \sum\_{bonds} \mathbf{K}\_r \left( \mathbf{r} - \mathbf{r}\_{eq} \right)^2 + \sum\_{angles} \mathbf{K}\_\theta \left( \theta - \theta\_{eq} \right)^2 + \sum\_{dihedrals} \frac{V\_n}{2} \left[ 1 + \cos \left( n\varphi - \varphi \right) \right] + \\ &\quad \sum\_{atoms \, i < j} \left[ \frac{\mathbf{A}\_{ij}}{r\_{ij}^{12}} - \frac{\mathbf{B}\_{ij}}{r\_{ij}^{6}} + \frac{q\_i q\_j}{\varepsilon r\_{ij}} \right] \end{aligned} \tag{2}$$

The first three summations incorporate harmonic or periodic potentials for bond vibrations, bond angle constraints and dihedral (bond rotation) constraints. The final term describes nonbonding interactions, including a van der Waals term that prevents atomic clashes, and an electrostatic term. A detailed explanation of the terms and coefficients for this objective energy function can be found in Cornell (Cornell *et al.* 1995).

Fig. 2. Charge-pair interactions can be inferred from the sequence for fibrous proteins with periodic structure. (LEFT) The seven-residue heptad of the repeat of the -helix coiled-coil places acidic (red) and basic (blue) amino acids adjacent in structure. In this case, the interaction between two chains of a three-chain homotrimeric protein are shown (Ogihara *et al.* 1997). (RIGHT) The collagen triple-helix is another type of periodic structure where charge-pairs adjacent in structure can be inferred directly from the sequence. A theoretical model structure of two chains in the triple helix are shown highlighting an extensive chargepair network. E = glutamic acid, D = aspartic acid, K = lysine.

The objective energy function can be used to predict the lowest energy configuration of a protein chain and to model molecular motions over short (nanosecond to microsecond) periods of time. Electrostatic interactions in the protein are calculated as the sum of all pairwise atomic interactions. In treating electrostatics this way, two assumptions are made: the partial charge is located at the center of mass of the atom (point charge approximation), and other terms of the multipole expansion beyond ion-ion interactions are disregarded. As 46 Electrostatics

Mechanical modeling of protein structure and dynamics requires a *force field*, a set of atomic and inter-atomic parameters that define how atoms interact in the context of the macromolecule. Among these parameters are the radius and partial charge of specific atom types, optimal bond lengths for atom pairs or optimal angles between groups of three atoms. These parameters are combined in an *objective energy function* which includes bonding (covalent) and nonbonding (electrostatics, van der Waals, hydrogen bonding)

*<sup>n</sup> r eq eq*

1 2

 

(2)

*ij ij i j*

*A B qq r r εr*

The first three summations incorporate harmonic or periodic potentials for bond vibrations, bond angle constraints and dihedral (bond rotation) constraints. The final term describes nonbonding interactions, including a van der Waals term that prevents atomic clashes, and an electrostatic term. A detailed explanation of the terms and coefficients for this objective

Fig. 2. Charge-pair interactions can be inferred from the sequence for fibrous proteins with periodic structure. (LEFT) The seven-residue heptad of the repeat of the -helix coiled-coil places acidic (red) and basic (blue) amino acids adjacent in structure. In this case, the interaction between two chains of a three-chain homotrimeric protein are shown (Ogihara *et al.* 1997). (RIGHT) The collagen triple-helix is another type of periodic structure where charge-pairs adjacent in structure can be inferred directly from the sequence. A theoretical model structure of two chains in the triple helix are shown highlighting an extensive charge-

The objective energy function can be used to predict the lowest energy configuration of a protein chain and to model molecular motions over short (nanosecond to microsecond) periods of time. Electrostatic interactions in the protein are calculated as the sum of all pairwise atomic interactions. In treating electrostatics this way, two assumptions are made: the partial charge is located at the center of mass of the atom (point charge approximation), and other terms of the multipole expansion beyond ion-ion interactions are disregarded. As

 

*atoms i j ij ij ij*

12 6

*<sup>V</sup> E Krr K cos n*

2 2

 

**3. Theory and modeling of electrostatics in protein engineering** 

interactions to reflect the stability of a specific protein configuration:

*bonds angles dihedrals*

energy function can be found in Cornell (Cornell *et al.* 1995).

pair network. E = glutamic acid, D = aspartic acid, K = lysine.

physical chemical methods for measuring protein structure, thermodynamic stability and molecular motions improve, the corresponding force fields that model these processes are updated, improving their predictive power. In protein engineering, the modeling task becomes significantly more difficult relative to molecular dynamics, as we are not only concerned with determining the optimal molecular configuration, but also varying the amino acid sequence to modulate properties of the protein. Even for small proteins, the number of possible sequences to consider is immense: for example, a 100-residue protein has 20 possible amino acid choices at each position, resulting in a total space of 20100 ≈ 10130 sequences. Combine this with the configurational degrees of freedom of the protein chain and it is clear that an enumeration of all possible states is computationally impossible. To circumvent this search problem, a number of simplifications or coarse graining approaches are used, and no single level of chemical accuracy is universally applied to all protein engineering problems. The requirements of the design problem dictate the level of theory to use. We present several models of electrostatics with varying levels of chemical accuracy that are employed in protein engineering.

The simplest electrostatics treatments do not incorporate atomic detail and assign discrete values to classes of interactions. This reduces a three-dimensional modeling problem to one dimension and is most useful in the design of molecules where positions that are adjacent in structure can be directly inferred from the amino acid sequence. This scenario is found in fibrous proteins such as -helical bundles and collagen fibrils (Spek *et al.* 1998). Due to the structural periodicity of the -helix and the collagen triple-helix, it is possible to anticipate which sequence positions are adjacent in structure (Figure 2). Using this information, a scoring function can be used to optimize these interactions. In many cases, the interactions are designed manually without computation(Berger *et al.* 1996; Lombardi *et al.* 1996; Bryson *et al.* 1998; Olson *et al.* 2001; Shi *et al.* 2001). Amino acids of opposite charge are introduced at adjacent positions such that the maximal number of charge pairs is satisfied. When the design goal is of sufficient complexity that computational intervention is required discrete scores are assigned to interactions (Nautiyal *et al.* 1995). One simple scoring function recently applied to both collagen and -helical proteins is:

$$\begin{aligned} \text{Arg} &/ \text{Arg} \quad + 2\\ \text{Glu} &/ \text{Glu} \quad + 3\\ \text{Arg} &/ \text{Glu} \quad -1 \end{aligned} \tag{3}$$

In this scenario, any structurally adjacent arginine (Arg) pairs are penalized by two kcals/mole. The penalty for adjacent glutamates (Glu) is greater in anticipation of their shorter side chains which bring repulsive charges in closer proximity. Only favorable Arg/Glu interactions are rewarded. The total energy for a given sequence is the sum of all residue-pair scores. If the number of sequences to sample is small, sequence-space can be fully searched. For larger design problems, Monte Carlo methods such as simulated evolution are often used (Hellinga and Richards 1994). Because they ignore molecular details, these scores are far from accurate, but they allow the rapid evaluation of large ensembles of sequences. The discrete scoring function in equation 3 has been used to design stable helical oligomers with specific composition – e.g. combining -helical chains A and B yielded an A2B2 tetramer, without forming A4 or B4 species (Summa *et al.* 2002). The same scoring function was recently extended to design of collagen heterotrimers where three

Electrostatics in Protein Engineering and Design 49

the use of quantum mechanics calculations is warranted. This approach has been used in the engineering of novel protein catalysts where the active site and substrate transition state are modeled using semi-empirical density functional methods (DFT), and the remaining protein treated using standard molecular mechanics and knowledge based potentials (Jiang *et al.*

Modeling hydrogen bonding with reasonable accuracy is an important challenge in protein design. Although primarily electrostatic in nature, hydrogen bonds also have partial covalent character which mediates their linearity in molecular structures. Such properties can only be modeled using quantum mechanical (QM) methods, which is computationally infeasible as these are distributed throughout the protein. Instead, empirical functions are often used that include both proximity and orientation terms. These have been refined using the extensive database of high-resolution protein structures to develop knowledge-based potentials that can capture subtle properties (Kortemme *et al.* 2003). QM methods can also be used to explore the role of other types of electrostatic interactions such as cation interactions between ions and aromatic amino acids. In the next sections, several examples of protein engineering of electrostatic properties are presented, highlighting the application

It was long thought that surface electrostatics do not make a significant contribution to protein stability because the interactions of polar residues with water in the unfolded state are as energetically favorable as their interactions with each other in the folded state. However, recent work has demonstrated that surface charge optimization can offer significant stability increases to a wide range of proteins (Schweiker and Makhatadze 2009). Surface charge optimization is an attractive option for protein engineering and design because surface positions are generally much more permissive to mutation compared to buried positions, where side chains are prone to clashing as they pack tightly into the protein core. Nature also takes advantage of this evolutionary flexibility at surface positions, modifying surface charge interactions to modulate energetic folding barriers (Halskau *et al.*

The hypothesis that surface electrostatics can be important for stability is supported by the observation that thermophilic proteins generally contain more charged surface residues than their mesophilic analogs (Kumar and Nussinov 2001). Thermophilic proteins have evolved to be active at high temperatures, and their structures must therefore be very stable. This stabilization is achieved through a number of different strategies, including enriching the sequences in charged surface residues and buried hydrophobic residues at the expense of polar residues. This adaptive response to evolutionary pressure for increased stability has been reproduced in computer simulations of simple lattice model proteins (Berezovsky *et al.* 2007). As a result, the number of salt bridges in a protein is correlated with the temperature of the environment in which its host organism lives (Kumar *et al.* 2000). In one study, mutations to two surface residues of a mesophilic cold shock protein (one of which eliminated an unfavorable electrostatic interaction) yielded a mutant that nearly matched the stability of the thermophilic version of the protein (Perl *et al.* 2000). The stability change was greatly reduced in the presence of 2M NaCl, confirming the importance of electrostatic interactions (which are sensitive to the screening effects of salt) in stabilizing the mutant.

of various levels of theory as needed to achieve the design objective.

**4. Surface charges in protein electrostatics** 

2008) and to stabilize thermophilic proteins.

2008; Rothlisberger *et al.* 2008).

peptides A, B and C combine specifically to form an ABC heterotrimer (Xu *et al.* 2011). Molecules such as these are now finding applications as synthetic biomaterials where the electrostatic control of self-assembly is responsible for directing the formation of protein fibers (Pandya *et al.* 2000; O'Leary *et al.* 2011).

When it is necessary to include some level of atomic detail in modeling ion-ion interactions, the simplest potential is Coulomb's law:

$$E = \text{332} \cdot \frac{q\_i q\_j}{\varepsilon \cdot r\_{ij}} \tag{4}$$

where the interactions of atoms *i* and *j* are a function of charge *q* and the distance of separation *r*. The constant of 332 converts the units of energy to kcals/mole. This can be applied to all atoms in the protein as described in equation 2, or restricted to side chains with a net formal charge. In full-atom implementations, the point charge is located at the center-of-mass of the atom, whereas residue-level charges are often placed at the center of the chemical moiety carrying the partial charge, i.e. the center of the guanidino group of the arginine sidechain. The choice of charge is determined by the force field used.

The strength of a charge-charge interaction is influenced by the polarity of the surrounding medium which is reflected by the choice of dielectric coefficient used. In cases where the structural context is known, often a fixed constant dielectric (e.g. 5-10 for the protein interior and ~78 for the surface) is used. One empirical approximation is to use a distancedependent dielectric (= 40·*rij*) based on the premise that the greater the separation between atoms, the more solvent can access the intervening space and screen electrostatic forces (Mayo *et al.* 1990; Gordon *et al.* 1999). In cases where it is desirable to include the effect of counterions, Debye-Huckel and Coulombic terms can be combined to include an ionic strength parameter (Lee *et al.* 2002).

In addition to charge-charge interactions within the protein, solvent-protein interactions are an important electrostatic component of the free energy of folding. Burial of charged side chains in the protein core comes at the cost of desolvating the sidechain ion. These energies can be modeled with reasonable accuracy using finite difference methods applied to the Poisson-Boltzmann equation (Sharp and Honig 1990), but are infrequently used in protein design applications due to the computational burden. Many software packages dedicated to protein design use an atom or residue-level solvation energy that scales with the fraction of accessible surface area buried upon folding. Although these are grossly approximate calculations, rapid algorithms for calculating solvent-exposed surface area make them attractive for evaluating large numbers of candidate sequences. A number of analytic and empirical methods continue to be developed that are finding applications in protein modeling and design (Flohil *et al.* 2002; Morozov *et al.* 2003; Pokala and Handel 2004; Jaramillo and Wodak 2005; am Busch *et al.* 2008). The assumption that atoms have point charges localized to the atom center of mass becomes problematic when designing proteins where electronic polarizability is important, such as the design of metalloproteins where the solvent reorganization energy around the metal can be important for tuning redox properties (Papoian *et al.* 2003), and enzymes where accurate modeling of the transition state and surrounding ligands is critical for an effective design (Tantillo *et al.* 1998). In this case, 48 Electrostatics

peptides A, B and C combine specifically to form an ABC heterotrimer (Xu *et al.* 2011). Molecules such as these are now finding applications as synthetic biomaterials where the electrostatic control of self-assembly is responsible for directing the formation of protein

When it is necessary to include some level of atomic detail in modeling ion-ion interactions,

332 *<sup>i</sup> <sup>j</sup>*

*q q <sup>E</sup>* 

where the interactions of atoms *i* and *j* are a function of charge *q* and the distance of separation *r*. The constant of 332 converts the units of energy to kcals/mole. This can be applied to all atoms in the protein as described in equation 2, or restricted to side chains with a net formal charge. In full-atom implementations, the point charge is located at the center-of-mass of the atom, whereas residue-level charges are often placed at the center of the chemical moiety carrying the partial charge, i.e. the center of the guanidino group of the

The strength of a charge-charge interaction is influenced by the polarity of the surrounding medium which is reflected by the choice of dielectric coefficient used. In cases where the structural context is known, often a fixed constant dielectric (e.g. 5-10 for the protein interior and ~78 for the surface) is used. One empirical approximation is to use a distancedependent dielectric (= 40·*rij*) based on the premise that the greater the separation between atoms, the more solvent can access the intervening space and screen electrostatic forces (Mayo *et al.* 1990; Gordon *et al.* 1999). In cases where it is desirable to include the effect of counterions, Debye-Huckel and Coulombic terms can be combined to include an ionic

In addition to charge-charge interactions within the protein, solvent-protein interactions are an important electrostatic component of the free energy of folding. Burial of charged side chains in the protein core comes at the cost of desolvating the sidechain ion. These energies can be modeled with reasonable accuracy using finite difference methods applied to the Poisson-Boltzmann equation (Sharp and Honig 1990), but are infrequently used in protein design applications due to the computational burden. Many software packages dedicated to protein design use an atom or residue-level solvation energy that scales with the fraction of accessible surface area buried upon folding. Although these are grossly approximate calculations, rapid algorithms for calculating solvent-exposed surface area make them attractive for evaluating large numbers of candidate sequences. A number of analytic and empirical methods continue to be developed that are finding applications in protein modeling and design (Flohil *et al.* 2002; Morozov *et al.* 2003; Pokala and Handel 2004; Jaramillo and Wodak 2005; am Busch *et al.* 2008). The assumption that atoms have point charges localized to the atom center of mass becomes problematic when designing proteins where electronic polarizability is important, such as the design of metalloproteins where the solvent reorganization energy around the metal can be important for tuning redox properties (Papoian *et al.* 2003), and enzymes where accurate modeling of the transition state and surrounding ligands is critical for an effective design (Tantillo *et al.* 1998). In this case,

arginine sidechain. The choice of charge is determined by the force field used.

*ij*

*<sup>r</sup>* (4)

fibers (Pandya *et al.* 2000; O'Leary *et al.* 2011).

the simplest potential is Coulomb's law:

strength parameter (Lee *et al.* 2002).

the use of quantum mechanics calculations is warranted. This approach has been used in the engineering of novel protein catalysts where the active site and substrate transition state are modeled using semi-empirical density functional methods (DFT), and the remaining protein treated using standard molecular mechanics and knowledge based potentials (Jiang *et al.* 2008; Rothlisberger *et al.* 2008).

Modeling hydrogen bonding with reasonable accuracy is an important challenge in protein design. Although primarily electrostatic in nature, hydrogen bonds also have partial covalent character which mediates their linearity in molecular structures. Such properties can only be modeled using quantum mechanical (QM) methods, which is computationally infeasible as these are distributed throughout the protein. Instead, empirical functions are often used that include both proximity and orientation terms. These have been refined using the extensive database of high-resolution protein structures to develop knowledge-based potentials that can capture subtle properties (Kortemme *et al.* 2003). QM methods can also be used to explore the role of other types of electrostatic interactions such as cation interactions between ions and aromatic amino acids. In the next sections, several examples of protein engineering of electrostatic properties are presented, highlighting the application of various levels of theory as needed to achieve the design objective.

#### **4. Surface charges in protein electrostatics**

It was long thought that surface electrostatics do not make a significant contribution to protein stability because the interactions of polar residues with water in the unfolded state are as energetically favorable as their interactions with each other in the folded state. However, recent work has demonstrated that surface charge optimization can offer significant stability increases to a wide range of proteins (Schweiker and Makhatadze 2009). Surface charge optimization is an attractive option for protein engineering and design because surface positions are generally much more permissive to mutation compared to buried positions, where side chains are prone to clashing as they pack tightly into the protein core. Nature also takes advantage of this evolutionary flexibility at surface positions, modifying surface charge interactions to modulate energetic folding barriers (Halskau *et al.* 2008) and to stabilize thermophilic proteins.

The hypothesis that surface electrostatics can be important for stability is supported by the observation that thermophilic proteins generally contain more charged surface residues than their mesophilic analogs (Kumar and Nussinov 2001). Thermophilic proteins have evolved to be active at high temperatures, and their structures must therefore be very stable. This stabilization is achieved through a number of different strategies, including enriching the sequences in charged surface residues and buried hydrophobic residues at the expense of polar residues. This adaptive response to evolutionary pressure for increased stability has been reproduced in computer simulations of simple lattice model proteins (Berezovsky *et al.* 2007). As a result, the number of salt bridges in a protein is correlated with the temperature of the environment in which its host organism lives (Kumar *et al.* 2000). In one study, mutations to two surface residues of a mesophilic cold shock protein (one of which eliminated an unfavorable electrostatic interaction) yielded a mutant that nearly matched the stability of the thermophilic version of the protein (Perl *et al.* 2000). The stability change was greatly reduced in the presence of 2M NaCl, confirming the importance of electrostatic interactions (which are sensitive to the screening effects of salt) in stabilizing the mutant.

Electrostatics in Protein Engineering and Design 51

Streptavidin and glutathione-S-transferase were also successfully supercharged to yield highly aggregation-resistant engineered variants. The supercharging process often involves a relatively large number of mutations, but because the hydrophobic core of the protein is undisturbed, protein folding is typically not significantly adversely affected. For example, the -7 net charge of superfolder GFP was pushed to extremes of +48 (by 36 mutations) or -30 (by 15 mutations). Remarkably, despite the repulsion that would be expected from gathering so many like charges on the surface of a protein, and the stability to be gained by optimizing surface charges demonstrated by the studies presented earlier, the supercharged GFPs were able to fold and fluoresce normally, with only slight decreases in thermodynamic stability. The destabilizing effect of the high concentration of like charges at the surface may be limited by equal or greater destabilization of competing states within the denatured state ensemble (Pace *et al.* 2000). The intuitive electrostatics-based supercharging strategy has already become a popular choice among protein engineers for stabilizing *de novo* designed

Another major limitation of peptide therapeutics is the difficulty of transporting peptides and proteins across the cell membrane. Currently, the leading strategy to improve cellular uptake is to express the target protein as a fusion with one of several polycationic amino acid sequences derived from natural cell-penetrating peptides (Heitz *et al.* 2009). In a recent study, positively supercharged GFP was shown to be capable of entering a range of mammalian cells, and of delivering fused protein payloads more effectively than the

standard cationic fusion tags Tat, Arg10, and penetratin (Cronican *et al.* 2010).

Fig. 3. Supercharging decreases the tendency of unfolded proteins to aggregate by

increasing like-charge repulsion. Thermally denatured green fluorescent protein (center) is capable of refolding into the fluorescent state (left) or aggregating with other unfolded chains (right). In the case of the wild-type protein (top), aggregation dominates. In contrast, a sample of a supercharged version of GFP with a net charge of +36 (bottom) regained 62% of its fluorescence following thermal denaturation. The like-charge repulsion between the positive charges on each denatured supercharged polypeptide mitigated aggregation.

proteins and therapeutic peptides against aggregation.

Designed and engineered proteins can also benefit from the stability gains that are possible by optimizing surface electrostatics. The generality of this strategy for protein stabilization was demonstrated experimentally in a study in which the surface residues of a diverse set of five proteins were modified (Strickler *et al.* 2006). A computational algorithm was used to search for mutations to surface positions that would provide the maximum improvement to the energy by adding favorable interactions or alleviating unfavorable ones. Because the combinatorial space of possible surface charges is too large to cover exhaustively, a genetic algorithm was used to search for near-optimal sequences. Genetic algorithms efficiently sample sequence space by mimicking the natural evolutionary process. A population of sequences is generated and evaluated with an energy function - in this case, the energy function was based on a solvent accessibility-corrected Tanford-Kirkwood model. The topscoring sequences are kept, multiplied, and diversified by random mutations within sequences and crossover or recombination events in which sections are swapped among multiple sequences. At the end of the process, sequences containing between three and eight mutations were selected. One to three designs were constructed for each target protein, synthesized, and purified. Protein unfolding was then monitored by circular dichroism spectroscopy. Remarkably, an increase in stability relative to the wild-type was observed for each of the designed sequences. The largest increase in stability was 4.4 kcal/mol. Another recent study applied this approach to the surface electrostatics optimization of two enzymes. The activity of enzymes is often highly sensitive to even small perturbations to the active site. Nonetheless, human acylphosphatase (AcPh) and human cell-division cycle 42 factor (Cdc42) were successfully stabilized by surface charge optimization with no loss in enzymatic activity (Gribenko *et al.* 2009). Mutant sequences were chosen that maximized the improvement in electrostatic energy while limiting the number of mutations from the wildtype sequence to ~5% of the total residues. The stability of each modified protein was ~10C higher than their corresponding wild-type protein, while the structures, monomeric nature, and enzymatic activities were retained. This study demonstrated the possibility of increasing the stability of an enzyme by making rational mutations to surface residues on the basis of electrostatic calculations, without disturbing the protein core or the enzymatic activity.

In addition to influencing the intramolecular stability of engineered proteins, electrostatics are important in intermolecular interactions. The balance of charged and hydrophobic residues in a protein sequence is important in determining the tendency of that sequence to aggregate when unfolded (Calamai *et al.* 2003; Chiti *et al.* 2003; Pawar *et al.* 2005). Charged residues within otherwise hydrophobic regions can act as "sequence breakers" that prevent those regions from aggregating. The ability of like-charge repulsive interactions to discourage aggregation is the basis of a surface electrostatics engineering strategy called "supercharging" (Lawrence *et al.* 2007). Amino acids at surface positions of a supercharged protein are mutated to charged residues so that the net charge of the protein is maximized. Net positive and net negative supercharged proteins have both been shown to be less prone to aggregation than their corresponding wild-types. For example, the green fluorescent protein (GFP) is unable to refold into a fluorescent state after thermal denaturation because of aggregation with neighboring unfolded chains. However, the extremely high net charges of supercharged GFP chains disfavor interactions with other unfolded chains of like charge (Figure 3). When a GFP variant supercharged to a net charge of +36 was thermally or chemically denatured, the sample was able to regain up to 62% of its initial fluorescence, confirming that the high net charge of the protein disfavored interchain aggregation. 50 Electrostatics

Designed and engineered proteins can also benefit from the stability gains that are possible by optimizing surface electrostatics. The generality of this strategy for protein stabilization was demonstrated experimentally in a study in which the surface residues of a diverse set of five proteins were modified (Strickler *et al.* 2006). A computational algorithm was used to search for mutations to surface positions that would provide the maximum improvement to the energy by adding favorable interactions or alleviating unfavorable ones. Because the combinatorial space of possible surface charges is too large to cover exhaustively, a genetic algorithm was used to search for near-optimal sequences. Genetic algorithms efficiently sample sequence space by mimicking the natural evolutionary process. A population of sequences is generated and evaluated with an energy function - in this case, the energy function was based on a solvent accessibility-corrected Tanford-Kirkwood model. The topscoring sequences are kept, multiplied, and diversified by random mutations within sequences and crossover or recombination events in which sections are swapped among multiple sequences. At the end of the process, sequences containing between three and eight mutations were selected. One to three designs were constructed for each target protein, synthesized, and purified. Protein unfolding was then monitored by circular dichroism spectroscopy. Remarkably, an increase in stability relative to the wild-type was observed for each of the designed sequences. The largest increase in stability was 4.4 kcal/mol. Another recent study applied this approach to the surface electrostatics optimization of two enzymes. The activity of enzymes is often highly sensitive to even small perturbations to the active site. Nonetheless, human acylphosphatase (AcPh) and human cell-division cycle 42 factor (Cdc42) were successfully stabilized by surface charge optimization with no loss in enzymatic activity (Gribenko *et al.* 2009). Mutant sequences were chosen that maximized the improvement in electrostatic energy while limiting the number of mutations from the wildtype sequence to ~5% of the total residues. The stability of each modified protein was ~10C higher than their corresponding wild-type protein, while the structures, monomeric nature, and enzymatic activities were retained. This study demonstrated the possibility of increasing the stability of an enzyme by making rational mutations to surface residues on the basis of

electrostatic calculations, without disturbing the protein core or the enzymatic activity.

In addition to influencing the intramolecular stability of engineered proteins, electrostatics are important in intermolecular interactions. The balance of charged and hydrophobic residues in a protein sequence is important in determining the tendency of that sequence to aggregate when unfolded (Calamai *et al.* 2003; Chiti *et al.* 2003; Pawar *et al.* 2005). Charged residues within otherwise hydrophobic regions can act as "sequence breakers" that prevent those regions from aggregating. The ability of like-charge repulsive interactions to discourage aggregation is the basis of a surface electrostatics engineering strategy called "supercharging" (Lawrence *et al.* 2007). Amino acids at surface positions of a supercharged protein are mutated to charged residues so that the net charge of the protein is maximized. Net positive and net negative supercharged proteins have both been shown to be less prone to aggregation than their corresponding wild-types. For example, the green fluorescent protein (GFP) is unable to refold into a fluorescent state after thermal denaturation because of aggregation with neighboring unfolded chains. However, the extremely high net charges of supercharged GFP chains disfavor interactions with other unfolded chains of like charge (Figure 3). When a GFP variant supercharged to a net charge of +36 was thermally or chemically denatured, the sample was able to regain up to 62% of its initial fluorescence, confirming that the high net charge of the protein disfavored interchain aggregation. Streptavidin and glutathione-S-transferase were also successfully supercharged to yield highly aggregation-resistant engineered variants. The supercharging process often involves a relatively large number of mutations, but because the hydrophobic core of the protein is undisturbed, protein folding is typically not significantly adversely affected. For example, the -7 net charge of superfolder GFP was pushed to extremes of +48 (by 36 mutations) or -30 (by 15 mutations). Remarkably, despite the repulsion that would be expected from gathering so many like charges on the surface of a protein, and the stability to be gained by optimizing surface charges demonstrated by the studies presented earlier, the supercharged GFPs were able to fold and fluoresce normally, with only slight decreases in thermodynamic stability. The destabilizing effect of the high concentration of like charges at the surface may be limited by equal or greater destabilization of competing states within the denatured state ensemble (Pace *et al.* 2000). The intuitive electrostatics-based supercharging strategy has already become a popular choice among protein engineers for stabilizing *de novo* designed proteins and therapeutic peptides against aggregation.

Another major limitation of peptide therapeutics is the difficulty of transporting peptides and proteins across the cell membrane. Currently, the leading strategy to improve cellular uptake is to express the target protein as a fusion with one of several polycationic amino acid sequences derived from natural cell-penetrating peptides (Heitz *et al.* 2009). In a recent study, positively supercharged GFP was shown to be capable of entering a range of mammalian cells, and of delivering fused protein payloads more effectively than the standard cationic fusion tags Tat, Arg10, and penetratin (Cronican *et al.* 2010).

Fig. 3. Supercharging decreases the tendency of unfolded proteins to aggregate by increasing like-charge repulsion. Thermally denatured green fluorescent protein (center) is capable of refolding into the fluorescent state (left) or aggregating with other unfolded chains (right). In the case of the wild-type protein (top), aggregation dominates. In contrast, a sample of a supercharged version of GFP with a net charge of +36 (bottom) regained 62% of its fluorescence following thermal denaturation. The like-charge repulsion between the positive charges on each denatured supercharged polypeptide mitigated aggregation.

Electrostatics in Protein Engineering and Design 53

Fig. 4. (A) The key placement of a buried titratable residue can enhance protein function. The cis-conformation of the indole ring of the Cerulean chromophore (shown in yellow) is stabilized by a substituted buried aspartate (shown in green) (PDB code 2q57). The chromophore is comprised of two rings, an indole and an imidazolinone, connected by a methylene bridge. The structure is further stabilized by a network of hydrogen bonding with backbone residues and bound water molecules surrounding the chromophore (not shown). (B) A buried ionized residue that is unable to form a salt-bridge can be stabilized by a hydrogen-bonding network. A charged aspartate residue (shown in green) is stabilized by a network of hydrogen bonding among three polar residues (shown in yellow) within the hydrophobic core of ribonuclease T1 (PDB code 9rnt). (C) The burial of a charged residue can be used to destabilize the protein structure. Glutamate is substituted for leucine (L50E; shown in green) within the hydrophobic core of ubiquitin (PDB code 1ubq). The buried glutamate is surrounded by a hydrophobic microenvironment (shown as yellow residues within 8 Å). The ionization of glutamate results in unfavorable conditions for the charged residue leading to local unfolding in the protein. This charge burial strategy was used to stabilize high-energy folding intermediates of ubiquitin. (D) Residues that become buried following protein-protein interaction can form stabilizing hydrogen-bonding networks. A buried two-carboxylate aspartate of Hsp90 C-terminal peptide (shown in green) is stabilized through its interactions with the polar residues of HOP (shown in yellow) at the protein interface (PDB code 1elr). The Hsp90 peptide is further stabilized along its length by hydrogen bonding with the side chains of the HOP helices (not shown). All figures are generated with PyMOL (Schrodinger, LLC) using a color scheme of red for oxygen and blue for nitrogen, and black dotted lines are used to indicate hydrogen bonding. Hydrogen atoms

are not explicitly shown.

**A B**

**C D**

#### **5. Electrostatics with buried polar or charged residues**

Proteins can tolerate the burial of ionizable residues when environmental modification of the pKa of the buried side chains prevents them from assuming the charged state. In a series of studies using 96 variants of an engineered form of staphylococcal nuclease, hydrophobic buried residues were individually mutated to lysine, glutamate, or aspartate (Isom *et al.* 2008; Isom *et al.* 2010; Isom *et al.* 2011). The apparent pKa values of these residues were determined by curve fitting plots of the changes in free energy associated with individually charging the mutants relative to a reference state as a function of pH. In general, the pKa values of these buried residues were shifted by the environment so that they existed in neutral form within the hydrophobic core.

Protein function can be improved by the burial of a polar residue if the conformation of an associated ligand can be stabilized by electrostatic interactions. Enhanced cyan fluorescent protein (ECFP) was optimized as a FRET1 donor molecule by mutating several of its residues – S72A2, Y145A and H148D (Rizzo *et al.* 2004; Malo *et al.* 2007). The new protein variant was called Cerulean. The authors describe the contribution of the H148D substitution of Cerulean in stabilizing a single conformation (i.e., the *cis*-form) of its associated chromophore. Unlike the histidine residue in ECFP, the buried aspartate side group stabilized the *cis*-conformation of the internal chromophore as part of an extended network of hydrogen bonding which included forming a bifurcated hydrogen bond with the indole nitrogen of the chromophore (Figure 4A). The pKa of the buried aspartate was estimated to be ~6 allowing the residue to remain protonated (i.e., neutral form) for hydrogen bonding, and the smaller size of the aspartate (relative to the histidine) aided in packing of the core. Other hydrogen-bonding interactions were made with nearby polar side groups and with bound water which provided a cage-like enclosure for the internal chromophore (not shown in the figure). The *cis*-conformation of the chromophore placed the six-membered ring of the indole in close proximity to the imidazolinone ring, enhancing energy transfer. The result of the H148D substitution was an engineered molecule that had relatively homogeneous exponential fluorescence emission decay, a property which is necessary for fluorescence-lifetime measurement studies.

The burial of an ionized residue in a protein is an unfavorable event that can be countered by stabilizing electrostatic interactions such as the formation of hydrogen bonding networks. The enzyme ribonuclease T1 is an example of a protein that contains an ionized buried residue, D76, that lacks an ion-pairing partner with which it can form a stabilizing salt bridge. The measured pKa of D76 is extremely low (pKa~0.5), ensuring that it always remains fully charged. As a result, it forms a hydrogen bonding network with nearby polar residues T91, Y11, and N9, and with bound water molecules in the protein (Giletto and Pace 1999). This local conformation is depicted in Figure 4B. The wild-type ribonuclease has been shown to have better thermal and chemical stability when compared to uncharged variants D76N, D76S and D76A of the enzyme. In this instance, having a buried charge within a polar microenvironment is advantageous.

<sup>1</sup>FRET = Förster Resonance Energy Transfer

<sup>2</sup>The standard one-letter code is used to designate the amino acid residues

52 Electrostatics

Proteins can tolerate the burial of ionizable residues when environmental modification of the pKa of the buried side chains prevents them from assuming the charged state. In a series of studies using 96 variants of an engineered form of staphylococcal nuclease, hydrophobic buried residues were individually mutated to lysine, glutamate, or aspartate (Isom *et al.* 2008; Isom *et al.* 2010; Isom *et al.* 2011). The apparent pKa values of these residues were determined by curve fitting plots of the changes in free energy associated with individually charging the mutants relative to a reference state as a function of pH. In general, the pKa values of these buried residues were shifted by the environment so that they existed in

Protein function can be improved by the burial of a polar residue if the conformation of an associated ligand can be stabilized by electrostatic interactions. Enhanced cyan fluorescent protein (ECFP) was optimized as a FRET1 donor molecule by mutating several of its residues – S72A2, Y145A and H148D (Rizzo *et al.* 2004; Malo *et al.* 2007). The new protein variant was called Cerulean. The authors describe the contribution of the H148D substitution of Cerulean in stabilizing a single conformation (i.e., the *cis*-form) of its associated chromophore. Unlike the histidine residue in ECFP, the buried aspartate side group stabilized the *cis*-conformation of the internal chromophore as part of an extended network of hydrogen bonding which included forming a bifurcated hydrogen bond with the indole nitrogen of the chromophore (Figure 4A). The pKa of the buried aspartate was estimated to be ~6 allowing the residue to remain protonated (i.e., neutral form) for hydrogen bonding, and the smaller size of the aspartate (relative to the histidine) aided in packing of the core. Other hydrogen-bonding interactions were made with nearby polar side groups and with bound water which provided a cage-like enclosure for the internal chromophore (not shown in the figure). The *cis*-conformation of the chromophore placed the six-membered ring of the indole in close proximity to the imidazolinone ring, enhancing energy transfer. The result of the H148D substitution was an engineered molecule that had relatively homogeneous exponential fluorescence emission decay, a property which is

The burial of an ionized residue in a protein is an unfavorable event that can be countered by stabilizing electrostatic interactions such as the formation of hydrogen bonding networks. The enzyme ribonuclease T1 is an example of a protein that contains an ionized buried residue, D76, that lacks an ion-pairing partner with which it can form a stabilizing salt bridge. The measured pKa of D76 is extremely low (pKa~0.5), ensuring that it always remains fully charged. As a result, it forms a hydrogen bonding network with nearby polar residues T91, Y11, and N9, and with bound water molecules in the protein (Giletto and Pace 1999). This local conformation is depicted in Figure 4B. The wild-type ribonuclease has been shown to have better thermal and chemical stability when compared to uncharged variants D76N, D76S and D76A of the enzyme. In this instance, having a buried charge within a

**5. Electrostatics with buried polar or charged residues** 

neutral form within the hydrophobic core.

necessary for fluorescence-lifetime measurement studies.

2The standard one-letter code is used to designate the amino acid residues

polar microenvironment is advantageous.

1FRET = Förster Resonance Energy Transfer

Fig. 4. (A) The key placement of a buried titratable residue can enhance protein function. The cis-conformation of the indole ring of the Cerulean chromophore (shown in yellow) is stabilized by a substituted buried aspartate (shown in green) (PDB code 2q57). The chromophore is comprised of two rings, an indole and an imidazolinone, connected by a methylene bridge. The structure is further stabilized by a network of hydrogen bonding with backbone residues and bound water molecules surrounding the chromophore (not shown). (B) A buried ionized residue that is unable to form a salt-bridge can be stabilized by a hydrogen-bonding network. A charged aspartate residue (shown in green) is stabilized by a network of hydrogen bonding among three polar residues (shown in yellow) within the hydrophobic core of ribonuclease T1 (PDB code 9rnt). (C) The burial of a charged residue can be used to destabilize the protein structure. Glutamate is substituted for leucine (L50E; shown in green) within the hydrophobic core of ubiquitin (PDB code 1ubq). The buried glutamate is surrounded by a hydrophobic microenvironment (shown as yellow residues within 8 Å). The ionization of glutamate results in unfavorable conditions for the charged residue leading to local unfolding in the protein. This charge burial strategy was used to stabilize high-energy folding intermediates of ubiquitin. (D) Residues that become buried following protein-protein interaction can form stabilizing hydrogen-bonding networks. A buried two-carboxylate aspartate of Hsp90 C-terminal peptide (shown in green) is stabilized through its interactions with the polar residues of HOP (shown in yellow) at the protein interface (PDB code 1elr). The Hsp90 peptide is further stabilized along its length by hydrogen bonding with the side chains of the HOP helices (not shown). All figures are generated with PyMOL (Schrodinger, LLC) using a color scheme of red for oxygen and blue for nitrogen, and black dotted lines are used to indicate hydrogen bonding. Hydrogen atoms are not explicitly shown.

Electrostatics in Protein Engineering and Design 55

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Native proteins can be unfolded by ionized residues buried within the hydrophobic core if there are no stabilizing electrostatic interactions to counter the charge. This notion was exploited in a charge burial strategy where "foldons" (regions of secondary structures that cooperatively unfold) of ubiquitin were selectively destabilized in order to trap high-energy intermediate folded states of the protein (Zheng and Sosnick 2010). A strategically located hydrophobic buried residue was substituted with glutamate which was subsequently ionized (Glu-) during pH titration. In the case of an L50E substitution located at the Cterminal end of the β5 strand, the Glu was placed in a microenvironment that was dominantly hydrophobic, with no nearby polar residues or backbone nitrogens to stabilize the charge (Figure 4C). Structural change within the protein caused by the Glu was monitored by nuclear magnetic resonance spectroscopy where the authors were able to detect the sequential unfolding of the β5 strand and an adjacent 310-helix. Interestingly, these unfolded intermediates could be stabilized by pH, and it was possible to refold the protein back to its native structure by neutralizing Glu-.

Protein-protein interfaces rely on electrostatic interactions to stabilize their previously exposed charged or polar residues. A study on the binding interaction between heat shock protein (Hsp)-organizing protein (HOP) domain TPR2A and the C-terminal end of Hsp90 (MEEVD) revealed the formation of an extensive network of hydrogen bonding between the ionized residues on Hsp90 and polar groups on TPR2A (Kajander *et al.* 2009). As an example, we illustrate the stabilization of the two carboxyl groups of the C-terminal aspartate residue, which is clamped by polar side chains from the TPR2A -helices and forms hydrogen bonds with K229, N233, Q298 and K301 (Figure 4D). Of these polar groups, N233 was found to be one of several significant binding surface residues that become buried. Similar electrostatic interactions were found along the length of the binding cavity, demonstrating how interfacial residues are stabilized.

#### **6. Acknowledgements**

This work was supported by grants from the National Institute of Health (5R21AI088627, 5R01GM089949, 1DP2OD006478 and 1F32GM099291) and the National Science Foundation (DMR0907273).

#### **7. References**


54 Electrostatics

Native proteins can be unfolded by ionized residues buried within the hydrophobic core if there are no stabilizing electrostatic interactions to counter the charge. This notion was exploited in a charge burial strategy where "foldons" (regions of secondary structures that cooperatively unfold) of ubiquitin were selectively destabilized in order to trap high-energy intermediate folded states of the protein (Zheng and Sosnick 2010). A strategically located hydrophobic buried residue was substituted with glutamate which was subsequently ionized (Glu-) during pH titration. In the case of an L50E substitution located at the Cterminal end of the β5 strand, the Glu- was placed in a microenvironment that was dominantly hydrophobic, with no nearby polar residues or backbone nitrogens to stabilize the charge (Figure 4C). Structural change within the protein caused by the Glu-

monitored by nuclear magnetic resonance spectroscopy where the authors were able to detect the sequential unfolding of the β5 strand and an adjacent 310-helix. Interestingly, these unfolded intermediates could be stabilized by pH, and it was possible to refold the protein

Protein-protein interfaces rely on electrostatic interactions to stabilize their previously exposed charged or polar residues. A study on the binding interaction between heat shock protein (Hsp)-organizing protein (HOP) domain TPR2A and the C-terminal end of Hsp90 (MEEVD) revealed the formation of an extensive network of hydrogen bonding between the ionized residues on Hsp90 and polar groups on TPR2A (Kajander *et al.* 2009). As an example, we illustrate the stabilization of the two carboxyl groups of the C-terminal aspartate residue, which is clamped by polar side chains from the TPR2A -helices and forms hydrogen bonds with K229, N233, Q298 and K301 (Figure 4D). Of these polar groups, N233 was found to be one of several significant binding surface residues that become buried. Similar electrostatic interactions were found along the length of the binding cavity,

This work was supported by grants from the National Institute of Health (5R21AI088627, 5R01GM089949, 1DP2OD006478 and 1F32GM099291) and the National Science Foundation

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Proceedings of the National Academy of Sciences of the United States of America


**Part 3** 

**Measurement and Instrumentation** 


## **Part 3**

**Measurement and Instrumentation** 

58 Electrostatics

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*UK* 

Jianyong Zhang *Teesside University* 

**Air-Solids Flow Measurement Using** 

Many industrial processes such as coal pulverising, flour making, cement production, and fertiliser processing involve moving bulk solids by means of pneumatic conveying. Almost all particles become electrically charged during pneumatic transportation, which can be hazardous in industrial environment. The primary sources of electrification are frictional contact charging between particles, between particles and the conducting facility, charge

Contact charging occurs at their common boundary when two dissimilar substances are brought into contact. On separation, each surface will carry an equal amount of charge with opposite polarity. Triboelectrification can be regarded as a complicated form of contact electrification in which there is transverse motion when two substances impinge or are rubbed together [1]. The transverse motion can in turn accentuate the charge transfer. Contact electrification occurs not only in pneumatic conveying, but also in milling, grinding,

Another source of electrostatic charge is induction. Charges will be induced on a conductor in an electrostatic field generated by charged particles. This conductor in turn changes the field distribution. If a conductor is insulated from the earth, its potential depends on the amount of charges, the permittivity of particles and their locations relative to the conductor [2]. The charge due to induction disappears when the charged particle moves away from the

Charges can be shared by two particles when they collide to each other, or when one particle

Electrostatic charge can be recombined, for example via the earth or by contact with an object holding opposite charge. However charge on non-conductive particles can be retained and the relaxation time depends on the volume resistivity of bulk solids. If the volume resistivity is high, the charge could be retained even if the solids are in an earthed container. For particles suspended in pure gases as in pneumatic conveying, particles can remain charged for a long period of time irrespective of the particle material's conductivity. Table 1 [3] shows the level of charge accumulation in particles, where the charge carried by unit mass of particle is given for

is settled on another. Charge sharing is more obvious between conductive particles.

transfer or sharing from one particle to another and charge induction.

vicinity or sensing volume of the conductor as in pneumatic conveyance.

solids of medium volume resistivity emerging from different processes.

**1.Introduction** 

sieving and screw feeding.

**1.1 Electrostatic charging and discharging** 

**Electrostatic Techniques** 

### **Air-Solids Flow Measurement Using Electrostatic Techniques**

Jianyong Zhang *Teesside University UK* 

#### **1.Introduction**

#### **1.1 Electrostatic charging and discharging**

Many industrial processes such as coal pulverising, flour making, cement production, and fertiliser processing involve moving bulk solids by means of pneumatic conveying. Almost all particles become electrically charged during pneumatic transportation, which can be hazardous in industrial environment. The primary sources of electrification are frictional contact charging between particles, between particles and the conducting facility, charge transfer or sharing from one particle to another and charge induction.

Contact charging occurs at their common boundary when two dissimilar substances are brought into contact. On separation, each surface will carry an equal amount of charge with opposite polarity. Triboelectrification can be regarded as a complicated form of contact electrification in which there is transverse motion when two substances impinge or are rubbed together [1]. The transverse motion can in turn accentuate the charge transfer. Contact electrification occurs not only in pneumatic conveying, but also in milling, grinding, sieving and screw feeding.

Another source of electrostatic charge is induction. Charges will be induced on a conductor in an electrostatic field generated by charged particles. This conductor in turn changes the field distribution. If a conductor is insulated from the earth, its potential depends on the amount of charges, the permittivity of particles and their locations relative to the conductor [2]. The charge due to induction disappears when the charged particle moves away from the vicinity or sensing volume of the conductor as in pneumatic conveyance.

Charges can be shared by two particles when they collide to each other, or when one particle is settled on another. Charge sharing is more obvious between conductive particles.

Electrostatic charge can be recombined, for example via the earth or by contact with an object holding opposite charge. However charge on non-conductive particles can be retained and the relaxation time depends on the volume resistivity of bulk solids. If the volume resistivity is high, the charge could be retained even if the solids are in an earthed container. For particles suspended in pure gases as in pneumatic conveying, particles can remain charged for a long period of time irrespective of the particle material's conductivity. Table 1 [3] shows the level of charge accumulation in particles, where the charge carried by unit mass of particle is given for solids of medium volume resistivity emerging from different processes.

Where

 --the electrical permittivity of particle K--the electrical conductivity of particle

V--the flow velocity in the region of the probe

*M* --the mass flow rate or flux of particle flow

 

constant, the above function can be simplified as

by TR-tech (now owned by Foster Wheeler) [11] [12].

AP--the cross-sectional area of probe

d--the particle diameter


groups such that

Air-Solids Flow Measurement Using Electrostatic Techniques 63

According to Pi Theorem [7], these variables may be expressed as four dimensionless

*P*

(1.2)

(1.3)

2 2 2 4 , , *<sup>n</sup>*

> 2 4 *n <sup>I</sup> <sup>f</sup> <sup>M</sup>*

*V* 

*I vd M <sup>f</sup> d V Kd A Kd* 

Then if the condition where only V and *M* vary, other parameters are assumed to be

At the time of research, the conclusion was that "the unique relation between probe current and the p.f. flux (mass flow rate) cannot be obtained, thus the electrostatic probe cannot be used as a means of determining the flux of p.f. in a pipe". The method proposed by Batch et al has been known as intrusive method which has been used by Hignett [8], and further explored by Soo [9] and King [10]. A commercial product based on the same principle, which is comprised of three probes installed with 120o gap apart, has been designed and manufactured

King [10] has used non-intrusive method which measures the induced voltage on an insulated pipe section (Pipe wall sensor), and he compared AC voltage measurement method to DC measurement method [13] [14]. The AC or noise measurement method, also known as "dynamic" method [15] takes the AC signal component to indicate flow concentration. Coulthard applied this technique to coal flow measurement in the Methil power station in Scotland [16], and Gajewski has employed it to measure dust in motion [17]. Gajewski further studied the measuring mechanism by combining the electrostatic field theory with electrical circuit analysis for a lined circular pipe wall sensor [18]. Since 1970s, positive and convincing results have been reported for measuring a range of different particles, e.g. pf, glass beads,

The mechanism of electrostatic metering system has been studied by many people, for example, Gajewski [19], Massen [20] and Hammer [21] studied filtering effect of the circular electrode, and Yan investigated charge induction based on free space electrostatic field

sands and polymer granules, which encouraged further study in this area.


Table 1. Charge build up on powder

Electrostatic charging can be a hazard if the charges are suddenly released via discharging to earth or another body, which produces a high local energy density and thus act as a possible ignition source. In section 7.2.4 of CENELEC and British Standard PD CLC/TR 50404: 2003, the discharges have been classified as "spark discharges", "brush discharges", "corona discharges", "propagating brush discharges", "cone discharges" and "lighting like discharges". Among them, spark discharges, brush discharge and lighting like discharges may occur in pneumatic conveying. The incendivity of discharge is very much depends on the energy stored and the minimum ignition energy (MIE). Therefore the hazardousness of discharge depends on the area classification (zones) and gas group of process environment.

Potential build-up on metal items (pipe lines, flanges, bolts and etc) can be avoided by earthing all these items. Sometimes pipe sections can become floated due to gaskets and other insulators. Therefore it is important to bond such sections to the earthed sections. Care must be taken when non-conductive pipes and hoses have to be used for pneumatic conveying, the maximum possible energy stored must not exceed the MIE. In some case, it is possible to choose the dense phase conveying which can reduce the risk of ignition inside pipe due to lack of air. Different from one's empirical knowledge, humidification is not effective as a means of dissipating the charge from a dust cloud. The precaution for electrostatic discharging is not the main focus in this chapter, more details can be found in British and CENELEC Standard PD CLC/TR 50404:2003, "Electrostatics—code of practice for the avoidance of hazards due to static electricity" [3].

#### **1.2 Brief history of electrostatic techniques for air solid flow measurement**

Electrostatic charging of flowing particles has long been known. The method to relate the magnitude of charge to the flow parameters was studied as early as in 1963. Batch, Dalmon and Hignett [4] used a pin probe to detect the current from the probe to earth in order to measure the mass flow rate of pneumatically conveyed pulverised fuel (p.f). The probe current depends on charge generated on probe due to contact electrification and induction.

The derivation of relations between the probe current and flow parameters was based on a model developed by Cooper [5] and Hignett [6] which was for electrification of liquids in motion.

Assume that the current flowing into the probe is I, the general relation is expressed as,

$$I = f\_n(\varepsilon, \mathcal{K}, d, V, \rho, \dot{M}, A\_P) \tag{1.1}$$

#### Where

62 Electrostatics

Electrostatic charging can be a hazard if the charges are suddenly released via discharging to earth or another body, which produces a high local energy density and thus act as a possible ignition source. In section 7.2.4 of CENELEC and British Standard PD CLC/TR 50404: 2003, the discharges have been classified as "spark discharges", "brush discharges", "corona discharges", "propagating brush discharges", "cone discharges" and "lighting like discharges". Among them, spark discharges, brush discharge and lighting like discharges may occur in pneumatic conveying. The incendivity of discharge is very much depends on the energy stored and the minimum ignition energy (MIE). Therefore the hazardousness of discharge depends on the area classification (zones) and gas group of process environment. Potential build-up on metal items (pipe lines, flanges, bolts and etc) can be avoided by earthing all these items. Sometimes pipe sections can become floated due to gaskets and other insulators. Therefore it is important to bond such sections to the earthed sections. Care must be taken when non-conductive pipes and hoses have to be used for pneumatic conveying, the maximum possible energy stored must not exceed the MIE. In some case, it is possible to choose the dense phase conveying which can reduce the risk of ignition inside pipe due to lack of air. Different from one's empirical knowledge, humidification is not effective as a means of dissipating the charge from a dust cloud. The precaution for electrostatic discharging is not the main focus in this chapter, more details can be found in British and CENELEC Standard PD CLC/TR 50404:2003, "Electrostatics—code of practice

Operation Mass charge density (C/kg)

Sieving 10-5-10-3 Pouring 10-3-10-1 Scroll feed transfer 10-2-1 Grinding 10-1-1 Micronising 10-1-102 Pneumatic conveying 10-1-103 Triboelectrical powder coating 103-104

for the avoidance of hazards due to static electricity" [3].

motion.

**1.2 Brief history of electrostatic techniques for air solid flow measurement** 

Electrostatic charging of flowing particles has long been known. The method to relate the magnitude of charge to the flow parameters was studied as early as in 1963. Batch, Dalmon and Hignett [4] used a pin probe to detect the current from the probe to earth in order to measure the mass flow rate of pneumatically conveyed pulverised fuel (p.f). The probe current depends on charge generated on probe due to contact electrification and induction. The derivation of relations between the probe current and flow parameters was based on a model developed by Cooper [5] and Hignett [6] which was for electrification of liquids in

Assume that the current flowing into the probe is I, the general relation is expressed as,

(, ,, , , , ) *n P I f*

 

*KdV M A* (1.1)

Table 1. Charge build up on powder


According to Pi Theorem [7], these variables may be expressed as four dimensionless groups such that

$$f\left(\frac{I^2}{\varepsilon d^2 \rho V^4}\right) = f\_n \left[ \left(\frac{\varepsilon v}{\mathcal{K}d}\right) \Big/ \left(\frac{d^2}{A\_P}\right) \Big/ \left(\frac{\dot{M}\varepsilon}{\mathcal{K}d\rho}\right) \right] \tag{1.2}$$

Then if the condition where only V and *M* vary, other parameters are assumed to be constant, the above function can be simplified as

$$\left(\frac{I^2}{V^4}\right) = f\_n\left(\dot{M}\right) \tag{1.3}$$

At the time of research, the conclusion was that "the unique relation between probe current and the p.f. flux (mass flow rate) cannot be obtained, thus the electrostatic probe cannot be used as a means of determining the flux of p.f. in a pipe". The method proposed by Batch et al has been known as intrusive method which has been used by Hignett [8], and further explored by Soo [9] and King [10]. A commercial product based on the same principle, which is comprised of three probes installed with 120o gap apart, has been designed and manufactured by TR-tech (now owned by Foster Wheeler) [11] [12].

King [10] has used non-intrusive method which measures the induced voltage on an insulated pipe section (Pipe wall sensor), and he compared AC voltage measurement method to DC measurement method [13] [14]. The AC or noise measurement method, also known as "dynamic" method [15] takes the AC signal component to indicate flow concentration. Coulthard applied this technique to coal flow measurement in the Methil power station in Scotland [16], and Gajewski has employed it to measure dust in motion [17]. Gajewski further studied the measuring mechanism by combining the electrostatic field theory with electrical circuit analysis for a lined circular pipe wall sensor [18]. Since 1970s, positive and convincing results have been reported for measuring a range of different particles, e.g. pf, glass beads, sands and polymer granules, which encouraged further study in this area.

The mechanism of electrostatic metering system has been studied by many people, for example, Gajewski [19], Massen [20] and Hammer [21] studied filtering effect of the circular electrode, and Yan investigated charge induction based on free space electrostatic field

Air-Solids Flow Measurement Using Electrostatic Techniques 65

For the convenience of calculation, assume a non-conductive, negatively charged particle

In order to know the induced charge Q on the electrode, the charge density on the inner

*<sup>S</sup> Q ds* 

where S stands for the entire inner surface area of the electrode, s is the surface area variable. According to the electrostatic theory, the surface charge density is equal to the electric flux

> *D*

 *D* 

> *D E*

is the gradient operator, E is the electrical field strength, , the relative permittivity of the

 *D E*

 ( ) 

( )0 ()0 ()0 *Pi t* (1.10)

(1.4)

(1.5)

(1.6)

(1.7)

(1.9)

(1.10)

*E* (1.8)

Fig. 1. Charge induction

surrounded by air as the only source of electrostatic field.

surface of the electrode, , needs to be found.

medium and refers to as the electrical potential.

Assume the following boundary conditions

From equations 1.5 to 1.8, Equation 1.9 and 1.10 can be derived,

density (electric displacement) D, i.e.

theory [15]. When the Finite Element Method became practically viable with the increased speed and capacity of computers, more in depth study of charge induction became possible. A model describing the relation between the induced charge on an insulated pipe section (electrode) and the charge carried by a particle with respect of its location, known as "Spatial sensitivity" was established by Cheng [22]. This model was developed based on electrostatic field theory using the Finite Element method (FEM). Based on the above model, and with further study, Zhang [23] has related flow concentration and solids mass flow rate to the charge level on electrodes by employing stochastic process theory. He also investigated and verified 2-D spatial sensitivity of ring-shaped electrode to "roping" flow [24] and effects of particle size [25] on the charge carried by particles. The effects of velocity and concentration have been studied experimentally and theoretically since then [26]. Cheng's model [22] has also been used by Xu [27], and an exploitation of frequency method for velocity measurement has also been based on the Cheng's model [28] [29]. The product operating according to dynamic electrostatic techniques with trade name PfMaster has been developed and manufactured by ABB Ltd.

#### **2. Charge induction and "dynamic" measurement method**

#### **2.1 Mathematic model for charge induction**

In this section, the main focus of the analysis will be on the meters with circular electrodes. For the analysis of other types of electrostatic sensors, the same principles apply. The method adopted here is based on the analysis conducted by Cheng [22].

Fig.1 depicts a simplified, schematic view of circular electrostatic meter. The circular metallic electrode is installed flush with, and electrically insulated from the inner surface of the earthed pipe, but exposed to the medium inside the pipe. This arrangement ensures that the electrode is sensitive to the charges carried by particles without restricting flow and can avoid severe charge build-up on non-conductive lining. It also minimises the electrode wear that can occur with intrusive probes [30]. The charge generated on the electrode is due to the following effects:


Since the speed of charged particle through the sensing volume is insignificant compared to the speed of light, so the electromagnetic effect generated by flowing charged solids can be neglected. The analysis therefore is under the assumption of a pure electrostatic electrical field. The electrode is narrow compared to the pipe diameter, so the contact charge between particle and electrode was not considered. The model presented here is also assumed a lean phase flow regime, under which effect of dielectric property of solids on the electrostatic field is neglected.

The principle of the measurement can be approximately (but not accurately) explained as follows: Regarding the entire conveying pipe (ignoring the insulator) as an enclosed system, the total charge induced on the inner surface of the system should equal to the charge carried by the source particle, but of opposite polarity. The portion of the total induced charge on the electrode varies with the location of the particle (although total charge induced on the inner surface of the enclosed system does not vary).

Fig. 1. Charge induction

64 Electrostatics

theory [15]. When the Finite Element Method became practically viable with the increased speed and capacity of computers, more in depth study of charge induction became possible. A model describing the relation between the induced charge on an insulated pipe section (electrode) and the charge carried by a particle with respect of its location, known as "Spatial sensitivity" was established by Cheng [22]. This model was developed based on electrostatic field theory using the Finite Element method (FEM). Based on the above model, and with further study, Zhang [23] has related flow concentration and solids mass flow rate to the charge level on electrodes by employing stochastic process theory. He also investigated and verified 2-D spatial sensitivity of ring-shaped electrode to "roping" flow [24] and effects of particle size [25] on the charge carried by particles. The effects of velocity and concentration have been studied experimentally and theoretically since then [26]. Cheng's model [22] has also been used by Xu [27], and an exploitation of frequency method for velocity measurement has also been based on the Cheng's model [28] [29]. The product operating according to dynamic electrostatic techniques with trade name PfMaster has been

In this section, the main focus of the analysis will be on the meters with circular electrodes. For the analysis of other types of electrostatic sensors, the same principles apply. The

Fig.1 depicts a simplified, schematic view of circular electrostatic meter. The circular metallic electrode is installed flush with, and electrically insulated from the inner surface of the earthed pipe, but exposed to the medium inside the pipe. This arrangement ensures that the electrode is sensitive to the charges carried by particles without restricting flow and can avoid severe charge build-up on non-conductive lining. It also minimises the electrode wear that can occur with intrusive probes [30]. The charge generated on the electrode is due to the

2. charge induction due to the presence of charged particles within the sensing volume. Since the speed of charged particle through the sensing volume is insignificant compared to the speed of light, so the electromagnetic effect generated by flowing charged solids can be neglected. The analysis therefore is under the assumption of a pure electrostatic electrical field. The electrode is narrow compared to the pipe diameter, so the contact charge between particle and electrode was not considered. The model presented here is also assumed a lean phase flow regime, under which effect of dielectric property of solids on the electrostatic

The principle of the measurement can be approximately (but not accurately) explained as follows: Regarding the entire conveying pipe (ignoring the insulator) as an enclosed system, the total charge induced on the inner surface of the system should equal to the charge carried by the source particle, but of opposite polarity. The portion of the total induced charge on the electrode varies with the location of the particle (although total charge

induced on the inner surface of the enclosed system does not vary).

developed and manufactured by ABB Ltd.

**2.1 Mathematic model for charge induction** 

1. particle contacting with the electrode, and

following effects:

field is neglected.

**2. Charge induction and "dynamic" measurement method** 

method adopted here is based on the analysis conducted by Cheng [22].

For the convenience of calculation, assume a non-conductive, negatively charged particle surrounded by air as the only source of electrostatic field.

In order to know the induced charge Q on the electrode, the charge density on the inner surface of the electrode, , needs to be found.

$$Q = \int\_{S} \sigma ds \tag{1.4}$$

where S stands for the entire inner surface area of the electrode, s is the surface area variable.

According to the electrostatic theory, the surface charge density is equal to the electric flux density (electric displacement) D, i.e.

$$D = \sigma \tag{1.5}$$

$$\nabla \cdot \mathbf{D} = \rho \tag{1.6}$$

$$D = \mathfrak{e}E \tag{1.7}$$

$$E = -\nabla \Phi \tag{1.8}$$

 is the gradient operator, E is the electrical field strength, , the relative permittivity of the medium and refers to as the electrical potential.

From equations 1.5 to 1.8, Equation 1.9 and 1.10 can be derived,

$$
\nabla \cdot D = \nabla \cdot \varepsilon E \tag{1.9}
$$

$$\nabla \cdot (\varepsilon \nabla \Phi) = -\rho \tag{1.10}$$

Assume the following boundary conditions

$$\Phi(\Gamma\_P) = 0 \cup \Phi(\Gamma\_i) = 0 \cup \Phi(\Gamma\_t) = 0 \tag{1.10}$$

Air-Solids Flow Measurement Using Electrostatic Techniques 67

Fig. 3 shows the relationship governed by Equation 1.11 for a given electrode W/R ratio (W/R=1/5) when the unit charge particle moves along the pipe axial direction (x

Fig. 4. compares the spatial sensitivity for the electrodes of different width as a particle

Fig. 5. [22] depicts the sensing volume of the electrodes with different width to radius ratios. In the figure, a minimum value of spatial sensitivity has been set. A point is within the

**2.2 Spatial sensitivity** 

coordinate) at different radial locations.

move along the pipe central line (r=0)

Fig. 3. Spatial Sensitivity for Particle passing along Different Axies.

Fig. 4. Spatial Sensitivity for Different Electrode Widhts

Fig. 5. Sensing Volume of electrodes of different width

where P, I, t represent the boundaries of the earthed conveying pipe, the insulator and the electrode respectively.

The conveying pipe is earthed, so the potential on it is zero. The electrode is usually connected to a charge amplifier in which the electrode is virtually earthed, the potential on electrode is very close to zero. It may be noticed that the potential on the insulator is hard to set. In the simulation, it was set as zero, and other low voltages (2, and 3Volts) on the insulator the similar results were obtained.

The problem becomes to find the potential . If the potential distribution is known, the charge density on the inner surface of electrode can be found, hence the induced charge on the electrode can be derived from Equation 1.4. This is a 3-D problem. The location of charged particle affects the amount of induced charge on the electrode. However in a cylindrical co-ordinate, if the particle only changes it angular co-ordinate with its radial (r) and axial (x) coordinates keeping unchanged, the induced charge on the electrode should not change due to the symmetrical configuration of the system. Consider also the superposition theorem in electrostatic field, a 2-D model is sufficient for solving this 3-D problem, and a ring-shaped charge situated with its axis coinciding with the pipe central line will produces the same induced charge on the electrode as a point particle carrying the same amount of charge at the same axial and radial locations. The equivalent ring was used by Cheng to calculate the charge induction as shown in Fig.2.

Fig. 2. Charge Induction

The detailed analysis can be found in [22]. Here provided is an equation relating the charge induction and source charge and its location, also known as "spatial sensitivity" which was obtained from FEM simulation and regression.

$$Q = Ae^{-kx^2} \tag{1.11}$$

where Q is the charge induced on the electrode due to a point charged particle carrying unit charge located at (x, r, ), but Q depends on r and x only. A and k are two parameters determined by electrode geography, namely W/R ratio (where W is the width of electrode, and R the radius of the sensor) and radial location r of the charged particle.

#### **2.2 Spatial sensitivity**

66 Electrostatics

where P, I, t represent the boundaries of the earthed conveying pipe, the insulator and the

The conveying pipe is earthed, so the potential on it is zero. The electrode is usually connected to a charge amplifier in which the electrode is virtually earthed, the potential on electrode is very close to zero. It may be noticed that the potential on the insulator is hard to set. In the simulation, it was set as zero, and other low voltages (2, and 3Volts) on the

The problem becomes to find the potential . If the potential distribution is known, the charge density on the inner surface of electrode can be found, hence the induced charge on the electrode can be derived from Equation 1.4. This is a 3-D problem. The location of charged particle affects the amount of induced charge on the electrode. However in a cylindrical co-ordinate, if the particle only changes it angular co-ordinate with its radial (r) and axial (x) coordinates keeping unchanged, the induced charge on the electrode should not change due to the symmetrical configuration of the system. Consider also the superposition theorem in electrostatic field, a 2-D model is sufficient for solving this 3-D problem, and a ring-shaped charge situated with its axis coinciding with the pipe central line will produces the same induced charge on the electrode as a point particle carrying the same amount of charge at the same axial and radial locations. The equivalent ring was used

The detailed analysis can be found in [22]. Here provided is an equation relating the charge induction and source charge and its location, also known as "spatial sensitivity" which was

where Q is the charge induced on the electrode due to a point charged particle carrying unit charge located at (x, r, ), but Q depends on r and x only. A and k are two parameters determined by electrode geography, namely W/R ratio (where W is the width of electrode,

and R the radius of the sensor) and radial location r of the charged particle.

<sup>2</sup> *kx Q Ae* (1.11)

electrode respectively.

Fig. 2. Charge Induction

obtained from FEM simulation and regression.

insulator the similar results were obtained.

by Cheng to calculate the charge induction as shown in Fig.2.

Fig. 3 shows the relationship governed by Equation 1.11 for a given electrode W/R ratio (W/R=1/5) when the unit charge particle moves along the pipe axial direction (x coordinate) at different radial locations.

Fig. 3. Spatial Sensitivity for Particle passing along Different Axies.

Fig. 4. compares the spatial sensitivity for the electrodes of different width as a particle move along the pipe central line (r=0)

Fig. 4. Spatial Sensitivity for Different Electrode Widhts

Fig. 5. Sensing Volume of electrodes of different width

Fig. 5. [22] depicts the sensing volume of the electrodes with different width to radius ratios. In the figure, a minimum value of spatial sensitivity has been set. A point is within the

Air-Solids Flow Measurement Using Electrostatic Techniques 69

Equations 1.11 1.12 and 1.13 provide the temporal and frequency spatial responses of a circular electrostatic meter to a charged particle. Zhang [23] extended these models to study the response to flow concentration and flow mass flow rate. In order to simplify the analysis, it is assumed that the particles of uniform size are evenly distributed over the sensing volume so that the volume concentration of solids is determined only by the number of particles per unit volume, i.e. N. Because the solids are fed or dropped into the conveying system at an upstream point, so that N and solids concentration can be regarded as a waveform travelling along pipe line at velocity V. The point of injection is the source of the wave. Hereafter the number of particles per unit volume and concentration will be respectively denoted as N(x,t) and Con(x,t), both of which depend on x, the axial distance and t, the time. The charge induced on the electrode, Q, is a function of N(x,t) in Equation

<sup>2</sup> 2 2 -k(r) x rA(r)N(x,t)e *<sup>n</sup>*

2 2 <sup>r</sup> -k(r) x A(r)Con(x,t)e ( )

(1.14)

(1.15)

(1.16)

*Q GD drdx*

*G r <sup>Q</sup> d dx D R*

where N(x, t) is the waveform of the number of particles which varies along the pipeline at a given time, and at any point of x, it varies with time; is a constant for a given diameter of sensor. A and k depend on radius r for an electrode of a given width; Gn and G are constants related to particle surface charge density and pipe geometry. Con(x,t) is the concentration

In order to find the unit impulse response of the electrode, let Con(x,t) be a delta function, i.e.

( ,) ( ) *<sup>x</sup> Con x t t <sup>V</sup>* 

thus there is only one non-zero point at any given time in the co-ordinate x which is V\*t ( or at given point x, the impulse arrives at time x/V). Under such a concentration, the induced

*G r <sup>r</sup> Q ht Are d <sup>D</sup> R R*

0

where Q() is the Fourier transfer function of the induced charge, and the Con() is the

*Q G r Ar <sup>r</sup> H e <sup>d</sup> Con DV R k r R*

2 2 <sup>1</sup>

() (\* )

<sup>1</sup> ( /) <sup>1</sup> <sup>1</sup> <sup>2</sup>

 

 (1.18)

(1.17)

2 2

4 ( )

*V k r*

charge is the unit impulse response of the electrode. From Equation1.15, we have

*m*

( ) ( ) ( ) ( ) ( )

*m*

 

0 ( ) ( ) *krV t*

*Vol*

*<sup>m</sup>* R *Vol*

**3. Measurement of velocity and mass flow rate 3.1 Unit impulse response of ring-shaped electrode** 

1.14 or a function of Con(x, t) as in Equation 1.15.

waveform, and R is the radius of the electrode.

and the Fourier transfer function of the electrode is

Fourier transfer function of the concentration.

sensing zone if the spatial sensitivity at that point is above this value. The shape of sensing volume depends on the geometry of the electrode.

If the particle movement along the axial direction is the main concern, the velocity of the particle in this direction can be related to Equation 1.11 by replacing x with Vt, where V is the particle velocity along the pipe axial direction, t is time.

$$Q = Ae^{-kV^2t^2} \tag{1.12}$$

This temporal expression relates the time, axial velocity and induced charge together, where for a given electrode, A and k vary with radius location r only. The radial velocity component of particle is not considered. The recent research on analysis of radial velocity can be found in [31].

The Fourier transform of Equation 1.12 provides the frequency property of the electrode to a point charge moving at velocity V along the pipe line.

$$Q(\rho o) = F\{Q\} = F\left\{ A e^{-kV^2 \hat{t}^2} \right\}$$

Therefore,

$$Q(\phi) = \frac{A}{V} \sqrt{\frac{\pi}{k}} e^{-\frac{1}{4k} \left(\frac{\phi}{V}\right)^2} \tag{1.13}$$

The analysis conducted by Cheng [22] is presented above. Different from the previous analysis, the model in Equation 1.11 has taken the presence of metal conveying pipe (earthed), the insulator between the conveyor and electrode, and the effect of resultant charge on the electrode into account. The significances of this model are that it allows studying the effects of sensor geometry and velocity on charge induction, and permits the frequency analysis. From this model, 2-D and 3-D spatial sensitivity profiles of a sensor can be derived. Equation 1.11 was the first such expression to be used for temporal and frequency domain analysis and it can be used as a guide for sensor design.

#### **2.3 Dynamic measurement**

As reviewed in section 1, King [10] compared AC and DC measurement methods for both circular sensor (he named it as "pipe sensor") and pin sensor (intrusive probe). In industrial environment, DC signal on electrodes is more prone to interference so the fluctuation of induced charge has been used for measurement. An electrostatic metering system which measures the signal fluctuation is termed unofficially "dynamic" measurement system, although in the author's view, the word "dynamic" has been misused. In such a system, it is the change or variation of the induced signal that matters. The fluctuation produced by airsolids flow is regarded as band-limited white noise [13] proportional to solids concentration [32]. The fluctuation in number of particles, random movement of particles, particle size and shape changes can also result in the random change in signal level. The signal level is dependent upon mass flow rate or concentration for given mean velocity, the distribution of particle size, humidity and etc.

68 Electrostatics

sensing zone if the spatial sensitivity at that point is above this value. The shape of sensing

If the particle movement along the axial direction is the main concern, the velocity of the particle in this direction can be related to Equation 1.11 by replacing x with Vt, where V is

This temporal expression relates the time, axial velocity and induced charge together, where for a given electrode, A and k vary with radius location r only. The radial velocity component of particle is not considered. The recent research on analysis of radial velocity

The Fourier transform of Equation 1.12 provides the frequency property of the electrode to a

2 2 *kV t Q F Q F Ae*

<sup>2</sup> 1

*<sup>A</sup>* <sup>4</sup>*k V Q e V k*

The analysis conducted by Cheng [22] is presented above. Different from the previous analysis, the model in Equation 1.11 has taken the presence of metal conveying pipe (earthed), the insulator between the conveyor and electrode, and the effect of resultant charge on the electrode into account. The significances of this model are that it allows studying the effects of sensor geometry and velocity on charge induction, and permits the frequency analysis. From this model, 2-D and 3-D spatial sensitivity profiles of a sensor can be derived. Equation 1.11 was the first such expression to be used for temporal and

As reviewed in section 1, King [10] compared AC and DC measurement methods for both circular sensor (he named it as "pipe sensor") and pin sensor (intrusive probe). In industrial environment, DC signal on electrodes is more prone to interference so the fluctuation of induced charge has been used for measurement. An electrostatic metering system which measures the signal fluctuation is termed unofficially "dynamic" measurement system, although in the author's view, the word "dynamic" has been misused. In such a system, it is the change or variation of the induced signal that matters. The fluctuation produced by airsolids flow is regarded as band-limited white noise [13] proportional to solids concentration [32]. The fluctuation in number of particles, random movement of particles, particle size and shape changes can also result in the random change in signal level. The signal level is dependent upon mass flow rate or concentration for given mean velocity, the distribution of

frequency domain analysis and it can be used as a guide for sensor design.

2 2 *kV t Q Ae* (1.12)

(1.13)

volume depends on the geometry of the electrode.

can be found in [31].

**2.3 Dynamic measurement** 

particle size, humidity and etc.

Therefore,

the particle velocity along the pipe axial direction, t is time.

point charge moving at velocity V along the pipe line.

#### **3. Measurement of velocity and mass flow rate**

#### **3.1 Unit impulse response of ring-shaped electrode**

Equations 1.11 1.12 and 1.13 provide the temporal and frequency spatial responses of a circular electrostatic meter to a charged particle. Zhang [23] extended these models to study the response to flow concentration and flow mass flow rate. In order to simplify the analysis, it is assumed that the particles of uniform size are evenly distributed over the sensing volume so that the volume concentration of solids is determined only by the number of particles per unit volume, i.e. N. Because the solids are fed or dropped into the conveying system at an upstream point, so that N and solids concentration can be regarded as a waveform travelling along pipe line at velocity V. The point of injection is the source of the wave. Hereafter the number of particles per unit volume and concentration will be respectively denoted as N(x,t) and Con(x,t), both of which depend on x, the axial distance and t, the time. The charge induced on the electrode, Q, is a function of N(x,t) in Equation 1.14 or a function of Con(x, t) as in Equation 1.15.

$$Q = G\_n \overline{D}^2 \iint\_{Vol} \mathbf{r} \mathbf{A}(\mathbf{r}) \mathbf{N}(\mathbf{x}, \mathbf{t}) \mathbf{e}^{\cdot \mathbf{k}(\mathbf{r}) \sigma^2 \mathbf{x}^2} \, dr d\mathbf{x} \tag{1.14}$$

$$Q = \frac{G}{\rho\_m \overline{D}} \iint \frac{\mathbf{r}}{R} \mathbf{A}(\mathbf{r}) \text{Con}(\mathbf{x}, \mathbf{t}) \mathbf{e}^{-\mathbf{k}(\mathbf{r}) \phi^2 \mathbf{x}^2} \, d(\frac{r}{R}) d\mathbf{x} \tag{1.15}$$

where N(x, t) is the waveform of the number of particles which varies along the pipeline at a given time, and at any point of x, it varies with time; is a constant for a given diameter of sensor. A and k depend on radius r for an electrode of a given width; Gn and G are constants related to particle surface charge density and pipe geometry. Con(x,t) is the concentration waveform, and R is the radius of the electrode.

In order to find the unit impulse response of the electrode, let Con(x,t) be a delta function, i.e.

$$\text{Con}(\mathbf{x}, t) = \mathcal{S}(t - \frac{\mathbf{x}}{V}) \tag{1.16}$$

thus there is only one non-zero point at any given time in the co-ordinate x which is V\*t ( or at given point x, the impulse arrives at time x/V). Under such a concentration, the induced charge is the unit impulse response of the electrode. From Equation1.15, we have

$$Q = h(t) = \frac{G}{\rho\_m D} \frac{1}{D} \int\_0^r A(r) e^{-k(r)V^2 \left(t^\*\phi\right)^2} d\frac{r}{R} \tag{1.17}$$

and the Fourier transfer function of the electrode is

$$H(\rho) = \frac{Q(\rho)}{\text{Con}(\rho)} = \frac{G\pi^{\frac{1}{2}}}{\rho\_m \overline{D} V} \Big| \frac{1}{R} \frac{A(r)}{\sqrt{k(r)}} e^{-\frac{\left(\pi/\rho\right)^2}{4V^2} \frac{1}{k(r)}} d\frac{r}{R} \tag{1.18}$$

where Q() is the Fourier transfer function of the induced charge, and the Con() is the Fourier transfer function of the concentration.

Air-Solids Flow Measurement Using Electrostatic Techniques 71

If the root mean square conrms of the flow noise con(t) is directly proportional to the mean solids concentration *Con t*( ) , Urms, the rms value of the sensor's output has a linear relationship with the mean solids concentration for given solids density and particle size. The above analysis assumes that the net charge carried by solids does not depend upon velocity. Hence the velocity in Equation 1.20 reflects the effect of velocity on characteristics of the sensing system only. The amount of net charge carried by particles is actually affected by velocity. Gajewski [17] has studied the this effect on the 'charging tendency' of PVC dust, and Masuda conducted the tests on several different materials and found that the

effect of velocity on the net charge has also been confirmed from many tests on pulverized

If we assume the net charge is proportional to the solids velocity over the range investigated, as it was suggested by Gajewski [17] for PVC dust over the velocity range

*rms rms*

where max is the signal frequency up limit, *M* is the solids mass flow rate. Hence the root mean square value of signal can be used to directly measure solids mass flow rate if the

In Equations 1.20 and 1.21, it can be seen that, to achieve accurate concentration or flow rate measurement result, the effect of velocity needs to be compensated for, therefore, the

There are several different ways to measure velocity of conveyed solids, however, the cross correlation method remains the most practical and viable one. Since late 1960s and early 1970s, the cross-correlation found its applications in flow measurement. Various sensors have been used to measure different types of flow. A cross correlatior detects the flow noise transit time, from which the mean velocity of flow can be derived. Beck [34], Coulthard [35], Cole [13] and King [10] used this method to measure velocity of multi-phase flow. Keech and Coulthard realised a microprocessor based electrostatic cross correlator for the ABB cable meter [36]. Cheng adopted "polarity cross correlation" to measure pulverised coal flow velocity in a blast furnace [37]. The technique has been further improved to

In electrostatic air-solids flow measurement system, usually two identical electrodes are mounted up and down stream with a known distance apart. If the flow concentration Con(t) is rectilinearly transferred from upstream to downstream at a velocity V, it can be expected that the signal from the downstream electrode is a delayed replica of the signal from the

Con t Con t L / V Con (t t) 21 1 , (1.22)

or 3/2

[14], where varied from 1.4-1.9. The

max max

(1.21)

*con <sup>V</sup> U VM*

electrostatic meter's output was proportional to V

2 2 3 max *rms rms con U V* 

below 20m/s, equation 1.20 becomes

effect of velocity has been compensated.

velocity measurement cannot be avoided.

accommodate multi-channel velocity measurement [38].

**3.4 Velocity measurement** 

upstream electrode, i.e.

coal [33].

#### **3.2 Equivalent circuit and charge amplifier**

The signal induced on the electrode has to be connected to a measuring equipment or a preamplifier. Usually the input impedance of a preamplifier or a measurement equipment is finite, therefore the characteristics of the sensor comprising the electrode and the connected electronics depend not only on the electronics but also on the internal impedance of the electrode due to "loading effect".

Fig. 6. Sensing system

Although there are various types of preamplifier circuits, the charge preamplifier is among those of most widely used for such systems. As shown in Fig.6 the electrode is at virtual earth potential. The capacitance CF in the feedback loop is used to suppress the effects of the wiring capacitance and the equivalent capacitance Cn, when the values of the wiring capacitance and Cn as well as their variation have to be considered. RF is used to provide a DC path, which also determines the lower cut-off frequency with CF.

The charge amplifier blocks DC component of the signal Q, so the system measures signal fluctuation, performing "dynamic" measurement.

The transfer function of the measurement system T() is

$$T(\alpha o) = \frac{\mathcal{U}\_{\boldsymbol{\phi}}(\alpha o)}{\operatorname{Con}(\alpha o)} = \frac{\mathcal{Q}(\alpha o)}{\operatorname{Con}(\alpha o)} \frac{\mathcal{U}\_{\boldsymbol{\phi}}(\alpha o)}{\mathcal{Q}(\alpha o)} = H(\alpha o)P(\alpha o) \tag{1.19}$$

where Uo() is the output voltage of the charge amplifier, P() is the transfer function of the charge amplifier. The loading effect of source is reflected in P() which depends on Cn and rn.

#### **3.3 Measuring solids mass flow rate**

Assume the concentration signal is a band-limited white noise, based on equation 1.19, Zhang [23] has used Parseval's formula to relate the rms of Uo to the fluctuation in concentration,

$$\text{U}\!\!\!\!\text{U}\_{rms}^{2}\propto\frac{\text{con}\_{rms}^{2}}{\varpi\_{\text{max}}}\,\text{V}\tag{1.20}$$

70 Electrostatics

The signal induced on the electrode has to be connected to a measuring equipment or a preamplifier. Usually the input impedance of a preamplifier or a measurement equipment is finite, therefore the characteristics of the sensor comprising the electrode and the connected electronics depend not only on the electronics but also on the internal impedance of the

Although there are various types of preamplifier circuits, the charge preamplifier is among those of most widely used for such systems. As shown in Fig.6 the electrode is at virtual earth potential. The capacitance CF in the feedback loop is used to suppress the effects of the wiring capacitance and the equivalent capacitance Cn, when the values of the wiring capacitance and Cn as well as their variation have to be considered. RF is used to provide a

The charge amplifier blocks DC component of the signal Q, so the system measures signal

() () ( ) ( ) ( )( ) () () () *U U o o <sup>Q</sup> <sup>T</sup> H P Con Con Q*

 

where Uo() is the output voltage of the charge amplifier, P() is the transfer function of the charge amplifier. The loading effect of source is reflected in P() which depends on Cn and rn.

Assume the concentration signal is a band-limited white noise, based on equation 1.19, Zhang [23] has used Parseval's formula to relate the rms of Uo to the fluctuation in

2

2

max *rms rms con U V* 

 

> 

(1.19)

(1.20)

DC path, which also determines the lower cut-off frequency with CF.

fluctuation, performing "dynamic" measurement.

**3.3 Measuring solids mass flow rate** 

concentration,

The transfer function of the measurement system T() is

**3.2 Equivalent circuit and charge amplifier** 

electrode due to "loading effect".

Fig. 6. Sensing system

If the root mean square conrms of the flow noise con(t) is directly proportional to the mean solids concentration *Con t*( ) , Urms, the rms value of the sensor's output has a linear relationship with the mean solids concentration for given solids density and particle size.

The above analysis assumes that the net charge carried by solids does not depend upon velocity. Hence the velocity in Equation 1.20 reflects the effect of velocity on characteristics of the sensing system only. The amount of net charge carried by particles is actually affected by velocity. Gajewski [17] has studied the this effect on the 'charging tendency' of PVC dust, and Masuda conducted the tests on several different materials and found that the electrostatic meter's output was proportional to V [14], where varied from 1.4-1.9. The effect of velocity on the net charge has also been confirmed from many tests on pulverized coal [33].

If we assume the net charge is proportional to the solids velocity over the range investigated, as it was suggested by Gajewski [17] for PVC dust over the velocity range below 20m/s, equation 1.20 becomes

$$\text{i } \mathcal{U}\_{rms}^2 \propto \frac{\text{con}\_{rms}^2}{\sigma\_{\text{max}}} V^3 \text{ or } \mathcal{U}\_{rms} \propto \frac{\text{con}\_{rms}}{\sqrt{\sigma\_{\text{max}}}} V^{3/2} = \dot{M} \sqrt{\frac{V}{\rho\_{\text{max}}}} \tag{1.21}$$

where max is the signal frequency up limit, *M* is the solids mass flow rate. Hence the root mean square value of signal can be used to directly measure solids mass flow rate if the effect of velocity has been compensated.

#### **3.4 Velocity measurement**

In Equations 1.20 and 1.21, it can be seen that, to achieve accurate concentration or flow rate measurement result, the effect of velocity needs to be compensated for, therefore, the velocity measurement cannot be avoided.

There are several different ways to measure velocity of conveyed solids, however, the cross correlation method remains the most practical and viable one. Since late 1960s and early 1970s, the cross-correlation found its applications in flow measurement. Various sensors have been used to measure different types of flow. A cross correlatior detects the flow noise transit time, from which the mean velocity of flow can be derived. Beck [34], Coulthard [35], Cole [13] and King [10] used this method to measure velocity of multi-phase flow. Keech and Coulthard realised a microprocessor based electrostatic cross correlator for the ABB cable meter [36]. Cheng adopted "polarity cross correlation" to measure pulverised coal flow velocity in a blast furnace [37]. The technique has been further improved to accommodate multi-channel velocity measurement [38].

In electrostatic air-solids flow measurement system, usually two identical electrodes are mounted up and down stream with a known distance apart. If the flow concentration Con(t) is rectilinearly transferred from upstream to downstream at a velocity V, it can be expected that the signal from the downstream electrode is a delayed replica of the signal from the upstream electrode, i.e.

$$\text{Con}\_2(\mathbf{t}) = \text{Con}\_1(\mathbf{t} - \mathbf{L} \,/\, \mathbf{V}) = \text{Con}\_1(\mathbf{t} - \mathbf{t}) \,. \tag{1.22}$$

Air-Solids Flow Measurement Using Electrostatic Techniques 73

from a mill and split into six or eight conveyors. Under normal conditions, the density of solids, moisture content and even flow profile are similar in different conveyors, but particle size distribution, mass flow rate and velocity vary from one conveyor to another. If the system can provide the signal proportional to the split (relative or percentage of overall mass flow rate) with velocity and particle size compensation, the mass flow rate in each conveyor can be

Fig. 8 presents a typical set of test results on a dynamic electrostatic meter. The tests were carried using the Teesside University 40mm diameter rig, and the material used was "Fillite", a commercial product made from fly ash. The air and solids mass flow rate were controlled to maintain the constant air to solids ratio (i.e., mass flow rate of air/mass flow rate of solids), hence under each of the ratios 3.86, 3.34, 2.88, 2.39 and 1.92, when the solids mass flow rate increases, the air flow rate is increased proportionally. For each air to solids ratio, the relationship between signal rms value and the solids mass flow rate was close to a second order polynomial due to combined effect of solids mass flow rate and velocity [26].

It is also clear that the higher air to solids ratio (means less solids, or lower concentration) resulted in higher signal for a given mass flow rate. It seems contradictory to the common sense, but again it is due to higher velocity, the hidden information in the graph. The signal is more sensitive to velocity than to any other parameters, and the effect of velocity requires

The signal depends on the combined effects of concentration, mass flow rate and velocity. From the above analysis, an algorithm given by Equation 1.27 was derived to relate the

where Uorms is the rms value of output voltage of the meter, A, B, C, D, E and F are constants, Ras represents air to solids ratio, and *M* is the solids mass flow rate. Fig. 9 shows

<sup>2</sup> ( )( ) *U AR B M CR D M ER F orms as as as* (1.27)

meter's output signal, solid mass flow rate and air to solids ratio (or concentration),

compensation for mass flow rate or concentration measurement.

given with reasonable accuracy because the overall loading entering the mill is known.

**4.1 Signal, concentration, mass flow rate and velocity** 

Fig. 8. Response of Electrostatic meter

where L is the distance between two electrode. is the transit time.

Because L is known, thus once is found, the velocity V can be determined from Equation 1.23

$$\mathbf{V} = \mathbf{L} \;/\; \mathbf{t}.\tag{1.23}$$

According to the definition, the cross-correlation function between Con1(t) and Con2(t) is equal to

$$R\_{c\_1c\_2}\left(\tau\right) = \lim\_{T \to \infty} \frac{1}{T} \Big| \text{Con}\_1\left(\mathbf{t} - \tau\right) \text{Con}\_2\left(\mathbf{t}\right) \text{d}\mathbf{t} \tag{1.24}$$

If the two signals are exactly identical,

$$R\_{c1c2}(\tau) = \lim\_{T \to \infty} \frac{1}{T} \Big| \text{Con}\_1(\mathbf{t} - \tau) \text{Con}\_1(\mathbf{t} - \mathbf{L} / \mathbf{V}) \text{d}\mathbf{t} \tag{1.25}$$

The cross correlation becomes a delayed version of auto correlation of Con1(t), as shown in Fig. 7.

Fig. 7. Cross Correlation

$$R\_{c\_1c\_2}(\tau) = R\_{c\_1c\_1}(\tau - L \;/\; V) = R\_{cc}(\tau - L \;/\; V) \tag{1.26}$$

In reality, two signals are not exactly identical, however the cross correlation efficient can be very high. Even for low cross correlation coefficient, say, 0.5, a cross correlator can still successfully capture the flow transit time and find the average flow velocity. The frequency band of the signal determines the measurement accuracy of transit time, which in turn affects the accuracy of velocity measurement.

#### **4. Relative measurement**

The response of an electrostatic meter for air solids flow measurement depends on density, particle size, velocity, mass flow rate and flow profile. Over the past ten years, the performance of dynamic electrostatic meters has been significantly improved, however the high measurement accuracy is still not achievable if all the above parameters vary over wide ranges.

In many cases, only two or three parameters vary and other parameters stay relatively stable. This is particularly true in coal-fired power station, where pulverised fuel comes 72 Electrostatics

Because L is known, thus once is found, the velocity V can be determined from Equation 1.23

According to the definition, the cross-correlation function between Con1(t) and Con2(t) is

 1 2 1 2 0

1 2 <sup>1</sup> <sup>1</sup>

The cross correlation becomes a delayed version of auto correlation of Con1(t), as shown in

1 2 1 1 () ( / ) ( / ) *R R LV R LV cc cc cc*

In reality, two signals are not exactly identical, however the cross correlation efficient can be very high. Even for low cross correlation coefficient, say, 0.5, a cross correlator can still successfully capture the flow transit time and find the average flow velocity. The frequency band of the signal determines the measurement accuracy of transit time, which in turn

The response of an electrostatic meter for air solids flow measurement depends on density, particle size, velocity, mass flow rate and flow profile. Over the past ten years, the performance of dynamic electrostatic meters has been significantly improved, however the high measurement accuracy is still not achievable if all the above parameters vary over wide

In many cases, only two or three parameters vary and other parameters stay relatively stable. This is particularly true in coal-fired power station, where pulverised fuel comes

 

(1.26)

<sup>1</sup> ( ) lim Con t Con t L / V dt

 

 *T*

<sup>1</sup> ( ) lim Con t Con t dt

 

V L / t. (1.23)

*T* (1.24)

*T* (1.25)

where L is the distance between two electrode. is the transit time.

*R*

*R*

affects the accuracy of velocity measurement.

*c c <sup>T</sup>*

If the two signals are exactly identical,

*c c <sup>T</sup>*

0

*T*

equal to

Fig. 7.

Fig. 7. Cross Correlation

**4. Relative measurement** 

ranges.

from a mill and split into six or eight conveyors. Under normal conditions, the density of solids, moisture content and even flow profile are similar in different conveyors, but particle size distribution, mass flow rate and velocity vary from one conveyor to another. If the system can provide the signal proportional to the split (relative or percentage of overall mass flow rate) with velocity and particle size compensation, the mass flow rate in each conveyor can be given with reasonable accuracy because the overall loading entering the mill is known.

#### **4.1 Signal, concentration, mass flow rate and velocity**

Fig. 8 presents a typical set of test results on a dynamic electrostatic meter. The tests were carried using the Teesside University 40mm diameter rig, and the material used was "Fillite", a commercial product made from fly ash. The air and solids mass flow rate were controlled to maintain the constant air to solids ratio (i.e., mass flow rate of air/mass flow rate of solids), hence under each of the ratios 3.86, 3.34, 2.88, 2.39 and 1.92, when the solids mass flow rate increases, the air flow rate is increased proportionally. For each air to solids ratio, the relationship between signal rms value and the solids mass flow rate was close to a second order polynomial due to combined effect of solids mass flow rate and velocity [26].

Fig. 8. Response of Electrostatic meter

It is also clear that the higher air to solids ratio (means less solids, or lower concentration) resulted in higher signal for a given mass flow rate. It seems contradictory to the common sense, but again it is due to higher velocity, the hidden information in the graph. The signal is more sensitive to velocity than to any other parameters, and the effect of velocity requires compensation for mass flow rate or concentration measurement.

The signal depends on the combined effects of concentration, mass flow rate and velocity. From the above analysis, an algorithm given by Equation 1.27 was derived to relate the meter's output signal, solid mass flow rate and air to solids ratio (or concentration),

$$
\dot{M}\_{\rm corrs} = (\dot{A}R\_{\rm as} + B)\dot{M}^2 + (\dot{C}R\_{\rm as} + D)\dot{M} + \dot{E}R\_{\rm as} + F \tag{1.27}
$$

where Uorms is the rms value of output voltage of the meter, A, B, C, D, E and F are constants, Ras represents air to solids ratio, and *M* is the solids mass flow rate. Fig. 9 shows

Air-Solids Flow Measurement Using Electrostatic Techniques 75

Fig. 10 was obtained from experiments using sieved materials [39]. For the given mass flow rate, velocity and concentration, the signal from a dynamic electrostatic meter decreased with particle size for the size above 250 µm, confirming Equation 1.15, 1.17 and 1.18. However for particles below that size, the signal reversed the trend, i.e. the smaller particle size resulted in lower signal. At the time of experiment, the signal drop for smaller particles was thought to be caused by the sudden change in solids flow rate. The recent research revealed that the signal drop for small particles could have been caused by flow regime change. When the size of particle is getting smaller, the flow becomes less turbulent. This effect outweighs the effect of total particle surface area increase so that overall signal level decreases.

In pneumatically conveyed air solid flow, the distribution of solid phase is often un-even. For example around bends and restrictive devices, the roping flow regime may be formed. The air solids flow profiles depend on conveying velocity, particle size, humidity and

The measurement results will be affected by flow regime unless a meter has a uniform

Fig.11 depicts the test results obtained on a 14" (356mm) diameter electrostatic meter [24]. A roping stream of constant flow rate was provided with an one-inch jet, the roping stream was parallel to the pipe axial central line and moved cross the pipe cross sectional area along its diameter. The material used was pulverised coal. The output voltage (rms) of the meter to the "roping" flow stream was recorded when the jet moved from the centre to a location very close to pipe wall. The signals on the wide electrode (Red W/R=0.5) and on the narrow electrode (Blue W/R=0.014) followed the same trend. It is clear that the signal increased with the flow stream getting closer to the pipe wall, and then it started to drop as the roping stream crossing about 70% of full radius, which is caused by combination of the increased sensitivity and the reduced sensitive volume of the sensor as shown in Figs 3, 4 and 5.

geometry of conveyor. The research in this area can be found elsewhere [40][41][42].

**4.3 Spatial sensitivity** 

spatial sensitivity.

Fig. 11. Spatial Sensitivity Test Results

the measured mass flow rate against the true mass flow rate for various velocities and air to solids ratios [26].

Fig. 9. Calibration Graph

#### **4.2 Effect of particle size**

As discussed from the beginning of this chapter, the induced charge on the insulated electrode is a function of several factors including particle size.

From Equations 1.15, 1.17, 1.18, it can be seen that induced charge on electrode is inversely proportional to particle size. It is due to the fact that the mass of solids is proportional to D3 and the total surface area of solids is proportional to D2 for spherical particles, where D is particle diameter. If surface charge density is a constant, larger surface of total particles will provide higher signal level when the particles are getting smaller [25].

Fig. 10. Signal Vs particle size

Fig. 10 was obtained from experiments using sieved materials [39]. For the given mass flow rate, velocity and concentration, the signal from a dynamic electrostatic meter decreased with particle size for the size above 250 µm, confirming Equation 1.15, 1.17 and 1.18. However for particles below that size, the signal reversed the trend, i.e. the smaller particle size resulted in lower signal. At the time of experiment, the signal drop for smaller particles was thought to be caused by the sudden change in solids flow rate. The recent research revealed that the signal drop for small particles could have been caused by flow regime change. When the size of particle is getting smaller, the flow becomes less turbulent. This effect outweighs the effect of total particle surface area increase so that overall signal level decreases.

#### **4.3 Spatial sensitivity**

74 Electrostatics

the measured mass flow rate against the true mass flow rate for various velocities and air to

As discussed from the beginning of this chapter, the induced charge on the insulated

From Equations 1.15, 1.17, 1.18, it can be seen that induced charge on electrode is inversely proportional to particle size. It is due to the fact that the mass of solids is proportional to D3 and the total surface area of solids is proportional to D2 for spherical particles, where D is particle diameter. If surface charge density is a constant, larger surface of total particles will

electrode is a function of several factors including particle size.

provide higher signal level when the particles are getting smaller [25].

solids ratios [26].

Fig. 9. Calibration Graph

**4.2 Effect of particle size** 

Fig. 10. Signal Vs particle size

In pneumatically conveyed air solid flow, the distribution of solid phase is often un-even. For example around bends and restrictive devices, the roping flow regime may be formed. The air solids flow profiles depend on conveying velocity, particle size, humidity and geometry of conveyor. The research in this area can be found elsewhere [40][41][42].

The measurement results will be affected by flow regime unless a meter has a uniform spatial sensitivity.

Fig. 11. Spatial Sensitivity Test Results

Fig.11 depicts the test results obtained on a 14" (356mm) diameter electrostatic meter [24]. A roping stream of constant flow rate was provided with an one-inch jet, the roping stream was parallel to the pipe axial central line and moved cross the pipe cross sectional area along its diameter. The material used was pulverised coal. The output voltage (rms) of the meter to the "roping" flow stream was recorded when the jet moved from the centre to a location very close to pipe wall. The signals on the wide electrode (Red W/R=0.5) and on the narrow electrode (Blue W/R=0.014) followed the same trend. It is clear that the signal increased with the flow stream getting closer to the pipe wall, and then it started to drop as the roping stream crossing about 70% of full radius, which is caused by combination of the increased sensitivity and the reduced sensitive volume of the sensor as shown in Figs 3, 4 and 5.

Air-Solids Flow Measurement Using Electrostatic Techniques 77

then be derived based on the integration of the product of pixel concentration and velocity

For any type of process tomography, the successful realization depends on sensing system design, signal conditioning, signal to noise (S/N) ratio, proper data acquisition system and

EST is a passive sensing system, which is one of its advantages [47] over ECT. However inherently, for the same number of sensors (electrodes), the resolution of EST is lower than that of ECT. Combined systems (dual modality) [48] can offer better resolution and reliability. Fig. 13 provides the simulation results for an EST and an EST/ECT combined systems [2]. In this figure, a uniform positive charge density distribution in a stratified flow at the bottom half of a pipe is assumed, the reconstructed image using information from the EST system only in Fig.13b is vague, the boundary is not clear. Compared to the image in Fig.13b, Fig.13c offers much better result which is obtained by combining the information

(a) (b) (c)

At present, the modelling of charge induction with consideration of particle dielectric property is the new development in this area [2]. The research to develop an overall model to relate the signal rms, solids velocity and solids mass flow rate is under way [26]. The electrostatic method used in square pipe lines [49] has also been investigated, and the study on the effect of radial velocity on flow measurement is useful for understanding of the mechanism of electrostatic meters [31]. The technique of combing ECT and EST for gas

At the time of writing this chapter, the electrostatic technique have been successful in some areas, for example in measuring flow split among pneumatic conveyors, in providing warning of blockage and for inferring primary air flow rate measurement . Some research outcomes are yet to be applied in practice. It is envisaged that the techniques will be further improved for flow measurement and flow regime diagnose not only in lean-phase conveying as in coal-fire power generation, but also for dense phase flow as in gasification

over a given cross sectional area.

from the EST and ECT of dual modality system.

Fig. 13. Image Reconstruction from an EST/ECT

solids flow measurement opened a new frontier.

**5. Current research in this area** 

and in blast furnace feeding.

efficient algorithm.

Fig. 12. Frequency Sensitivity of a circular Dynamic Electro static meter

Fig.11 provides a temporal spatial sensitivity. The corresponding frequency spatial behaviour of circular electrostatic meter is shown in Fig. 12 [24], In this figure, r is the radial coordinate from the pipe central line, R is the radius of the sensor. The vertical coordinate represents normalised output signal when a roping flow stream in parallel with pipe axial central line, passes through the sensor at different radial positions. It can be seen that the meter produces the signals with higher magnitude and wider frequency band when the stream is at r/R=0.8, compared that with the roping stream passes the central line (r/R=0). The figure does not provide the response to the roping stream passing from the location where r/R is greater than 0.8. However from Fig.11, it can be predicted, the magnitude will be lower, and the frequency band will be wider.

Theoretically, if the frequency components can be split and weighted according to where the flow stream passing through, a uniform sensitivity of meter can be achieved, which is one of possible solutions for non-uniform sensitivity compensation.

#### **4.4 EST (Electrostatic Tomography)**

Represented by Capacitance Tomography (ECT), "Process tomography" has attracted great attention since 1980s [43] [44] [45], and the research in this area has made significant progress. Besides ECT, there are many different types of tomographic techniques such as Electrical impedance tomography (EIT), optical tomography and Electrostatic Tomography [46]. "Procee tomography" uses an array of sensors mounted on the boundary of a vessel or a pipe to detect the pixel flow concentration and velocity in process. The flow profile can be reconstructed based on the information obtained from the sensor array. Theoretically, this is an ideal method to solve the problems caused by non even solids distribution in air solids two-phase flow.

As the name suggests, electrostatic tomography (EST) uses an array of electrostatic sensors to detect the distribution of charges carried by particles and particle velocities. If the amount of charge carried by particles to concentration ratio is constant, the flow rate of solids can 76 Electrostatics

Fig.11 provides a temporal spatial sensitivity. The corresponding frequency spatial behaviour of circular electrostatic meter is shown in Fig. 12 [24], In this figure, r is the radial coordinate from the pipe central line, R is the radius of the sensor. The vertical coordinate represents normalised output signal when a roping flow stream in parallel with pipe axial central line, passes through the sensor at different radial positions. It can be seen that the meter produces the signals with higher magnitude and wider frequency band when the stream is at r/R=0.8, compared that with the roping stream passes the central line (r/R=0). The figure does not provide the response to the roping stream passing from the location where r/R is greater than 0.8. However from Fig.11, it can be predicted, the magnitude will

Theoretically, if the frequency components can be split and weighted according to where the flow stream passing through, a uniform sensitivity of meter can be achieved, which is one of

Represented by Capacitance Tomography (ECT), "Process tomography" has attracted great attention since 1980s [43] [44] [45], and the research in this area has made significant progress. Besides ECT, there are many different types of tomographic techniques such as Electrical impedance tomography (EIT), optical tomography and Electrostatic Tomography [46]. "Procee tomography" uses an array of sensors mounted on the boundary of a vessel or a pipe to detect the pixel flow concentration and velocity in process. The flow profile can be reconstructed based on the information obtained from the sensor array. Theoretically, this is an ideal method to solve the problems caused by non even solids distribution in air solids

As the name suggests, electrostatic tomography (EST) uses an array of electrostatic sensors to detect the distribution of charges carried by particles and particle velocities. If the amount of charge carried by particles to concentration ratio is constant, the flow rate of solids can

Fig. 12. Frequency Sensitivity of a circular Dynamic Electro static meter

be lower, and the frequency band will be wider.

**4.4 EST (Electrostatic Tomography)** 

two-phase flow.

possible solutions for non-uniform sensitivity compensation.

then be derived based on the integration of the product of pixel concentration and velocity over a given cross sectional area.

For any type of process tomography, the successful realization depends on sensing system design, signal conditioning, signal to noise (S/N) ratio, proper data acquisition system and efficient algorithm.

EST is a passive sensing system, which is one of its advantages [47] over ECT. However inherently, for the same number of sensors (electrodes), the resolution of EST is lower than that of ECT. Combined systems (dual modality) [48] can offer better resolution and reliability. Fig. 13 provides the simulation results for an EST and an EST/ECT combined systems [2]. In this figure, a uniform positive charge density distribution in a stratified flow at the bottom half of a pipe is assumed, the reconstructed image using information from the EST system only in Fig.13b is vague, the boundary is not clear. Compared to the image in Fig.13b, Fig.13c offers much better result which is obtained by combining the information from the EST and ECT of dual modality system.

Fig. 13. Image Reconstruction from an EST/ECT

#### **5. Current research in this area**

At present, the modelling of charge induction with consideration of particle dielectric property is the new development in this area [2]. The research to develop an overall model to relate the signal rms, solids velocity and solids mass flow rate is under way [26]. The electrostatic method used in square pipe lines [49] has also been investigated, and the study on the effect of radial velocity on flow measurement is useful for understanding of the mechanism of electrostatic meters [31]. The technique of combing ECT and EST for gas solids flow measurement opened a new frontier.

At the time of writing this chapter, the electrostatic technique have been successful in some areas, for example in measuring flow split among pneumatic conveyors, in providing warning of blockage and for inferring primary air flow rate measurement . Some research outcomes are yet to be applied in practice. It is envisaged that the techniques will be further improved for flow measurement and flow regime diagnose not only in lean-phase conveying as in coal-fire power generation, but also for dense phase flow as in gasification and in blast furnace feeding.

Air-Solids Flow Measurement Using Electrostatic Techniques 79

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to 25 2006.

#### **6. References**


78 Electrostatics

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**Part 4** 

**Mathematical Modelling** 


## **Part 4**

**Mathematical Modelling** 

80 Electrostatics

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Toshko Boev

 *Bulgaria* 

**Mathematical Models for** 

 **Electrostatics of Heterogeneous Media** 

Since most of twenty years intensive investigations, in Physical Chemistry and applied Physics, have been directed to various basic processes for complexly structured material systems, with a strong accent on surface phenomena problems. Such systems, known as heterogeneous, consist typically of different bulk phases, separated by specific interfaces, which can be also realized as containing distinct but near by sub-phases. Additionally, the common line contours, of 2D sub-phases, are treated as materially autonomous (1D) phases as well. One of the main directions in said topics consider surface nucleation phenomena in gas-liquid systems. The interest here has been provoked mainly from the open questions for the mechanism of the surface nucleation – in particular in lipid systems. Said questions essentially concern basic topics of Physical Chemistry, related also to ecological applications. As a second class, note the problems on structure building of semiconductor films of aircrystal media, via an actual technological interest: it primarily concerns the main factors governing the growth and roughness of semiconducting surface films. Another (third) class of related topics is shown by the recent studies on cell biology problems (e.g. [4]). This class includes also mathematical models for detecting of anomalies in the human organic systems – for instance, the blood circulatory system and that of the white liver oxygen transfer.

The electrostatic properties of matter have been taken as the basic framework for investigations of surface phenomena problems. Especially, adequate expressions have been sought for the electric potential – as the key quantity, integrating the basic electrostatic parameters of medium. Our main goal here consists in finding such expressions, primarily concerning the interface potential of complex (3-2-1D) heterogeneous systems. Said aim yields the key question how to construct a proper mathematical model of the matter electrostatics, which introduces a correct problem for the electric potential. Secondly, it is necessary to do the main steps in the mathematical analysis of the relevant problem. Here we propose an answer of the above question, taking into account two required basic steps. The first one consists in introducing the object called heterogeneous media, when in reality we have given a material system of different bulk (3D) phase, for instance – gas and liquid, with a relatively thin transition layer (say emulsion). In our treatment an additional stage of heterogeneity is presumed: the bulk transition layer consists also of two near-by sub-phases (of differing matter). According to the Gibbs idealizing approach ([6]), we have to consider

**1. Introduction** 

*Department of Differential Equations, University of Sofia, Sofia,* 

### **Mathematical Models for Electrostatics of Heterogeneous Media**

Toshko Boev

*Department of Differential Equations, University of Sofia, Sofia, Bulgaria* 

#### **1. Introduction**

Since most of twenty years intensive investigations, in Physical Chemistry and applied Physics, have been directed to various basic processes for complexly structured material systems, with a strong accent on surface phenomena problems. Such systems, known as heterogeneous, consist typically of different bulk phases, separated by specific interfaces, which can be also realized as containing distinct but near by sub-phases. Additionally, the common line contours, of 2D sub-phases, are treated as materially autonomous (1D) phases as well. One of the main directions in said topics consider surface nucleation phenomena in gas-liquid systems. The interest here has been provoked mainly from the open questions for the mechanism of the surface nucleation – in particular in lipid systems. Said questions essentially concern basic topics of Physical Chemistry, related also to ecological applications. As a second class, note the problems on structure building of semiconductor films of aircrystal media, via an actual technological interest: it primarily concerns the main factors governing the growth and roughness of semiconducting surface films. Another (third) class of related topics is shown by the recent studies on cell biology problems (e.g. [4]). This class includes also mathematical models for detecting of anomalies in the human organic systems – for instance, the blood circulatory system and that of the white liver oxygen transfer.

The electrostatic properties of matter have been taken as the basic framework for investigations of surface phenomena problems. Especially, adequate expressions have been sought for the electric potential – as the key quantity, integrating the basic electrostatic parameters of medium. Our main goal here consists in finding such expressions, primarily concerning the interface potential of complex (3-2-1D) heterogeneous systems. Said aim yields the key question how to construct a proper mathematical model of the matter electrostatics, which introduces a correct problem for the electric potential. Secondly, it is necessary to do the main steps in the mathematical analysis of the relevant problem. Here we propose an answer of the above question, taking into account two required basic steps. The first one consists in introducing the object called heterogeneous media, when in reality we have given a material system of different bulk (3D) phase, for instance – gas and liquid, with a relatively thin transition layer (say emulsion). In our treatment an additional stage of heterogeneity is presumed: the bulk transition layer consists also of two near-by sub-phases (of differing matter). According to the Gibbs idealizing approach ([6]), we have to consider

<sup>1</sup> ( ) 1/ 

[ ] 

problems).

case of semiconductor interface.

problem (0.1) – (0.3) remains however as open one.

Mathematical Models for Electrostatics of Heterogeneous Media 85

In this chapter we give the main steps of deriving and solve two sub-cases of the general problem (0.1) – (0.3), related to heterogeneous systems with flat interfaces, respectively of semiconductor and organic nature. A straight line contour *l* enters in both the models as 1D phase of anomalies. Note that the model, with a defect line on the semiconductor interface, is closely related to real experimental data, found by scanning tunneling microscopy. On the other hand, in the case of organic interface (considered in our second model as the known *lamina basale*), the lamina folio is supposed cleft in two sub-phases by a (straight) line of functionally anomalous intercellular spaces (holes). In both the models bulk charge densities

[ ] *u* are replaced with their linear approximations, however charge quantities [ ]

*<sup>l</sup> u* enter nonlinearly in the cases, respectively of semiconductor and organic interfaces. By transforming relations (0.2), (0.3), we derive and solve the relevant integral equations for the surface potential *uS* . We obtain also effective formulas for certain approximations of *uS* (in the case of *S* – a semiconductor folio), and – for the exact potential (in the case of organic *S* ). Recall here that the contemporary problems primarily focus in detecting the surface values of the electric potential. As a consequence of finding the surface potential, we express in addition, by the classical Dirichlet problem, the bulk potentials *u* as well. Thus we get expressions for potential *u* , valid far from the interface, which ensure in particular important diagnostic analyses (made, for instance, by parametric identification inverse

In Sect. 2 we give phenomenology comments and a common derivation of the two considered models. The basic results on the surface potential are presented in Sect. 3. The question for determining of explicit approximations to *uxyz* (,,) is discussed in Sect. 4, in

Let us give firstly some phenomenology comment on said two classes of heterogeneous media. Beginning with the case of organic interface, we should note the following. The interest of tools for biomedical detections of anomalies in the human circulatory system, via the walls-structure of the blood vessels, is directly motivated from the quite specific ruling function of the wall-layers. To recall and clarify the main (simplified) viewpoints here (cf. e.g. [11]), we assume a stretched location of the wall, as the flat surface ( 0 *z* ) on Fig. 1, below. Said construction is introduced as an admissible version of the real situation: a 3D localization is made to a capillary (practically cylindrical) vessel in the human white liver and the vessel-wall is (functionally) identified with its middle layer (called lamina basale). In reality the wall is a 3D organic threefold layer, deep not less than 120-180 nm and the midmost (just the lamina basale) is of corpulence about 40-60 nm. The upper (external) and lower (internal) layers, built – as a short description – respectively of endothelian and adventitale cells, are neglected. They are considered with a secondary role (compared to lamina basale). As known, acting as a typical bio-membrane of polysaccharide-matter, with

**2. Elements of phenomenology and mathematical modeling** 

*<sup>s</sup> <sup>u</sup>* and

 . Mathematically, relations (0.1) – (0.3) present a new type of transmission problem (cf. the Colton – Kress monograph, [3]). In the above generality of formulation,

said transition layer as a material 2D formation *S* – the common surface boundary of the two bulk phases. Thus the first stage of introducing heterogeneous media results in the (Gibbs idealized) heterogeneous system {*B SB* } , consisting in two bulk phases – *B* , *B* , and the 2D phase *S* – as an interface. The above mentioned stage (of second order heterogeneity) should be noted as a first point of new elements in our results: we assume the interface in the form { } *S S lS* , with two 2D sub-phases – *S* , *S* (of generally differing 2D materials), and a line material component *l* – the common boundary of *S* , *S* . Component *l* is assumed homogeneous and introduced by applying again the Gibbs idealizing approach, taken now on the interface. Next, let us comment the second required basic step of modeling. Because the aim is to model electrostatics, we have to deal with the Maxwell electrostatic system, as a constituting-phenomena low, applied however for the totally heterogeneous medium {*B S lS B* } . Note here the specific detail concerning the charge density (which essentially enters in the Maxwell system): via an electrochemical principle, we should presume depending on the electric potential *u* , i.e. it generally holds [ ] *u* . Moreover, function [ ] *u* takes the form of the known Boltzmann distribution, for instance in case of electrolytes. Because the Gibbs idealization assumes a step transition across the interface (consequently again such transition, but of lower dimension, is assumed across the phase contour*l* ), the next appearing problem reads: how to formalize said step transitions, in order to use effectively the Maxwell system. A useful suggestion for the first transition stage (across the interface) can be found in the monograph of D. Bedeaux and J. Vlieger ([2]). According to Bedeaux – Vlieger, we introduce the relevant material characteristics across the interface by a (first level) decomposition scheme of singularities, using Heaviside step functions regarding respectively the bulk phases *B* , and Dirac delta function *<sup>s</sup>* , supported on the interface. As a second new point of our modeling, we introduce analogous decomposition scheme on the surface *S* , using in particular delta function supported on the contour *l* . After a technical procedure of solving the Maxwell electrostatic system by singular solutions, it can be established the following final form of our electrostatic model for the class of heterogeneous media {*B S lS B* } :

$$\nabla^2 \mu = -\varepsilon\_0^{-1} (\varepsilon^{\pm})^{-1} \, \rho^{\pm} [\mu] \text{ (in } B^{\pm} \text{)}\tag{1.1}$$

$$J\_S[\mu] + \varepsilon\_s^{\pm} \nabla\_S^2 \mu = -\varepsilon\_0^{-1} \rho\_s^{\pm}[\mu] \text{ (on } S^{\pm} \text{)}\tag{1.2}$$

$$J\_l[\![\mu] = -\beta\_l[\![\mu] \text{ (on } l\text{)}] \tag{1.3}$$

Above *u uxyz* (,,) is the electric potential, which is sough as bounded continuous function, regular enough in the relevant 3D and 2D phases (domains) of the material system; <sup>2</sup> is the 3D Laplace operator and <sup>2</sup> *S* is a tangential to surface *S* Laplace operator; *SJ* is a jump type operator acting on the normal to *S* derivative of potential *u* , and, by analogy – for operator *<sup>l</sup> J* , concerning contour *l* ; and *s* are the charge density terms, respectively for the bulk ( *B* ) and surface ( *S* ) phases, and *<sup>l</sup>* is an analogous quantity, for 1D phase *l* ; 0 8.85 / *pF m* is the known absolute dielectric permittivity, and *<sup>s</sup>* are the (relative) dielectric permitivities, respectively for the matter of phases *B* and *S* , with <sup>1</sup> 0 0 1 /, 84 Electrostatics

said transition layer as a material 2D formation *S* – the common surface boundary of the two bulk phases. Thus the first stage of introducing heterogeneous media results in the (Gibbs idealized) heterogeneous system {*B SB* } , consisting in two bulk phases – *B* , *B* , and the 2D phase *S* – as an interface. The above mentioned stage (of second order heterogeneity) should be noted as a first point of new elements in our results: we assume the interface in the form { } *S S lS* , with two 2D sub-phases – *S* , *S* (of generally differing 2D materials), and a line material component *l* – the common boundary of *S* , *S* . Component *l* is assumed homogeneous and introduced by applying again the Gibbs idealizing approach, taken now on the interface. Next, let us comment the second required basic step of modeling. Because the aim is to model electrostatics, we have to deal with the Maxwell electrostatic system, as a constituting-phenomena low, applied however for the totally heterogeneous medium {*B S lS B* } . Note here the specific detail

distribution, for instance in case of electrolytes. Because the Gibbs idealization assumes a step transition across the interface (consequently again such transition, but of lower dimension, is assumed across the phase contour*l* ), the next appearing problem reads: how to formalize said step transitions, in order to use effectively the Maxwell system. A useful suggestion for the first transition stage (across the interface) can be found in the monograph of D. Bedeaux and J. Vlieger ([2]). According to Bedeaux – Vlieger, we introduce the relevant material characteristics across the interface by a (first level) decomposition scheme of

modeling, we introduce analogous decomposition scheme on the surface *S* , using in particular delta function supported on the contour *l* . After a technical procedure of solving the Maxwell electrostatic system by singular solutions, it can be established the following final form of our electrostatic model for the class of heterogeneous

> 

2 1 <sup>0</sup> [ ] [ ] *S sS s Ju u u*

> [] [] *l l Ju u*

dielectric permitivities, respectively for the matter of phases *B* and *S* , with <sup>1</sup>

 

Above *u uxyz* (,,) is the electric potential, which is sough as bounded continuous function, regular enough in the relevant 3D and 2D phases (domains) of the material system; <sup>2</sup> is the 3D Laplace operator and <sup>2</sup> *S* is a tangential to surface *S* Laplace operator; *SJ* is a jump type operator acting on the normal to *S* derivative of potential *u* , and, by analogy – for operator

2 11 <sup>0</sup> *u u* 

(which essentially enters in the Maxwell system): via an

, supported on the interface. As a second new point of our

( ) [] (in *B* ) (1.1)

(on *S* ) (1.2)

are the charge density terms, respectively for the bulk

*<sup>l</sup>* is an analogous quantity, for 1D phase *l* ;

 and *<sup>s</sup>* 

are the (relative)

1 /

0 0

,

(on *l* ) (1.3)

depending on the electric potential *u* , i.e.

regarding respectively the bulk phases *B* ,

[ ] *u* takes the form of the known Boltzmann

concerning the charge density

and Dirac delta function *<sup>s</sup>*

media {*B S lS B* } :

*<sup>l</sup> J* , concerning contour *l* ;

0 

( *B* ) and surface ( *S* ) phases, and

 

it generally holds

electrochemical principle, we should presume

singularities, using Heaviside step functions

 and *s*

8.85 / *pF m* is the known absolute dielectric permittivity,

[ ] *u* . Moreover, function

<sup>1</sup> ( ) 1/ . Mathematically, relations (0.1) – (0.3) present a new type of transmission problem (cf. the Colton – Kress monograph, [3]). In the above generality of formulation, problem (0.1) – (0.3) remains however as open one.

In this chapter we give the main steps of deriving and solve two sub-cases of the general problem (0.1) – (0.3), related to heterogeneous systems with flat interfaces, respectively of semiconductor and organic nature. A straight line contour *l* enters in both the models as 1D phase of anomalies. Note that the model, with a defect line on the semiconductor interface, is closely related to real experimental data, found by scanning tunneling microscopy. On the other hand, in the case of organic interface (considered in our second model as the known *lamina basale*), the lamina folio is supposed cleft in two sub-phases by a (straight) line of functionally anomalous intercellular spaces (holes). In both the models bulk charge densities [ ] *u* are replaced with their linear approximations, however charge quantities [ ] *<sup>s</sup> <sup>u</sup>* and [ ] *<sup>l</sup> u* enter nonlinearly in the cases, respectively of semiconductor and organic interfaces. By transforming relations (0.2), (0.3), we derive and solve the relevant integral equations for the surface potential *uS* . We obtain also effective formulas for certain approximations of *uS* (in the case of *S* – a semiconductor folio), and – for the exact potential (in the case of organic *S* ). Recall here that the contemporary problems primarily focus in detecting the surface values of the electric potential. As a consequence of finding the surface potential, we express in addition, by the classical Dirichlet problem, the bulk potentials *u* as well. Thus we get expressions for potential *u* , valid far from the interface, which ensure in particular important diagnostic analyses (made, for instance, by parametric identification inverse problems).

In Sect. 2 we give phenomenology comments and a common derivation of the two considered models. The basic results on the surface potential are presented in Sect. 3. The question for determining of explicit approximations to *uxyz* (,,) is discussed in Sect. 4, in case of semiconductor interface.

#### **2. Elements of phenomenology and mathematical modeling**

Let us give firstly some phenomenology comment on said two classes of heterogeneous media. Beginning with the case of organic interface, we should note the following. The interest of tools for biomedical detections of anomalies in the human circulatory system, via the walls-structure of the blood vessels, is directly motivated from the quite specific ruling function of the wall-layers. To recall and clarify the main (simplified) viewpoints here (cf. e.g. [11]), we assume a stretched location of the wall, as the flat surface ( 0 *z* ) on Fig. 1, below. Said construction is introduced as an admissible version of the real situation: a 3D localization is made to a capillary (practically cylindrical) vessel in the human white liver and the vessel-wall is (functionally) identified with its middle layer (called lamina basale). In reality the wall is a 3D organic threefold layer, deep not less than 120-180 nm and the midmost (just the lamina basale) is of corpulence about 40-60 nm. The upper (external) and lower (internal) layers, built – as a short description – respectively of endothelian and adventitale cells, are neglected. They are considered with a secondary role (compared to lamina basale). As known, acting as a typical bio-membrane of polysaccharide-matter, with

Fig. 1.

Mathematical Models for Electrostatics of Heterogeneous Media 87

a fine fibers structure, lamina basale is the main factor for the oxygen transfer to the blood. Via the Gibbs approach (and taking into account the ratio of the wall corpulence to the radius of the capillary vessel, which is 1 ), we consider lamina basale layer as infinitely thin; thus we get an interface film of organic matter in an air-blood (vacuum-blood) heterogeneous 3D media. On the lamina (2D) film it is uniformly distributed a set of points – presenting the holes (tunnels, in reality radial to the vessel axis and known as intercellular spaces), which provide the oxygen contact to the blood; they are assumed however of certain functional anomaly, extremely activated on a relatively narrow (cylindrical, in reality) strip, interpreted, following Gibbs, as the middle circumference of the strip, across to the vessel-axis. Thus a specific homogeneous 1D matter phase (of the extreme anomalies) has appeared. Stretching (locally) the curved anomaly-line (and the surrounding cylindrical surface, together), we get the above mentioned (flat-interfaced) construction. Now the organic lamina-film can be presented as the plain 0 *z* (regarding a Cartesian (,,) *xyz* coordinate system), where said 1D contour, of defective air permeability, has already shaped as a straight line. We shall take this line as the *Oy* - axis. This manner the laminafilm is cleft in two electrostatic equivalent 2D (sub-) phases by the anomaly contour and we have given a typical case of 3-2-1 D heterogeneous system, schematically shown on Fig. 1, below. The system consists in upper and lower 3D (bulk) phases, respectively of air and liquid, and a complex-structured organic interface (with a special role of a line phase). The bulk phases *B* , *B* fill the subspaces *z* 0 ( *B* ), 0 *z* ( *B* ) and their common 2D boundary – the organic interface *S* – is given (as already noted) by the equation 0 *z* ; *S* consists in the two neighbouring surface phases *S* ( *x z* 0, 0 ), *S* ( *x z* 0, 0 ), separated by the anomaly line *l Oy* , as an autonomous phase of 1D matter. To forecast certain influence of vessel-zones, relatively far from the phase contour *l* , surface phases *S* , *S* are presumed with prescribed asymptotic values ( , ) of the electric potential.

It is possible however a sharp variant of anomalies: an air volume can leave involved between the blood and the surface, shaping an internal (lower) air bulk phase. Such presence of two-side bulk air phases is due to the anomalous air transfer: the outgoing stage gets blocked, after a previous air invasion. We would have then a heterogeneous system with upper and lower air (vacuum) bulk phases, a two-phased lamina-interface (as a 2D film) and a separating the surface phases homogeneous 1D material (straight line) phase.

Another class of heterogeneous media is that including a semiconductor interface. The model under consideration relates to electrostatics for the specific case of air-gas matter, with a semiconductor separating surface (interface), which includes moreover a defect straight line (considered below as the *Oy* - axis, see Fig. 2). In said case of systems the importance of the interface electrostatics is motivated, as already noted, from actual technological questions (e.g. [5], [7]).The structure of such a system can be explained as a space location of given electronic device. For a short description we will take into account the following. By a teen boundary wall of semiconductor-matter it is closed a volume of gas, and the external medium is of air. This boundary, generally curved, will be treated here as flat (observing a small part of it). Thus the system possesses two bulk phases – of internal (gas) and external (air) media, and a flat semiconducting interface, with a fine surface 86 Electrostatics

a fine fibers structure, lamina basale is the main factor for the oxygen transfer to the blood. Via the Gibbs approach (and taking into account the ratio of the wall corpulence to the radius of the capillary vessel, which is 1 ), we consider lamina basale layer as infinitely thin; thus we get an interface film of organic matter in an air-blood (vacuum-blood) heterogeneous 3D media. On the lamina (2D) film it is uniformly distributed a set of points – presenting the holes (tunnels, in reality radial to the vessel axis and known as intercellular spaces), which provide the oxygen contact to the blood; they are assumed however of certain functional anomaly, extremely activated on a relatively narrow (cylindrical, in reality) strip, interpreted, following Gibbs, as the middle circumference of the strip, across to the vessel-axis. Thus a specific homogeneous 1D matter phase (of the extreme anomalies) has appeared. Stretching (locally) the curved anomaly-line (and the surrounding cylindrical surface, together), we get the above mentioned (flat-interfaced) construction. Now the organic lamina-film can be presented as the plain 0 *z* (regarding a Cartesian (,,) *xyz* coordinate system), where said 1D contour, of defective air permeability, has already shaped as a straight line. We shall take this line as the *Oy* - axis. This manner the laminafilm is cleft in two electrostatic equivalent 2D (sub-) phases by the anomaly contour and we have given a typical case of 3-2-1 D heterogeneous system, schematically shown on Fig. 1, below. The system consists in upper and lower 3D (bulk) phases, respectively of air and liquid, and a complex-structured organic interface (with a special role of a line phase). The bulk phases *B* , *B* fill the subspaces *z* 0 ( *B* ), 0 *z* ( *B* ) and their common 2D boundary – the organic interface *S* – is given (as already noted) by the equation 0 *z* ; *S* consists in the two neighbouring surface phases *S* ( *x z* 0, 0 ), *S* ( *x z* 0, 0 ), separated by the anomaly line *l Oy* , as an autonomous phase of 1D matter. To forecast certain influence of vessel-zones, relatively far from the phase contour *l* , surface phases *S* , *S* are

> 

It is possible however a sharp variant of anomalies: an air volume can leave involved between the blood and the surface, shaping an internal (lower) air bulk phase. Such presence of two-side bulk air phases is due to the anomalous air transfer: the outgoing stage gets blocked, after a previous air invasion. We would have then a heterogeneous system with upper and lower air (vacuum) bulk phases, a two-phased lamina-interface (as a 2D film) and a separating the surface phases homogeneous 1D material (straight line)

Another class of heterogeneous media is that including a semiconductor interface. The model under consideration relates to electrostatics for the specific case of air-gas matter, with a semiconductor separating surface (interface), which includes moreover a defect straight line (considered below as the *Oy* - axis, see Fig. 2). In said case of systems the importance of the interface electrostatics is motivated, as already noted, from actual technological questions (e.g. [5], [7]).The structure of such a system can be explained as a space location of given electronic device. For a short description we will take into account the following. By a teen boundary wall of semiconductor-matter it is closed a volume of gas, and the external medium is of air. This boundary, generally curved, will be treated here as flat (observing a small part of it). Thus the system possesses two bulk phases – of internal (gas) and external (air) media, and a flat semiconducting interface, with a fine surface

) of the electric potential.

presumed with prescribed asymptotic values ( ,

phase.

Fig. 1.

Fig. 2.

Mathematical Models for Electrostatics of Heterogeneous Media 89

a) . D

permitivity for the relevant part of the medium (in particular *<sup>b</sup>*

vectors nabla and D, i.e. . D *div* D;

 ; b) D 0

is the charge density;

Here is the nabla operator, D is the vector of the electric induction ([12]), called also (in Electrochemistry, e.g. [9]) electric displacement, . D is the formal scalar product of the

*z x* 0, 0 ); *u* is the electric potential, *u grad u*( ), where ( ) *u* represents the electric field, propagated in the whole 3D material system. Equations (2.1) hold for the total (3D) system and, as known, potential *u* is a continuous function of (,,) *xyz* , in spite of the various material phases; the heterogeneity of the system is indicated however mainly by the

*u* . (2.1)

is the relative dielectric

 , at

, at 0 *<sup>z</sup>* , *<sup>s</sup>*

 

roughness, as a straight line defect. The bulk phases are considered as materially equivalent (3D) sub-domains of vacuum. The separating boundary is of indium-phosphorus, InP(110), semiconductor: the real corpulence of the InP(110)-wall is neglected and the wall is identified with its external surface film. The defect line, playing the role of a homogeneous detachment, i.e. of an electrostatic autonomous material 1D phase, separates said interface in two surface (2D) phases, denoted as before by *S* ( *x z* 0, 0 ), *S* ( *x z* 0, 0 ). It is posed again a typical case of 3-2-1 D heterogeneity – by a material system, interpreted as vacuum-semiconductor-vacuum (Fig. 2). The InP(110) surface film is presumed as the plain *z* 0 as well (see Fig. 2) and the *Oy* - axis is oriented on the defect straight line. The vacuum bulk phases fill the upper and lower semi-spaces, 0 *z* and 0 *z* , respectively. Each 2D phase (on 0 *z* ) is characterized by an essentially dominating distribution of positively charged phosphorus vacancies, while, as a key anomaly, the line phase *l* ( *l Oy* ) enters in the surface electrostatics symmetrically surrounded by an extremely narrow strip of width 2*d* (0 *d* ). The whole this band is denuded of phosphorus vacancies ([5], [7]). The above construction is essentially supported by real experimentally found data. A credible visual result of [5] and [7] (see Fig. 2, [7]) has been found by the so-called scanning tunneling microscopy. The picture (Fig. 2, [7]) shows the surface structure, fixed after annealing of InP(110) samples at temperatures up to 480 K, followed by heat normalizing. The scanned image includes two near by surface domains (let us denote them by , *P P* - see the semiplanes *x dx d* , on Fig. 2, below; clearly *P S* ). These zones are materially equivalent and separated by a transition strip *T* . Its breadth ( 2*d* ) really is less than 10 nm ([7]). The strip surrounds symmetrically a straight line *l* (see *Oy* , Fig. 2). Each of , *P P* is filled by positively charged (+1e) phosphorus vacancies, with about 5.5 nm ([5]) mean distance between them. Note however that strip *T* is free of vacancies, but remains generally charged with about +2e (per spacing of 0.6 nm, [7]) mean magnitude of charges on the axis of symmetry (the line *l* ). This way *T* enters as an electrostatic autonomous surface component. On the other hand the ratio {[area]( *T* ) / [area]( *P TP* )} is negligible to consider *T* as an equipollent (say to , *P P* ) 2D surface component. Moreover, the two relations - that of *T* -wide (10 nm) to the above density unit (5.5 nm), and the other – of possible (averaged) electrostatic impact of *T* to the influence of its middle axis *l* , allow some identifying of the strip *T* with the axis *l* ; thus *l* takes the role of an intrinsic 1D phase. Let us note that, via the Gibbs approach, the semi-zones ( ( 0), ( 0) *Tx Tx* ) of *T* seem to complicate additionally the surface heterogeneity, imposing – as a (Gibbs) principle – new line phases: the contours { ; 0} *dl x dz* and { ; 0} *dl x dz* . These (new) phases however enter also negligibly in the surface electrostatics: across , *d d l l* the surface electric field stays continuous, under equal permitivity-values *<sup>s</sup>* . So-described picture (of a smooth, flat 2D film) represents a real surface layer with certain nanoscopic roughness, due to step defects. In reality *l* is actually the edge of a step, deep about 4 nm ([7], Fig. 1), *P* and *P* are the terraces (lower and upper, say) of the step, and *T* is a space construction, divided by *l* in two halves, *T* (lower) and *T* (upper), marking off the edge from the relevant terraces.

The next stage of this section is to sketch the basic step of modeling. Via the introduced framework (see Sect. 1) the key tool for description of electrostatic phenomena in complex media relates to the Maxwell system (in case of dielectrics, e.g. [10], [12]):

Fig. 2.

88 Electrostatics

roughness, as a straight line defect. The bulk phases are considered as materially equivalent (3D) sub-domains of vacuum. The separating boundary is of indium-phosphorus, InP(110), semiconductor: the real corpulence of the InP(110)-wall is neglected and the wall is identified with its external surface film. The defect line, playing the role of a homogeneous detachment, i.e. of an electrostatic autonomous material 1D phase, separates said interface in two surface (2D) phases, denoted as before by *S* ( *x z* 0, 0 ), *S* ( *x z* 0, 0 ). It is posed again a typical case of 3-2-1 D heterogeneity – by a material system, interpreted as vacuum-semiconductor-vacuum (Fig. 2). The InP(110) surface film is presumed as the plain *z* 0 as well (see Fig. 2) and the *Oy* - axis is oriented on the defect straight line. The vacuum bulk phases fill the upper and lower semi-spaces, 0 *z* and 0 *z* , respectively. Each 2D phase (on 0 *z* ) is characterized by an essentially dominating distribution of positively charged phosphorus vacancies, while, as a key anomaly, the line phase *l* ( *l Oy* ) enters in the surface electrostatics symmetrically surrounded by an extremely narrow strip of width 2*d* (0 *d* ). The whole this band is denuded of phosphorus vacancies ([5], [7]). The above construction is essentially supported by real experimentally found data. A credible visual result of [5] and [7] (see Fig. 2, [7]) has been found by the so-called scanning tunneling microscopy. The picture (Fig. 2, [7]) shows the surface structure, fixed after annealing of InP(110) samples at temperatures up to 480 K, followed by heat normalizing. The scanned image includes two near by surface domains (let us denote them by , *P P* - see the semiplanes *x dx d* , on Fig. 2, below; clearly *P S* ). These zones are materially equivalent and separated by a transition strip *T* . Its breadth ( 2*d* ) really is less than 10 nm ([7]). The strip surrounds symmetrically a straight line *l* (see *Oy* , Fig. 2). Each of , *P P* is filled by positively charged (+1e) phosphorus vacancies, with about 5.5 nm ([5]) mean distance between them. Note however that strip *T* is free of vacancies, but remains generally charged with about +2e (per spacing of 0.6 nm, [7]) mean magnitude of charges on the axis of symmetry (the line *l* ). This way *T* enters as an electrostatic autonomous surface component. On the other hand the ratio {[area]( *T* ) / [area]( *P TP* )} is negligible to consider *T* as an equipollent (say to , *P P* ) 2D surface component. Moreover, the two relations - that of *T* -wide (10 nm) to the above density unit (5.5 nm), and the other – of possible (averaged) electrostatic impact of *T* to the influence of its middle axis *l* , allow some identifying of the strip *T* with the axis *l* ; thus *l* takes the role of an intrinsic 1D phase. Let us note that, via the Gibbs approach, the semi-zones ( ( 0), ( 0) *Tx Tx* ) of *T* seem to complicate additionally the surface heterogeneity, imposing – as a (Gibbs) principle

– new line phases: the contours { ; 0} *dl x dz*

relevant terraces.

electric field stays continuous, under equal permitivity-values *<sup>s</sup>*

phases however enter also negligibly in the surface electrostatics: across , *d d l l*

media relates to the Maxwell system (in case of dielectrics, e.g. [10], [12]):

smooth, flat 2D film) represents a real surface layer with certain nanoscopic roughness, due to step defects. In reality *l* is actually the edge of a step, deep about 4 nm ([7], Fig. 1), *P* and *P* are the terraces (lower and upper, say) of the step, and *T* is a space construction, divided by *l* in two halves, *T* (lower) and *T* (upper), marking off the edge from the

The next stage of this section is to sketch the basic step of modeling. Via the introduced framework (see Sect. 1) the key tool for description of electrostatic phenomena in complex

and { ; 0} *dl x dz*

. These (new)

. So-described picture (of a

the surface

$$\begin{array}{c} \text{(a)} \; \nabla. \; \mathbf{D} = \rho \; ; \mathbf{b} \; \text{)} \; \mathbf{D} = -\varepsilon\_{0} \varepsilon \nabla \mu \; . \end{array} . \tag{2.1}$$

Here is the nabla operator, D is the vector of the electric induction ([12]), called also (in Electrochemistry, e.g. [9]) electric displacement, . D is the formal scalar product of the vectors nabla and D, i.e. . D *div* D; is the charge density; is the relative dielectric permitivity for the relevant part of the medium (in particular *<sup>b</sup>* , at 0 *<sup>z</sup>* , *<sup>s</sup>* , at *z x* 0, 0 ); *u* is the electric potential, *u grad u*( ), where ( ) *u* represents the electric field, propagated in the whole 3D material system. Equations (2.1) hold for the total (3D) system and, as known, potential *u* is a continuous function of (,,) *xyz* , in spite of the various material phases; the heterogeneity of the system is indicated however mainly by the

Mathematical Models for Electrostatics of Heterogeneous Media 91

(a given quantity):

*u* – the electric charge density (supposed depending on potential *u* )

 

is the asymptotic surface power;

*q u* , while a linear approximation (regarding

, 0 *<sup>b</sup> <sup>k</sup>* . (2.7)

*s s s s s ll*

8.85 / *pF m* is the mentioned absolute dielectric permitivity. In

is the space-jump operator,

. Here

*k u* , where

and

 

(at *x x* 0, 0 ),

*l l u* enters as

*<sup>s</sup> <sup>u</sup>* ,

are real, generally different constants; the parameter [ ]

and in the right hand side of (2.4) we have

while for semiconductors we shall use 0 2 [] ( [ ]) <sup>2</sup> *s d ss u k sg q us*

  

*<sup>x</sup>* is the sign function; 1 1 2 \*2 2 \*3 [ ] [ ( )] [ ( )] 2! 3! *<sup>s</sup> s s s s qu k u*

. In the case of organic interface we shall suppose [ ] 0

is the rescaled characteristic function of the unit interval, i.e.

. The cubic nonlinear charge density (of the surface 0 *<sup>z</sup>* ) is preferred just on

Via the phenomenology-essence potential *u* will be searched for a bounded function (condition (2.3)), continuous in <sup>3</sup> *R* , classically regular in the sets 0 *z* and *x z* 0 ( 0) , with continuous gradients *uz* , *ux* , respectively at 0 *z* , 0 *z* (for *uz* ), and *x x* 0, 0 ( 0) *z* – for *ux* . Now

Definition. A function *uxz* (,), with the above noted regularity, shall be called *classical solution* to problem (2.2) – (2.6) if satisfies the additional property 2 *ux L* ( ,0)

From the vacuum assumption for the upper (external) air phase ( 0 *z* ) we suppose from

The central results below relate to determination of the surface potential (possibly by an explicit approximation) – as the key first step in solving the full (2.2) – (2.6) – problem.

Let us now sketch the main steps for derivation of the final mathematical models, starting from the Maxwell system (2.1). Via the presumed complex heterogeneity, we shall seek solutions (D, *u*) of system (2.1) by decompositions in two levels (bulk and surface), of the

 ; (2.8)

relations (2.2) – (2.6). ( *L*2 is the well known space of the squared-integrable functions.)

potential are prescribed by the parameter

equation (2.4.b) [ ] [ ]( ,0)

 

the strip | |*x d* – by the rest term <sup>0</sup> [ ] *d s*

1 *<sup>b</sup>*

 a) ( ) *bbs z* ; b)

potential *u*) is assumed adequate to reality out of the strip.

we can define the needed space of regular solutions to (2.2) – (2.6).

[ ]( ,0) ( , 0) ( , 0) *bz b z u x ux ux b z*

*x*

 

*d d*

 ( ) 1,| | 1; ( ) 0,| | 1 *xx xx* ;

   

 *bz bz u ux* 

where , 

*l* 

\*

*sg sg x*

now on:

following type:

0 *l*

 by [ ] *l l* 

upon the line phase; 0

2 <sup>0</sup> [ ] ( ( ) [ ])

*<sup>s</sup> u ku u s s <sup>s</sup>*

00 0 <sup>1</sup> ( ) <sup>2</sup> *d d*

0 0

( ) | | *x*

 

2 

*x*

 

quantities D and (note that the permitivity enters in these quantities). Next, from the singular decompositions, mentioned in Sect.1, applied below for quantities D and , we get the following problem. Find the (admissibly regular) solutions (D, *u*) to (2.1), corresponding to the said singular decompositions.

Both the considered cases of heterogeneous systems are however homogeneous on the *y* – direction, due to assumed homogeneity of the 1D phase *l* , and the electric potential *u uxyz* (,,) will actually depend on *x z*, , i.e. *u uxz* (,). Applying systematically a double decomposition scheme in reworking of the Maxwell system (see below), we shall establish the following final formulation to the sought mathematical models:

$$
\nabla^2 \mu = \kappa\_b^2 \mu \text{ (}\!z \neq 0\text{)}, \propto \in \mathbb{R}^1 \text{ ;}\tag{2.2}
$$

$$\|\|u\|\leq \text{const.} \,, \text{(x,z)}\in \mathbb{R}^{2} \;\;\text{;}\tag{2.3}$$

$$
\mu(\mathbf{x}, +0) = \mu(\mathbf{x}, -0), \; \mathbf{x} \in \mathbb{R}^1; \tag{2.4a}
$$

$$
\Delta \{ \varepsilon\_b \mu\_z \} + \varepsilon\_s \mu\_{\text{xx}} = \varepsilon\_s k\_s^2 (\mu - \varphi\_\alpha) + \rho\_s^\* [\mu], \text{ x} \neq 0 \; ; \tag{2.4b}
$$

$$
\mu(-\infty,0) = \phi\_{w,\prime}^- \text{ } \mu(+\infty,0) = \phi\_w^+ \text{ } \tag{2.5}
$$

$$
\mu(-0,0) = \mu(+0,0) \; ; \tag{2.6a}
$$

$$
\varepsilon\_s^+ \mu\_x(+0,0) - \varepsilon\_s^- \mu\_x(-0,0) = -\beta\_l[\mu].\tag{2.6b}
$$

In (2.2) <sup>222</sup> *x z* is the Laplace operator; , , *uuu x z xx* are first or second order derivatives regarding the relevant variable; *u x*( , 0) , *u x*( , 0) are respectively the limits (supposed finite) 0 lim ( , ) *z uxz* (at 0 *z* or 0 *z* ), and, by analogy – for ( , 0) *u x <sup>z</sup>* , ( , 0) *u x <sup>z</sup>* ; 0 ( 0,0) lim ( ,0) *<sup>x</sup> u ux* and 0 ( 0,0) lim ( ,0) *x x <sup>x</sup> u ux* , respectively at *x x* 0, 0 , both – for *<sup>u</sup>*( 0,0) and ( 0,0) *ux* ; ( ,0) lim ( ,0) *<sup>x</sup> <sup>u</sup> u x* . (Above *<sup>m</sup> R* is the real *m* – dimensional Euclidean space, *m* 1,2,... .) As known, parameters *<sup>b</sup> k* , *<sup>b</sup>* and , *s s k* are the main factors of the system-electrostatic nature; they are given step constants: ( 0) *b b <sup>z</sup>* , ( 0) *b b z* , , *b b* – positive (and generally different); ( 0) *b b k kz* , ( 0) *b b k kz* are nonnegative constants in (2.2); ( 0) *s s k kx* , ( 0) *s s k kx* – in (2.4), with positive *<sup>s</sup> <sup>k</sup>* , *<sup>s</sup> <sup>k</sup>* ; by analogy, ( 0) *s s <sup>x</sup>* , ( 0) *s s <sup>x</sup>* – in (2.4), (2.6), with 0 *<sup>s</sup>* , 0 *<sup>s</sup>* – constants. The material meaning of parameter *<sup>s</sup> k* (by analogy from that of *<sup>b</sup> k* ) is expressed by the quantity <sup>1</sup> 1 *s s k k* , known as the surface Debye length (e.g. [13], or the surface screening length (e.g. [7]). Parameters *<sup>b</sup>* , *<sup>s</sup>* are respectively the bulk and surface dielectric permitivities, with ( ) *b b* , ( ) *s s* - for the relevant bulk and surface phases. The asymptotic values of the 90 Electrostatics

the following problem. Find the (admissibly regular) solutions (D, *u*) to (2.1), corresponding

Both the considered cases of heterogeneous systems are however homogeneous on the *y* – direction, due to assumed homogeneity of the 1D phase *l* , and the electric potential *u uxyz* (,,) will actually depend on *x z*, , i.e. *u uxz* (,). Applying systematically a double decomposition scheme in reworking of the Maxwell system (see below), we shall establish

2 2 <sup>1</sup> ( 0), *u uz x R*

<sup>2</sup> [ ] ( ) [] *b z s xx s s u u ku u <sup>s</sup>*

( 0,0) ( 0,0) [ ] *sx sx uu ul*

In (2.2) <sup>222</sup> *x z* is the Laplace operator; , , *uuu x z xx* are first or second order derivatives regarding the relevant variable; *u x*( , 0) , *u x*( , 0) are respectively the limits (supposed

 – positive (and generally different); ( 0) *b b k kz* , ( 0) *b b k kz* are nonnegative constants in (2.2); ( 0) *s s k kx* , ( 0) *s s k kx* – in (2.4), with positive *<sup>s</sup> <sup>k</sup>* , *<sup>s</sup> <sup>k</sup>* ; by analogy,

meaning of parameter *<sup>s</sup> k* (by analogy from that of *<sup>b</sup> k* ) is expressed by the quantity

, known as the surface Debye length (e.g. [13], or the surface screening length (e.g.


 

 

singular decompositions, mentioned in Sect.1, applied below for quantities D and

enters in these quantities). Next, from the

*<sup>b</sup>* ; (2.2)

<sup>2</sup> | | ., ( , ) *u const x z R* ; (2.3)

<sup>1</sup> *ux ux x R* ( , 0) ( , 0), ; (2.4a)

; (2.5)

*u u* ( 0,0) ( 0,0) ; (2.6a)

, 0 *<sup>x</sup>* ; (2.4b)

 . (2.6b)

(at 0 *z* or 0 *z* ), and, by analogy – for ( , 0) *u x <sup>z</sup>* , ( , 0) *u x <sup>z</sup>* ;

, respectively at *x x* 0, 0 , both – for

 and , *s s k* 

> 

are the main factors of

 *<sup>z</sup>* , ( 0) *b b z* ,

– constants. The material

. (Above *<sup>m</sup> R* is the real *m* – dimensional

 , 0 *<sup>s</sup>* 

are respectively the bulk and surface dielectric permitivities, with

, we get

(note that the permitivity

the following final formulation to the sought mathematical models:

 

0 ( 0,0) lim ( ,0) *x x <sup>x</sup> u ux*

the system-electrostatic nature; they are given step constants: ( 0) *b b*

*<sup>x</sup>* – in (2.4), (2.6), with 0 *<sup>s</sup>*

*<sup>u</sup> u x*

Euclidean space, *m* 1,2,... .) As known, parameters *<sup>b</sup> k* , *<sup>b</sup>*

( ,0) , ( ,0) *u u*

quantities D and

finite) 0

, *b b* 

 

*k*

( 0) *s s*

*s*

[7]). Parameters *<sup>b</sup>*

*k*

<sup>1</sup> 1 *s*

( ) *b b* , ( ) *s s* 

lim ( , ) *z uxz*

0 ( 0,0) lim ( ,0) *<sup>x</sup> u ux*

and

 *<sup>x</sup>* , ( 0) *s s* 

> , *<sup>s</sup>*

*<sup>u</sup>*( 0,0) and ( 0,0) *ux* ; ( ,0) lim ( ,0) *<sup>x</sup>*

to the said singular decompositions.

potential are prescribed by the parameter (a given quantity): (at *x x* 0, 0 ), where , are real, generally different constants; the parameter [ ] *l l u* enters as 0 *l l* by [ ] *l l u* – the electric charge density (supposed depending on potential *u* ) upon the line phase; 0 8.85 / *pF m* is the mentioned absolute dielectric permitivity. In equation (2.4.b) [ ] [ ]( ,0) *bz bz u ux* is the space-jump operator, [ ]( ,0) ( , 0) ( , 0) *bz b z u x ux ux b z* and in the right hand side of (2.4) we have 2 <sup>0</sup> [ ] ( ( ) [ ]) *<sup>s</sup> u ku u s s <sup>s</sup>* . In the case of organic interface we shall suppose [ ] 0 *<sup>s</sup> <sup>u</sup>* , while for semiconductors we shall use 0 2 [] ( [ ]) <sup>2</sup> *s d ss u k sg q us* . Here 00 0 <sup>1</sup> ( ) <sup>2</sup> *d d x x d d* is the rescaled characteristic function of the unit interval, i.e. 0 0 ( ) 1,| | 1; ( ) 0,| | 1 *xx xx* ; is the asymptotic surface power; ( ) | | *x sg sg x <sup>x</sup>* is the sign function; 1 1 2 \*2 2 \*3 [ ] [ ( )] [ ( )] 2! 3! *<sup>s</sup> s s s s qu k u k u* , where \* 2 . The cubic nonlinear charge density (of the surface 0 *<sup>z</sup>* ) is preferred just on the strip | |*x d* – by the rest term <sup>0</sup> [ ] *d s q u* , while a linear approximation (regarding potential *u*) is assumed adequate to reality out of the strip.

Via the phenomenology-essence potential *u* will be searched for a bounded function (condition (2.3)), continuous in <sup>3</sup> *R* , classically regular in the sets 0 *z* and *x z* 0 ( 0) , with continuous gradients *uz* , *ux* , respectively at 0 *z* , 0 *z* (for *uz* ), and *x x* 0, 0 ( 0) *z* – for *ux* . Now we can define the needed space of regular solutions to (2.2) – (2.6).

Definition. A function *uxz* (,), with the above noted regularity, shall be called *classical solution* to problem (2.2) – (2.6) if satisfies the additional property 2 *ux L* ( ,0) and relations (2.2) – (2.6). ( *L*2 is the well known space of the squared-integrable functions.)

From the vacuum assumption for the upper (external) air phase ( 0 *z* ) we suppose from now on:

$$
\varepsilon\_b^+ = \mathbb{1} \,, \ k\_b^+ = 0 \, . \tag{2.7}
$$

The central results below relate to determination of the surface potential (possibly by an explicit approximation) – as the key first step in solving the full (2.2) – (2.6) – problem.

Let us now sketch the main steps for derivation of the final mathematical models, starting from the Maxwell system (2.1). Via the presumed complex heterogeneity, we shall seek solutions (D, *u*) of system (2.1) by decompositions in two levels (bulk and surface), of the following type:

$$\begin{array}{c} \text{a) } \rho = \rho\_b^- \eta^- + \rho\_b^+ \eta^+ + \rho\_s \delta(\mathbf{z}) : \mathbf{b} \rangle \ \rho\_s = \rho\_s^- \eta\_s^- + \rho\_s^+ \eta\_s^+ + \rho\_l \delta\_l ; \end{array} \tag{2.8}$$

Mathematical Models for Electrostatics of Heterogeneous Media 93

Boltzmann distribution (see e.g. [9]) for the bulk phases, and in the case of organic interface

phenomena, the surface charges should also depend on the space variables by potential *u*,

polysaharide matter of the interface admits to consider it as a lipid medium, where the potential-magnitude can be assumed relatively smaller than the basic ratio (*RT0*)/*F*, which yields that linear approximations become acceptable (*F, R, T0* are – as follows – the so-called Faraday and gas constants, and the absolute temperature). The Boltzmann principle, applied

> 

parametric expression, known as Fermi-Dirac integral (e.g. [1]). Said dependence is of exponential type regarding the potential and relates well enough to the simpler one,

for the surface density of phosphorus vacancies. From the above-noted viewpoint we shall chose a truncation of exponential dependence for the surface phases in the following form (taking into account the total electro-neutrality of the considered material system and the

> 2 0 2 0 <sup>0</sup> ( ) [ ] <sup>2</sup> *s ss dss d s ku k sg q <sup>u</sup>*

 

 

neutrality of the considered material system, via the upper and lower vacuum phases (by

potential, different from zero (far from the specific edge *l* ). To forecast the more complicated impact of the vacancy-denuded zone *T T* , with { 0; 0} *T dx z* and

<sup>0</sup> {(1 ) exp[ ( )] 1 exp[ ( )] 1 } *<sup>s</sup> d ss d ss*

motivation issues from the observation on the interaction energy between phosphorus vacancies on the surface (see [7]) – this energy seems to be relatively small. (It has been estimated in [7] with a maximal value of 65 15 meV at a vacancy separation of 1.2 nm.)

 

*s d s ds* 

*<sup>s</sup>*[ ]~ *t t* , at 0 *t* , such an expression takes into account the total electro-

0 2 0 2\*

 

*k u k u* . (2.15)

 exp( ) *k u* , used for the so-called screened Coulomb potential in the bulk phases (see [5], [7] and the literature therein). It is important that the same expression has been

 

 

exponential dependence for the surface phases: <sup>2</sup>

analogy to the case of gas-lamina-liquid media). The difference *u*

*T x dz* {0 ; 0} , we introduce the modified dependence:

 

On the other hand we cut off the infinite exponential sum for <sup>1</sup>

exponential term from the assumption to have given asymptotic values

<sup>0</sup> ( ) *s s*

 

*u* . Such type of dependence is well known for

*u* is expressed by the so-called

*k u* . Via the real

. In the case of

*k* ), to the analysis

is present in the

*s* up to the cubic term,

of the surface

*<sup>s</sup>* could be derived from a

<sup>0</sup> *b b* 

*u* to the surface phases; i.e. we can take

*<sup>s</sup> u* 

*k u* instead of [ ]

*k* instead respectively of ,

 

*<sup>s</sup>* (in said framework) we start from the following type of

 

 

<sup>0</sup> exp[ ( )] 1 *<sup>s</sup> s s*

 *k u* . Under the

 

*<sup>s</sup>* , instead. For undertaking that, the basic

(for the sake of shortness), we shall deal

. (2.14)

*u* with its linear approximation <sup>2</sup>

*u* . A preliminary motivation to do that follows from the argument that the

established certain relation [ ]

we can replace in (2.11.a) [ ]

2

specific inclusion of component *T* ):

To get the above expression for

 

i.e. [ ] *s s* 

0 

behaviour

 

electrolytes by the Gouy-Chapmann theory, where [ ]

*s s* 

semiconductors however a nonlinear dependence [ ] *u u*

for surfaces, suggests dependence [ ]

again the relevant linear approximation <sup>2</sup>

experimentally examined in [5], [7] (with , *s s*

 

> 

When rewrite the above as 0 0 01 (1 ) 

with the linear approximation of density <sup>0</sup>

$$\mathbf{D} = (\mathbf{D}\_b)^- \eta^-(z) + (\mathbf{D}\_b)^+ \eta^+(z) + \mathbf{D}\_\* \delta(z) \,. \tag{2.9}$$

$$\mathbf{D}\_s = (\mathbf{D}\_s)^- \eta\_s^{\bar{\phantom{a}}} + (\mathbf{D}\_s)^+ \eta\_s^{\bar{\phantom{a}}} + \mathbf{D}\_l \delta\_{\bar{l}}.\tag{2.10}$$

In the above relations ( ) *z* / () *z* are respectively the Heaviside forward/backward functions (i.e. ( ) 1 *z* , at 0 *z* , ( ) 0 *z* , at 0 *z* , ( ) ( ) *z z* ) and ( ) *z* is the Dirac delta-function, supported at 0 *z* ; 1 *s* , at 0 *z* , 0 *x* and 0 *s* , at 0 *z* , 0 *x* , by analogy: 1 *s* , at 0 *z* , 0 *x* and 0 *s* , at 0 *<sup>z</sup>* , 0 *<sup>x</sup>* ; next, *<sup>l</sup>* is delta-function, supported on the line *lx z* : 0 ( 0) , and we shall also use the notation ( ) *x* , for *<sup>l</sup>* . Relations (2.8.a), (2.8.b) and (2.9), (2.10) just illustrate, respectively for the charge density and the electric induction, the essential generalization, in two levels (see [13]), of the Bedeoux-Vlieger ([2]) step formalism to the bulk-surface-bulk transitions. *Remark:* terms like ( ) *<sup>b</sup> z* do not enter in the right hand side of (2.8.a) in the case of semiconductor interface because of the vacuum hypothesis ( 0 *b b* ). In (2.9) ( ) **<sup>D</sup>***b* and ( ) **<sup>D</sup>***b* are at least smooth (vector) functions of ( *x z*, ), respectively at 0 *z* and 0 *z* , with finite but generally different limit values at 0 *z* , fixed *x* and D*s* is a vector function of *x* , assumed in the form of (2.10). Analogous presumptions hold to ( ) **<sup>D</sup>***<sup>s</sup>* , ( ) **<sup>D</sup>***s* – in (2.10), as functions actually of *x* (with finite and different limit values at 0 *x* ), and to (scalar) functions ( ) *s s x* (considered as at least continuous respectively at 0 *x* and 0 *x* ); *<sup>l</sup>* and D*l* enter in (2.8.b) and (2.10) respectively as constant scalar and vector. Substituting from (2.8) – (2.10) into electrostatic equations (2.1), we get (with 1 *<sup>b</sup>* ):

$$
\nabla.\left(\mathbf{D}\_b\right)^+ = 0 \ (z > 0) \/\text{V}.\left(\mathbf{D}\_b\right)^- = \rho\_b^-[\mu] \ (z < 0);\tag{2.11a}
$$

$$\left(\mathbf{D}\_{b}\right)^{+} = -\varepsilon\_{0}\varepsilon\_{b}^{+}.\nabla\mu\left(z>0\right),\ \left(\mathbf{D}\_{b}\right)^{-} = -\varepsilon\_{0}\varepsilon\_{b}^{-}.\nabla\mu\left(z<0\right);\tag{2.11b}$$

$$D\_{+}^{z}(\mathbf{x},0) - D\_{-}^{z}(\mathbf{x},0) + \nabla\_{s}.\mathrm{D}\_{s} = \rho\_{s}[\![\mathbf{u}\!]\!] \text{ (}z = 0\text{)}, \text{x} \neq 0\text{);}\tag{2.12.a}$$

$$\mathbf{D}\_s = -\varepsilon\_0 \varepsilon\_s \nabla\_s \mu \left( z = 0, \mathbf{x} \neq 0 \right);\tag{2.12.b}$$

$$\left(\mathbf{D}\_s\right)^{x\_{\cdot}+} - \left(\mathbf{D}\_s\right)^{x\_{\cdot}-} = \rho\_{\cdot}\,.\tag{2.13}$$

Here we have denoted by *s* the tangential (to 0 *z* ) component of the nabla operator ; <sup>0</sup> ( ,0) lim ( , ) *z z <sup>z</sup> D x D xz* , and, by analogy – for ( ,0) *<sup>z</sup> D x* , where *<sup>z</sup> <sup>D</sup>* is the normal to 0 *<sup>z</sup>* component of vector D, and the limits are supposed finite, *x* ; *x*, *s* **D** , *x*, *s* **D** are the relevant limits (also assumed finite), at 0 *<sup>x</sup>* , for the normal to *l* component *<sup>l</sup>* **<sup>D</sup>***s* of D*s*. Let us note (calculating the results of said substituting) that the normal to 0 *<sup>z</sup>* component *<sup>z</sup> Ds* of vector Ds is found to vanish ( <sup>0</sup> *<sup>z</sup> Ds* ), i.e. Ds presents a flat (planar) vector field (see e.g. [13] for details). *Remark:* the used derivations of the Heaviside and Dirac delta-functions, necessary to get system (2.11) – (2.13), are taken in the Schwartz distributions meaning (e.g. [8]).

Now we will discuss the charge density terms [ ] *<sup>b</sup> <sup>u</sup>* , [ ] *<sup>s</sup> u* , respectively in (2.11.a), (2.12.a), especially that of *<sup>s</sup>* . For the vacuum-semiconductor-vacuum systems we have to take [] 0 *<sup>b</sup> <sup>u</sup>* (from the vacuum hypothesis). For vacuum-lamina-liquid systems it is 92 Electrostatics

+ ( ) **<sup>D</sup>***<sup>s</sup>*

Relations (2.8.a), (2.8.b) and (2.9), (2.10) just illustrate, respectively for the charge density and the electric induction, the essential generalization, in two levels (see [13]), of the Bedeoux-Vlieger ([2]) step formalism to the bulk-surface-bulk transitions. *Remark:* terms like ( ) *<sup>b</sup>*

do not enter in the right hand side of (2.8.a) in the case of semiconductor interface because of

functions of ( *x z*, ), respectively at 0 *z* and 0 *z* , with finite but generally different limit values at 0 *z* , fixed *x* and D*s* is a vector function of *x* , assumed in the form of (2.10). Analogous presumptions hold to ( ) **<sup>D</sup>***<sup>s</sup>* , ( ) **<sup>D</sup>***s* – in (2.10), as functions actually of *x* (with

respectively as constant scalar and vector. Substituting from (2.8) – (2.10) into electrostatic

*<sup>b</sup> u z* , ( ) **<sup>D</sup>***b* = 0 . ( 0)

( ) *<sup>z</sup>* + D*<sup>s</sup>*

 *z* , at 0 *z* , ( ) ( ) 

, at 0 *z* , 0 *x* and 0

, at 0 *<sup>z</sup>* , 0 *<sup>x</sup>* ; next, *<sup>l</sup>*

). In (2.9) ( ) **<sup>D</sup>***b* and ( ) **<sup>D</sup>***b* are at least smooth (vector)

 

*s* + D*<sup>l</sup> <sup>l</sup>* 

*z* are respectively the Heaviside forward/backward

 *z z* ) and

> *s*

( ) *z* , (2.9)

. (2.10)

 

*<sup>l</sup>* and D*l* enter in (2.8.b) and (2.10)

*<sup>b</sup> <sup>u</sup>* (0 *<sup>z</sup>* ); (2.11a)

*<sup>b</sup> u z* ; (2.11b)

*<sup>s</sup> u* ( 0, 0 *z x* ); (2.12.a)

*s s uz x* ; (2.12.b)

*<sup>z</sup> D x* , where *<sup>z</sup> D* is the normal to 0 *z*

*s* **D** , *x*, *s*

*<sup>s</sup> u* , respectively in (2.11.a), (2.12.a),

**D** are the

**D D** . (2.13)

, at 0 *z* , 0 *x* , by

( ) *z* is the Dirac

is delta-function,

( ) *x* , for *<sup>l</sup>*

 *z*

*x* (considered

.

( ) *<sup>z</sup>* + ( ) **<sup>D</sup>***<sup>b</sup>*

*s* 

D = ( ) **<sup>D</sup>***<sup>b</sup>*

 *z* / () 

*s*

> *s*

supported on the line *lx z* : 0 ( 0) , and we shall also use the notation

finite and different limit values at 0 *x* ), and to (scalar) functions ( ) *s s*

 

component of vector D, and the limits are supposed finite, *x* ; *x*,

get system (2.11) – (2.13), are taken in the Schwartz distributions meaning (e.g. [8]).

 *x x* , , *s sl*

Here we have denoted by *s* the tangential (to 0 *z* ) component of the nabla operator ;

relevant limits (also assumed finite), at 0 *<sup>x</sup>* , for the normal to *l* component *<sup>l</sup>* **<sup>D</sup>***s* of D*s*. Let us note (calculating the results of said substituting) that the normal to 0 *<sup>z</sup>* component *<sup>z</sup> Ds* of vector Ds is found to vanish ( <sup>0</sup> *<sup>z</sup> Ds* ), i.e. Ds presents a flat (planar) vector field (see e.g. [13] for details). *Remark:* the used derivations of the Heaviside and Dirac delta-functions, necessary to

*<sup>b</sup> <sup>u</sup>* (from the vacuum hypothesis). For vacuum-lamina-liquid systems it is

*<sup>b</sup> <sup>u</sup>* , [ ] 

*<sup>s</sup>* . For the vacuum-semiconductor-vacuum systems we have to take

*z* , at 0 *z* , ( ) 0

, at 0 *z* , 0 *x* and 0

<sup>D</sup>*s* = ( ) **<sup>D</sup>***<sup>s</sup>*

delta-function, supported at 0 *z* ; 1

the vacuum hypothesis ( 0

equations (2.1), we get (with 1 *<sup>b</sup>*

<sup>0</sup> ( ,0) lim ( , ) *z z*

especially that of

[] 0

() **<sup>D</sup>***b* = 0 . ( 0)

 *b b*

as at least continuous respectively at 0 *x* and 0 *x* );

):

. ( ) **<sup>D</sup>***b* = 0 ( 0 *<sup>z</sup>* ), . ( ) **<sup>D</sup>***b* = [ ]

 

( ,0) ( ,0) . *z z Dx Dx <sup>s</sup>* D*s* = [ ]

*<sup>z</sup> D x D xz* , and, by analogy – for ( ,0)

Now we will discuss the charge density terms [ ]

D*s* = 0 . ( 0, 0)

In the above relations ( )

functions (i.e. ( ) 1 

analogy: 1 *s*

established certain relation [ ] *u* . Such type of dependence is well known for electrolytes by the Gouy-Chapmann theory, where [ ] *u* is expressed by the so-called Boltzmann distribution (see e.g. [9]) for the bulk phases, and in the case of organic interface we can replace in (2.11.a) [ ] *u* with its linear approximation <sup>2</sup> <sup>0</sup> *b b k u* . Via the real phenomena, the surface charges should also depend on the space variables by potential *u*, i.e. [ ] *s s u* . A preliminary motivation to do that follows from the argument that the polysaharide matter of the interface admits to consider it as a lipid medium, where the potential-magnitude can be assumed relatively smaller than the basic ratio (*RT0*)/*F*, which yields that linear approximations become acceptable (*F, R, T0* are – as follows – the so-called Faraday and gas constants, and the absolute temperature). The Boltzmann principle, applied for surfaces, suggests dependence [ ] *s s u* to the surface phases; i.e. we can take again the relevant linear approximation <sup>2</sup> <sup>0</sup> ( ) *s s k u* instead of [ ] *<sup>s</sup> u* . In the case of semiconductors however a nonlinear dependence [ ] *u u <sup>s</sup>* could be derived from a parametric expression, known as Fermi-Dirac integral (e.g. [1]). Said dependence is of exponential type regarding the potential and relates well enough to the simpler one, 2 0 exp( ) *k u* , used for the so-called screened Coulomb potential in the bulk phases (see [5], [7] and the literature therein). It is important that the same expression has been experimentally examined in [5], [7] (with , *s s k* instead respectively of , *k* ), to the analysis for the surface density of phosphorus vacancies. From the above-noted viewpoint we shall chose a truncation of exponential dependence for the surface phases in the following form (taking into account the total electro-neutrality of the considered material system and the specific inclusion of component *T* ):

$$
\rho\_s \rho\_s = \varepsilon\_0 \left( -\varepsilon\_s k\_s^2 (\mu - \rho\_\infty) - \alpha\_d^0 \varepsilon\_s k\_s^2 \frac{\Delta \rho\_\infty}{2} \text{sg} + \alpha\_d^0 q\_s [\mu] \right). \tag{2.14}
$$

To get the above expression for *<sup>s</sup>* (in said framework) we start from the following type of exponential dependence for the surface phases: <sup>2</sup> <sup>0</sup> exp[ ( )] 1 *<sup>s</sup> s s k u* . Under the behaviour *<sup>s</sup>*[ ]~ *t t* , at 0 *t* , such an expression takes into account the total electroneutrality of the considered material system, via the upper and lower vacuum phases (by analogy to the case of gas-lamina-liquid media). The difference *u* is present in the exponential term from the assumption to have given asymptotic values of the surface potential, different from zero (far from the specific edge *l* ). To forecast the more complicated impact of the vacancy-denuded zone *T T* , with { 0; 0} *T dx z* and *T x dz* {0 ; 0} , we introduce the modified dependence:

$$\rho\_s = \varepsilon\_0 \left| (1 - \alpha\_d^0) \left( \exp[-\varepsilon\_s k\_s^2 (\mu - \rho\_w)] - 1 \right) + \alpha\_d^0 \left( \exp[-\varepsilon\_s k\_s^2 (\mu - \rho\_w^\*)] - 1 \right) \right| \tag{2.15}$$

When rewrite the above as 0 0 01 (1 ) *s d s ds* (for the sake of shortness), we shall deal with the linear approximation of density <sup>0</sup> *<sup>s</sup>* , instead. For undertaking that, the basic motivation issues from the observation on the interaction energy between phosphorus vacancies on the surface (see [7]) – this energy seems to be relatively small. (It has been estimated in [7] with a maximal value of 65 15 meV at a vacancy separation of 1.2 nm.) On the other hand we cut off the infinite exponential sum for <sup>1</sup> *s* up to the cubic term,

(Above ˆ

*L*[ ] 

We have denoted by

 

( 0) ,

the quantity [ ] *L*

 

> 

components of operator [ ] *Fs*

structure of [ ] *Us*

for auxiliary function

ˆ () . () 

The posed

limits 

() () *x x* 

( ) ,

0 

is the Fourier image of *L*[ ]

presented as a linear operator : [ ] *L L*

()

 and

 by [ ] *L* 

:

2

Multiplying the Fourier image of [ ] *Fs*

the form: <sup>1</sup> ( ) .exp( | |) [ ] *s s*

*x c kx U*

:

 *k L* 

( 0) ,

( 0) ,

\* [ ] [ ] . [1] . [ ] <sup>2</sup> *L L L L sg*

 

Mathematical Models for Electrostatics of Heterogeneous Media 95

1 1 *H R*( ) (we refer e.g. to [8], for the *<sup>k</sup> H* -spaces of Sobolev). It admits to separate the key part (2.4) – (2.6) from the full system (2.2) – (2.6) in an autonomous *boundary transmission problem*:

[ ( 0) ( 0)] *s l*

*i sg i* 1 , and [1] 0 *L* . Now, from (3.3), we get the next reduced problem

 and *b b* 

, acting from <sup>1</sup>

 

( 0) ( 0) 

> 

respectively the first and second derivative of

2 

> *s s*

. (3.5)

 

 by factor 2 21 ( ) *<sup>s</sup>* 

(at 0 *x* ), with a constant *c* . To clarify the

, let us introduce the following auxiliary functions, related to the relevant

 

 and

, 0 *x* ; (3.3a)

. (3.3c)

( 0) – the relevant limits. Taking the substitution

 ). (3.6)

*k* we find a single bounded

*s U* 

 , we get

); (3.4)

, which

of (3.4) – from the

<sup>2</sup> *L R*( ); in

. Then from

, *b b k k* .) Thus said jump term is

; (3.3b)

. Let us express firstly

, where 0

( ) *x* , and by

( ) *x* :

<sup>2</sup> *L R*( ) into the Sobolev space

<sup>2</sup> [] ( ) [] *L k*

*s ss s*

,

, the problem (3.3) reduces into a simpler one for

. It is directly seen that 0 *L sg x x* [ ]( ) 2 ( )

0

 

[ ( 0) ( 0)] *s l*


addition they are assumed to have the classical regularity at 0 *x* , with finite values of the

 ( <sup>0</sup> [] [] . *F L s s* 

the general formula for the (bounded) solutions of (3.6), we can directly get a presentation in

 , 0 *x* ( []

; using notations \*

 

<sup>1</sup> ([ ] . ) *s s*

<sup>2</sup> <sup>1</sup> [ ], 0 *s s s w kw F x*

*s*

 

( 0) . To our next step, observe before that, given a solution

solution of (3.6) (see below for some details). Denote this solution by <sup>1</sup> [ ] *<sup>s</sup>*

are continuous in ( , 0], [ 0, ) , tend to zero, at | |*x* and belong to <sup>1</sup>

said class, we actually have a suitably regular and bounded solution to the equation:

 

*s*

assuming secondary the impact of the higher powers. Thus we shall presume in relation (2.12.a) the given one in (2.14), for the surface charge density of semiconductors. On the linear (1D) phase, the contour *l Oy* , we assume *l l* <sup>0</sup> , with *<sup>l</sup>* – given constant, for the semiconductor case, while in the organic case we prefer a nonlinear Boltzmann type model [ ] *l l u* , forecasting possible unknown complications, close to the line contour.

The next main step of modeling consists in some reworking to system (2.11) – (2.13). By the right hand sides from (2.11.b) we firstly express ( ) **<sup>D</sup>***<sup>b</sup>* , ( ) **<sup>D</sup>***b* in (2.11.a) and come to the Helmholtz–Laplace equations from (2.2). As noted, condition (2.3) corresponds to the physical nature of the potential (to be a space-bounded and continuous quantity). Going to the next relations, (2.4.a), (2.6.a), they show that potential stays continuous across the transition surfaces and lines. On the other hand condition (2.5) introduces the asymptotic value of the surface potential *u x*( ,0) – they are considered as experimentally known (gauged) data. Afterwards we replace D*s* in (2.12.a) by the right hand side of (2.12.b) and use that 0 ( ,0) ( , 0) *<sup>z</sup> Dx ux b z* , <sup>0</sup> ( ,0) ( , 0) *<sup>z</sup> Dx ux b z* (with 1 *<sup>b</sup>* , see (2.7)). In addition we rearrange the right hand side of (2.12.a) respectively by the nonlinear density (2.8) or the linear expression <sup>2</sup> <sup>0</sup> ( ) *s s k u* . This way we get, from (2.12), the complicated jump condition (2.4.b). For the second jump-condition on the electric field (see (2.6.b)), it is enough to recall that *l l* <sup>0</sup> and , <sup>0</sup> ( 0,0) *<sup>x</sup> s s x <sup>u</sup>* **<sup>D</sup>** .

This completes the sketch of derivation to final form (2.2) – (2.6) of our mathematical models.

#### **3. The basic integral equation and finding of surface potentials**

We shall reduce in this section problem (2.2) – (2.6) to a corresponding (nonlinear) integral equation – as a background for finding of explicit type presentations to the surface electric potential. Recall firstly the supposed electrostatic equivalence of the surface phases, which yields that *sss* , *sss kkk* .

For the needed technical reworks the *x* - Fourier transformation is systematically taken into account below – by well known conventional expressions (e.g. as in [8]). By the *x* -Fourier transformation to the relations in (2.2) we find ordinary differential equations (regarding *z*), which yield the following presentation (for a classical solution *u uxz* (,) to problem (2.2) – (2.6)):

$$\|\hat{u}(\xi, z) = \hat{\phi}(\xi) \exp(-z \left|\xi\right|), z > 0 \, \cdot \, \hat{u}(\xi, z) = \hat{\phi}(\xi) \exp(z \sqrt{\xi^2 + \kappa\_b^2}), z < 0 \, \,. \tag{3.1}$$

It is denoted here by *u z* ˆ(,) the (partial) Fourier transformation of *uxz* (,) - with respect to *x* .

In (3.1) ( ) ( ,0) *x ux* and ˆ is the Fourier image of . The jump term in (2.4.b) can be then expressed in the next form:

$$
\mu\_z(\mathbf{x}, +0) - \mu\_z(\mathbf{x}, -0) = L[\varphi] \cdot \hat{L}[\varphi](\xi) = -\lambda(\xi)\hat{\rho}(\xi) \,, \ \lambda(\xi) = \left| \xi \right| + \varepsilon\_b \sqrt{\xi^2 + k\_b^2} \, . \tag{3.2}
$$

94 Electrostatics

assuming secondary the impact of the higher powers. Thus we shall presume in relation (2.12.a) the given one in (2.14), for the surface charge density of semiconductors. On the

semiconductor case, while in the organic case we prefer a nonlinear Boltzmann type model

The next main step of modeling consists in some reworking to system (2.11) – (2.13). By the right hand sides from (2.11.b) we firstly express ( ) **<sup>D</sup>***<sup>b</sup>* , ( ) **<sup>D</sup>***b* in (2.11.a) and come to the Helmholtz–Laplace equations from (2.2). As noted, condition (2.3) corresponds to the physical nature of the potential (to be a space-bounded and continuous quantity). Going to the next relations, (2.4.a), (2.6.a), they show that potential stays continuous across the transition surfaces and lines. On the other hand condition (2.5) introduces the asymptotic value of the surface potential *u x*( ,0) – they are considered as experimentally known (gauged) data. Afterwards we replace D*s* in (2.12.a) by the right hand side of (2.12.b) and use

*u* , forecasting possible unknown complications, close to the line contour.

*<sup>z</sup> Dx ux b z* 

<sup>0</sup> ( 0,0) *<sup>x</sup>*

 *s x <sup>u</sup>* **<sup>D</sup>** .

**3. The basic integral equation and finding of surface potentials** 

*l l* <sup>0</sup> , with

(with 1 *<sup>b</sup>*

we rearrange the right hand side of (2.12.a) respectively by the nonlinear density (2.8) or the

condition (2.4.b). For the second jump-condition on the electric field (see (2.6.b)), it is enough

This completes the sketch of derivation to final form (2.2) – (2.6) of our mathematical

We shall reduce in this section problem (2.2) – (2.6) to a corresponding (nonlinear) integral equation – as a background for finding of explicit type presentations to the surface electric potential. Recall firstly the supposed electrostatic equivalence of the surface phases, which

For the needed technical reworks the *x* - Fourier transformation is systematically taken into account below – by well known conventional expressions (e.g. as in [8]). By the *x* -Fourier transformation to the relations in (2.2) we find ordinary differential equations (regarding *z*), which yield the following presentation (for a classical solution *u uxz* (,) to problem (2.2) –

*L*[ ]( ) ( ) ( )

 

 

*k u* . This way we get, from (2.12), the complicated jump

2 2 ˆ ˆ ( ) ( )exp( ), 0 *uz z z <sup>b</sup>*

the (partial) Fourier transformation of *uxz* (,) - with respect

 ˆ , 2 2 ( ) *b b* 

 *k* . (3.2)

 . (3.1)

. The jump term in (2.4.b) can be then

*<sup>l</sup>* – given constant, for the

, see (2.7)). In addition

linear (1D) phase, the contour *l Oy* , we assume

, <sup>0</sup> ( ,0) ( , 0)

 

*s*

<sup>0</sup> ( ) *s s*

and ,

*uz z z* ˆ ˆ ( ) ( )exp( ), 0

( , 0) ( , 0) [ ] *ux ux L z z*

 ;

> ; <sup>ˆ</sup>

ˆ is the Fourier image of

 

 

[ ]

that 0 ( ,0) ( , 0) *<sup>z</sup> Dx ux b z* 

linear expression <sup>2</sup>

*l l* <sup>0</sup> 

yields that *sss* 

 , *sss kkk* .

( ) ( ,0) *x ux* and

It is denoted here by *u z* ˆ(,)

expressed in the next form:

 

to recall that

models.

(2.6)):

to *x* .

In (3.1)

*l l* 

(Above ˆ *L*[ ] is the Fourier image of *L*[ ] and *b b* , *b b k k* .) Thus said jump term is presented as a linear operator : [ ] *L L* , acting from <sup>1</sup> <sup>2</sup> *L R*( ) into the Sobolev space 1 1 *H R*( ) (we refer e.g. to [8], for the *<sup>k</sup> H* -spaces of Sobolev). It admits to separate the key part (2.4) – (2.6) from the full system (2.2) – (2.6) in an autonomous *boundary transmission problem*:

$$L[\varphi] + \varepsilon\_s \varphi^\* = \varepsilon\_s k\_s^2 (\varphi - \varphi\_\circ) + \rho\_s^\* [\varphi] \; , \; \mathbf{x} \neq \mathbf{0} \; ; \tag{3.3a}$$

$$
\varphi(\pm\infty) = \varphi\_{\alpha}^{\pm}, \ \varphi(+0) = \varphi(-0) \ ; \tag{3.3b}
$$

$$
\omega\_s[\varphi'(+0) - \varphi'(-0)] = -\beta\_l \,. \tag{3.3c}
$$

We have denoted by and respectively the first and second derivative of ( ) *x* , and by ( ) , ( 0) , ( 0) , ( 0) , ( 0) – the relevant limits. Taking the substitution () () *x x* , the problem (3.3) reduces into a simpler one for . Let us express firstly the quantity [ ] *L* by [ ] *L* ; using notations \* 2 and , we get \* [ ] [ ] . [1] . [ ] <sup>2</sup> *L L L L sg* . It is directly seen that 0 *L sg x x* [ ]( ) 2 ( ) , where 0 ( ) *x* : 0 ˆ () . () *i sg i* 1 , and [1] 0 *L* . Now, from (3.3), we get the next reduced problem for auxiliary function :

$$\mu \prime \nu \prime - k\_s^2 \nu = -\frac{1}{\varepsilon\_s} (\mathrm{L}[\mu \prime] + \Delta \rho\_w \, \sigma\_0 - \rho\_s^\* \,) \; , \; \ge \neq 0 \; \left( \, \rho\_s^\* = \rho\_s^\* [\![\nu \prime + \rho\_w \, ]\right) \prime \tag{3.4}$$

$$
\omega\_s [\wp'(+0) - \wp'(-0)] = -\beta\_l. \tag{3.5}
$$

The posed -problem ((3.4)-(3.5)) is considered on the space of the real functions , which are continuous in ( , 0], [ 0, ) , tend to zero, at | |*x* and belong to <sup>1</sup> <sup>2</sup> *L R*( ); in addition they are assumed to have the classical regularity at 0 *x* , with finite values of the limits ( 0) . To our next step, observe before that, given a solution of (3.4) – from the said class, we actually have a suitably regular and bounded solution to the equation:

$$
\delta w'' - k\_s^2 w = -\frac{1}{\mathcal{E}\_s} F\_s[\nu], \; \ge \neq 0 \; \text{ (} F\_s[\nu] \equiv \text{L}[\nu] + \Delta \varphi\_v, \sigma\_0 - \rho\_s^\* \text{)}. \tag{3.6}$$

Multiplying the Fourier image of [ ] *Fs* by factor 2 21 ( ) *<sup>s</sup> k* we find a single bounded solution of (3.6) (see below for some details). Denote this solution by <sup>1</sup> [ ] *<sup>s</sup> s U* . Then from the general formula for the (bounded) solutions of (3.6), we can directly get a presentation in the form: <sup>1</sup> ( ) .exp( | |) [ ] *s s s x c kx U* (at 0 *x* ), with a constant *c* . To clarify the structure of [ ] *Us* , let us introduce the following auxiliary functions, related to the relevant components of operator [ ] *Fs* :

Mathematical Models for Electrostatics of Heterogeneous Media 97

operator. Reworking this way (3.10) we get the following expression, via the functions from

<sup>0</sup> <sup>0</sup> 0,1 \* [ ]\* [ ] . .

 

0 , with <sup>0</sup>

*sg x*

<sup>0</sup> 0,1 \* [] . .

*s s l ss s s*

*k k* is the Fourier transform of the inverse

. (3.14)

 and 0,1 <sup>1</sup> ( 0) <sup>2</sup> *s s* 

 *l ss s s* . (3.16)

> *t t*

 , i.e. ,0 <sup>0</sup> 2 2 1

> : 1 \*2 ,0 <sup>4</sup> | | <sup>27</sup> *l s*

*<sup>p</sup> <sup>k</sup>*

() ( ) *<sup>s</sup>*

.

 

of the surface potential, arbitrary

 

 

). Note that <sup>0</sup>

*<sup>s</sup>* instead of

.)

*<sup>s</sup>* (playing a key

*<sup>s</sup>* is the (unique)

) to use – for the

 *<sup>l</sup>* ); this

. (3.15)

, with coefficient

 

 (0) 0 ,

> )is

*s s d*

*p* , there exists a

0 ) root of the equation

 

*x* is the (Schwartz) first order derivative of function <sup>0</sup>

0,1 0,1 <sup>1</sup> ( ) ( ) [ ]( ) exp( | |) <sup>2</sup> *<sup>s</sup> L s <sup>s</sup>*

*xW x k x*

Equation (3.14) is the sought basic integral equation related to problem (2.2) – (2.6). In the case of vacuum – liquid heterogeneous system, with organic interface, (3.14) takes the

solution of the linear canonical version of problem (3.4), (3.5) (with 0, 1

 

<sup>0</sup> () [ ] () . () . () *ls s s <sup>s</sup>*

 

<sup>1</sup> () ( ) *<sup>z</sup> u x z x t dt*

   

*<sup>l</sup>* . The main results for said class of charge densities are summarized in the following

,0 (0) *s s p* 

0 0,1 \*

0 ) or negative (at \*

determined as the unique (classical) solution of problem (2.2) – (2.6), by the next formulas:

. In addition, the relevant space potential *uxz* (,)(with *ux x* ( ,0) ( )

2 2

*z t*

*l*

 *x tx x x* , 0 *x* ; (3.17)

, 0 *<sup>z</sup>* , <sup>1</sup> *x R* ; (3.18)

( ) *x* , satisfying (3.16), such that \*

*<sup>s</sup>* , for the behaviour of potential

 

*s*

(In particular (3.15) yields the finite limits 0,1 <sup>1</sup> ( 0) <sup>2</sup> *s s*

(3.10), using that 2 2 2 21 ( )[ ( ) ( )] *s s s s* 

> 

admits, differentiating as before (3.10) (at

*<sup>s</sup>* – the integral relation:

 *s* ):

assertion. Below we shall use the quantity 0

For arbitrary non zero asymptotic mean value \*

 

 

parameters 0, 0, 0, 0 *b b ss*

<sup>0</sup>*t* is either the positive (at \*

unique continuous bounded potential

determined by the formula

3 \* ,0 0 *s l p tt*

 

 

We shall study the above equation for nonlinear functions <sup>3</sup> [ ] *l l*

*k k* , and coefficient

(3.13):

Above 0,1 0,1( ) *s s* 

analysis of 0,1 

0 

**3.1 Proposition** 

role, together with \*

 

specialized form (with [ ] 0

 

$$\log(\mathbf{x}) = -\frac{1}{\pi} \int\_0^\infty \frac{\sin(\mathbf{x}\xi)}{k\_s^2 + \xi^2} d\xi \quad ; \tag{3.7}$$

$$\mathcal{W}\_{L}[\boldsymbol{\wp}\,](\mathbf{x}) = \frac{1}{2k\_{s}} L[\boldsymbol{\wp}\,]^{\*} \exp(-k\_{s} \, |\cdot|)(\mathbf{x})\,,\tag{3.8}$$

$$R\_s^{\sigma}[\varphi](\mathbf{x}) = \frac{1}{2k\_s} \rho\_s^\* [\varphi + \varphi\_\sigma]^\* \exp(-k\_s |\cdot|) (\mathbf{x}) \,. \tag{3.9}$$

Above *F* \* is the convolution of two (Schwartz) distributions, *F* and (see e.g. [8]), and [ ]( ) *W x <sup>L</sup>* is a bounded function. In addition [ ]( ) *R x <sup>s</sup>* is also a bounded function (because <sup>0</sup> ( )( ) *<sup>d</sup> sg x* and <sup>0</sup> ( [ ])( ) *d s q x* are compactly supported, while *g x*( ) is evidently bounded. Now, for [ ] *Us* , it can be found: [ ] . [ ] *U W gR s L s* and therefore function (the solution of (3.4)) satisfies the equation:

$$\Psi = c.exp(-k\_s \mid .) + \frac{1}{\varepsilon\_s} \left( \mathcal{W}\_L[\nu \mid + \Delta \varphi\_v, \mathcal{g} - \mathcal{R}\_s^\sigma[\nu \mid ) \right). \tag{3.10}$$

It can be easily seen that the Schwartz derivative of [ ] *Us* is in <sup>1</sup> <sup>2</sup> *L R*( ) and that of exp( | |) *<sup>s</sup> c kx* is in 1 1 *H R*( ) , belonging in addition to 2 *L x*( 0) , 2 *L x*( 0) . Consequently, differentiating in (3.10), we conclude that 1 1 *H R*( ) and 2 *L x*( 0) . This yields the next distribution-relation: 1 () () () *xx x* (where <sup>1</sup> 1 2 *L R*( ) and ( ) *x* is the Dirac-function). Then 1 ( [ ]) ( ) [ ]( ) [ ]( ) [ ]( ) *W xW xW x W x L LL L* . On the other hand – for the rest components of [ ] *Us* – it is not difficult to get as follows: ( ) [ ]( ) *<sup>L</sup> gx W x* ; ( [ ]) ( 0) ( [ ]) ( 0) *R R s s* . Thus we find, for derivative (at 0 *x* ):

$$\boldsymbol{\nu}^{\prime}(\mathbf{x}) = -c k\_s \text{sg}(\mathbf{x}) \exp(-k\_s \, |\, \mathbf{x} \, |) + \frac{1}{\mathcal{E}\_s} \Big( \mathcal{W}\_L[\boldsymbol{\nu}\_1](\mathbf{x}) + (\mathcal{R}\_s^{\prime\prime}[\boldsymbol{\nu} \boldsymbol{\nu}])^{\prime}(\mathbf{x}) \Big). \tag{3.11}$$

However it also holds 1 1 [ ]( 0) [ ]( 0) *W W L L* and, substituting from (3.11) in the jump relation (3.5), we determine the free constant *c* , as *<sup>s</sup> c c* , with

$$\mathcal{L}\_s = \frac{1}{2} \left( \frac{\mathcal{J}\_l}{\mathcal{e}\_s k\_s} - \Delta \phi\_\infty \, \text{sg}(\mathbf{x}) \right). \tag{3.12}$$

In order to modify (3.10) into an integral equation regarding the surface potential , we introduce also the next two functions:

$$\boldsymbol{\psi}\_{s}^{\prime 0}(\mathbf{x}) = \frac{1}{\pi} \Big|\_{0}^{\infty} \frac{\cos(\mathbf{x}\boldsymbol{\xi})d\boldsymbol{\xi}}{\lambda(\boldsymbol{\xi}) + \varepsilon\_{s}(\boldsymbol{k}\_{s}^{2} + \boldsymbol{\xi}^{2})} \; ; \; \boldsymbol{\psi}\_{s}^{\*}(\mathbf{x}) = -\frac{1}{\pi} \Big|\_{0}^{\infty} \frac{\sin(\mathbf{x}\boldsymbol{\xi})d\boldsymbol{\xi}}{\lambda(\boldsymbol{\xi}) + \varepsilon\_{s}(\boldsymbol{k}\_{s}^{2} + \boldsymbol{\xi}^{2})} \; . \tag{3.13}$$

Now, going back to (3.10), with *<sup>s</sup> c c* (see (3.12)), via formulas (3.7) – (3.9) and relation , we shall obtain the basic integral equation for surface potential . As a preliminary step we apply the inverse operator of <sup>1</sup> *L s I W* ( *<sup>I</sup>* – the identity) to equation (3.10), using that 2 2 2 21 ( )[ ( ) ( )] *s s s s k k* is the Fourier transform of the inverse operator. Reworking this way (3.10) we get the following expression, via the functions from (3.13):

$$
\delta\varphi - \varphi\_{\alpha} + \rho\_s^\* [\![\varphi]\!]^\* \varphi\_s^0 = \beta\_l [\![\varphi]\!] \nu\_s^0 + \varepsilon\_s \Delta \varphi\_{\alpha} \nu\_s^{0,1} + \Delta \varphi\_{\alpha} \nu\_s^\* \,. \tag{3.14}
$$

Above 0,1 0,1( ) *s s x* is the (Schwartz) first order derivative of function <sup>0</sup> *<sup>s</sup>* (playing a key role, together with \* *<sup>s</sup>* , for the behaviour of potential ). Note that <sup>0</sup> *<sup>s</sup>* is the (unique) solution of the linear canonical version of problem (3.4), (3.5) (with 0, 1 *<sup>l</sup>* ); this admits, differentiating as before (3.10) (at 0 , with <sup>0</sup> *<sup>s</sup>* instead of ) to use – for the analysis of 0,1 *<sup>s</sup>* – the integral relation:

$$\boldsymbol{\Psi}\_{s}^{0,1}(\mathbf{x}) = \frac{1}{\varepsilon\_{s}} \Big( \boldsymbol{\mathcal{W}}\_{L}[\boldsymbol{\nu}\_{s}^{0,1}](\mathbf{x}) - \frac{s \mathbf{g}(\mathbf{x})}{2} \exp(-k\_{s} \|\mathbf{x}\|) \Big) . \tag{3.15}$$

(In particular (3.15) yields the finite limits 0,1 <sup>1</sup> ( 0) <sup>2</sup> *s s* and 0,1 <sup>1</sup> ( 0) <sup>2</sup> *s s* .)

Equation (3.14) is the sought basic integral equation related to problem (2.2) – (2.6). In the case of vacuum – liquid heterogeneous system, with organic interface, (3.14) takes the specialized form (with [ ] 0 *s* ):

$$
\delta \varphi - \varphi\_{\alpha} = \beta\_l [\![\varphi\!] \!] \nu\_s^0 + \varepsilon\_s \Lambda \varphi\_{\alpha}. \nu\_s^{0,1} + \Lambda \varphi\_{\alpha}. \nu\_s^\* \ . \tag{3.16}
$$

We shall study the above equation for nonlinear functions <sup>3</sup> [ ] *l l t t* , with coefficient 0 *<sup>l</sup>* . The main results for said class of charge densities are summarized in the following assertion. Below we shall use the quantity 0 ,0 (0) *s s p* , i.e. ,0 <sup>0</sup> 2 2 1 () ( ) *<sup>s</sup> s s d <sup>p</sup> <sup>k</sup>* .

#### **3.1 Proposition**

96 Electrostatics

0 1 sin( ) ( ) *s*

> 

<sup>1</sup> [ ]( ) [ ] \* exp( | |)( ) <sup>2</sup> *L s*

<sup>1</sup> [ ]( ) [ ] \* exp( | |)( ) <sup>2</sup> *ss s*

Above *F* \* is the convolution of two (Schwartz) distributions, *F* and (see e.g. [8]), and

*s*

 () () () *xx x* (where <sup>1</sup>

. Thus we find, for derivative

<sup>1</sup> ( ) ( )exp( | |) [ ]( ) ( [ ]) ( ) *s s Ls s*

<sup>1</sup> . () <sup>2</sup>

*x*

Now, going back to (3.10), with *<sup>s</sup> c c* (see (3.12)), via formulas (3.7) – (3.9) and relation

, we shall obtain the basic integral equation for surface potential

*s s c sg x k* 

In order to modify (3.10) into an integral equation regarding the surface potential

 

*x ck sg x kx W x R x*

 

 

*<sup>l</sup> <sup>s</sup>*

*c k W gR*

<sup>1</sup> .exp( |.|) [ ] . [ ] *sL s*

exp( | |) *<sup>s</sup> c kx* is in 1 1 *H R*( ) , belonging in addition to 2 *L x*( 0) , 2 *L x*( 0) . Consequently,

*q x* are compactly supported, while *g x*( ) is evidently bounded.

 *s* and therefore function

. (3.10)

*H R*( ) and 2

 

. (3.11)

*R x k x*

*g x d k*

*s Wx L k x k*

*s*

 

*k*

, it can be found: [ ] . [ ] *U W gR s L*

is a bounded function. In addition [ ]( ) *R x <sup>s</sup>*

It can be easily seen that the Schwartz derivative of [ ] *Us*

 

Dirac-function). Then 1 ( [ ]) ( ) [ ]( ) [ ]( ) [ ]( ) *W xW xW x W x L LL L*

 

differentiating in (3.10), we conclude that 1 1

; ( [ ]) ( 0) ( [ ]) ( 0) *R R s s*

relation (3.5), we determine the free constant *c* , as *<sup>s</sup> c c* , with

0 2 2 1 cos( ) ( ) () ( ) *<sup>s</sup>*

; \*

 

preliminary step we apply the inverse operator of <sup>1</sup>

*s s x d*

*k* 

 

[ ]( ) *W x <sup>L</sup>* 

<sup>0</sup> ( )( ) *<sup>d</sup>* 

Now, for [ ] *Us*

( ) [ ]( ) *<sup>L</sup> gx W x* 

 *sg x* and <sup>0</sup> ( [ ])( ) *d s* 

solution of (3.4)) satisfies the equation:

next distribution-relation: 1

introduce also the next two functions:

0

  *x*

hand – for the rest components of [ ] *Us*

However it also holds 1 1 [ ]( 0) [ ]( 0) *W W L L* 

2 2

*x*

 

; (3.7)

; (3.8)

 

 

. On the other

<sup>1</sup>

and, substituting from (3.11) in the jump

1 2 

> 

0 2 2

*s s x d*

*k* 

 

( *<sup>I</sup>* – the identity) to equation

. (3.13)

1 sin( ) ( ) () ( ) *<sup>s</sup>*

 

*L s I W* 

is also a bounded function (because

is in <sup>1</sup>

*L R*( ) and

– it is not difficult to get as follows:

(the

( ) *x* is the

, we

. As a

<sup>2</sup> *L R*( ) and that of

*L x*( 0) . This yields the

. (3.12)

(at 0 *x* ):

. (3.9)

For arbitrary non zero asymptotic mean value \* of the surface potential, arbitrary parameters 0, 0, 0, 0 *b b ss k k* , and coefficient *l* : 1 \*2 ,0 <sup>4</sup> | | <sup>27</sup> *l s p* , there exists a unique continuous bounded potential ( ) *x* , satisfying (3.16), such that \* (0) 0 , determined by the formula

$$\boldsymbol{\varphi}(\mathbf{x}) = \boldsymbol{\varphi}\_{\boldsymbol{\alpha}} + \beta\_{l}[t\_{0}]\boldsymbol{\nu}\_{s}^{0}(\mathbf{x}) + \boldsymbol{\varepsilon}\_{s}\Delta\boldsymbol{\varphi}\_{\boldsymbol{\alpha}}\boldsymbol{\omega}\boldsymbol{\nu}\_{s}^{0,1}(\mathbf{x}) + \boldsymbol{\Delta}\boldsymbol{\varphi}\_{\boldsymbol{\alpha}}\boldsymbol{\nu}\_{s}^{\*}(\mathbf{x}) \; , \; \mathbf{x} \neq \mathbf{0} \; ; \tag{3.17}$$

<sup>0</sup>*t* is either the positive (at \* 0 ) or negative (at \* 0 ) root of the equation 3 \* ,0 0 *s l p tt* . In addition, the relevant space potential *uxz* (,)(with *ux x* ( ,0) ( ) )is determined as the unique (classical) solution of problem (2.2) – (2.6), by the next formulas:

$$\mu(\mathbf{x}, z) = \frac{1}{\pi} \int\_{-\infty}^{+\infty} \varphi(\mathbf{x} - t) \frac{z}{z^2 + t^2} dt \; \; \; z > 0 \; \; \; \mathbf{x} \in \mathbb{R}^1 \; ; \tag{3.18}$$

1 2 *s s r k*

, [ ]

for ( ) *Br* 

equality <sup>0</sup>


> 1 2

3 4 

, having radius *r* ; here ( ) *Br*

that under the norm ||.|| ( ) *Br*

 , [ ] [ ] *<sup>s</sup> Qd s* 

observing that 0 0 <sup>1</sup> || || (0) <sup>2</sup> *s s*

<sup>0</sup> | [ ]( )| || [ ]||. (0) *<sup>s</sup> Qx q d ss*

*x dx*

Under condition <sup>3</sup> <sup>1</sup>

 

() 1 *<sup>d</sup>*

*<sup>r</sup>* we find for [ ] *<sup>s</sup> <sup>q</sup>*

*r* , we can fix the magnitude of *r* :

Then we find the next inequality, for || [ ]|| *<sup>s</sup> q*

bounded continuous functions <sup>1</sup> *vx x R* ( ), , such that || || *v r*

*s s k*

, by the obvious triangle inequality:

 || || || [ ]|| || || *<sup>s</sup> Qd s* 

. By the inverse Fourier transformation of

(via the given definition of [ ] *<sup>s</sup> Qd*

*q* 

(3.25) can be easily reworked till a convenient estimate for || [ ]|| *<sup>s</sup> q*

a sake of simplicity, also a restriction in the form <sup>1</sup>

2 . Thus we can suppose from now on that

), consequently it holds: <sup>1</sup> || [ ]|| || [ ]||2

*q*

3 4 *s s*

*k k*

*s k*

<sup>4</sup> *<sup>s</sup> <sup>k</sup>* (applied below as a contraction requirement to [ ]

). For a large class of semiconductors (including these in [5], [7]) it is enough to take

<sup>3</sup> <sup>1</sup> <sup>2</sup>

that

*r*

:

<sup>2</sup> 9 3 || [ ]|| 1 32 8 *s s <sup>s</sup>*

 *f* , for ( )

radius *<sup>r</sup>* , we shall take into account that [ ] 0 *<sup>s</sup> Qd*

 

canonical form to problem (3.4), (3.5) (possessing <sup>0</sup>

Mathematical Models for Electrostatics of Heterogeneous Media 99

( *r* - a small positive parameter). We begin the estimation to the image deviation

 

 *Br* 

is the closure, regarding the norm ||.||, to the set of the

is a complete metric space. We have to study the map

. For choosing a proper magnitude of

*s s f* 

, for a fixed *<sup>r</sup>* . It is clear

*<sup>s</sup>* as the unique solution). Thus,

 ( [ ] *<sup>s</sup> Q d Q* 

, formula (3.13) – for <sup>0</sup>

*s s*

. Next, from

. (3.25)

. We shall introduce, for

(with an arbitrary constant

) inequality

in case of the linear

) we have

*<sup>s</sup>* and

. Choosing now

and <sup>0</sup>

, we shall fix below a final choice of *r* in the form

*f* , (3.23)

ˆ *<sup>Q</sup>*( ) 

*d s*

*Q q <sup>k</sup>*

*kr kr*

<sup>4</sup> *<sup>s</sup> <sup>k</sup>* . (3.26)

 

 

*<sup>k</sup>* . (3.24)

*s*

22 2 ( ) || [ ]|| 1 2 3 *s s s s <sup>s</sup>*

$$\mu(\mathbf{x}, z) = \frac{1}{\pi} \int\_{-\infty}^{+\infty} \rho(\mathbf{x} - t) \frac{(-1)z}{z^2 + t^2} K\_1^0(\kappa\_b \sqrt{z^2 + t^2}) dt \text{ , } z < 0 \text{ , } \mathbf{x} \in \mathbb{R}^1. \tag{3.19}$$

(Above <sup>0</sup> 1 1 *K x xK x* () () , where 1 *K x*( ) is the McDonald function.)

*Proof.* Suppose ( ) *x* is a real, continuous solution of (3.16) and let for instance 0 *x* (in (3.16)), using that 0,1 <sup>1</sup> ( 0) <sup>2</sup> *s s* (see (3.15)). Because [ ] [ (0)] *l l* , we get then relation 03 \* (0) (0) (0) 0 *s l* , i.e. *t* (0) is (by necessity) a real solution of the algebraic equation 3 \* ,0 0 *s l p tt* . Assumptions \* 0 , 1 \*2 ,0 <sup>4</sup> | | <sup>27</sup> *l s p* easily yields existence of a unique positive root 0 0 *t t* of said equation (when \* 0 ), and the same for the negative one, 0 0 *t t* (when \* 0 ). Conversely, let for instance \* 0 and take in (3.17) 0 0 *t t* . Function ( ) *x* given now by formula (3.17) is bounded and continuous on <sup>1</sup> *R* (which is not difficult to be verified) and, letting 0 *x* (in (3.17)), we find \* 3 ,0 0 (0) *s l p t* ; i.e. 0 (0) *t* , 0 [ ] [ (0)] [ ] *ll l t* , and (3.17) shows that ( ) *x* satisfies integral equation (3.16). Having the surface values *ux x* ( ,0) ( ) , it remains to solve the following two Dirichlet problems (as already noted in Sect.1, above): 2 1 { 0, 0; ( ,0) ( ), } *u z ux x x R* and 2 1 { , 0; ( ,0) ( ), } *u ku z ux x x R <sup>b</sup>* . As it generally known, the relevant solutions are determined respectively by (3.18), (3.19).

Consider now the case of vacuum – vacuum heterogeneous system, with a semiconductor interface; then (3.14) is written as:

0 0 0,1 \* [ ]\* . . *s s ls s s s* ; (3.20)

here *<sup>l</sup>* is a given constant. Recall that 0 2 [] ( [ ]) <sup>2</sup> *s d ss u ks <sup>s</sup> g q <sup>u</sup>* .

For equation (3.20) we shall establish existence of a unique continuous and bounded solution, via the contraction mapping argument. Let us introduce the notations 0 0 [ ]( ) (( [ ]) ) ( ) *<sup>s</sup> Q x <sup>d</sup> ds s q x* and

$$f\_s(\mathbf{x}) = \Lambda \varphi\_o \left( \frac{\text{sg}(\mathbf{x})}{2} [1 - \exp(-k\_s \lfloor \mathbf{x} \rfloor)] + \boldsymbol{\nu}\_s^\*(\mathbf{x}) - \frac{\mathcal{E}\_s k\_s^2}{2} (\{\boldsymbol{\alpha}\_d^0 \cdot \mathbf{g}\} \cdot \boldsymbol{\nu}\_s^0)(\mathbf{x}) + \mathcal{W}\_L \left[ \boldsymbol{\nu}\_s^{0,1} \right](\mathbf{x}) \right) + \beta \boldsymbol{\mu}\_s^0(\mathbf{x}) \cdot \text{(3.21)}$$

Now, substituting 0,1 *<sup>s</sup>* in (3.20) with the right hand side of (3.15), equation (3.20) takes the form

$$
\boldsymbol{\varphi} - \boldsymbol{\varphi}\_{\boldsymbol{\varphi}}^{\*} - Q\_{d}^{\boldsymbol{s}}[\boldsymbol{\varphi}] = \boldsymbol{f}\_{\boldsymbol{s}}\,. \tag{3.22}
$$

To analyze the above equation, we shall use the norm || || sup | ( )| *w wx <sup>x</sup>* , <sup>1</sup> *x R* , for continuous, bounded functions *w x*( ) on <sup>1</sup> *R* , and shall deal with balls ( ) *Br* , centered at 98 Electrostatics

 

(see (3.15)). Because [ ] [ (0)]

2 2 1 1 ( 1) () ( ) ( ) *<sup>b</sup> <sup>z</sup> u x z x t K z t dt z t*

, where 1 *K x*( ) is the McDonald function.)

. Assumptions \*

of a unique positive root 0 0 *t t* of said equation (when \*

 

(Above <sup>0</sup>

 

here 

form

*Proof.* Suppose

1 1 *K x xK x* () () 

(3.16)), using that 0,1 <sup>1</sup> ( 0) <sup>2</sup> *s s* 

 *s l* , i.e. *t*

> 

,0 0 *s l p tt*

2 1 { 0, 0; ( ,0) ( ), } *u z ux x x R*

interface; then (3.14) is written as:

0 0 [ ]( ) (( [ ]) ) ( ) *<sup>s</sup> Q x <sup>d</sup> ds s*

 *q x* and

Now, substituting 0,1

[] *<sup>s</sup>*

03 \* (0) (0) (0) 0

equation 3 \*

0 0 *t t* . Function

\* 3 ,0 0 (0) *s l*

 *p t* ; i.e. 0

 

negative one, 0 0 *t t* (when \*

0 22

( ) *x* is a real, continuous solution of (3.16) and let for instance 0 *x* (in

*l l* 

0 , 1 \*2

( ) *x* given now by formula (3.17) is bounded and continuous on <sup>1</sup> *R*

and 2 1 { , 0; ( ,0) ( ), } *u ku z ux x x R <sup>b</sup>*

; (3.20)

> 

*<sup>s</sup>* in (3.20) with the right hand side of (3.15), equation (3.20) takes the

.

 

 

0 ). Conversely, let for instance \*

(which is not difficult to be verified) and, letting 0 *x* (in (3.17)), we find

the following two Dirichlet problems (as already noted in Sect.1, above):

Consider now the case of vacuum – vacuum heterogeneous system, with a semiconductor

 

For equation (3.20) we shall establish existence of a unique continuous and bounded solution, via the contraction mapping argument. Let us introduce the notations

<sup>2</sup> 0 0 0,1 <sup>0</sup> ( ) ( ) [1 exp( | |)] ( ) (( . ) )( ) [ ]( ) ( ) 2 2 *s s <sup>s</sup> s s d s Ls l s sg x <sup>k</sup> f x kx x sg x W x x* 

> *Q f d s*

To analyze the above equation, we shall use the norm || || sup | ( )| *w wx <sup>x</sup>* , <sup>1</sup> *x R* , for

 

0 0 0,1 \* [ ]\* . .

*s s ls s s s*

 

(0) *t* , 0 [ ] [ (0)] [ ] *ll l*

generally known, the relevant solutions are determined respectively by (3.18), (3.19).

*t*

*<sup>l</sup>* is a given constant. Recall that 0 2 [] ( [ ]) <sup>2</sup> *s d ss u ks <sup>s</sup> g q <sup>u</sup>*

continuous, bounded functions *w x*( ) on <sup>1</sup> *R* , and shall deal with balls ( ) *Br*

satisfies integral equation (3.16). Having the surface values *ux x* ( ,0) ( )

, 0 *<sup>z</sup>* , <sup>1</sup> *x R* . (3.19)

(0) is (by necessity) a real solution of the algebraic

, and (3.17) shows that

 

. (3.22)

 

, centered at

. (3.21)

. As it

,0 <sup>4</sup> | | <sup>27</sup> *l s*

 

, we get then relation

0 ), and the same for the

0 and take in (3.17)

, it remains to solve

( ) *x*

*p* easily yields existence

 , having radius *r* ; here ( ) *Br* is the closure, regarding the norm ||.||, to the set of the bounded continuous functions <sup>1</sup> *vx x R* ( ), , such that || || *v r* , for a fixed *<sup>r</sup>* . It is clear that under the norm ||.|| ( ) *Br* is a complete metric space. We have to study the map , [ ] [ ] *<sup>s</sup> Qd s f* , for ( ) *Br* . For choosing a proper magnitude of radius *<sup>r</sup>* , we shall take into account that [ ] 0 *<sup>s</sup> Qd* and <sup>0</sup> *s s f* in case of the linear canonical form to problem (3.4), (3.5) (possessing <sup>0</sup> *<sup>s</sup>* as the unique solution). Thus, observing that 0 0 <sup>1</sup> || || (0) <sup>2</sup> *s s s s k* , we shall fix below a final choice of *r* in the form 1 *r*

2 *s s k* ( *r* - a small positive parameter). We begin the estimation to the image deviation , [ ] , by the obvious triangle inequality:

$$\mid \mid \Phi - \boldsymbol{\sigma}\_v^\* \mid \mid \le \mid \mid Q\_d^s[\boldsymbol{\sigma}] \mid \mid + \mid f\_s \mid \mid \; \tag{3.23}$$

for ( ) *Br* . By the inverse Fourier transformation of ˆ *<sup>Q</sup>*( ) ( [ ] *<sup>s</sup> Q d Q* ) we have <sup>0</sup> | [ ]( )| || [ ]||. (0) *<sup>s</sup> Qx q d ss* (via the given definition of [ ] *<sup>s</sup> Qd* , formula (3.13) – for <sup>0</sup> *<sup>s</sup>* and equality <sup>0</sup> () 1 *<sup>d</sup> x dx* ), consequently it holds: <sup>1</sup> || [ ]|| || [ ]||2 *s d s s s Q q <sup>k</sup>* . Next, from || || *<sup>r</sup>* we find for [ ] *<sup>s</sup> <sup>q</sup>* that 22 2 ( ) || [ ]|| 1 2 3 *s s s s <sup>s</sup> kr kr q* . Choosing now 1 2 *r* , we can fix the magnitude of *r* :

$$r = \frac{3}{4\varepsilon\_s k\_s}.\tag{3.24}$$

Then we find the next inequality, for || [ ]|| *<sup>s</sup> q* :

$$|\parallel\!\!\!/q\_s[\rho]\vert \mid \vert \le \frac{9k\_s^2}{32} \left(1 + \frac{3k\_s}{8}\right). \tag{3.25}$$

Under condition <sup>3</sup> <sup>1</sup> <sup>4</sup> *<sup>s</sup> <sup>k</sup>* (applied below as a contraction requirement to [ ] ) inequality (3.25) can be easily reworked till a convenient estimate for || [ ]|| *<sup>s</sup> q* . We shall introduce, for a sake of simplicity, also a restriction in the form <sup>1</sup> *s k* (with an arbitrary constant 3 4 ). For a large class of semiconductors (including these in [5], [7]) it is enough to take 2 . Thus we can suppose from now on that

$$\frac{3}{4} \le k\_s^{-1} \le 2 \cdot \tag{3.26}$$

Mathematical Models for Electrostatics of Heterogeneous Media 101

By (3.28) the above found for || || *sf* modifies to inequality || || (14| | 2| |) <sup>3</sup> *s l*


*<sup>l</sup>* satisfies condition

<sup>1</sup> 7| | | | <sup>2</sup> 

Summarizing (3.23), (3.27) and (3.29), we conclude the following. For any data

*Br* we have to estimate difference 2 1 , [ ] *j j*

we shall consider difference of [ ] *<sup>s</sup> Qd*

2 1 21 1 21 <sup>0</sup> [ ] [ ] ( ) [ ( )] *s s Q Q dd d s <sup>s</sup>*

2 1 2,1 <sup>1</sup> <sup>ˆ</sup> || [ ] [ ]|| | [ ]( )|.| ( )| <sup>ˆ</sup> <sup>2</sup> *s s <sup>s</sup> QQ R d d d s*

> (at \* 1 2 , ()

1 21 <sup>0</sup>

(We have used in (3.32) relation (3.26), via (3.24), and the above notations <sup>0</sup> ˆ

 

 

 

 ;

1 0 2,1 2 1 1 21 <sup>0</sup> ˆ || [ ]|| || || ( ) | [ ( )]( )|

*<sup>s</sup> Rd d s x q x d dx* ;

 

<sup>3</sup> || [ ( )]|| <sup>2</sup> *<sup>s</sup> q d*

 

satisfying (3.26), (3.28) and (3. 30) relation [ ]

<sup>1</sup> 0 0

*dt* , with 1 1 2 \*2 2 \*3 [ ] [ ( )] [ ( )] 2! 3! *s ss s s qt k t* 

the right hand side of (3.31) and take into account the next several

 

> 

 

( ) *<sup>s</sup>* 

 

*q d* . (3.31)

0

*Br* ), i.e. 3 3 <sup>3</sup> || [ ]|| (1 ) 4 82

. (3.32)

 *x* , 2,1 [ ]( ) *<sup>s</sup> R x <sup>d</sup>* 

This will give the estimate

and parameters *<sup>s</sup> k* , *<sup>s</sup>*

\*

21 2 1 [] [] *s s Q Q d d* 

 

1 2 , ()

by <sup>0</sup> 2,1 [ ] *<sup>s</sup> Rd s* 

inequalities:

\* ( ) *Br* 

 

It holds when the sum 7| | | |

, with *r* as in (3.24), into itself.

<sup>2</sup> || [ ]|| (1 ) <sup>2</sup> *s s s ss k r q kr*

 

(Above [ ] *<sup>s</sup> q t* is derivative [ ] *<sup>s</sup> dq <sup>t</sup>*

Now we shall study the contraction property of [ ]

  

 

2

1

taken for the Fourier images respectively to functions <sup>0</sup>

3 2 *s s* 

*k* . (3.28)

*<sup>r</sup> <sup>f</sup>* . (3.29)

*<sup>l</sup>* . (3.30)

*<sup>r</sup> <sup>f</sup>*

 , , *l*

maps the ball

( *j* 1,2 ). From

. For arbitrary two elements

*<sup>j</sup>* ( *j* 1,2 ), using relation

 *k t* .) Denote

> *d*

*s s <sup>s</sup> k k q* ;

 

.) Then we have:

 *<sup>s</sup>* , 2,1 <sup>ˆ</sup> [ ] *<sup>s</sup> Rd* 

are

 .

Now, reworking (3.25) we get: <sup>3</sup> || [ ]|| <sup>4</sup> *<sup>s</sup> <sup>q</sup>* . Consequently [ ] *<sup>s</sup> Qd* is estimated as

$$|\left|\left|Q\_d^s[\rho]\right|\right|| \le \frac{r}{2}.\tag{3.27}$$

The obvious next step is to establish the analogous estimate for *sf* , to get this way the needed property for [ ] ( see (3.23)). For relevant terms with in expression (3.21) it holds as follows. Function ( ) *<sup>s</sup> x* satisfies (as directly shows (3.13)) inequality <sup>0</sup> || || (0) *s s* , therefore <sup>1</sup> || || <sup>2</sup> *<sup>s</sup> s s k* . Next, estimating the term 0 0 ( .) *d s sg* by analogy to [ ] *<sup>s</sup> Qd* , we have:

$$|\ll| (\mathcal{O}\_d^0 \text{sg}) \ast \mathcal{W}\_s^0 \mid | \le \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \mathcal{O}\_d^0(\mathbf{x}) d\mathbf{x} \int\_{-\infty}^{+\infty} |\hat{\varphi}\_s^0(\xi)| \, d\xi \le \frac{1}{2\mathcal{E}\_s k\_s} \dots$$

Concerning the element 0,1 [ ] *WL s* (in said expression for *sf* ) we shall firstly introduce inequality

$$|\mid \mathcal{W}\_{\boldsymbol{L}}[\boldsymbol{\psi}\_{s}^{0,1}] \mid \mid \leq \frac{1}{2\pi} \mid \mid \mathcal{Z} \mid \boldsymbol{\xi} \mid (k\_{s}^{2} + \boldsymbol{\xi}^{2})^{-1} \mid \mid\_{\boldsymbol{L}\_{2}} \mid \mid \boldsymbol{\psi}\_{s}^{0,1} \mid \mid\_{\boldsymbol{L}\_{2}} \mid$$

(where 2 ||.||*L* is the norm in <sup>1</sup> <sup>2</sup> *L R*( ) ). Then we use that 2 1/2 2 21 <sup>2</sup> ||2| |( ) || *s L s k k* and, because of (3.15), we have 222 0,1 0,1 1 1 0,1 ˆ ˆ || || || ( [ ])|| ||( .exp( |.|)|| ˆ ˆ <sup>2</sup> *s L s Ls L s L s s W sg k* (with ˆ ( ) as the Fourier transformation of ). The above yields inequality 0,1 1 3 || [ ]|| <sup>2</sup> <sup>4</sup> *L s s s s s W k k* , taking into account that 2 1/2 <sup>ˆ</sup> ||( .exp( |.|)|| 2 *s L s sg k <sup>k</sup>* ; i.e. 0,1 || [ ]|| *W r L s* (see (3.24)). Then, from inequality (see (3.21))

$$|\parallel\rangle \, |f\_s\rangle \mid \leq |\Delta \phi\_w\rangle \left(\frac{1}{2} + |\ |\psi\_s^\*\rangle\| + \frac{\varepsilon\_s k\_s^2}{2} \mid |\left(o\_d^0 \text{.sg}\right)\* \psi\_s^0\rangle\ |+ |\ |\mathcal{W}\_{\mathcal{L}}[\psi\_s^{0,1}]\| \mid \right) + |\mathcal{J}\_l\rangle \mid |\psi\_s^0\rangle\ |\ i\_s\rangle$$

it follows the next one: <sup>2</sup> 1 2 <sup>2</sup> || || | | || 23 4 <sup>3</sup> *s s s l s s <sup>k</sup> <sup>r</sup> f rr <sup>k</sup>* . Consequently, at 2 1 *r* , i.e. (via (3.24)) introducing condition (3.28), below, we find firstly that <sup>2</sup> 8 2 || || [ ]| | | | 33 3 *s s s l <sup>k</sup> f r* .

100 Electrostatics


The obvious next step is to establish the analogous estimate for *sf* , to get this way the

( see (3.23)). For relevant terms with

. Consequently [ ] *<sup>s</sup> Qd*

. (3.27)

 *sg* 

*s s*

2 21 <sup>2</sup> ||2| |( ) || *s L*

 

*s s s s*

(see (3.24)). Then,

 

*k k*

, taking

*k*

  *k*

*x* satisfies (as directly shows (3.13)) inequality

 

(in said expression for *sf* ) we shall firstly introduce

2 2

 *s L sL*

. Next, estimating the term 0 0 ( .) *d s*

 

<sup>2</sup> *L R*( ) ). Then we use that 2

*W*

; i.e. 0,1 || [ ]|| *W r L s* 

> 

 is estimated as

in expression (3.21) it

by analogy

1/2

*s*

*k* 

,

. Consequently, at

 

 

*s s k*

00 0 0 1 1 ||( . ) || ( ) | ( )| <sup>ˆ</sup> 2 2 *ds d s*

0,1 <sup>1</sup> 2 21 0,1 || [ ]|| ||2| |( ) || || || <sup>ˆ</sup> <sup>2</sup>

and, because of (3.15), we have

(with ˆ ( ) as the Fourier

1/2

*s*

<sup>2</sup> <sup>1</sup> 0 0 0,1 <sup>0</sup> || || | | || || ||( . ) || || [ ]|| | ||| || 2 2 *s s <sup>s</sup> <sup>s</sup> d s Ls l s*

 

> 

2 1 *r* , i.e. (via (3.24)) introducing condition (3.28), below, we find firstly that

 

<sup>2</sup> 1 2 <sup>2</sup> || || | | || 23 4 <sup>3</sup> *s s s l s s <sup>k</sup> <sup>r</sup> f rr <sup>k</sup>* 

 

 

 

 .

*sg x dx d*

 

*W k L s*

222

*W sg k*

 

transformation of ). The above yields inequality 0,1 1 3 || [ ]|| <sup>2</sup> <sup>4</sup> *L s*

0,1 0,1 1 1 0,1 ˆ ˆ || || || ( [ ])|| ||( .exp( |.|)|| ˆ ˆ <sup>2</sup> *s L s Ls L s L s s*

<sup>ˆ</sup> ||( .exp( |.|)|| 2 *s L*

 

*sg k <sup>k</sup>*

*<sup>k</sup> <sup>f</sup> sg <sup>W</sup>* 

.

  Now, reworking (3.25) we get: <sup>3</sup> || [ ]|| <sup>4</sup> *<sup>s</sup> <sup>q</sup>*

holds as follows. Function ( ) *<sup>s</sup>*

, therefore <sup>1</sup> || || <sup>2</sup> *<sup>s</sup>*

Concerning the element 0,1 [ ] *WL s*

(where 2 ||.||*L* is the norm in <sup>1</sup>

into account that 2

<sup>2</sup> 8 2 || || [ ]| | | | 33 3 *s s s l*

 

from inequality (see (3.21))

it follows the next one:

*<sup>k</sup> f r*

needed property for [ ]

, we have:

<sup>0</sup> || || (0) 

to [ ] *<sup>s</sup> Qd* 

inequality

 *s s*

$$
\varepsilon\_s k\_s \le \frac{3}{2}.\tag{3.28}
$$

By (3.28) the above found for || || *sf* modifies to inequality || || (14| | 2| |) <sup>3</sup> *s l <sup>r</sup> <sup>f</sup>* . This will give the estimate

$$\|\|f\_s\|\|\leq \frac{r}{3}.\tag{3.29}$$

It holds when the sum 7| | | | *<sup>l</sup>* satisfies condition

$$\left| \nabla \left| \Delta \boldsymbol{\uprho}\_{\boldsymbol{\uprho}} \right| + \left| \boldsymbol{\uprho}\_{l} \right| \right| \leq \frac{1}{2} \,. \tag{3.30}$$

Summarizing (3.23), (3.27) and (3.29), we conclude the following. For any data , , *l* and parameters *<sup>s</sup> k* , *<sup>s</sup>* satisfying (3.26), (3.28) and (3. 30) relation [ ] maps the ball \* ( ) *Br* , with *r* as in (3.24), into itself.

Now we shall study the contraction property of [ ] . For arbitrary two elements \* 1 2 , () *Br* we have to estimate difference 2 1 , [ ] *j j* ( *j* 1,2 ). From 21 2 1 [] [] *s s Q Q d d* we shall consider difference of [ ] *<sup>s</sup> Qd <sup>j</sup>* ( *j* 1,2 ), using relation

$$Q\_d^s[\varphi\_2] - Q\_d^s[\varphi\_1] = \left(\alpha\_d^0(\varphi\_2 - \varphi\_1)\right)^1\_0 q\_s'[\varphi\_1 + \tau(\varphi\_2 - \varphi\_1)]d\tau\right) \* \nu\_s^0 \,. \tag{3.31}$$

(Above [ ] *<sup>s</sup> q t* is derivative [ ] *<sup>s</sup> dq <sup>t</sup> dt* , with 1 1 2 \*2 2 \*3 [ ] [ ( )] [ ( )] 2! 3! *s ss s s qt k t k t* .) Denote by <sup>0</sup> 2,1 [ ] *<sup>s</sup> Rd s* the right hand side of (3.31) and take into account the next several inequalities:

$$\begin{aligned} \left| \mid Q\_d^s[\varphi\_2] - Q\_d^s[\varphi\_1] \mid \right| &\leq \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \left| \hat{R}\_s^s[\varphi\_{2,1}](\xi) \mid \right| \mid \dot{\varphi}\_s^0(\xi) \mid d\xi \; \forall \; \varphi\_s^0 \; \middle| \; \begin{aligned} \left| \varphi\_2 \right| \mid \quad &\leq \frac{1}{2} \\\\ \left| \mid \hat{R}\_d^s[\varphi\_{2,1}] \mid \right| \mid \leq \left| \mid \; \varphi\_2 - \varphi\_1 \mid \mid \; \int\_{-\infty}^{+\infty} \alpha\_d^0(\mathbf{x}) \int\_0^1 \mid q\_s^\*[\varphi\_1 + \tau(\varphi\_2 - \varphi\_1)](\mathbf{x}) \mid d\tau d\mathbf{x} \; \forall \; \varphi\_2 \right| \end{aligned} \\\\ \left| \mid \; \left| q\_s^\prime[\cdot] \right| \mid \leq \varepsilon\_s k\_s^2 r \left( 1 + \frac{\varepsilon\_s k\_s^2 r}{2} \right) \text{ (at  $\varphi\_1, \varphi\_2 \in B\_r(\rho\_\infty^+)$ ), i.e. } \mid \; \left| q\_s^\prime[\cdot] \right| \mid \leq \frac{3k\_s}{4} (1 + \frac{3k\_s}{8}) \leq \frac{3}{2}; \\\\ \left| \right| \mid \; q\_s^\prime[\varphi\_1 + \tau(\varphi\_2 - \varphi\_1)] \mid \; \mid \; d\tau \leq \frac{3}{2}. \end{aligned}$$

(We have used in (3.32) relation (3.26), via (3.24), and the above notations <sup>0</sup> ˆ *<sup>s</sup>* , 2,1 <sup>ˆ</sup> [ ] *<sup>s</sup> Rd* are taken for the Fourier images respectively to functions <sup>0</sup> ( ) *<sup>s</sup> x* , 2,1 [ ]( ) *<sup>s</sup> R x <sup>d</sup>* .) Then we have:

Mathematical Models for Electrostatics of Heterogeneous Media 103

<sup>0</sup> (0) ( [ (0)]) (0) *ls s*

<sup>0</sup> 3 0 <sup>2</sup> <sup>0</sup> ( ) ( ) (0) ( ) (0) ( ) (0) 0 6 2 *s s s s <sup>s</sup> <sup>s</sup> l s*

*z z z*

 

6 2 (0) *<sup>l</sup> ss s*

quadratic polynomial does not have real roots (via (3.26) and the known inequality

*k*

 

(0) 

0

 

 

> ( ) *x* and

 

, from (3.20) – with

 

 *<sup>q</sup>* ;

2 0 0

0

*d sd*

 

> 

> > *s*

.

*x Q x fx Q x*

<sup>0</sup> 0 0 (.; ) [ (.; )] [ ] [ ] *s ss d dd*

0 00 || (.; ) || || (.; ) || || [ ]|| *<sup>s</sup>*

0 0 <sup>1</sup> || (.; ) || || [ ]|| <sup>1</sup>

*<sup>d</sup> d Q r*

*d rd Q* ,

 

*d Q dQ Q*

<sup>0</sup> 0 0 0,1 ( ) ( [ ]) ( ) ( ( ) ( )) *ls s s ss*

*x qx x x*

2 *s s t k*

> 

. (4.4)

. Formula (4.4) then directly shows that <sup>0</sup>

. (4.5)

(.; ) *d* , we evidently get:

 

*d*

 

in the above given contraction estimate – for

. (4.6)

.

 

. (4.2)

2

*<sup>s</sup>* ). This yields existence of a unique real solution

. (4.3)

 

*<sup>t</sup> gt t*

<sup>1</sup> ( ) <sup>2</sup> (0) *ss s*

 *<sup>k</sup>* . The found

2 0

(;) *x d* . Let us introduce the difference

( ) *x* is a

 

(0) is a real solution to equation

( ) *s s t kz* 

i.e. 

Setting <sup>2</sup>

<sup>0</sup> <sup>1</sup> (0) <sup>2</sup> *<sup>s</sup>*

solution to equation

0

*s s k*

*t* of (4.2) and we set

Now from (4.1) we get the function

<sup>0</sup> [ ]( ) [ ]( ) [ (0)]) ( ) *s s Q xQ xq x d d ss*

 

Subtracting (4.5), with <sup>0</sup>

Putting afterwards 2

2 1 [] [] *s s Q Q d d* 

consequently

The next step will be the comparison of <sup>0</sup>

 

 

 

 

, we find directly that

It presents actually the unique solution of equation (4.1).

 

( ) [ ]( ) ( ) [ ]( ) *s s*

 (.; ) *d* and <sup>0</sup> 1 

 

2 3 2 2

 

*k k*

we rewrite this equation in the form:

3 2

*tt t*

Derivative of the left hand side (denote it by *g t*( ) ) is

, recall (3.13) concerning <sup>0</sup>

$$|\langle \mid Q\_d^s[\varphi\_2] - Q\_d^s[\varphi\_1] \vert \mid \le \frac{3}{2} \vert \mid \varphi\_2 - \varphi\_1 \vert \mid \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \vert \hat{\psi}\_s^0(\xi) \vert \, d\xi \le \frac{3}{4\varepsilon\_s k\_s} \vert \mid \varphi\_2 - \varphi\_1 \vert \mid \, . \le$$

Consequently,

$$|\ |\ |\Phi\_2 - \Phi\_1\ |\ |\ | \le r\ |\ |\phi\_2 - \phi\_1\ |\ |\ |$$

(because of (3.24)), and under estimates (3.27), (3.29) (valid at (3.30)) the considered map [ ] is a contraction in the ball \* ( ) *Br* , for *<sup>r</sup>* <sup>1</sup> , i.e. <sup>3</sup> <sup>4</sup> *s s k* . Combining the latter inequality with (3.28), we come to condition

$$\frac{3}{4} < \left. \varepsilon\_s k\_s \right| \le \frac{3}{2}. \tag{3.33}$$

Thus we have given the proof of the following basic result.

#### **3.2 Proposition**

For arbitrary values of positive parameters *<sup>s</sup>* , *<sup>s</sup> k* , each asymptotic data of the surface potential and line charges *l l* <sup>0</sup> , such that conditions (3.26), (3.33) and (3.30) hold, equation (3.20) possesses a unique continuous, bounded solution (;) *x d* , satisfying the estimate

$$\sup\_{\mathbf{x}} |\varphi(\mathbf{x}; d) - \boldsymbol{\rho}\_{\boldsymbol{\alpha}}^{\*}| \le \frac{5}{8\varepsilon\_{s}k\_{s}} \;/\forall d > 0 \;. \tag{3.34}$$

#### **4. Explicit approximations in the case of semiconductor interface**

Via the possible applications, it is important to ask for a suitable approximation <sup>0</sup> ( ) *x* to the interface data *u x*( ,0) , well enough at small | |*d* and explicitly determined. To that goal, suppose a sequence { } *<sup>n</sup>* of solutions to (3.20), () (;) *<sup>n</sup> x xd* , at *<sup>n</sup> d d* , with 0 *<sup>n</sup> d* ( *n* ), is convergent (in a distribution sense) to a bounded continuous function ( ) *<sup>x</sup>* , 1 *x R* . Putting *<sup>n</sup> d d* in (3.20) and letting *<sup>n</sup>* , we can conclude that ( ) *x* is a solution to equation

$$
\varphi(\mathbf{x}) = \varphi\_{\boldsymbol{\alpha}} + (\beta\_l + q\_s[\varphi(\mathbf{0})]) \boldsymbol{\upmu}\_s^0(\mathbf{x}) + \boldsymbol{\upmu}\varphi\_{\boldsymbol{\alpha}}(\boldsymbol{\upmu}\_s^\*(\mathbf{x}) + \boldsymbol{\upmu}\_s \boldsymbol{\upmu}\_s^{0,1}(\mathbf{x})) \,. \tag{4.1}
$$

For finding (4.1) we have taken into account that <sup>0</sup> [ (.; )]( ) [ (0)] ( ) *<sup>s</sup> Q dx q x d ss* and 0 0 [( . ) \* ]( ) 0 *d s sg x* , <sup>1</sup> *x R* , at *<sup>n</sup> d d* ( *n* ). Next we shall study equation (4.1). Note first of all the necessary condition to have a continuous solution ( ) *x* :

$$
\rho^-\_{\alpha\flat} + \varepsilon\_s \Delta \varphi\_{\alpha\flat} \nu^{0,1}\_s(-0) = \rho^+\_{\alpha\flat} + \varepsilon\_s \Delta \varphi\_{\alpha\flat} \nu^{0,1}\_s(+0) \nmid \rho^+\_{\alpha\flat}
$$

it is fulfilled, because of (3.15). If ( ) *x* is a continuous solution to (4.1), for value (0) we obtain (from (4.1), at 0 *x* or 0 *x* ) the algebraic equation

$$
\varphi(0) = \varphi\_\circ^\* + (\beta\_l + q\_s[\varphi(0)])\nu\_s^0(0) \; ;
$$

i.e. (0) is a real solution to equation

102 Electrostatics

*d d s*

*Q Q d*

 

*s s*

 

is a contraction in the ball \* ( ) *Br*

Thus we have given the proof of the following basic result.

*l l* <sup>0</sup> 

equation (3.20) possesses a unique continuous, bounded solution

inequality with (3.28), we come to condition

For arbitrary values of positive parameters *<sup>s</sup>*

   

first of all the necessary condition to have a continuous solution

obtain (from (4.1), at 0 *x* or 0 *x* ) the algebraic equation

 

it is fulfilled, because of (3.15). If

Consequently,

[ ] 

**3.2 Proposition** 

estimate

to equation

0 0 [( . ) \* ]( ) 0 *d s*

 

potential and line charges

suppose a sequence { }

2 1 21 2 1 31 3 || [ ] [ ]|| || || | ( )| || || <sup>ˆ</sup> 22 4


(because of (3.24)), and under estimates (3.27), (3.29) (valid at (3.30)) the considered map

3 3 4 2 *s s* 

\* <sup>5</sup> sup| ( ; ) | *<sup>x</sup>* 8 *s s*

 

*x d*

**4. Explicit approximations in the case of semiconductor interface** 

 

Via the possible applications, it is important to ask for a suitable approximation <sup>0</sup>

interface data *u x*( ,0) , well enough at small | |*d* and explicitly determined. To that goal,

( ) *<sup>x</sup>* , 1 *x R* . Putting *<sup>n</sup> d d* in (3.20) and letting *<sup>n</sup>* , we can conclude that

<sup>0</sup> 0,1 ( ) ( [ (0)]) ( ) ( ( ) ( )) *ls s s ss*

*sg x* , <sup>1</sup> *x R* , at *<sup>n</sup> d d* ( *n* ). Next we shall study equation (4.1). Note

 

;

*s s s s*

*<sup>n</sup>* of solutions to (3.20), () (;) *<sup>n</sup>*

( *n* ), is convergent (in a distribution sense) to a bounded continuous function

For finding (4.1) we have taken into account that <sup>0</sup> [ (.; )]( ) [ (0)] ( )

0

, *<sup>s</sup> k* , each asymptotic data

*k*

 

0,1 0,1 ( 0) ( 0)

*x q x xx* . (4.1)

 

> 

> > ( ) *x* :

( ) *x* is a continuous solution to (4.1), for value

, such that conditions (3.26), (3.33) and (3.30) hold,

, *d* 0 . (3.34)

 

 

, for *<sup>r</sup>* <sup>1</sup> , i.e. <sup>3</sup>

 .

*s s*

 

<sup>4</sup> *s s* 

*k* . (3.33)

*k*

 

*k* . Combining the latter

of the surface

(;) *x d* , satisfying the

*x xd* , at *<sup>n</sup> d d* , with 0 *<sup>n</sup> d*

*<sup>s</sup> Q dx q x d ss*

 and

( ) *x* to the

( ) *x* is a solution

(0) we

$$\int \wp\_s^0(0) \frac{(\varepsilon\_s k\_s^2)^3}{6} (z - \wp\_\alpha^\*)^3 - \wp\_s^0(0) \frac{(\varepsilon\_s k\_s^2)^2}{2} (z - \wp\_\alpha^\*)^2 + z - \wp\_\alpha^\* - \beta \wp\_s^0(0) = 0 \cdot \frac{1}{2}$$

Setting <sup>2</sup> ( ) *s s t kz* we rewrite this equation in the form:

$$\frac{t^3}{6} - \frac{t^2}{2} + \frac{t}{\varepsilon\_s k\_s^2 \nu\_s^0(0)} - \beta\_l = 0 \,\,\,\,\tag{4.2}$$

Derivative of the left hand side (denote it by *g t*( ) ) is 2 2 0 <sup>1</sup> ( ) <sup>2</sup> (0) *ss s <sup>t</sup> gt t <sup>k</sup>* . The found quadratic polynomial does not have real roots (via (3.26) and the known inequality <sup>0</sup> <sup>1</sup> (0) <sup>2</sup> *<sup>s</sup> s s k* , recall (3.13) concerning <sup>0</sup> (0) *<sup>s</sup>* ). This yields existence of a unique real solution 0 *t* of (4.2) and we set

$$
\varphi^0 = \varphi^\*\_{\varphi} + \frac{t^0}{\varepsilon\_s k\_s^2} \cdot \tag{4.3}
$$

Now from (4.1) we get the function

$$
\boldsymbol{\phi}\_{\*}^{0}(\mathbf{x}) \equiv \boldsymbol{\uprho}\_{\boldsymbol{\upalpha}} + \left(\boldsymbol{\upbeta}\_{l} + \boldsymbol{\uprho}\_{s}[\boldsymbol{\uprho}^{0}]\right) \boldsymbol{\uprho}\_{s}^{0}(\mathbf{x}) + \boldsymbol{\upLambda} \boldsymbol{\uprho}\_{\boldsymbol{\upalpha}}(\boldsymbol{\upmu}\_{s}^{\*}(\mathbf{x}) + \boldsymbol{\upvarepsilon}\_{s} \boldsymbol{\upmu}\_{s}^{0,1}(\mathbf{x})) \,. \tag{4.4}
$$

It presents actually the unique solution of equation (4.1).

The next step will be the comparison of <sup>0</sup> ( ) *x* and (;) *x d* . Let us introduce the difference <sup>0</sup> [ ]( ) [ ]( ) [ (0)]) ( ) *s s Q xQ xq x d d ss* . Formula (4.4) then directly shows that <sup>0</sup> ( ) *x* is a solution to equation

$$
\varphi(\mathbf{x}) - \boldsymbol{\varphi}\_v^\* - Q\_d^\flat[\boldsymbol{\varphi}](\mathbf{x}) = f\_s(\mathbf{x}) - \Lambda Q\_d^\flat[\boldsymbol{\varphi}](\mathbf{x}) \,. \tag{4.5}
$$

Subtracting (4.5), with <sup>0</sup> , from (3.20) – with (.; ) *d* , we evidently get:

$$
\log(\colon d) - \varphi^0\_\* = Q^s\_d[\varphi(\colon d)] - Q^s\_d[\varphi^0\_\*] + \Delta Q^s\_d[\varphi^0\_\*] \ .
$$

Putting afterwards 2 (.; ) *d* and <sup>0</sup> 1 in the above given contraction estimate – for 2 1 [] [] *s s Q Q d d* , we find directly that

$$|\ l \mid \left\| \boldsymbol{\varrho}(.; \boldsymbol{d}) - \boldsymbol{\varrho}\_{\ast}^{0} \mid \mid \leq \ r \mid \mid \left\| \boldsymbol{\varrho}(.; \boldsymbol{d}) - \boldsymbol{\varrho}\_{\ast}^{0} \mid \mid + \mid \left\| \boldsymbol{\Delta} \boldsymbol{Q}\_{d}^{s}[\boldsymbol{\varrho}\_{\ast}^{0}] \right\| \mid \vdash \right\|$$

consequently

$$<\langle \mid \boldsymbol{\varrho}(\cdot; \boldsymbol{d}) - \boldsymbol{\varrho}^{0}\_{\*} \mid \mid \leq \frac{1}{1 - r} \mid \mid \Delta Q^{s}\_{d}[\boldsymbol{\varrho}^{0}\_{\*}] \mid \mid. \tag{4.6}$$

Mathematical Models for Electrostatics of Heterogeneous Media 105

 

 *q* for

0 0 00 0 0,1 ( ) ( ) ( [ ])[ ( ) (0)] [ ( ) ( )] <sup>2</sup> *ls s s s ss x sg <sup>x</sup> <sup>q</sup> x xx*

0 0 00 0 ( ) ( )[1 exp( | |)] ( [ ])[ ( ) (0)] <sup>2</sup> *s ls s s <sup>x</sup> sg x k x q x*

0,1 0,1 0,1

0,1 0,1


0,1 ( ) || || <sup>ˆ</sup>*<sup>s</sup> L R*

*<sup>d</sup> <sup>W</sup>*


2 2 1/ 2 || 1 1 1 || 1 | ( )| log(1 ) log(1 ) <sup>2</sup> *<sup>s</sup> <sup>d</sup> <sup>s</sup> <sup>s</sup> s s ss <sup>x</sup> x d x d*

 

*x x <sup>x</sup> x dd*

 

0,1 || [ ]|| *Ls d*

1/

 

> 

2

 

2 |sin( )| 1 |sin ( )| | [ ]|( ) | ( )| ˆ ˆ .|| || ( ) *L s <sup>s</sup> <sup>s</sup> L R s s*

*k k*

0,1 ( ( ) [ ]( )) *s Ls*

 *xW x* .

,

 

(4.10)

0 0 2 2 2 22 ( )

1/2 <sup>2</sup>

0 2 ( )

*s s k* 

*s s*

0 2 2 1/ 2 2

*x k k*

;

*k d k d*

.

  *k*

.

.

 

*q d* . (4.9.b)

 

 

*<sup>d</sup>* in (4.9.a). By (4.4)

(0) (see the initial form of

; it is continuous

 

1 2

 

 

1/2 2 2

1 2

and the well known relation

. (4.11)

   

 

 

 

<sup>0</sup> 0 0,1 || [ ]|| | [ ]|.|| || <sup>2</sup> *s s I*

 

and the algebraic equation <sup>0</sup> (0) ( [ (0)]) (0) *ls s* 

*<sup>s</sup>* by (3.15) we get

and odd, and we can use the following inequalities:

Recall here the already found estimate 1

(4.2)) we firstly have

and expressing 0,1

Therefore:

2 0 2 sin

For ( ) *<sup>s</sup>* 

i.e.:

*d* 

2

. Then it follows:

*x* we start with the inequality

 

 

 

To final reworking of (4.8.a) we shall estimate the quantity 0 0 || ||

 

> 

Next we analyze the terms in (4.10) beginning with function 0,1 [ ]( ) *W x L s*

*x x W x <sup>d</sup> <sup>d</sup>*

*<sup>x</sup> W x <sup>d</sup>*

 

By the known definition 0 0 [ ] ( [ ]) *<sup>s</sup> Q q d ds s* , perturbation term <sup>0</sup> [ ] *<sup>s</sup> Qd* can be easily presented as

$$\begin{split} \Delta Q\_d^s[\varphi\_\*^0](\mathbf{x}) &= \int\_{-\infty}^{+\infty} \alpha^0(\tau) (q\_s[\varphi\_\*^0(\tau d)] - q\_s[\varphi^0]) \wp\_s^0(\mathbf{x} - \tau d) d\tau + \\ &+ q\_s[\varphi^0] \int\_{-\infty}^{+\infty} \alpha^0(\tau) [\wp\_s^0(\mathbf{x} - \tau d) - \wp\_s^0(\mathbf{x})] d\tau \end{split} \tag{4.7}$$

Denote the first and second integrals in (4.7) respectively by <sup>0</sup> <sup>1</sup>*I x* [ ]( ) and <sup>0</sup> <sup>2</sup>*I x* [ ]( ) ; they satisfy the following inequalities:

$$\mathbb{P}\left|\left|I\_1[\boldsymbol{\varrho}\_\*^0]\right|\right| \leq 2\left|\left|\boldsymbol{\varphi}\_s^0\right|\right| \sup\_{|y| \leq d} \left|q\_s[\boldsymbol{\varrho}\_\*^0(y)] - q\_s[\boldsymbol{\varrho}^0]\right|;\tag{4.8.a}$$

$$\left| \mid \left| I\_2[\phi\_\*^0] \right| \mid \right| \le \left| \eta\_s[\phi^0] \right| \left| \int\_{-\infty}^{+\infty} \phi^0(\tau) \sup\_{\mathbf{x}} \left| \psi\_s^0(\mathbf{x} - \tau d) - \psi\_s^0(\mathbf{x}) \right| d\tau \right.\tag{4.8.b}$$

To rework estimate (4.8.a), taking the arguments, known from the analysis of (3.29) (see above), we use at the beginning that

$$|\langle \ | \neq\_s [\varphi\_\*^{0}] - q\_s [\varphi^0] \rangle| \ |\_{d} \leq \varepsilon\_s k\_s^2 (1 + \frac{\varepsilon\_s k\_s^2}{2}) |\ |\ |\varphi\_\*^{0} - \varphi^0 \ |\ |\_{d} \ |$$

where

$$\|\mid \boldsymbol{\varrho}^{0}\_{\ast} - \boldsymbol{\varrho}^{0}\|\mid\_{d} = \sup\_{\|\boldsymbol{y}\| \leq d} \|\boldsymbol{\varrho}^{0}\_{\ast}(\boldsymbol{y}) - \boldsymbol{\varrho}^{0}\|\text{ ; i.e.}$$

$$1 \mid \left| q\_s[\varphi\_\*^0] - q\_s[\varphi^0] \right| \mid \downarrow\_d \le \frac{3k\_s}{2} (1 + \frac{3}{4}k\_s) \mid \left| \varphi\_\*^0 - \varphi^0 \right| \mid \downarrow\_d \le \left| 4 \mid \left| \varphi\_\*^0 - \varphi^0 \right| \mid \downarrow\_d \ldots \right| $$

via (3.26), (3.33). Substituting then in (4.8.a), we have:

$$\left| \mid \left| I\_1[\varphi\_\*^0] \right| \mid \mid \le \ 8 \mid \left| \varphi\_s^0 \right| \mid \mid \left| \mid \varphi\_\*^0 - \varphi^0 \right| \mid \mid\_d \right. \tag{4.9.a}$$

(We shall show afterwards that 0 0 || || 1 *<sup>d</sup>* at small enough *d* .) Next, for reworking of (4.8.b), we shall firstly apply equality

$$
\left\|\boldsymbol{\nu}\_s^0(\boldsymbol{\chi}-\boldsymbol{y})-\boldsymbol{\nu}\_s^0(\boldsymbol{\chi}) = -\boldsymbol{y}\right\|\_0^1 \boldsymbol{\nu}\_s^{0,1}(\boldsymbol{\chi}-\boldsymbol{t}\boldsymbol{y})dt\,.
$$

(Via formula (3.13) for <sup>0</sup> ( ) *<sup>s</sup> x* it is not difficult to verify validity of the above.) Consequently <sup>0</sup> <sup>0</sup> 0,1 sup| ( ) ( )| | |.|| || *ss s x xy x y* , and, because of (4.8.b), we find inequality

$$|\left| \mid I\_2[\varphi^0\_\*] \right| \mid \mid \le \left| \: \eta\_s[\varphi^0] \right| . |\: \mid \: \left| \: \nu^{0,1}\_s \right| \mid \: \left| d \right|\_{-\varphi}^{+\circ} \mid \: \tau \mid o^0(\tau) d\tau \mid \: \mid$$

which actually yields that:

$$<\langle \ | I\_2[\varphi\_\*^0] \vert \ | \ \leq \langle \ \eta\_s[\varphi^0] \vert \ | \cdot \ \vert \ \psi\_s^{0,1} \vert \ \vert \ \vert d \ . \tag{4.9.b}$$

To final reworking of (4.8.a) we shall estimate the quantity 0 0 || || *<sup>d</sup>* in (4.9.a). By (4.4) and the algebraic equation <sup>0</sup> (0) ( [ (0)]) (0) *ls s q* for (0) (see the initial form of (4.2)) we firstly have

$$
\log \boldsymbol{\uprho}\_{s}^{0}(\mathbf{x}) - \boldsymbol{\uprho}^{0} = \frac{\boldsymbol{\Delta} \boldsymbol{\uprho}\_{\boldsymbol{\uprho}}}{2} \text{sg}(\mathbf{x}) + (\boldsymbol{\upbeta}\_{l} + \boldsymbol{q}\_{s}[\boldsymbol{\uprho}^{0}]) [\boldsymbol{\upmu}\_{s}^{0}(\mathbf{x}) - \boldsymbol{\upmu}\_{s}^{0}(\mathbf{0})] + \boldsymbol{\upLambda} \boldsymbol{\uprho}\_{\boldsymbol{\uprho}} [\boldsymbol{\upmu}\_{s}^{\*}(\mathbf{x}) + \boldsymbol{\upomega}\_{s} \boldsymbol{\upmu}\_{s}^{0,1}(\mathbf{x})], \dots, \boldsymbol{\upmu}\_{s}^{0}]
$$

and expressing 0,1 *<sup>s</sup>* by (3.15) we get

$$
\Delta \boldsymbol{\phi}\_{\*}^{0}(\mathbf{x}) - \boldsymbol{\varphi}^{0} = \frac{\Delta \boldsymbol{\uprho}\_{\boldsymbol{\alpha}}}{2} \text{sg}(\mathbf{x}) [1 - \exp(\mathbf{k}\_{s} \, | \, \mathbf{x} \, | \, \mathbf{l} \, )] + (\boldsymbol{\beta}\_{l} + q\_{s} [\boldsymbol{\varphi}^{0} \, ]) [\boldsymbol{\upnu}\_{s}^{0}(\mathbf{x}) - \boldsymbol{\upnu}\_{s}^{0}(\mathbf{0})] + \\
\tag{4.10}
$$

$$
+ \Delta \boldsymbol{\uprho}\_{\boldsymbol{\alpha}} (\boldsymbol{\upmu}\_{s}^{\*}(\mathbf{x}) + \boldsymbol{\upnu}\_{L} [\boldsymbol{\upmu}\_{s}^{0,1}](\mathbf{x})) \,.
$$

Next we analyze the terms in (4.10) beginning with function 0,1 [ ]( ) *W x L s* ; it is continuous and odd, and we can use the following inequalities:

$$\mathbb{P}\left|\mathcal{W}\_{\mathbb{L}}[\boldsymbol{\nu}\_{s}^{0,1}]\right|(\mathbf{x}) \leq \frac{2}{\pi} \int\_{0}^{+\infty} \frac{\xi^{\varepsilon} \left|\sin(\mathbf{x}\xi)\right|}{k\_{s}^{2} + \xi^{2}} \left|\boldsymbol{\hat{\nu}}\_{s}^{0,1}(\xi)\right| d\xi \leq \frac{1}{\pi} \left(\int\_{0}^{+\infty} \frac{\xi^{2} \left|\sin^{2}(\mathbf{x}\xi)\right|}{(k\_{s}^{2} + \xi^{2})^{2}} d\xi\right)^{1/2} . \left|\boldsymbol{\hat{\nu}}\_{s}^{0,1}\right| \left|\boldsymbol{\hat{\nu}}\_{\mathbb{L}\_{2}(\mathbb{R}^{1})}\right| \cdot \frac{\xi^{2} \left|\sin^{2}(\mathbf{x}\xi)\right|}{k\_{s}^{2} + \xi^{2}}\right) d\xi$$

Therefore:

104 Electrostatics

0 0 0 00

 

*d s ss*

 

 

*<sup>k</sup> qq k*

[ ]( ) ( )( [ ( )] [ ]) ( )

0 0 00

*x*


To rework estimate (4.8.a), taking the arguments, known from the analysis of (3.29) (see

00 2 0 0 || [ ] [ ]|| (1 )|| || <sup>2</sup> *s s s s d ss <sup>d</sup>*

> 0 0 0 0 | | || || sup| ( ) | *<sup>d</sup> y d*

0 0 <sup>3</sup> <sup>3</sup> 0 0 0 0 || [ ] [ ]|| (1 )|| || 4|| || 2 4 *<sup>s</sup> s sd s d <sup>d</sup>*

<sup>0</sup> 0 00 || [ ]|| 8|| ||.|| || <sup>1</sup> *s d I*

 

1 0 0 0,1 <sup>0</sup> ( ) () ( ) *ss s*

*xy x y* , and, because of (4.8.b), we find inequality

<sup>0</sup> 0 0,1 <sup>0</sup> || [ ]|| | [ ]|.|| || | | ( ) <sup>2</sup> *s s I*

 

,

*k*

 *y* ; i.e.

 

 

0 00 0 0 || [ ]|| | [ ]| ( )sup| ( ) ( )| <sup>2</sup> *<sup>s</sup> s s*

 

 *q yq* 

*Q x q d q x dd*

, perturbation term <sup>0</sup> [ ]

 

<sup>1</sup>*I x* [ ]( ) 

 

 

 

 

 ,

 *x d xd* . (4.8.b)

2

 

 *x y x y x ty dt* .

*x* it is not difficult to verify validity of the above.) Consequently

 

*d d*

 

; (4.8.a)

*<sup>s</sup> Qd* 

and <sup>0</sup>

 

 

,

. (4.9.a)

*<sup>d</sup>* at small enough *d* .) Next, for reworking of

can be easily

. (4.7)

<sup>2</sup>*I x* [ ]( ) 

; they

*<sup>s</sup> Q q d ds s*

0 00 0

*s ss*

 

*k*

Denote the first and second integrals in (4.7) respectively by <sup>0</sup>

 

[ ] ( )[ ( ) ( )]

*q x d xd*

 

 

 

1

*I q* 

*I* 

By the known definition 0 0 [ ] ( [ ])

*s*

satisfy the following inequalities:

above), we use at the beginning that

*q q* 

 

via (3.26), (3.33). Substituting then in (4.8.a), we have:

( ) *<sup>s</sup>* 

> 

*q*

(We shall show afterwards that 0 0 || || 1

(4.8.b), we shall firstly apply equality

<sup>0</sup> <sup>0</sup> 0,1 sup| ( ) ( )| | |.|| || *ss s*

(Via formula (3.13) for <sup>0</sup>

which actually yields that:

*x*

presented as

where

$$|\mathcal{W}\_{\boldsymbol{L}}[\boldsymbol{\psi}\_{s}^{0,1}]|\left(\boldsymbol{x}\right) \leq \frac{\sqrt{|\boldsymbol{x}|}}{\pi} \left(\int\_{0}^{+\infty} \frac{\sin^{2}\theta}{\theta^{2}} d\theta\right)^{1/2} . |\, |\, \boldsymbol{\hat{\psi}}\_{s}^{0,1}|\, |\_{\boldsymbol{L}\_{2}(\mathbb{R}^{1})}\dots|$$

Recall here the already found estimate 1 2 0,1 ( ) || || <sup>ˆ</sup>*<sup>s</sup> L R s s k* and the well known relation 2 0 2 sin *d* . Then it follows:

$$\|\|\|\mathcal{W}\_{L}[\boldsymbol{\psi}\_{s}^{0,1}]\|\|\_{d} \leq \frac{\sqrt{d}}{\varepsilon\_{s}\sqrt{k\_{s}}}.\tag{4.11}$$

For ( ) *<sup>s</sup> x* we start with the inequality

2

$$|\left|\boldsymbol{\psi}\_{s}^{\star}(\mathbf{x})\right| \leq \frac{|\boldsymbol{\chi}|}{\pi \boldsymbol{\varepsilon}\_{s}} \int\_{0}^{1/d} \left|\frac{\sin(\mathbf{x}\boldsymbol{\xi})}{\mathbf{x}\boldsymbol{\xi}}\right| \frac{\boldsymbol{\xi}}{k\_{s}^{2} + \boldsymbol{\xi}^{2}} d\boldsymbol{\xi} + \frac{1}{\pi \boldsymbol{\varepsilon}\_{s}} \int\_{1/d}^{\infty} \frac{|\sin(\mathbf{x}\boldsymbol{\xi})|}{k\_{s}^{2} + \boldsymbol{\xi}^{2}} d\boldsymbol{\xi} \right| \boldsymbol{\xi}$$

i.e.:

$$|\left|\boldsymbol{\nu}\_{s}^{\*}(\mathbf{x})\right| \leq \frac{|\boldsymbol{\chi}|}{2\pi\varepsilon\_{s}}\log(1+\frac{1}{k\_{s}^{2}d^{2}}) + \frac{1}{\pi\varepsilon\_{s}}\int\_{1/4}^{\infty}\frac{1}{\boldsymbol{\xi}^{2}}d\boldsymbol{\xi} \leq \frac{|\boldsymbol{\chi}|}{\pi\varepsilon\_{s}}\log(1+\frac{1}{k\_{s}d}) + \frac{d}{\pi\varepsilon\_{s}}\cdot\frac{1}{\varepsilon\_{s}}$$

Mathematical Models for Electrostatics of Heterogeneous Media 107

0 0 || [ ]|| 8(4| | 2| | 3| [ ]|) *<sup>s</sup> Qd ls s*

0 0 <sup>8</sup> || (.; ) || (4| | 3| [ ]| 2| |) <sup>1</sup> *s ls <sup>d</sup> <sup>q</sup> dk*

Now let us introduce (for a simplicity sake) the restriction 0.5 0.9 *r* , equivalent to

5 3 6 2 *s s* 

0 0 || (.; ) || 80(2| | 3| [ ]| 4| |) *l s <sup>s</sup>*

explicitly determined by formula (4.4) and satisfying estimate (4.19), for parameters ,

Here we accent on the approximating solutions, in several applicable variants, via the convenience of solution determination by effective formulas (see positions 2) – 5), below). Note, as a principle, that the possible explicit solutions (presenting for instance the interface electric potential) are necessary for examination of relevant numerical methods, and the same holds for the explicit approximations to the exact implicit solutions. Below we start

> 6 *s s*

classes of semiconductors analyzed in [5], [7] the values of parameter <sup>1</sup>

<sup>1</sup> 0.75 2 *nm k nm <sup>s</sup>*

with some dimensional remarks, related in particular to known experimental data.

 


*k nm* , <sup>3</sup> <sup>1</sup>

2 *s s* 

*k* varying in the compact set determined by (3.26), (4.18).

   

 *q dk* . (4.17)

> 

, and, from the above inequality for <sup>0</sup>

 *d q dk* . (4.19)

*k* . (4.18)

(;) *x d* of equation (3.20), at 0 *d* ,

<sup>4</sup> *<sup>s</sup> k nm* and <sup>1</sup> <sup>2</sup> *<sup>s</sup> k nm* (as the bottoms

. (4.20)

*k nm* (as the thighs). For the

(.; ) *d* ,

> *l* ,

*<sup>s</sup> <sup>k</sup>* are not

(;) *x d* ,at 0 *d* :

 

.

 

*r*

1 *r*

By the above arguments we have actually proven the following assertion:

we obtain the needed estimate, approximating for the exact solution

( ) *x* is an approximation to solution

trapezoid, with contours – the straight lines: <sup>1</sup> <sup>3</sup>

greater than 1 nm, satisfying thus condition (3.26) in the form

of trapezoid, vertically situated), and <sup>5</sup> <sup>1</sup>

Applying the above two inequalities to (4.16), we find that

Finally, from (4.6), (4.17) we directly get:

Then 1 0.1 *<sup>r</sup>* , consequently <sup>8</sup> <sup>80</sup>

condition

**4.1 Proposition**  Function <sup>0</sup>

1. In a <sup>1</sup> ( ,) *s s k*

which fulfill (3.30) and (,) *s s*

**5. Concluding remarks** 

Evidently, 1 1 log(1 ) *<sup>s</sup> <sup>s</sup> k d k d* , and we find:

$$<\langle \, \vert \, \vert \, \vert \nu\_s^\* \vert \, \vert\_d \le \frac{1}{\pi \varepsilon\_s} (d + \frac{\sqrt{d}}{\sqrt{k\_s}}) \,. \tag{4.12}$$

The estimate for 0 0 ( ) (0) *s s x* is a consequence from the above one for 0 0 ( ) () *s s x y x* , thus we come to inequality

$$d \mid \left| \boldsymbol{\nu}\_s^0 - \boldsymbol{\nu}\_s^0(\mathbf{0}) \mid \downarrow\_d \le d \mid \left| \boldsymbol{\nu}\_s^{0,1} \right| \mid \downarrow \tag{4.13}$$

and for 1 exp( | |) *<sup>s</sup> k x* we can take the next obviously one:

$$\|\left|\left|1-\exp(k\_s\left|x\right|)\right|\right|\|\_{d} \leq k\_s d \exp(k\_s d) \,. \tag{4.14}$$

Now (4.10) and (4.11) – (4.14) yield:

$$\|\|\|\varphi^0\_\* - \varphi^0\|\|\|\_d \le \|\Lambda \varphi\_w\| (\frac{\sqrt{k\_s d}}{2} \exp(k\_s d) + \frac{4d\sqrt{d}}{9\varepsilon\_s^2 \sqrt{k\_s}}) + (\|\|\beta\_l\| + \|q\_s[\varphi^0]\|) \|\|\varphi^{0,1}\_s\|\|\|d\dots$$

The above simplifies, at 1 *<sup>s</sup> dk* , to inequality

$$\|\left(\lfloor\rho\_\*^0 - \rho^0\rfloor\right)\|\_d \le |\Delta\rho\_\omega| \left(\frac{3}{2} + \frac{20}{9}r\right)\sqrt{dk\_s} + (|\lfloor\beta\_l\rfloor| + |\lfloor\eta\_s[\rho^0]\rfloor|) \frac{3d}{4\varepsilon\_s} (1 + \frac{1}{\varepsilon\_s k\_s}) \vdots$$

i.e.

$$\|\|\|\boldsymbol{\varrho}\_{\star}^{0} - \boldsymbol{\varrho}^{0}\|\|\|\_{d} \leq \left[\mathsf{G} \|\Delta \boldsymbol{\varrho}\_{\boldsymbol{\varrho}}\| + (1 + \frac{4}{3})(\|\, \boldsymbol{\beta}\_{l}\| + \|\, \boldsymbol{q}\_{s}[\boldsymbol{\varrho}^{0}]\|)\right] \boldsymbol{r} \sqrt{d\boldsymbol{k}\_{s}} \; \forall \; l$$

and

$$\|\|\|\boldsymbol{\varrho}^{0}\_{\*} - \boldsymbol{\varrho}^{0}\|\|\|\_{d} \leq \Im(\mathfrak{D}\,\vert \,\Delta \boldsymbol{\varrho}\_{\boldsymbol{\varpi}}\,\vert + \,\|\,\boldsymbol{\beta}\_{\boldsymbol{l}}\,\vert + \,\|\,\boldsymbol{q}\_{s}[\boldsymbol{\varrho}^{0}\,\vert]\,\vert)\sqrt{d\boldsymbol{k}\_{s}}\,\,.\tag{4.15}$$

Going to expression (4.7), we have firstly 0 00 || [ ]|| || [ ]|| || [ ]|| 1 2 *<sup>s</sup> Q II <sup>d</sup>* and applying afterwards (4.9.a), combined with (4.15) and (4.9.b), we establish the estimate:

$$\left| \parallel \left| \Delta Q\_d^s[\varphi\_\*^0] \right| \mid \leq 8 \mid \left| \varphi\_s^0 \right| \mid \left| . \right| \mid \left| \varphi\_\*^0 - \varphi^0 \right| \mid \left| \kern-1. \left| \left. q\_s[\varphi^0] \right| \right| \mid \left| \left. \psi\_s^{0,1} \right| \mid \left| \end{ \right| \right| \;} \right| \; \tag{4.16}$$

Next, for the relevant quantities in (4.16) it can be easily established (via (3.26), (3.33)) as follows:

$$\begin{aligned} \left| \{ \left| \boldsymbol{\nu}\_{s}^{0} \right| \, | \, | \, |\, |\boldsymbol{\rho}\_{s}^{0} - \boldsymbol{\rho}^{0} \, | \, |\_{d} \leq \frac{12}{\mathcal{E}\_{s}k\_{s}} (2 \left| \, \Delta \boldsymbol{\rho}\_{\boldsymbol{x}} \right| + \left| \, \boldsymbol{\rho}\_{l}^{0} \right| + \left| q\_{s}[\boldsymbol{\rho}^{0}] \right|) \sqrt{d\boldsymbol{k}\_{s}} \leq 16 (2 \left| \, \Delta \boldsymbol{\rho}\_{\boldsymbol{x}} \right| + \left| \, \boldsymbol{\rho}\_{l}^{0} \right| + \left| q\_{s}[\boldsymbol{\rho}^{0}] \right|) \sqrt{d\boldsymbol{k}\_{s}}; \\\\ \left| q\_{s}[\boldsymbol{\rho}^{0}] \right| \, | \, | \, |\boldsymbol{\nu}\_{s}^{0,1}| \; | \, d \leq (1 + \frac{4}{3}) \left| q\_{s}[\boldsymbol{\rho}^{0}] \right| \frac{3d}{4 \varepsilon\_{s}} \leq 3 \left| q\_{s}[\boldsymbol{\rho}^{0}] \right| \left| \sqrt{d\boldsymbol{k}\_{s}} \right|. \end{aligned}$$

Applying the above two inequalities to (4.16), we find that

$$|\langle \ | \ | \Delta Q\_d^s[\rho\_\*^0] \ | \ | \le 8(4 \ | \ \Delta \phi\_w \ | \ + 2 \ | \ \beta\_1 \ | \ + 3 \ | \ q\_s[\rho^0] \ | \ ) \sqrt{d\mathbb{k}\_s} \ . \tag{4.17}$$

Finally, from (4.6), (4.17) we directly get:

$$|\ll|\varrho(\cdot;d)-\wp^0\_\*|\ll|\le\frac{8}{1-r}(4\mid\Delta\wp\_\circ\mid+\mathfrak{Z}\mid\eta\_s[\!\wp^0\!\!] \mid+\mathfrak{Z}\mid\mathcal{J}\_l\mid)\sqrt{dk\_s}\ .|$$

Now let us introduce (for a simplicity sake) the restriction 0.5 0.9 *r* , equivalent to condition

$$
\frac{5}{6} \le \varepsilon\_s k\_s \le \frac{3}{2}.\tag{4.18}
$$

Then 1 0.1 *<sup>r</sup>* , consequently <sup>8</sup> <sup>80</sup> 1 *r* , and, from the above inequality for <sup>0</sup> (.; ) *d* , we obtain the needed estimate, approximating for the exact solution (;) *x d* ,at 0 *d* :

$$\mathbb{P}\left(\mid\varphi(.;d)-\varphi^{0}\_{\*}\mid\mid\leq 80(2\mid\beta\_{l}\mid+3\mid\eta\_{s}[\rho^{0}]\mid+4\mid\Delta\varphi\_{\sigma}\mid\}\sqrt{dk\_{s}}\;.\tag{4.19}$$

By the above arguments we have actually proven the following assertion:

#### **4.1 Proposition**

106 Electrostatics

<sup>1</sup> || || ( ) *s d*

0 0 0,1 || (0)|| || || *ss d s*

0 0 0 0,1 2

0 0 3 20 <sup>0</sup> 3 1 || || | |( ) (| | | [ ]|) (1 ) 2 9 <sup>4</sup> *<sup>d</sup> s ls*

0 0 <sup>4</sup> <sup>0</sup> || || [6| | (1 )(| | | [ ]|)] <sup>3</sup> *<sup>d</sup> ls s*

0 0 <sup>0</sup> || || 3(2| | | | | [ ]|) *<sup>d</sup> ls s*

 

Going to expression (4.7), we have firstly 0 00 || [ ]|| || [ ]|| || [ ]|| 1 2

applying afterwards (4.9.a), combined with (4.15) and (4.9.b), we establish the estimate:

 

<sup>0</sup> 0 00 0 0,1 || [ ]|| 8|| ||.|| || | [ ]|.|| || *<sup>s</sup> Qd <sup>s</sup> ds s*

Next, for the relevant quantities in (4.16) it can be easily established (via (3.26), (3.33)) as

0 00 <sup>12</sup> <sup>0</sup> <sup>0</sup> 8|| ||.|| || (2| | | | | [ ]|) 16(2| | | | | [ ]|) ; *<sup>s</sup> <sup>d</sup> ls s ls s*

0 0,1 4 3 <sup>0</sup> <sup>0</sup> | [ ]|.|| || (1 )| [ ]| 3| [ ]| 3 4 *s s s ss*

 *d q q* 

 

 

 

;

*q r dk* ,

 

*d s ls s s s*

.

*k*

 

<sup>4</sup> || || | |( exp( ) ) (| | | [ ]|)|| || <sup>2</sup> <sup>9</sup>

*s s <sup>d</sup> <sup>d</sup> k*

*x* is a consequence from the above one for 0 0

 

. (4.12)

*d* , (4.13)


*k d d d k d <sup>q</sup> <sup>d</sup>*

*<sup>d</sup> r dk q <sup>k</sup>*

 

> 

*<sup>s</sup> Q II <sup>d</sup>*

 

 *dk*

 *q dk* . (4.15)

> 

*q dk q dk <sup>k</sup>*

*s d*

.

*q d* . (4.16)

 

*s ss*

 

and

> 

( ) () *s s*

 *x y x* ,

Evidently, 1 1 log(1 )

The estimate for 0 0

thus we come to inequality

Now (4.10) and (4.11) – (4.14) yield:

i.e.

and

follows:

 

> *q*

*<sup>s</sup> <sup>s</sup> k d k d*

( ) (0) *s s*

 

and for 1 exp( | |) *<sup>s</sup> k x* we can take the next obviously one:

 

The above simplifies, at 1 *<sup>s</sup> dk* , to inequality

*s s*

 

, and we find:

*s*

 

> Function <sup>0</sup> ( ) *x* is an approximation to solution (;) *x d* of equation (3.20), at 0 *d* , explicitly determined by formula (4.4) and satisfying estimate (4.19), for parameters , *l* , which fulfill (3.30) and (,) *s s k* varying in the compact set determined by (3.26), (4.18).

#### **5. Concluding remarks**

Here we accent on the approximating solutions, in several applicable variants, via the convenience of solution determination by effective formulas (see positions 2) – 5), below). Note, as a principle, that the possible explicit solutions (presenting for instance the interface electric potential) are necessary for examination of relevant numerical methods, and the same holds for the explicit approximations to the exact implicit solutions. Below we start with some dimensional remarks, related in particular to known experimental data.

1. In a <sup>1</sup> ( ,) *s s k* - coordinate system, scaled in nanometers, the above mentioned compact is trapezoid, with contours – the straight lines: <sup>1</sup> <sup>3</sup> <sup>4</sup> *<sup>s</sup> k nm* and <sup>1</sup> <sup>2</sup> *<sup>s</sup> k nm* (as the bottoms of trapezoid, vertically situated), and <sup>5</sup> <sup>1</sup> 6 *s s k nm* , <sup>3</sup> <sup>1</sup> 2 *s s k nm* (as the thighs). For the classes of semiconductors analyzed in [5], [7] the values of parameter <sup>1</sup> *<sup>s</sup> <sup>k</sup>* are not greater than 1 nm, satisfying thus condition (3.26) in the form

$$0.75\,\mathrm{nm} \le k\_s^{-1} \le 2\,\mathrm{nm}\,\mathrm{.}\tag{4.20}$$

Mathematical Models for Electrostatics of Heterogeneous Media 109

*t* (at <sup>0</sup> *t t* ) can be neglected in (4.2), and we can take the value of *s l* ,0 *p*

*k z x w xz dzR*

The found formula conveniently shows that the (nonlinear) impact of the vacancy denuded

5. A special variant is presented by the case of weakly charged contour { 0, 0} *z x* ,

 

*w xz x d z*

view point (c.f. [5], [7]) said situation seems to be another open question. Now we can

exp( | | )(1 )sin( ) (,) () , 0 2( )

. Then neglecting term 0 0 ( [ ]) ( ) *ls s*

*t tt*

. Modified potential 0, *w xz* (,) presents the

on the space potential distribution.

The important extension of the Bedeaux – Vlieger formal scheme into the larger one – for decomposing of different dimensional singularities, is due to Prof. B. Radoev (University of Sofia, Bulgaria, Dept. of Physical Chemistry). The author is grateful to Dr. Plamen Georgiev and Dr. Emil Molle (University of Sofia, Bulgaria, Faculty of Biology) for the useful comments on cell biology concepts and the assistance in preparing the

This study was partially supported by grant No DDVU 02/90 of the Bulgarian National

function 0, *w xz* (,) , below, as a next approximation of the exact potential *uxz* (,).

0 2 *s s t* 

*<sup>k</sup>* (using (4.2)). It gives that

). Thus we can consider the

1

) is compatible to the perturbation

. From experimental

( )*t* ) and find

*q x* in the expression

,0

 *k p* , and take

*l s*, with 242 2

 

*s s k*

 

(instead of <sup>0</sup>

*s*

 

*z x*

 

 

*s s*

*k*

 

> 

. (4.26)

 

. (4.24)

 

*s s*

*k*

1 2 *l sss l*

*<sup>l</sup>* , terms with

(with

samples, say ([5], [7]). Then we would have relatively small values of | |

) as an approximation of

. Replace in (4.23)

(via the known estimate <sup>0</sup> <sup>1</sup> (0) <sup>2</sup> *<sup>s</sup>*

122

 

*l sl*

0, <sup>0</sup> 2 2 8 . exp( | | )cos( ) (,) , 2( )

(with 0 1

1

combined with a relatively higher asymptotic surface power

 ( ) *x* , we insert in (4.22) 0,1 ( ) ( ( ) ( )) *s ss* 

0, <sup>0</sup> 2 2

 *s ss* 

2 *t* and <sup>3</sup>

2 2

> for <sup>0</sup>

*s l*

<sup>0</sup> <sup>1</sup> (0) <sup>2</sup> *<sup>s</sup>*

0 ,0 (0) *s s p* 

0 242 2

*s s k*

<sup>1</sup> [ ] <sup>2</sup> *s sss l q kp*

sub-strips { 0, | | } *z dxd* 

0 2 2 exp( | | )cos( ) , <sup>8</sup> 2( )

*s s <sup>k</sup> z x dzR k*

> assume that || | | *<sup>l</sup>*

the following expression:

 

0, ( ,0) ( ) ( ( ) ( )) *wx x x x*

 

Here 0,1

impact of asymptotic power

**6. Acknowledgement** 

illustrations.

Science Foundation.

 

.

,0

 

The key non-dimensional quantity in the surface electrostatics is given by the product *s s k* , and the same holds for the above used *<sup>s</sup> dk* . On the other hand, quantity <sup>1</sup> ( ) *s s k* can be automatically provided with a (preferable) voltage dimension (see also expression (3.24) of *r* ). Recall that such a mechanism has been suggested by the estimate 0 0 <sup>1</sup> || || (0) <sup>2</sup> *s s s s k* of canonical surface potential <sup>0</sup> *<sup>s</sup>* . This allows, for mathematical reworks, to use the product *<sup>s</sup> k r* (in the important factor <sup>2</sup> . *ss ss s k r k kr* ) as nondimensional.

2. The proposed model (2.2) – (2.6), with 1, 0 *b b k* , admits explicit approximations 0 \* ( ) *x* and <sup>0</sup> \* *u xz* (,) ,

$$\begin{aligned} \varphi(\mathbf{x};d) &= \phi^0\_\star(\mathbf{x}) + O(\sqrt{dk\_s} \ ) \ (d \to 0), \ \forall \mathbf{x} \in \mathbb{R}^1 \ ; \\\\ u(\mathbf{x}, \mathbf{z}) &= \mu^0\_\star(\mathbf{x}, \mathbf{z}) + O(\sqrt{dk\_s} \ ) \ (d \to 0), \ \forall (\mathbf{x}, \mathbf{z}) \in \mathbb{R}^2 \ . \end{aligned}$$

They satisfy estimates (4.19) and

$$\sup\_{z} |\lfloor u(.,z) - \mu^0(.,z) \rfloor| \le 80(2\lfloor \beta \rfloor + 3\lfloor q\_s[\rho^0] \rfloor + 4\lfloor \Delta \rho\_w \rfloor) \sqrt{dk\_s} \tag{4.21}$$

In addition, said approximations are determined respectively by formulas (4.4) and

$$u\_\*^{0}(x,z) = \begin{cases} \left| \frac{z}{\pi} \right| \int\_{-\infty}^{+\infty} \frac{\phi\_\*^{0}(t)}{\left(x - t\right)^2 + z^2} dt, \; z \neq 0;\\ \qquad \phi\_\*^{0}(x), \; z = 0 \end{cases} . \tag{4.22}$$

3. In case of relatively small , i.e. | | | | , the term 0,1 ( ( ) ( )) *s ss x x* can be neglected in representation (4.4) and we can use the simplified approximation 0 0 0, ( ) ( [ ]) ( ) *ls s x qx* of (;) *x d* , instead of <sup>0</sup> ( ) *x* . This yields the simpler approximation 0, *u xz* (,) to the space potential *uxz* (,), with

$$\mu\_{0,\*} (\mathbf{x}, z) = \wp\_{\sigma} (\mathbf{x}) + \frac{\beta\_{l,s}}{\pi} \int\_0^{+\infty} \frac{\exp(-|z \mid \xi) \cos(\mathbf{x}\xi)}{2\xi + \varepsilon\_s (k\_s^2 + \xi^2)} d\xi, \; z \in \mathbb{R}^1. \tag{4.23}$$

Above <sup>0</sup> , [ ] *ls l s q* . At 0 *z* formula (4.23) evidently gives 0, 0, *ux x* ( ,0) ( ) . Here it should be specially noted that known real situations (see for instance in [5], [7]) are contained in the case 0 .

4. The case 0 (then 0 and 0 ) covers the experimental models in [5], [7]. Now it seems to be an open question whether the line phase charges can get essentially smaller values than these for the surface zones (called terraces) { 0, | |} *zdx* - after specific annealing of indium-phosphorus semiconductor

108 Electrostatics

The key non-dimensional quantity in the surface electrostatics is given by the product *s s*

automatically provided with a (preferable) voltage dimension (see also expression (3.24) of *r* ). Recall that such a mechanism has been suggested by the estimate

\* ( ; ) ( ) ( ) ( 0), *<sup>s</sup>*

0 0 \* sup|| (., ) (., )|| 80(2| | 3| [ ]| 4| |) *l s <sup>s</sup>*

In addition, said approximations are determined respectively by formulas (4.4) and

0 2 2

 

 , i.e. | | | | 

*u xz x dzR*

should be specially noted that known real situations (see for instance in [5], [7]) are

[7]. Now it seems to be an open question whether the line phase charges can get essentially smaller values than these for the surface zones (called terraces) { 0, | |} *zdx* - after specific annealing of indium-phosphorus semiconductor

 0 and 0 

*u xz x t z*

0, <sup>0</sup> 2 2 exp( | | )cos( ) (,) () , 2( )

*l s*

approximation 0, *u xz* (,) to the space potential *uxz* (,), with

0 1

0 2 \* ( , ) ( , ) ( ) ( 0), ( , ) *u x z u x z O dk d x z R <sup>s</sup>* .

0

*z t dt z*

( ), 0

(;) *x d* , instead of <sup>0</sup>

, 1

*z x*

 

*s s*

*q* . At 0 *z* formula (4.23) evidently gives 0, 0, *ux x* ( ,0) ( )

*k*

*x z*

0

be neglected in representation (4.4) and we can use the simplified approximation


 

*q dk* . (4.21)

*x d x O dk d x R* ;

and the same holds for the above used *<sup>s</sup> dk* . On the other hand, quantity <sup>1</sup> ( ) *s s*

reworks, to use the product *<sup>s</sup> k r* (in the important factor <sup>2</sup> . *ss ss s*

of canonical surface potential <sup>0</sup>

 

2. The proposed model (2.2) – (2.6), with 1, 0 *b b*

*uz u z*

\*

0 0

 

*x qx* of

0 0 <sup>1</sup> || || (0) <sup>2</sup> *s s*

 

dimensional.

0 \* 

( ) *x* and <sup>0</sup>

*s s k*

\* *u xz* (,) ,

They satisfy estimates (4.19) and

3. In case of relatively small

 

Above <sup>0</sup>

contained in the case

4. The case 0 

 

, [ ] *ls l s*

 

> 0 .

(then

0, ( ) ( [ ]) ( ) *ls s*

 

*z*

*k* ,

*k* can be

*k r k kr* ) as non-

*<sup>s</sup>* . This allows, for mathematical

 

. (4.22)

 *x x* can

( ) *x* . This yields the simpler

. Here it

 , the term 0,1 ( ( ) ( )) *s ss* 

 

 

. (4.23)

) covers the experimental models in [5],

*k* , admits explicit approximations

samples, say ([5], [7]). Then we would have relatively small values of | | *<sup>l</sup>* , terms with 2 *t* and <sup>3</sup> *t* (at <sup>0</sup> *t t* ) can be neglected in (4.2), and we can take the value of *s l* ,0 *p* (with 0 ,0 (0) *s s p* ) as an approximation of 0 2 *s s t <sup>k</sup>* (using (4.2)). It gives that 0 242 2 ,0 <sup>1</sup> [ ] <sup>2</sup> *s sss l q kp* . Replace in (4.23) *l s*, with 242 2 ,0 1 2 *l sss l k p* , and take <sup>0</sup> <sup>1</sup> (0) <sup>2</sup> *<sup>s</sup> s s k* (via the known estimate <sup>0</sup> <sup>1</sup> (0) <sup>2</sup> *<sup>s</sup> s s k* ). Thus we can consider the function 0, *w xz* (,) , below, as a next approximation of the exact potential *uxz* (,).

$$w\_{0,\*} (\mathbf{x}, z) = \frac{\rho\_l + 8^{-1} \lambda\_s^2 \rho\_l^2}{\pi} \int\_0^{+\infty} \frac{\exp(-|z \mid \xi) \cos(\mathbf{x} \xi)}{2\xi + \varepsilon\_s (\mathbf{k}\_s^2 + \xi^2)} d\xi, \; z \in \mathbb{R}^1. \tag{4.24}$$

The found formula conveniently shows that the (nonlinear) impact of the vacancy denuded sub-strips { 0, | | } *z dxd* (with 0 1 ) is compatible to the perturbation 2 2 1 0 2 2 exp( | | )cos( ) , <sup>8</sup> 2( ) *s l s s <sup>k</sup> z x dzR k* .

5. A special variant is presented by the case of weakly charged contour { 0, 0} *z x* , combined with a relatively higher asymptotic surface power . From experimental view point (c.f. [5], [7]) said situation seems to be another open question. Now we can assume that || | | *<sup>l</sup>* . Then neglecting term 0 0 ( [ ]) ( ) *ls s q x* in the expression for <sup>0</sup> ( ) *x* , we insert in (4.22) 0,1 ( ) ( ( ) ( )) *s ss t tt* (instead of <sup>0</sup> ( )*t* ) and find the following expression:

$$w\_{0,w}(\mathbf{x},z) = \varphi\_w(\mathbf{x}) - \frac{\Lambda \varphi\_w}{\pi} \int\_0^{+\infty} \frac{\exp(-|z|\,\xi)(1+\varepsilon\_s\xi)\sin(\mathbf{x}\xi)}{2\,\xi+\varepsilon\_s(k\_s^2+\xi^2)}d\xi, \; z \neq 0 \,\,\,\,\tag{4.26}$$

Here 0,1 0, ( ,0) ( ) ( ( ) ( )) *wx x x x s ss* . Modified potential 0, *w xz* (,) presents the impact of asymptotic power on the space potential distribution.

#### **6. Acknowledgement**

The important extension of the Bedeaux – Vlieger formal scheme into the larger one – for decomposing of different dimensional singularities, is due to Prof. B. Radoev (University of Sofia, Bulgaria, Dept. of Physical Chemistry). The author is grateful to Dr. Plamen Georgiev and Dr. Emil Molle (University of Sofia, Bulgaria, Faculty of Biology) for the useful comments on cell biology concepts and the assistance in preparing the illustrations.

This study was partially supported by grant No DDVU 02/90 of the Bulgarian National Science Foundation.

**Part 5** 

**Nanoelectronics** 

#### **7. References**


### **Part 5**

**Nanoelectronics** 

110 Electrostatics

[1] Ashcroft, N.W., Mermin, N.D., Solid States Physics, Holt, Rinehart and Winston – New

[2] Bedeaux, D., Vlieger, J., Optical Properties of Surfaces, Imperial College Press,

[3] Colton, D., Kress, R., Integral Equation Methods in Scattering Theory, John Wiley &

[4] Cook, B., Kazakova, T., Madrid, P., Neal, J., Pauletti, M., Zhao, R., Cell-foreign Particle Interaction, IMA Preprint Series # 2133-3 (Sept. 2006), Univ. of Minnesota. [5] Ebert, Ph., Hun Chen, Heinrich, M., Simon, M., Urban, K., Lagally, M.G., Direct

[7] Heinrich, M., Ebert, Ph., Simon, M., Urban, K., Lagally, M.G., Temperature Dependent

[8] Hörmander, L., The Analysis of Linear Partial Differential Operators, v. I-IV, Springer-

[11] Junqueira, Z., Carneiro, J., Kelly, R.O., Basic Histology, A. ZANCE Medical Book (1995). [12] Landau, L., Lifschitz, S., Lectures on Modern Physics, vol. VIII Electrodynamics of

[13] Radoev, B., Boev, T., Avramov, M. Electrostatics of Heterogeneous Monolayers, Adv. In

[9] Israelishvili, J., Intermolecular and Surface Forces, Academic Press, London (1991). [10] Jackson, J.D., Classical Electrodynamics, John Wiley & Sons, New York, (1962).

Determination of the Interaction between Vacancies on InP(110) Surfaces,

Vacancy Concentrations on InP(110) Surfaces, J. Vac. Sci. Technol. A. 13(3),

**7. References** 

York (1975).

London(2001).

May/Jun. 1995.

Verlag, Berlin (1983).

Sons, New York (1983).

Phys.Rev.Lett., 76, (№ 12), 18 March 1996. [6] Gibbs, J., The Scientific Papers, 1, Dover, New York (1961).

Solids, Nauka (Moskow) 1982 (in Russian).

Colloid and Interface Sci., 114-115 (2005) 93-101.

*1,2,3France* 

**Nanowires: Promising Candidates** 

Dura Julien1,2, Martinie Sébastien2, Munteanu Daniela2, Triozon François1,

The microelectronics activity regroups the study, design, and manufacturing of very small electronic components. These devices are essentially based on interconnected transistors, sort of "switches" which allow controlling the electric current, and are made of semiconductor materials. Depending on the voltage applied to its "gate" electrode, a transistor is in ON state (high current) or OFF state (smallest possible current and low power consumption). Since the invention of the first transistor in 1948, technological progress allowed miniaturizing drastically electronic circuits, and the industry grew fast up to now. For example, the first microprocessor of INTEL (the "4004") contained 2300 transistors while the Pentium 4 in the early 2000's got 55 millions of transistors and the dual core more than 150 millions. To have a clear idea on the fast growing of this industry, in the 60's and 70's, the number of transistors in integrated circuits was doubled every year. Since the 80's, the standard rule is a factor 2 every 18 months. This evolution is more known as the "Moore's law". Of course, such an industry implies several companies. Microelectronics is become very competitive in performances as well as for economical aspects. The price of 1 million of transistor was 75000 € in 1973, while it was of 6 cents in 2000 then 0.5 cent in 2005. The common objective in microelectronics is so to go ahead with the improvement of

To follow this endless race, the well-known concept of downscaling is required consisting in continuously shrinking the geometrical dimensions of the transistor. However, for small device length, the electrostatics of the device is affected, which degrades the control of the electric current. So, to keep the performances under control, the device architectures have evolved by, for example, improving the gate (controlling electrode) or using thin-film transistor. This article focuses on MOSFETs (Metal-Oxide-Semiconductor Field-Effect Transistor) made on silicon, since it is the technology used since decades for microprocessors. The main part of the MOSFET is its semiconducting "channel" coupled to conducting "source" and "drain" regions, and surrounded by one or several gate electrodes

transistor in all aspects (electronic performances and economical).

**1. Introduction** 

Barraud Sylvain1, Niquet Yann-Michel3 and Autran Jean-Luc2

 **Future Nanoelectronic Devices** 

 *Institute for Nanosciences and Cryogenics (INAC), Grenoble,* 

 **for Electrostatic Control in** 

*1CEA-LETI MINATEC and 3CEA-UJF,* 

*2IM2NP-CNRS, UMR CNRS 6242, Marseille,* 

### **Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices**

Dura Julien1,2, Martinie Sébastien2, Munteanu Daniela2, Triozon François1, Barraud Sylvain1, Niquet Yann-Michel3 and Autran Jean-Luc2 *1CEA-LETI MINATEC and 3CEA-UJF, Institute for Nanosciences and Cryogenics (INAC), Grenoble, 2IM2NP-CNRS, UMR CNRS 6242, Marseille, 1,2,3France* 

#### **1. Introduction**

The microelectronics activity regroups the study, design, and manufacturing of very small electronic components. These devices are essentially based on interconnected transistors, sort of "switches" which allow controlling the electric current, and are made of semiconductor materials. Depending on the voltage applied to its "gate" electrode, a transistor is in ON state (high current) or OFF state (smallest possible current and low power consumption). Since the invention of the first transistor in 1948, technological progress allowed miniaturizing drastically electronic circuits, and the industry grew fast up to now. For example, the first microprocessor of INTEL (the "4004") contained 2300 transistors while the Pentium 4 in the early 2000's got 55 millions of transistors and the dual core more than 150 millions. To have a clear idea on the fast growing of this industry, in the 60's and 70's, the number of transistors in integrated circuits was doubled every year. Since the 80's, the standard rule is a factor 2 every 18 months. This evolution is more known as the "Moore's law". Of course, such an industry implies several companies. Microelectronics is become very competitive in performances as well as for economical aspects. The price of 1 million of transistor was 75000 € in 1973, while it was of 6 cents in 2000 then 0.5 cent in 2005. The common objective in microelectronics is so to go ahead with the improvement of transistor in all aspects (electronic performances and economical).

To follow this endless race, the well-known concept of downscaling is required consisting in continuously shrinking the geometrical dimensions of the transistor. However, for small device length, the electrostatics of the device is affected, which degrades the control of the electric current. So, to keep the performances under control, the device architectures have evolved by, for example, improving the gate (controlling electrode) or using thin-film transistor. This article focuses on MOSFETs (Metal-Oxide-Semiconductor Field-Effect Transistor) made on silicon, since it is the technology used since decades for microprocessors. The main part of the MOSFET is its semiconducting "channel" coupled to conducting "source" and "drain" regions, and surrounded by one or several gate electrodes

Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 115

Technology Roadmap of Semiconductor [ITRS], 2009) and regarding the advanced processing technologies, the literature provides a wide range of devices based on nanowires, stacked (Dupre & al., 2008), twin (Hwi Cho & al., 2007) or single Ω-FET nanowires (Tachi & al., 2009). In the following, a complete study of the electrostatics of nanowire MOSFETs is performed including all the ultimate physical phenomena which can occur in future

Standard silicon layers used in microelectronics are crystallographic. Silicon atoms are disposed in a periodical lattice similar to the diamond structure: each atom is tetrahedrally bonded to its four neighbours (see figure 2). The cubic unit cell parameter *a0* equals 5.43 Å, corresponding to an interatomic distance of 2.34 Å. Ideal silicon nanowires are thus periodic along their axis, and the length *L* of their unit cell depends on the crystallographic orientation:

The orientation and diameter of the nanowire determines its electronic structure, from which result its electrical and optical properties. In the following of this work, we will consider

The electronic structure of bulk silicon is expressed by the dispersion relations *En(k)*, which give the energy of an electron wavefunction with wavevector *k* in band *n*. A schematic of low energy electrons in the conduction band is shown in figure 3. It represents the isoenergy surfaces in the first Brillouin zone (wavevector space). For conduction bands, we can count six energy minima, named the six "Δ valleys" of bulk silicon. Each valley is characterized by an effective mass *ml* along its orientation axis and *mt* along its transverse directions. The longitudinal mass is *ml* equal to *0.919m0* and *mt* is equal to 0.196m0 where *m0*

In the following, we consider transport along the x-axis. So, projecting bulk valleys on the nanowire axis (x for the transport, y and z for the perpendicular direction), we can define two different valleys of the nanowire characterized by a conduction and a confinement masses. The valleys 1 and 2 of the figure 3 correspond to the longitudinal valley while the valleys 3, 4, 5 and 6 refer to the transverse valley. The table 1 gives the corresponding masses of the two nanowire valleys. The confinement mass of the transverse valleys is approximated by a

Conduction mass Confinement mass

2. .

*l t*

*m m*

( )

*m m*

*l t*

"cylindrical mass", which preserves cylindrical symmetry in the calculations.

Transverse valley *mt*

Longitudinal valley *ml mt*

Table 1. Definition of the longitudinal and the transverse masses for the <100> oriented

cylindrical nanowires oriented along the <100> axis, as the one represented in figure 2.

electronic devices.

**2. The electronic structure of silicon nanowires** 

 *L=a0* for <100> oriented nanowires, *L=a0 /* 2 for <110> oriented nanowires, *L=a0* 3 for <111> oriented nanowires.

is the free electron mass.

nanowire.

which control the current through the channel. The gate is separated from the channel by a thin insulating oxide. Figure 1 shows transmission electron microscopy (TEM) images of several MOSFET architectures (a) and schematics of these devices and of their potential for channel length reduction (b).

Fig. 1. a) Different MOSFET architectures observed by TEM (Fully-Depleted Silicon on Insulator FDSOI (Barral & al., 2007a), Double Gate (Barral & al., 2007b), finFET (Dupre & al., 2008) and stacked nanowires (Dupre & al., 2008)); b) schematics of the downscaling concept.

The essential parameter used to analyse the electrostatic behaviour (so to compare MOSFET architectures) is the natural length λ (Collinge, 2007). It represents the perturbation induced by the transistor source and drain junctions on the gate control. Numerical simulations establish that a device is relatively free of electrostatic perturbations if λ has a value smaller than 5–10 times the gate length.

$$\mathcal{A} = \sqrt{\frac{\varepsilon\_{si} \, t\_{si} \, t\_{\alpha x}}{N \, \mathcal{E}\_{\alpha x}}} \tag{1}$$

where εsi and εox are the silicon and oxide permittivity, tsi and tox the silicon and oxide thickness and N represents the number of gates of the architecture. Thus, for a given value of silicon thickness and oxide (tsi=10 nm and tox=1.5 nm) the corresponding minimum length for the bulk, thin BOX FDSOI, and nanowire are 20 nm, 15 nm and 10 nm respectively. That is why ITRS recommends nanowires for technology node sub-22nm (International Technology Roadmap of Semiconductor [ITRS], 2009) and regarding the advanced processing technologies, the literature provides a wide range of devices based on nanowires, stacked (Dupre & al., 2008), twin (Hwi Cho & al., 2007) or single Ω-FET nanowires (Tachi & al., 2009). In the following, a complete study of the electrostatics of nanowire MOSFETs is performed including all the ultimate physical phenomena which can occur in future electronic devices.

### **2. The electronic structure of silicon nanowires**

Standard silicon layers used in microelectronics are crystallographic. Silicon atoms are disposed in a periodical lattice similar to the diamond structure: each atom is tetrahedrally bonded to its four neighbours (see figure 2). The cubic unit cell parameter *a0* equals 5.43 Å, corresponding to an interatomic distance of 2.34 Å. Ideal silicon nanowires are thus periodic along their axis, and the length *L* of their unit cell depends on the crystallographic orientation:

*L=a0* for <100> oriented nanowires,

114 Electrostatics

which control the current through the channel. The gate is separated from the channel by a thin insulating oxide. Figure 1 shows transmission electron microscopy (TEM) images of several MOSFET architectures (a) and schematics of these devices and of their potential for

Fig. 1. a) Different MOSFET architectures observed by TEM (Fully-Depleted Silicon on Insulator FDSOI (Barral & al., 2007a), Double Gate (Barral & al., 2007b), finFET (Dupre & al., 2008) and stacked nanowires (Dupre & al., 2008)); b) schematics of the downscaling concept.

The essential parameter used to analyse the electrostatic behaviour (so to compare MOSFET architectures) is the natural length λ (Collinge, 2007). It represents the perturbation induced by the transistor source and drain junctions on the gate control. Numerical simulations establish that a device is relatively free of electrostatic perturbations if λ has a value smaller

> . . . *si si ox ox t t N*

where εsi and εox are the silicon and oxide permittivity, tsi and tox the silicon and oxide thickness and N represents the number of gates of the architecture. Thus, for a given value of silicon thickness and oxide (tsi=10 nm and tox=1.5 nm) the corresponding minimum length for the bulk, thin BOX FDSOI, and nanowire are 20 nm, 15 nm and 10 nm respectively. That is why ITRS recommends nanowires for technology node sub-22nm (International

(1)

channel length reduction (b).

than 5–10 times the gate length.


The orientation and diameter of the nanowire determines its electronic structure, from which result its electrical and optical properties. In the following of this work, we will consider cylindrical nanowires oriented along the <100> axis, as the one represented in figure 2.

The electronic structure of bulk silicon is expressed by the dispersion relations *En(k)*, which give the energy of an electron wavefunction with wavevector *k* in band *n*. A schematic of low energy electrons in the conduction band is shown in figure 3. It represents the isoenergy surfaces in the first Brillouin zone (wavevector space). For conduction bands, we can count six energy minima, named the six "Δ valleys" of bulk silicon. Each valley is characterized by an effective mass *ml* along its orientation axis and *mt* along its transverse directions. The longitudinal mass is *ml* equal to *0.919m0* and *mt* is equal to 0.196m0 where *m0* is the free electron mass.

In the following, we consider transport along the x-axis. So, projecting bulk valleys on the nanowire axis (x for the transport, y and z for the perpendicular direction), we can define two different valleys of the nanowire characterized by a conduction and a confinement masses. The valleys 1 and 2 of the figure 3 correspond to the longitudinal valley while the valleys 3, 4, 5 and 6 refer to the transverse valley. The table 1 gives the corresponding masses of the two nanowire valleys. The confinement mass of the transverse valleys is approximated by a "cylindrical mass", which preserves cylindrical symmetry in the calculations.


Table 1. Definition of the longitudinal and the transverse masses for the <100> oriented nanowire.

Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 117

This effective mass approach is only valid for thick nanowires (diameter > 5 nm) and low electron energy (a few tenths eV). The band structure of silicon is described more accurately by atomistic models, which allow modelling thinner nanowires. A tight-binding model is used here. It consists in developing the wavefunctions on an atomic orbital basis set. The sp3 model developed by (Niquet & al., 2000) is used here and in previous studies (see section 3.4). This model is fitted on first principles calculations based on the density functional theory (DFT) and so-called "GW" corrections for the bandgap. It contains one *s* orbital and three *p* orbitals per silicon atom and describes accurately electron (conduction band) and hole (valence band) dispersion relations. When studying nanostructures, the surface is passivated with hydrogen atoms (see figure 2). This model choice is due to the lack of a convenient tight-binding model for the Si/SiO2 interface. Passivation avoids unrealistic surface states and should not modify much the electronic structure of the nanowire. The tight-binding calculations are performed with the code TB\_Sim (TBSIM, 2011), which solves the Schrödinger equation in nanostructures containing up to 107 atoms (Niquet & al., 2006). For thin nanowires (diameter < 5 nm), the obtained electronic structure differs from the effective mass calculation (see section 3.4). TB\_Sim also allows Poisson-Schrödinger calculations, which give the repartition of the charge density in the nanowire under the influence of the gate voltage. Again, corrections of the effective mass approach are needed

**3. Modelling of the electrostatics in MOSFETs based on nanowires** 

Poisson equation, given here in cylindrical coordinates, is solved:

As said previously, the MOSFET transistor is defined by two states (ON or OFF) depending on the voltage applied at the gate. In fact, this polarization creates an electric field in the active region of the transistor which makes carriers concentrate near the interface with the oxide. Increasing the gate voltage, conduction bands start to fill with carriers from lower energy bands to higher energy bands up to saturation. In this regime, the semiconductor is then analog to a metal and forms a conduction layer between contacts (source and drain) of the transistor. Commonly, the threshold voltage is so defined as the frontier of the two states of the transistor and represents its capacity to switch from one state to the other. It is essentially dependent on the electrostatic characteristics. That is why, it is necessary to fully describe the potential ψ everywhere in the device active region. For this purpose, the

> ² 1. () <sup>²</sup> *<sup>A</sup> Si*

where NA is the channel doping, n is the electron density, εSi the silicon permittivity, and q the elementary charge. Note that *n* depends on via the Fermi-Dirac occupation of

Two approaches can be used to obtain the solution of such an equation. The first one is the double integration solving (Jimenez & al., 2004; Yu & al., 2007); however the solution is not totally analytical which is not convenient in our case. The second approach is to make an assumption on the potential description along the nanowire radius. In the following, we assume a parabolic potential along the radius of the nanowire cross-section described as:

(2)

*d d <sup>q</sup> N n*

 

*dr r dr* 

for thin nanowires.

electronic states.

**3.1 Definition of the threshold voltage** 

Fig. 2. Atomistic representation of a <100>-oriented nanowire. Blue: silicon atoms, grey: hydrogen atoms (necessary to passivate the surface in the tight-binding model), red: highlight of the tetrahedral structure of silicon. The nanowire is 1.5nm thick and 5nm long.

Fig. 3. Iso-energy surfaces of the conduction band of bulk silicon in wavevector space, and definition of longitudinal and transverse valleys for transport along a <100>-oriented nanowire.

This effective mass approach is only valid for thick nanowires (diameter > 5 nm) and low electron energy (a few tenths eV). The band structure of silicon is described more accurately by atomistic models, which allow modelling thinner nanowires. A tight-binding model is used here. It consists in developing the wavefunctions on an atomic orbital basis set. The sp3 model developed by (Niquet & al., 2000) is used here and in previous studies (see section 3.4). This model is fitted on first principles calculations based on the density functional theory (DFT) and so-called "GW" corrections for the bandgap. It contains one *s* orbital and three *p* orbitals per silicon atom and describes accurately electron (conduction band) and hole (valence band) dispersion relations. When studying nanostructures, the surface is passivated with hydrogen atoms (see figure 2). This model choice is due to the lack of a convenient tight-binding model for the Si/SiO2 interface. Passivation avoids unrealistic surface states and should not modify much the electronic structure of the nanowire. The tight-binding calculations are performed with the code TB\_Sim (TBSIM, 2011), which solves the Schrödinger equation in nanostructures containing up to 107 atoms (Niquet & al., 2006). For thin nanowires (diameter < 5 nm), the obtained electronic structure differs from the effective mass calculation (see section 3.4). TB\_Sim also allows Poisson-Schrödinger calculations, which give the repartition of the charge density in the nanowire under the

influence of the gate voltage. Again, corrections of the effective mass approach are needed

#### **3. Modelling of the electrostatics in MOSFETs based on nanowires**

#### **3.1 Definition of the threshold voltage**

for thin nanowires.

116 Electrostatics

Fig. 2. Atomistic representation of a <100>-oriented nanowire. Blue: silicon atoms, grey: hydrogen atoms (necessary to passivate the surface in the tight-binding model), red: highlight of the tetrahedral structure of silicon. The nanowire is 1.5nm thick and 5nm long.

Fig. 3. Iso-energy surfaces of the conduction band of bulk silicon in wavevector space, and definition of longitudinal and transverse valleys for transport along a <100>-oriented

nanowire.

As said previously, the MOSFET transistor is defined by two states (ON or OFF) depending on the voltage applied at the gate. In fact, this polarization creates an electric field in the active region of the transistor which makes carriers concentrate near the interface with the oxide. Increasing the gate voltage, conduction bands start to fill with carriers from lower energy bands to higher energy bands up to saturation. In this regime, the semiconductor is then analog to a metal and forms a conduction layer between contacts (source and drain) of the transistor. Commonly, the threshold voltage is so defined as the frontier of the two states of the transistor and represents its capacity to switch from one state to the other. It is essentially dependent on the electrostatic characteristics. That is why, it is necessary to fully describe the potential ψ everywhere in the device active region. For this purpose, the Poisson equation, given here in cylindrical coordinates, is solved:

$$\frac{d^2\psi}{dr^2} + \frac{1}{r} \cdot \frac{d\psi}{dr} = \frac{q}{\varepsilon\_{Si}}(N\_A + n) \tag{2}$$

where NA is the channel doping, n is the electron density, εSi the silicon permittivity, and q the elementary charge. Note that *n* depends on via the Fermi-Dirac occupation of electronic states.

Two approaches can be used to obtain the solution of such an equation. The first one is the double integration solving (Jimenez & al., 2004; Yu & al., 2007); however the solution is not totally analytical which is not convenient in our case. The second approach is to make an assumption on the potential description along the nanowire radius. In the following, we assume a parabolic potential along the radius of the nanowire cross-section described as:

potential.

Si/SiO2 interface:

2010).

obtained as:

with

intrinsic Fermi level as illustrated in figure 4(b).

Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 119

where Qi,lin is the charge integrated along the nanowire radius and ψ*<sup>s</sup>* is the surface

It is important to note that the parabolic assumption is valid at threshold, but could lose its validity in other operation regimes (for example in the strong inversion regime). Figure 4(a) shows the schematics of the nanowire with the longitudinal polarization (VDS) along the xaxis (transport). Figure 4(b) shows the band diagram along a transverse cutline in the nanowire for an applied gate voltage VGS. The potentials are defined with respect to the

The starting point of the threshold voltage modelling is the boundary condition at the

. *Si GS FB S SF ox*

potential and *Cox* is the oxide capacitance in cylindrical coordinates expressed in (Dura & al.,

In our case, we consider the threshold voltage defined as the gate voltage for which the

, . *ith lin ox kT Q C*

where *k* is the Boltzmann constant and *T* is the temperature. Under this condition, the surface potential reaches its threshold value, called ψ*s,th*. The threshold voltage is then

.

confinement, short channel effect, and band structure effect.

**3.2 Quantum mechanical confinement (QE)** 

, . . *Si th FB s th F ox VV D C* 

> 4 . *A ox Si Si q N kT C*

In equation (9), only the surface potential is unknown and has to be modeled taking into account the physical phenomena specific to nanowire MOSFETs described below: quantum

In silicon nanostructures such as nanowires, the wavefunctions related to the different valleys are modified and kinetic energy is quantized along the confinement directions, leading to a set of energy subbands for each valley. Previous works highlight the necessity to consider quantum confinement in the transport modelling of planar architectures (Munteanu & al., 2005), for which the confinement is one-dimensional. For nanowire

*q D*

 

 

(10)

  (7)

(9)

*<sup>q</sup>* (8)

*<sup>F</sup>* is the Fermi

*C* 

*V V*

where VFB is the flat-band voltage, *ζS* is the electric field at the interface,

inversion charge reaches its threshold value fixed to (Munteanu & al., 2005):

$$
\psi(r, \mathbf{x}) = \beta\_1 + \beta\_2.r + \beta\_3.r^2 \tag{3}
$$

where the parabolic terms βi are x-dependent functions.

To define these terms, the general expression of the potential (eq. 3) is injected in boundary conditions specific to nanowires (eq. 4): the potential at the position r=D/2 is equal to the potential at the position r=–D/2 (symmetry condition) and is defined as the surface potential ψs:

$$\Psi\left(\mathbf{D}\;/\;\mathcal{D},\mathbf{x}\right) = \Psi\left(-\mathbf{D}\;/\;\mathcal{D},\mathbf{x}\right) = \psi\_s(\mathbf{x})\tag{4}$$

Equation (3) becomes:

*D*

Fig. 4. (a) Schematics of a nanowire device (for a better view the gate oxide and material are not shown), indicating the specific area used in the Gauss law. (b) Band diagram along a vertical cut-line in the nanowire.

The term *β3* is found by including (5) into the Poisson equation (2) and integrating along the nanowire radius from 0 to D/2:

$$
\beta = \beta\_3 = \frac{q.N\_A}{4\varepsilon\_{Si}} + \frac{Q\_{i,lin}}{\varepsilon\_{Si}.D} \tag{6}
$$

where Qi,lin is the charge integrated along the nanowire radius and ψ*<sup>s</sup>* is the surface potential.

It is important to note that the parabolic assumption is valid at threshold, but could lose its validity in other operation regimes (for example in the strong inversion regime). Figure 4(a) shows the schematics of the nanowire with the longitudinal polarization (VDS) along the xaxis (transport). Figure 4(b) shows the band diagram along a transverse cutline in the nanowire for an applied gate voltage VGS. The potentials are defined with respect to the intrinsic Fermi level as illustrated in figure 4(b).

The starting point of the threshold voltage modelling is the boundary condition at the Si/SiO2 interface:

$$V\_{GS} - V\_{FB} = \frac{\mathcal{E}\_{Si}}{\mathcal{C}\_{ox}} \mathcal{L}\_S + \mathcal{\nu}\_S + \phi\_{\text{F}} \tag{7}$$

where VFB is the flat-band voltage, *ζS* is the electric field at the interface, *<sup>F</sup>* is the Fermi potential and *Cox* is the oxide capacitance in cylindrical coordinates expressed in (Dura & al., 2010).

In our case, we consider the threshold voltage defined as the gate voltage for which the inversion charge reaches its threshold value fixed to (Munteanu & al., 2005):

$$Q\_{ith,lit} = \frac{kT}{q} \mathcal{L}\_{ox} \tag{8}$$

where *k* is the Boltzmann constant and *T* is the temperature. Under this condition, the surface potential reaches its threshold value, called ψ*s,th*. The threshold voltage is then obtained as:

$$V\_{\rm flt} = V\_{\rm FB} + \frac{\mathcal{E}\_{\rm Si}}{\mathcal{C}\_{\rm ox}} \mathcal{J} \mathcal{D} + \mathcal{\mathcal{W}}\_{s,\rm flt} + \phi\_{\rm f} \tag{9}$$

with

118 Electrostatics

(, ) . .² *rx r r* 

To define these terms, the general expression of the potential (eq. 3) is injected in boundary conditions specific to nanowires (eq. 4): the potential at the position r=D/2 is equal to the potential at the position r=–D/2 (symmetry condition) and is defined as the surface

( /2, ) ( /2, ) ( )

<sup>²</sup> (, ) ( ) . .² <sup>4</sup> *<sup>s</sup>*

**<sup>0</sup> -(D/2+tox) -D/2 D/2 D/2+tox**

Fig. 4. (a) Schematics of a nanowire device (for a better view the gate oxide and material are not shown), indicating the specific area used in the Gauss law. (b) Band diagram along a

The term *β3* is found by including (5) into the Poisson equation (2) and integrating along the

.

4 . *A i lin Si Si q N Q*

 

3

  ,

*D*

(6)

Ψ<sup>s</sup> Ψ

VFB

VGS

vertical cut-line in the nanowire.

nanowire radius from 0 to D/2:

 

 *Dx Dx x* 

where the parabolic terms βi are x-dependent functions.

potential ψs:

Equation (3) becomes:

12 3

 

> 

3 3

ζs

 

**x x+dx**

ζ(x) ζ(x+dx)

*D*

(3)

*<sup>s</sup>* (4)

VDS

**r**

EF

*rx x r* (5)

**x**

Ec

EFi

Ev

$$\beta = \frac{q\_i N\_A}{4\varepsilon\_{Si}} + \frac{kT}{q} \frac{\mathbf{C}\_{ox}}{\varepsilon\_{Si} D} \tag{10}$$

In equation (9), only the surface potential is unknown and has to be modeled taking into account the physical phenomena specific to nanowire MOSFETs described below: quantum confinement, short channel effect, and band structure effect.

#### **3.2 Quantum mechanical confinement (QE)**

In silicon nanostructures such as nanowires, the wavefunctions related to the different valleys are modified and kinetic energy is quantized along the confinement directions, leading to a set of energy subbands for each valley. Previous works highlight the necessity to consider quantum confinement in the transport modelling of planar architectures (Munteanu & al., 2005), for which the confinement is one-dimensional. For nanowire

Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 121

included in the previous analytical model of threshold voltage in order to assess their

, ( ,,) *i lin <sup>c</sup> j i*

<sup>1</sup> ( , , ) 2 ( , ). ( ). *c c D c E*

where the factor 2 accounts for the number of equivalent valleys, *j* is the valley index, *i* is the

*E E f E dE*

1 2

*kT*

( . ( ). ). . . <sup>²</sup> 1

*q*

. \* , .

*<sup>S</sup> q*

*q* 

1 1 2.

1 exp *<sup>F</sup> f E E E*

where *mj* is the 1D density-of-states effective mass in valley *j* and *EF* is the Fermi level. The

*j i lin q Eg j i r E*

*<sup>m</sup> kT r <sup>Q</sup> q gj dr*

Under non-degenerate condition (valid at threshold), the Fermi-Dirac distribution can be approximated by a simple exponential (corresponding to the Boltzmann distribution). The

*kT Q Qe i lin*

( ) \* <sup>2</sup> <sup>1</sup> <sup>2</sup> ( . ( ). ). . . <sup>²</sup>

where Eg is the silicon bandgap, and g(j) is the degeneracy of the valley j (equal to 2 for each valley). The charge is then obtained by a sum over the different silicon valleys (index j) and a sum over all quantized levels (index i) of each valley (in practice limited to 5). The quantum energy levels are needed in (17) and have to be calculated analytically. For a

*<sup>m</sup> kT Q q gj <sup>e</sup>*

*<sup>j</sup>* is the subband bottom energy, *ρ1D(E,Ec)* is the 1D density-of-states of

*c*

0.5

*i j S*

(16)

*i j*

 (17)

0 2

(15)

*e*

*<sup>q</sup> Eg <sup>E</sup> <sup>j</sup> kT*

*kT*

*E E*

1 2

1 2

*j i*

*D c* (, ) . .

<sup>1</sup> ( )

*mj E E*

*nE i j*

the subband, and *f(E)* is the Fermi-Dirac distribution function:

general expression of the charge is then:

,

expression of the charge becomes (Autran & al., 2004):

*Q nE i <sup>j</sup>* (11)

(12)

(13)

(14)

For this purpose, the 1D quantum charge integrated along the radius Qi,lin has to be used:

impact up to circuit performances.

with

with

subband index, *Ec = Ei*

devices, quantum confinement is two-dimensional (leading to a 1D electronic gas) and its impact is expected to be stronger (Autran & al., 2005). Figure 5 shows the wavefunction in the first five subbands for the longitudinal valley and for two different nanowire diameters (5 and 10nm). These wavefunctions are calculated using an effective mass Schrödinger-Poisson solver (TBSIM, 2011). The associated energy represents the difference between each energy subband and the first subband level.

Fig. 5. Square modulus of the wavefunction for the five first levels in the longitudinal valley of a 5 and 10nm nanowire diameter. Energy increase of each subband with respect to the first level.

To extract numerically the impact of the diameter on the quantization of carriers, the same calculation has been done for the tranverse valley and for different nanowire diameters. We can note that for low diameters, the quantized levels are higher. So, regarding the targeted MOSFET downscaling to a few nanometers, quantum-mechanical effects have to be included in the previous analytical model of threshold voltage in order to assess their impact up to circuit performances.

For this purpose, the 1D quantum charge integrated along the radius Qi,lin has to be used:

$$Q\_{i,lin} = \sum\_{j} \sum\_{i} n(E\_{c'} i, j) \tag{11}$$

with

120 Electrostatics

devices, quantum confinement is two-dimensional (leading to a 1D electronic gas) and its impact is expected to be stronger (Autran & al., 2005). Figure 5 shows the wavefunction in the first five subbands for the longitudinal valley and for two different nanowire diameters (5 and 10nm). These wavefunctions are calculated using an effective mass Schrödinger-Poisson solver (TBSIM, 2011). The associated energy represents the difference between each

Fig. 5. Square modulus of the wavefunction for the five first levels in the longitudinal valley of a 5 and 10nm nanowire diameter. Energy increase of each subband with respect to the

To extract numerically the impact of the diameter on the quantization of carriers, the same calculation has been done for the tranverse valley and for different nanowire diameters. We can note that for low diameters, the quantized levels are higher. So, regarding the targeted MOSFET downscaling to a few nanometers, quantum-mechanical effects have to be

energy subband and the first subband level.

first level.

$$m(E\_c, i, j) = 2 \int\_{E\_c}^{+\infty} \rho\_{1D}(E, E\_c) f(E) \, dE \tag{12}$$

where the factor 2 accounts for the number of equivalent valleys, *j* is the valley index, *i* is the subband index, *Ec = Ei <sup>j</sup>* is the subband bottom energy, *ρ1D(E,Ec)* is the 1D density-of-states of the subband, and *f(E)* is the Fermi-Dirac distribution function:

$$\rho\_{1D}(E, E\_c) = \frac{1}{\pi} \cdot \left(\frac{2.m j}{\hbar^2}\right)^{\frac{1}{2}} \cdot \frac{1}{\sqrt{E - E\_c}}\tag{13}$$

$$f(E) = \frac{1}{1 + \exp\left(\frac{E - E\_F}{kT}\right)}\tag{14}$$

where *mj* is the 1D density-of-states effective mass in valley *j* and *EF* is the Fermi level. The general expression of the charge is then:

$$Q\_{i,lin} = q \sum\_{j} \sum\_{i} (\frac{1}{\pi} \cdot g(j) \cdot \sqrt{\frac{2m\_j}{\hbar^2}}) \cdot \sqrt{\frac{kT}{q}} \cdot \int\_0^v \frac{r^{-0.5}}{1 + e^{-\frac{q}{kT} \left(E\_j^i - \frac{E\_\theta}{2} - \mu\_\theta\right)}} dr \tag{15}$$

Under non-degenerate condition (valid at threshold), the Fermi-Dirac distribution can be approximated by a simple exponential (corresponding to the Boltzmann distribution). The expression of the charge becomes (Autran & al., 2004):

$$Q\_{i,lin} \approx Q^\* e^{\frac{q\_s y\_s}{kT}} \tag{16}$$

with

$$\mathbf{Q}^\* = q \sum\_{j} \sum\_{i} (\frac{1}{\pi} \cdot \mathbf{g}(j) \cdot \sqrt{\frac{2m\_j}{\hbar^2}}) \cdot \sqrt{\frac{kT}{q}} \cdot \sqrt{\pi} \, e^{-\frac{q}{kT}(E\_j^i + \frac{E\_\theta}{2})} \tag{17}$$

where Eg is the silicon bandgap, and g(j) is the degeneracy of the valley j (equal to 2 for each valley). The charge is then obtained by a sum over the different silicon valleys (index j) and a sum over all quantized levels (index i) of each valley (in practice limited to 5). The quantum energy levels are needed in (17) and have to be calculated analytically. For a

Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 123

diameter is reduced. Figure 6 also shows a very good agreement between the analytical model and data obtained by a Schrödinger-Poisson numerical solving (ATLAS, 2010).

As said in the introduction, the downscaling of transistor is required. That is why the gate length is continuously reduced. However, from a certain dimension, the transistor junctions (source and drain) have an impact on the electrostatic control of the device. Previously 1D (only the gate voltage), it becomes 2D because gate and drain polarizations compete to control the device. And this strongly affects the device characteristics. Thus, MOSFET architectures are considered to be impacted by the short channel effect when the channel length is the same order of magnitude as the depletion-layer widths of the source and drain junction. The main result is the modification of the threshold voltage (or the loss of electrostatic control) due to the shortening of the channel length. It is attributed to two phenomena: SCE (Short Channel Effect) and DIBL (Drain Induced Barrier Lowering). The first one is coming from the superposition of the depletion-layer widths of the source and drain junction. The second phenomenon is a secondary effect on the charge sharing due to higher drain voltage. Nanowire transistors being expected for ultimate technology node, consideration of short channel effect is required in a realistic modeling of this architecture. To fully describe 2D electrostatic effects (SCE and DIBL) in short channel devices, we propose a full analytical model describing the threshold voltage impacted by SCE and DIBL. The x-dependence (transport direction) of the surface potential has to be know. Applying the Gauss law on a nanowire slice as illustrated in figure 4(a), the following equation is

2 2 <sup>2</sup> . .. ( ). . ( ). . ( ). . . <sup>4</sup> <sup>4</sup> 4.

( ) () . *<sup>S</sup> d x*

η is a fitting parameter which models the lateral electric field variation (Banna & al., 1995). It depends on the channel doping, the channel length, the nanowire diameter and the polarization. An empirical formula for η, obtained from numerical simulations, will be presented below to include these both effect. Introducing (21) in (20), we find a second order

<sup>²</sup> <sup>2</sup> 4 . [. . 2 ( )] ² .. .. 2

*s ox <sup>s</sup> A ox GS FB F*

. . 3 1 2 . . <sup>²</sup>

 

*x x*

*<sup>K</sup> Ke Ke*

   

 

*x*

*D D qN D x x dx x D dx*

*S*

*dx* 

 

(20)

*d C <sup>D</sup> qN C V V dx D D* (22)

*A*

(21)

*Si*

(23)

**3.3 Short-Channel Effect (SCE)** 

obtained (Munteanu & al., 2005):

equation for the surface potential:

where ζ is the electric field expressed by:

 

The solution of this equation is given by (Munteanu & al., 2005):

*Si Si*

*s*

 

> 4. . .

 

*C D*

*ox Si*

cylindrical cross-section of the nanowire, the analytical expressions of the transversal (index t) and longitudinal (index l) quantum energy levels are given by (Baccarani, 2008):

$$\begin{aligned} E\_t^i = E\_{1,2}^i &= \frac{(\alpha \,\,\hbar \,\mathrm{i})^2}{4.q.(\mathrm{D} \,\, / \,\mathrm{2})^2} \cdot (\frac{1}{m\_t} + \frac{1}{m\_l})\\ E\_l^i = E\_3^i &= \frac{(\alpha \,\,\hbar \,\mathrm{i})^2}{2.q.(\mathrm{D} \,\, / \,\mathrm{2})^2} \cdot \frac{1}{m\_t} \end{aligned} \tag{18}$$

where α is a numerical parameter (Baccarani, 2008). We can note a good agreement between (18) with α=2.1 and a self-consistent cylindrical 1D Schrödinger-Poisson solver (ATLAS, 2010) for the first energy level (transversal and longitudinal) (Dura & al., 2010).

From equation (16), the surface potential at threshold voltage is given by:

$$\mathcal{W}\_{s, \text{fl}} = \frac{kT}{q} . \ln(\frac{Q\_{i\text{fl}, \text{lim}}}{Q^\*}) \tag{19}$$

with Qith,lin the inversion charge at threshold defined by (8).

Fig. 6. Threshold voltage shift between quantum (Vth,q) and classical (Vth,cl) approaches versus nanowire diameter in long channel transistors. Comparison between the analytical model and data obtained from Schrödinger-Poisson numerical solving (ATLAS, 2010).

Including this expression in (9), the quantum threshold voltage for long channel transistor is easily obtained. Figure 6 plots the difference between quantum and classical threshold voltage versus the nanowire diameter; as expected, this difference increases when reducing the nanowire diameter, due to a stronger quantization of carrier energy when the nanowire diameter is reduced. Figure 6 also shows a very good agreement between the analytical model and data obtained by a Schrödinger-Poisson numerical solving (ATLAS, 2010).

#### **3.3 Short-Channel Effect (SCE)**

122 Electrostatics

cylindrical cross-section of the nanowire, the analytical expressions of the transversal (index

( . . )² 1 . 2. .( /2)²

where α is a numerical parameter (Baccarani, 2008). We can note a good agreement between (18) with α=2.1 and a self-consistent cylindrical 1D Schrödinger-Poisson solver (ATLAS,

, \* .ln( ) *ith lin*

*kT Q q Q*

*qD m*

( . . )² 1 1 .( ) 4. .( / 2)²

*q D mm*

*t*

,

**Analytical model Numerical simulation**

**0 5 10 15 20**

Diameter D (nm)

Fig. 6. Threshold voltage shift between quantum (Vth,q) and classical (Vth,cl) approaches versus nanowire diameter in long channel transistors. Comparison between the analytical model and data obtained from Schrödinger-Poisson numerical solving (ATLAS, 2010).

Including this expression in (9), the quantum threshold voltage for long channel transistor is easily obtained. Figure 6 plots the difference between quantum and classical threshold voltage versus the nanowire diameter; as expected, this difference increases when reducing the nanowire diameter, due to a stronger quantization of carrier energy when the nanowire

*t l*

(18)

(19)

t) and longitudinal (index l) quantum energy levels are given by (Baccarani, 2008):

1,2

*<sup>i</sup> E E*

*<sup>i</sup> E E*

2010) for the first energy level (transversal and longitudinal) (Dura & al., 2010).

*s th*

*i i t*

*i i l*

3

From equation (16), the surface potential at threshold voltage is given by:

with Qith,lin the inversion charge at threshold defined by (8).

**0**

**0.1**

**0.2**

V


th,cl (V)

th,q

**0.3**

**0.4**

**0.5**

**0.6**

**0.7**

As said in the introduction, the downscaling of transistor is required. That is why the gate length is continuously reduced. However, from a certain dimension, the transistor junctions (source and drain) have an impact on the electrostatic control of the device. Previously 1D (only the gate voltage), it becomes 2D because gate and drain polarizations compete to control the device. And this strongly affects the device characteristics. Thus, MOSFET architectures are considered to be impacted by the short channel effect when the channel length is the same order of magnitude as the depletion-layer widths of the source and drain junction. The main result is the modification of the threshold voltage (or the loss of electrostatic control) due to the shortening of the channel length. It is attributed to two phenomena: SCE (Short Channel Effect) and DIBL (Drain Induced Barrier Lowering). The first one is coming from the superposition of the depletion-layer widths of the source and drain junction. The second phenomenon is a secondary effect on the charge sharing due to higher drain voltage. Nanowire transistors being expected for ultimate technology node, consideration of short channel effect is required in a realistic modeling of this architecture.

To fully describe 2D electrostatic effects (SCE and DIBL) in short channel devices, we propose a full analytical model describing the threshold voltage impacted by SCE and DIBL. The x-dependence (transport direction) of the surface potential has to be know. Applying the Gauss law on a nanowire slice as illustrated in figure 4(a), the following equation is obtained (Munteanu & al., 2005):

$$-\zeta(\mathbf{x}).\pi.\frac{D^2}{4} + \zeta(\mathbf{x} + d\mathbf{x}).\pi.\frac{D^2}{4} - \zeta\_S(\mathbf{x}).\pi.D.d\mathbf{x} = -\frac{q.N\_A.\pi.D^2}{4.\mathcal{E}\_{Si}}\tag{20}$$

where ζ is the electric field expressed by:

$$
\zeta'(\mathbf{x}) = -\eta. \frac{d\boldsymbol{\upmu}\_S(\mathbf{x})}{d\mathbf{x}} \tag{21}
$$

η is a fitting parameter which models the lateral electric field variation (Banna & al., 1995). It depends on the channel doping, the channel length, the nanowire diameter and the polarization. An empirical formula for η, obtained from numerical simulations, will be presented below to include these both effect. Introducing (21) in (20), we find a second order equation for the surface potential:

$$4\frac{d^2\nu\_s}{d\mathbf{x}^2} - 4\frac{\mathbb{C}\_{\rm ox}}{\eta\_r \varepsilon\_{\rm Si} D} \cdot \nu\_s = \frac{2}{\eta\_r \varepsilon\_{\rm Si} D} [\boldsymbol{\eta} \cdot \boldsymbol{N}\_A \cdot \frac{D}{2} - 2\mathbb{C}\_{\rm ox} (V\_{\rm GS} - V\_{\rm FB} - \phi\_{\rm F})] \tag{22}$$

The solution of this equation is given by (Munteanu & al., 2005):

$$\begin{aligned} \mathcal{W}\_s &= K\_1.e^{\mathcal{V}^{.\infty}} + K\_2.e^{-\mathcal{V}^{.\infty}} - \frac{K\_3}{\mathcal{V}^2} \\ \mathcal{V} &= \sqrt{\frac{4.C\_{ox}}{\eta\_r \mathcal{E}\_{Si} \mathcal{D}}} \end{aligned} \tag{23}$$

Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 125

At threshold, VGS=Vth in (26) and, considering (33), K3 will depend on ΔVth. Moreover K1 and K2 depend on K3, they will also depend on ΔVth. Then, developing (33) leads to a second

. 2.[1 . ] 2.[1 . ].[2. ] 2. .[1 . ].[ ]

*C C*

*C H ch L H V V*

4 4

4

*C DS DS*

*C DS*

<sup>2</sup> 4. .

 

*b th long FB F*

2. *th B B AC <sup>V</sup> A*

We can note that to find this term, we have to take into account the long-channel threshold voltage (see coefficient D in eq.35) which includes the dependence on the quantum confinement. Consequently, this model includes both the impact of quantum confinement on the long channel and the threshold voltage roll-off. The most common model (such as references (Banna & al., 1995; Suzuki & al., 1996)) does not include the effect of quantum confinement on the evolution of the short channel effect in analytical modeling which becomes dominant in nanoscale device such as nanowire (this aspect will be detailed later in

**0 5 10 15 20 25 30 35 40**

Channel Length LC

Fig. 7. Threshold voltage roll-off versus channel length for low (VDS=50mV) and high (0.7V) drain voltage obtained by the analytical model and TCAD simulations for a 5 nm nanowire

**Analytical model Numerical simulation**

Diameter 5nm

(nm)

2

*A sh L ch L B ch L H V*

,

*DV V V*

Finally, the threshold voltage roll-off is the solution of (34) given by:

order equation of ΔVth as:

with

paragraph 4).

diameter; tox=1 nm.

**-0.5**

**-0.4**

VDS=0.5V

**-0.3**

**-0.2**

Threshold voltage roll-off (V)

**-0.1**

**0**

VDS=50mV

**0.1**

VVV th th th,lon g (33)

<sup>2</sup> .. 0 *AV BV C th th* (34)

2

(36)

(35)

where K1, K2 and K3 are functions resulting from the Poisson equation solving:

$$K\_1 = \frac{(1 - e^{-\gamma \cdot L\_c}) \cdot (V\_b + \bigvee\_{\gamma^2}^{K\_3}) + V\_{DS}}{2 \operatorname{sh}(\gamma \cdot L\_c)}\tag{24}$$

$$K\_2 = -\frac{(1 - e^{+\gamma \cdot L\_c}) \cdot (V\_b + \bigvee\_{\mathcal{V}}^2) + V\_{DS}}{2 \operatorname{sh}(\mathcal{V} \cdot L\_c)}\tag{25}$$

$$K\_3 = \frac{2}{\eta\_\* \varepsilon\_{Si} D} [\eta\_\* N\_A \cdot \frac{D}{2} - 2C\_{ox} (V\_{GS} - V\_{FB} - \phi\_F)] \tag{26}$$

where Vb is the built-in potential depending on the channel doping NA, the source/drain doping NSD and the intrinsic carrier density ni as:

$$V\_{\phantom{a}b} = \frac{kT}{q} . \ln\left(\frac{N\_A N\_{SD}}{n\_i^2}\right) \tag{27}$$

The position where the surface potential is minimum xmin and the value of ψS (ψs,min) at the position xmin are obtained by forcing the first derivative of equation (23) to be equal to zero (Munteanu & al., 2005):

$$\mathbf{x}\_{\text{min}} = \frac{1}{2\mathcal{Z}}.\text{In} \left| \frac{K\_2}{K\_1} \right|\tag{28}$$

$$\mathcal{\mathcal{W}}\_s = \mathcal{\mathcal{W}}\_{s,\text{min}} = 2\sqrt{\mathcal{K}\_1 \mathcal{K}\_2} - \begin{cases} \mathcal{K}\_3 \\ \mathcal{Y}^2 \end{cases} \tag{29}$$

We assume that the transistor switches on when ψs,min= ψs,th. By inserting this expression in the general expression of the threshold voltage (9), we obtain:

$$V\_{\rm th} = V\_{\rm FB} + \frac{\varepsilon\_{\rm Si}}{C\_{\rm ox}} \beta D + \phi\_{\rm F} - \frac{K\_3}{\gamma^2} \Big/ \frac{1}{\gamma^2} + 2\sqrt{K\_1 K\_2} \tag{30}$$

We can, by analogy to (Suzuki & al., 1996), distinguish two different terms. The first one (independent from the channel length) refers to the long-channel threshold voltage Vth,long; the second term, which tends to zero for long channel length, represents the threshold voltage roll-off and describes the impact of SCE/DIBL:

$$V\_{\rm fb,long} = V\_{\rm FB} + \frac{\mathcal{E}\_{\rm Si}}{\mathcal{C}\_{\rm ox}} \mathcal{J} \mathcal{D} + \phi\_{\rm F} - \begin{aligned} &K\_3 \\ &\gamma^2 \end{aligned} \tag{31}$$

$$
\Delta V\_{\text{fl}} = -2\sqrt{K\_1 K\_2} \tag{32}
$$

where the variation of the threshold voltage is defined as:

$$
\Delta \mathbf{V}\_{\rm th} = \mathbf{V}\_{\rm th} - \mathbf{V}\_{\rm th, long} \tag{33}
$$

At threshold, VGS=Vth in (26) and, considering (33), K3 will depend on ΔVth. Moreover K1 and K2 depend on K3, they will also depend on ΔVth. Then, developing (33) leads to a second order equation of ΔVth as:

$$A\Delta V\_{\text{fl}}^2 + B\Delta V\_{\text{fl}} + C = 0\tag{34}$$

with

124 Electrostatics

. 3

2. ( . )

*sh L*

. 3

2. ( . )

<sup>2</sup> [. . 2 ( )] .. 2

*<sup>A</sup> ox GS FB F*

where Vb is the built-in potential depending on the channel doping NA, the source/drain

The position where the surface potential is minimum xmin and the value of ψS (ψs,min) at the position xmin are obtained by forcing the first derivative of equation (23) to be equal to zero

,min 1 2 <sup>2</sup> 2 . *s s*

We assume that the transistor switches on when ψs,min= ψs,th. By inserting this expression in

 

We can, by analogy to (Suzuki & al., 1996), distinguish two different terms. The first one (independent from the channel length) refers to the long-channel threshold voltage Vth,long; the second term, which tends to zero for long channel length, represents the threshold

> <sup>3</sup> , <sup>2</sup> . .

 *Si th long FB F ox <sup>K</sup> VV D*

*<sup>K</sup> VV D K K*

*Si*

*K K*

*kT N N <sup>V</sup> q n* 

*sh L*

*<sup>K</sup> eV V*

(1 ).( )

*Lc*

*<sup>K</sup> eV V*

*c*

*c*

2 . .ln *<sup>A</sup> SD*

*i*

2

1 <sup>1</sup> .ln( ) 2. *K*

3 <sup>2</sup> 1 2 . . 2 .

 

*C* (31)

1 2 2 . *V KK th* (32)

3

*K*

(1 ).( )

*Lc*

2

2

*b DS*

(24)

(25)

*<sup>K</sup>* (28)

(29)

*C* (30)

(27)

*<sup>D</sup> K qN C V V D* (26)

*b DS*

where K1, K2 and K3 are functions resulting from the Poisson equation solving:

1

*K*

2

 

*Si*

*b*

min

*x*

 

*th FB F ox*

 

the general expression of the threshold voltage (9), we obtain:

voltage roll-off and describes the impact of SCE/DIBL:

where the variation of the threshold voltage is defined as:

*K*

3

doping NSD and the intrinsic carrier density ni as:

(Munteanu & al., 2005):

$$\begin{cases} A = sh\left(\boldsymbol{\gamma} \, \boldsymbol{L}\_{\mathbb{C}}\right)^{2} + 2. \left[1 - ch\left(\boldsymbol{\gamma} \, \boldsymbol{L}\_{\mathbb{C}}\right)\right] \\ B = 2. \left[1 - ch\left(\boldsymbol{\gamma} \, \boldsymbol{L}\_{\mathbb{C}}\right)\right] \left[2. \boldsymbol{H}\_{4} + \boldsymbol{V}\_{DS}\right] \\ C = 2. \boldsymbol{H}\_{4} \, \left[1 - ch\left(\boldsymbol{\gamma} \, \boldsymbol{L}\_{\mathbb{C}}\right)\right] \left[\boldsymbol{H}\_{4} + \boldsymbol{V}\_{DS}\right] + \boldsymbol{V}\_{DS}^{2} \\ D = \boldsymbol{V}\_{b} - \boldsymbol{V}\_{\text{fl},\text{long}} + \boldsymbol{V}\_{FB} + \boldsymbol{\phi}\_{\text{F}} \end{cases} \tag{35}$$

Finally, the threshold voltage roll-off is the solution of (34) given by:

$$
\Delta V\_{\rm th} = \frac{-B + \sqrt{B^2 - 4.A.C}}{2.A} \tag{36}
$$

We can note that to find this term, we have to take into account the long-channel threshold voltage (see coefficient D in eq.35) which includes the dependence on the quantum confinement. Consequently, this model includes both the impact of quantum confinement on the long channel and the threshold voltage roll-off. The most common model (such as references (Banna & al., 1995; Suzuki & al., 1996)) does not include the effect of quantum confinement on the evolution of the short channel effect in analytical modeling which becomes dominant in nanoscale device such as nanowire (this aspect will be detailed later in paragraph 4).

Fig. 7. Threshold voltage roll-off versus channel length for low (VDS=50mV) and high (0.7V) drain voltage obtained by the analytical model and TCAD simulations for a 5 nm nanowire diameter; tox=1 nm.

**1.5**

**-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0**

**0.9 0.95 1 1.05 1.1 1.15 1.2 1.25**

Relative longitudianal mass

down: Valence band.

**Energy (eV)**

**0 0.1 0.2 0.3 0.4**

**0 0.1 0.2 0.3 0.4**

**k (x 2/L)**

**Model**

**Numerical Simulation Analytical model Numerical simulation**

**0 5 10 15 20**

obtained in (Sarrazin & al., 2009).

Diameter D (nm)

**2**

**Energy (eV)**

**2.5**

Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 127

**1.1 1.15 1.2 1.25 1.3 1.35 1.4**

**0**

**-0.25**

**-0.2**

**-0.15**

**Energy (eV)**

Fig. 8. Silicon nanowire band structure obtained with a tight-binding Schrödinger-Poisson solver (TBSIM, 2011) for two different diameters (2 and 10 nm). Up: Conduction band;

**0.15**

Fig. 9. Variation of the silicon band gap (a) and relative longitudinal and transversal masses (b) with respect to the silicon nanowire diameter. Comparison with atomistic simulations

**0.2**

**0.25**

Relative transversal mass

Band gap (eV)

**1**

**1.5**

**2**

**2.5**

**3**

**3.5**

**4**

**0.3**

**0.35**

**0.4**

**-0.1**

**-0.05**

**Energy (eV)**

conduction

D=2nm D=10nm

valence

**0 0.1 0.2 0.3 0.4**

**0 0.1 0.2 0.3 0.4**

**k (x 2/L)**

**Analytical model Numerical simulation**

**0 5 10 15 20**

Diameter D (nm)

The threshold voltage roll-off is represented in figure 7 for nanowire diameters of 5 nm (calculated in the classical case, i.e. without quantum confinement). TCAD numerical simulations have been done for a cylindrical structure using a drift-diffusion model in order to obtain drain current characteristics as a function of the gate voltage. The threshold voltage is extracted from these current-voltage characteristics using the classical constant current method. The threshold voltage data extracted from TCAD simulations for different diameters have been used to derive an empirical expression of the parameter *η* including the dependence on the channel length, nanowire diameter and drain to source voltage:

$$\eta = \frac{D}{f\_0 + D} + V\_{DS} \cdot \left[ f\_1 L\_{\odot} + f\_2 L\_{\odot}{}^2 \right] \tag{37}$$

where f0, f1 and f2 are constant fitting parameters, calibrated on numerical simulations. Equation (37) is valid for a wide range of nanowire diameter (down to 2 nm) and channel lengths (down to channel length equal to the nanowire diameter). The results in figure 7 show a good agreement between the analytical model and threshold voltage data obtained from TCAD simulations.

#### **3.4 Band Structure Effect (BSE)**

Advanced atomistic numerical simulations (Neophytou & al., 2008; Niquet & al., 2000, 2006; Sarrazin & al., 2009; Nehari & al., 2006) have shown that a strong reduction of the silicon thickness impacts the material properties by modifying the band structure. Indeed, the dimensions targeted in ultra-scaled devices are those of a few tens atomic layers (several nanometers). At these dimensions, the electronic properties differ from the calculations shown in section 3.2 and based on the bulk effective masses. In (Sarrazin & al., 2009), atomistic tight-binding (TB) Schrödinger-Poisson simulations have been performed for the case of [001] oriented silicon nanowire in order to highlight the variation of the band structure with the nanowire diameter. The code TB\_Sim (TBSIM, 2011) has been used with a sp3 tight-binding model (Niquet & al., 2000). Figure 8 shows the valence and the conduction bands for Si nanowire width of 2nm and 10nm. We can note that when thinning the silicon film the minimum of the conduction band is increased and the general shape of bands becomes smoother (Sarrazin & al., 2009). However, the bandgap increase is smaller than the effective mass result of section 2.4.

In order to include these modifications in the previous threshold voltage modelling, analytical expressions of parameters affected by the band structure effect (band gap and effective masses) are proposed here. Diameter-dependent analytical functions (fitted on numerical simulations as illustrated in figure 9) are found for the bandgap and effective masses (inspired from (Niquet & al., 2000):

$$E\_{\mathcal{g}} = E\_{\mathcal{g},bulk} + \frac{K\_1}{D^2 + A\_1 D + B\_1} \tag{37}$$

$$m\_{t\{l\}} = m\_{t\{l\},bulk} + \frac{K\_2}{D^2 + A\_2, D + B\_2} \tag{38}$$

where A, B and K are fitting constants.

126 Electrostatics

The threshold voltage roll-off is represented in figure 7 for nanowire diameters of 5 nm (calculated in the classical case, i.e. without quantum confinement). TCAD numerical simulations have been done for a cylindrical structure using a drift-diffusion model in order to obtain drain current characteristics as a function of the gate voltage. The threshold voltage is extracted from these current-voltage characteristics using the classical constant current method. The threshold voltage data extracted from TCAD simulations for different diameters have been used to derive an empirical expression of the parameter *η* including the

*DS C C* .. . *<sup>D</sup> V f L fL f D*

where f0, f1 and f2 are constant fitting parameters, calibrated on numerical simulations. Equation (37) is valid for a wide range of nanowire diameter (down to 2 nm) and channel lengths (down to channel length equal to the nanowire diameter). The results in figure 7 show a good agreement between the analytical model and threshold voltage data obtained

Advanced atomistic numerical simulations (Neophytou & al., 2008; Niquet & al., 2000, 2006; Sarrazin & al., 2009; Nehari & al., 2006) have shown that a strong reduction of the silicon thickness impacts the material properties by modifying the band structure. Indeed, the dimensions targeted in ultra-scaled devices are those of a few tens atomic layers (several nanometers). At these dimensions, the electronic properties differ from the calculations shown in section 3.2 and based on the bulk effective masses. In (Sarrazin & al., 2009), atomistic tight-binding (TB) Schrödinger-Poisson simulations have been performed for the case of [001] oriented silicon nanowire in order to highlight the variation of the band structure with the nanowire diameter. The code TB\_Sim (TBSIM, 2011) has been used with a sp3 tight-binding model (Niquet & al., 2000). Figure 8 shows the valence and the conduction bands for Si nanowire width of 2nm and 10nm. We can note that when thinning the silicon film the minimum of the conduction band is increased and the general shape of bands becomes smoother (Sarrazin & al., 2009). However, the bandgap increase is smaller than the

In order to include these modifications in the previous threshold voltage modelling, analytical expressions of parameters affected by the band structure effect (band gap and effective masses) are proposed here. Diameter-dependent analytical functions (fitted on numerical simulations as illustrated in figure 9) are found for the bandgap and effective

> <sup>1</sup> , <sup>2</sup> 1 1 . *g g bulk*

2 2 . *t l t l bulk*

*D AD B*

2

*K*

*D AD B*

*<sup>K</sup> E E*

( ) ( ), 2

*m m*

2

(37)

(37)

(38)

1 2

dependence on the channel length, nanowire diameter and drain to source voltage:

0

from TCAD simulations.

**3.4 Band Structure Effect (BSE)** 

effective mass result of section 2.4.

masses (inspired from (Niquet & al., 2000):

where A, B and K are fitting constants.

Fig. 8. Silicon nanowire band structure obtained with a tight-binding Schrödinger-Poisson solver (TBSIM, 2011) for two different diameters (2 and 10 nm). Up: Conduction band; down: Valence band.

Fig. 9. Variation of the silicon band gap (a) and relative longitudinal and transversal masses (b) with respect to the silicon nanowire diameter. Comparison with atomistic simulations obtained in (Sarrazin & al., 2009).

Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 129

In the following, we investigate the impact of the quantum confinement on SCE, then the impact of BSE on SCE. As stated previously, the threshold voltage roll-off depends on quantum confinement through the long-channel threshold voltage which includes quantum

> **Classical Quantum Quantum + BSE Numerical simulation**

> > L C =D

V DS =0.5V

**0 5 10 15 20**

Diameter D = Channel length LC

Fig. 11. Impact of band structure variations on SCE. Threshold voltage roll-off versus nanowire diameter for a channel length equal to the nanowire diameter. Comparison with data extracted from numerical simulations using a cylindrical Schrödinger-Poisson solver

Figure 11 shows the impact of quantum effect and BSE on SCE as a function of the nanowire diameter. The curves plot the threshold voltage roll-off for a channel length equal to the nanowire diameter for the three approaches: classical (i.e., without quantum confinement and BSE), quantum without BSE, and quantum with BSE. Quantum threshold voltage obtained using the analytical model is validated in Fig. 11 with numerical simulation data obtained with a cylindrical Schrödinger-Poisson solver (Munteanu & Autran, 2003; Zervos & Feiner, 2004). We can note that the quantum confinement tends to limit SCE. This is due to the enhanced electrostatics control of the active area due to carrier energy quantization. As expected, the difference between quantum and classical approaches increases when reducing the nanowire diameter (due to the increase of energy quantum level for thinner films). When considering quantum confinement, the carrier energy is higher than for classical approach. That is why it is less affected by the longitudinal source to drain electric field, which generally strongly impacts the transistors performances at these channel length values. Moreover, figure 11 shows that BSE tend to amplify the impact of quantum effects on SCE: the threshold voltage roll-off is reduced when considering quantum confinement with BSE compared to the case when only quantum confinement is considered. For long channels, the threshold voltage decrease when considering BSE was due to the increase of

**4.1.2 Short-channel transistors** 

**-0.3**

**-0.25**

**-0.2**

**-0.15**

Threshold voltage roll-off (V)

(Munteanu & Autran, 2003).

**-0.1**

**-0.05**

**0**

confinement effects.

#### **4. Results and discussion**

We have just presented the modeling of all the physical phenomena which affect the electrostatics of nanowire MOSFETs. In this part, the impact of each mechanism is assessed at different levels of interest: threshold voltage, drain current and small-circuits performance.

#### **4.1 Impact on the threshold voltage**

#### **4.1.1 Long-channel transistors**

Figure 10 shows the long-channel threshold voltage increase due to quantum confinement, without BSE (i.e., considering bulk value for the band gap and the conduction masses) and with BSE (i.e., considering equations (37) and (38)). The analytical model has been compared to both numerical simulations (Sarrazin & al., 2009) and experimental data (Suk & al., 2007).

Fig. 10. Difference between quantum (Vth,q) and classical threshold voltage (Vth,cl) with and without BSE versus nanowire diameter (long channel transistors). Comparison between the analytical model, atomistic simulations (Sarrazin & al., 2009) and experimental data (Suk & al. 2007).

We can note that the band structure effects tend to limit the impact of the quantum confinement on the threshold voltage of the nanowire. This is coherent with equation (18). Increasing the effective masses, the quantum energy levels are lowered and the energy quantization decreases; then the quantum threshold voltage is lower. Figure 10 highlights the importance of considering BSE especially for thin film where the difference between threshold voltage with BSE and threshold voltage without BSE increases when reducing the nanowire diameter.

#### **4.1.2 Short-channel transistors**

128 Electrostatics

We have just presented the modeling of all the physical phenomena which affect the electrostatics of nanowire MOSFETs. In this part, the impact of each mechanism is assessed at different levels of interest: threshold voltage, drain current and small-circuits

Figure 10 shows the long-channel threshold voltage increase due to quantum confinement, without BSE (i.e., considering bulk value for the band gap and the conduction masses) and with BSE (i.e., considering equations (37) and (38)). The analytical model has been compared to both numerical simulations (Sarrazin & al., 2009) and experimental data (Suk & al., 2007).

> **Analytical model Numerical simulation Experimental data**

without Band Structure effects

with Band Structure effects

**0 2 4 6 8 10 12 14**

Fig. 10. Difference between quantum (Vth,q) and classical threshold voltage (Vth,cl) with and without BSE versus nanowire diameter (long channel transistors). Comparison between the analytical model, atomistic simulations (Sarrazin & al., 2009) and experimental data (Suk &

We can note that the band structure effects tend to limit the impact of the quantum confinement on the threshold voltage of the nanowire. This is coherent with equation (18). Increasing the effective masses, the quantum energy levels are lowered and the energy quantization decreases; then the quantum threshold voltage is lower. Figure 10 highlights the importance of considering BSE especially for thin film where the difference between threshold voltage with BSE and threshold voltage without BSE increases when reducing the

Diameter D (nm)

**4. Results and discussion** 

**4.1 Impact on the threshold voltage** 

**0**

**0.1**

**0.2**

V

al. 2007).

nanowire diameter.


th,cl (V)

th,q

**0.3**

**0.4**

**0.5**

**0.6**

**0.7**

**4.1.1 Long-channel transistors** 

performance.

In the following, we investigate the impact of the quantum confinement on SCE, then the impact of BSE on SCE. As stated previously, the threshold voltage roll-off depends on quantum confinement through the long-channel threshold voltage which includes quantum confinement effects.

Fig. 11. Impact of band structure variations on SCE. Threshold voltage roll-off versus nanowire diameter for a channel length equal to the nanowire diameter. Comparison with data extracted from numerical simulations using a cylindrical Schrödinger-Poisson solver (Munteanu & Autran, 2003).

Figure 11 shows the impact of quantum effect and BSE on SCE as a function of the nanowire diameter. The curves plot the threshold voltage roll-off for a channel length equal to the nanowire diameter for the three approaches: classical (i.e., without quantum confinement and BSE), quantum without BSE, and quantum with BSE. Quantum threshold voltage obtained using the analytical model is validated in Fig. 11 with numerical simulation data obtained with a cylindrical Schrödinger-Poisson solver (Munteanu & Autran, 2003; Zervos & Feiner, 2004). We can note that the quantum confinement tends to limit SCE. This is due to the enhanced electrostatics control of the active area due to carrier energy quantization. As expected, the difference between quantum and classical approaches increases when reducing the nanowire diameter (due to the increase of energy quantum level for thinner films). When considering quantum confinement, the carrier energy is higher than for classical approach. That is why it is less affected by the longitudinal source to drain electric field, which generally strongly impacts the transistors performances at these channel length values. Moreover, figure 11 shows that BSE tend to amplify the impact of quantum effects on SCE: the threshold voltage roll-off is reduced when considering quantum confinement with BSE compared to the case when only quantum confinement is considered. For long channels, the threshold voltage decrease when considering BSE was due to the increase of

Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 131

In previous works, we have demonstrated the analytical model of drain current in GAA nanowire MOSFETs in the ballistic transport regime (without interactions). We remind that this ballistic drain current is derived from the flux method initiated by McKelvey *et al*  (McKelevey & al., 1961), doing a balance in the active region between the different carrier fluxes. In the degenerate case, the ballistic drain current is given by the following

1

*I DC V V <sup>v</sup> q V*

where D is the nanowire diameter, vth is the thermal velocity discussed above, VGS is the drain to source voltage, VDS is the drain to source voltage, Cox is the oxide capacitance, ηF is the Fermi level, <sup>0</sup> and 1/2 are the Fermi integral of order 0 and -1/2 respectively and Vt

*D ox GS t th*

is the threshold voltage modeled above.

**10-12**

**10-10**

**10-8**

**Ballistic drain current (A)**

**10-6**

**10-4**

**D=5nm**

0

*F DS*

**QE with BSE QE without BSE**

**Numerical simulations**

**Lc =20nm V**DS**=0.4V**

*F*

*F*

 

. ( )

*q V kT*

*DS*

( )

*F*

*kT*

(40)

2 1 2

0 0

1

1

*F F*

2 1

**0 0.2 0.4 0.6 0.8 1**

**D=3nm**

**Gate Voltage (V)**

Fig. 13. Impact of the band structure effect on the ballistic drain current. Comparison with numerical simulations (deterministic Wigner equation solving (Barraud & al., 2009)).

Figure 13 shows the result at the device level for a long channel transistor and for two different nanowire diameters (3 and 5nm) (Dura & al., 2011). The ballistic drain current model is compared to numerical simulations based on a deterministic Wigner equation solver (Barraud & al., 2009). We can note a strong impact of BSE on the current in the sub-

( ) ( ) . . .( ). . .( ) ( ) . ( )

**4.3 Impact on ballistic drain current of nanowire MOSFET** 

expression:

the effective masses which lowered the quantized levels. In the case of short channels, the reduction of SCE when BSE are taken into account is the consequence of the band gap increase. A wider band gap means a higher energetic barrier, leading to a better electrostatics control which is less impacted by source-channel and drain-channel junctions when reducing the nanowire channel length. Moreover, we can note that below a certain diameter (depending on the modeling approach), the diameter thinning has a stronger impact on the threshold voltage roll-off than the channel length reduction. For the same channel length to diameter ratio, the threshold voltage roll-off is higher for D=5 nm than for D=2 nm. Indeed, for ultra-thin films, the quantization of carrier energy is very strong and the carrier concentration is mainly controlled by quantum confinement. In the case of D=2 nm, the strong electrostatic control due to the ultra-thin diameter completely overcomes the increase of SCE expected for these ultra-short channel lengths.

#### **4.2 Impact on the injection velocity**

Another parameter affected by the BSE is the thermal velocity which depends on masses along the transport direction. Indeed, in our case, for a transport along the (001) direction, the expression of thermal velocity is:

$$v\_{th} = \sqrt{\frac{2kT}{\pi \cdot m\_t}}\tag{39}$$

Figure 12 (Dura & al., 2011) shows the thermal velocity evolution with respect to the nanowire diameter. We can note a non-negligible reduction for ultra-thin nanowires up to a 20% decrease for D = 2 nm.

Fig. 12. Impact of the band structure effect on the thermal velocity with respect to the silicon nanowire diameter.

#### **4.3 Impact on ballistic drain current of nanowire MOSFET**

130 Electrostatics

the effective masses which lowered the quantized levels. In the case of short channels, the reduction of SCE when BSE are taken into account is the consequence of the band gap increase. A wider band gap means a higher energetic barrier, leading to a better electrostatics control which is less impacted by source-channel and drain-channel junctions when reducing the nanowire channel length. Moreover, we can note that below a certain diameter (depending on the modeling approach), the diameter thinning has a stronger impact on the threshold voltage roll-off than the channel length reduction. For the same channel length to diameter ratio, the threshold voltage roll-off is higher for D=5 nm than for D=2 nm. Indeed, for ultra-thin films, the quantization of carrier energy is very strong and the carrier concentration is mainly controlled by quantum confinement. In the case of D=2 nm, the strong electrostatic control due to the ultra-thin diameter completely overcomes the

Another parameter affected by the BSE is the thermal velocity which depends on masses along the transport direction. Indeed, in our case, for a transport along the (001) direction,

> 2. . *th*

Figure 12 (Dura & al., 2011) shows the thermal velocity evolution with respect to the nanowire diameter. We can note a non-negligible reduction for ultra-thin nanowires up to a

*v*

*t kT*

**4 8 12 16 20**

**Diameter D (nm)**

Fig. 12. Impact of the band structure effect on the thermal velocity with respect to the silicon

**With band structure effect Without band structure effect**

*<sup>m</sup>* (39)

increase of SCE expected for these ultra-short channel lengths.

**4.2 Impact on the injection velocity** 

the expression of thermal velocity is:

**8.5**

**9**

**9.5**

**10**

**10.5**

**Thermal velocity (104 m/s)**

nanowire diameter.

**11**

**11.5**

**12**

**12.5**

20% decrease for D = 2 nm.

In previous works, we have demonstrated the analytical model of drain current in GAA nanowire MOSFETs in the ballistic transport regime (without interactions). We remind that this ballistic drain current is derived from the flux method initiated by McKelvey *et al*  (McKelevey & al., 1961), doing a balance in the active region between the different carrier fluxes. In the degenerate case, the ballistic drain current is given by the following expression:

$$I\_D = \pi . D.C\_{ox}.(V\_{GS} - V\_t). \frac{\mathfrak{T}\_0(\eta\_F)}{\mathfrak{T}\_{-\frac{1}{2}\xi}(\eta\_F)} . \upsilon\_{\text{flv}} (\frac{1 - \frac{\mathfrak{T}\_0(\eta\_F)}{kT})}{1 + \frac{\mathfrak{T}\_{-\frac{1}{2}\xi}(\eta\_F - \frac{q\_\cdot V\_{DS}}{kT})}{\mathfrak{T}\_{-\frac{1}{2}\xi}(\eta\_F)}}\tag{40}$$

where D is the nanowire diameter, vth is the thermal velocity discussed above, VGS is the drain to source voltage, VDS is the drain to source voltage, Cox is the oxide capacitance, ηF is the Fermi level, <sup>0</sup> and 1/2 are the Fermi integral of order 0 and -1/2 respectively and Vt is the threshold voltage modeled above.

Fig. 13. Impact of the band structure effect on the ballistic drain current. Comparison with numerical simulations (deterministic Wigner equation solving (Barraud & al., 2009)).

Figure 13 shows the result at the device level for a long channel transistor and for two different nanowire diameters (3 and 5nm) (Dura & al., 2011). The ballistic drain current model is compared to numerical simulations based on a deterministic Wigner equation solver (Barraud & al., 2009). We can note a strong impact of BSE on the current in the sub-

Nanowires: Promising Candidates for Electrostatic Control in Future Nanoelectronic Devices 133

**10**

**1**

In this chapter, the potential of silicon nanowires for microelectronics applications was evaluated. Regarding the evolution of transistor architectures, they appear as the best configuration for the gate control of the device. The particular shape with a surrounding gate provides an ideal electrostatic control to immunize transistors against perturbations generated by the scaling down of dimensions. In this work, we have developed a complete model of the electrostatic of transistors based on nanowires. The physical phenomena affecting the electrostatics was considered: short-channel effects due to the channel length reduction or the quantum mechanical effects due to the diameter thinning. Moreover, ultimate mechanisms as the modification of the band shape of silicon material is studied based on advanced simulations (essentially tight-binding Schrodinger-Poisson solving). All this physics (thanks to analytical model development) is transposed to higher simulation levels as characteristics of transistor or small-circuit performances. Following this idea, we have seen the impact of short-channel, quantum or band structure effect on the threshold voltage. For ultra-thin nanowires, we highlighted the necessity to consider all these phenomena to be as close as possible to experimental data. Then, a study of their impact on transport was performed with the analysis of ballistic drain current of single transistor and performances of inverters or ring oscillators. In all cases, the evaluation of performances is inaccurate if quantum or band structure effects are not considered. For example, one decade and a half of difference on OFF-state current or a reduction of factor 2 or 3 on the oscillation frequency show the importance of the electrostatics (and so a realistic modeling) if we

Fig. 15. Impact of BSE on the ring oscillator frequency versus the nanowire diameter. Comparison between classical, quantum (QE) and low dimensions effects (QE+BSE).

envisage nanowires as the future technological solution in microelectronics.

Autran, J.L.; Munteanu, D.; Tintori, O.; Harrison, S.; Decarre, E. & Skotnickin, T. (2004).

Quantum-mechanical analytical modeling of threshold voltage in long-channel

**2 3 4 5 6 7 8 9 10**

**Diameter D (nm)**

**Vdd=1.5V Lc=30nm**

**Classical QE QE+BSE**

**1 1 1** CL

**5. Conclusion** 

**6. References** 

ATLAS. (2010). Users Manual, SILVACO, 2010.

threshold regime for 3nm-diameter due to the Vt variation while the ON-state current stays almost unchanged. From this graph, we can highlight the necessity to take into account the correction due to bands variations in the modeling if we expect to provide predictive devices performances. Indeed, at 3nm, the off-state current is increased by more than one decade when BSE is considered.

#### **4.4 Impact on performances of small circuits based on nanowire MOSFETs**

After implementation in a Verilog-A environment, the model presented above has been used to simulate a CMOS inverter and then a complete 11 stages-ring oscillator. In order to build-up the CMOS inverter a p-type nanowire MOSFET is considered symmetrically to the n-type transistor in the inverter setup. The impact of BSE can be addressed at the circuit level through the study of the commutation characteristics of the inverter or the oscillation frequency of a ring oscillator.

Figure 14 shows the input/output characteristics of the inverter for the classical case (i.e., without QE), with quantum effects (QE) and with band structure effects (QE+BSE). We can note that the inverter characteristic is more abrupt when considering only QE. Similarly to the results obtained for the threshold voltage, BSE tends to limit the impact of quantum confinement by smoothing the CMOS inverter switch.

Fig. 14. Impact of BSE on the inverter characteristic. Comparison between classical (i.e, without QE), quantum (QE) and low dimensions effects (QE+BSE).

Regarding the ring oscillator, the results seem opposite to the inverter case. The better performances are for the classical case and introducing quantum confinement reduces the oscillation frequency. This is due to the fact that the ring-oscillator frequency is directly proportional to the ON-state current in strong inversion regime Vdd=1.5V (far from the threshold voltage). QE increases Vth and consequently reduces the current. The injection velocity also impacts directly the current and then the oscillation frequency is affected. The result is a reduction of the oscillation frequency when BSE are taken into account.

Fig. 15. Impact of BSE on the ring oscillator frequency versus the nanowire diameter. Comparison between classical, quantum (QE) and low dimensions effects (QE+BSE).

#### **5. Conclusion**

132 Electrostatics

threshold regime for 3nm-diameter due to the Vt variation while the ON-state current stays almost unchanged. From this graph, we can highlight the necessity to take into account the correction due to bands variations in the modeling if we expect to provide predictive devices performances. Indeed, at 3nm, the off-state current is increased by more than one

After implementation in a Verilog-A environment, the model presented above has been used to simulate a CMOS inverter and then a complete 11 stages-ring oscillator. In order to build-up the CMOS inverter a p-type nanowire MOSFET is considered symmetrically to the n-type transistor in the inverter setup. The impact of BSE can be addressed at the circuit level through the study of the commutation characteristics of the inverter or the oscillation

Figure 14 shows the input/output characteristics of the inverter for the classical case (i.e., without QE), with quantum effects (QE) and with band structure effects (QE+BSE). We can note that the inverter characteristic is more abrupt when considering only QE. Similarly to the results obtained for the threshold voltage, BSE tends to limit the impact of quantum

> **Classical QE QE+BSE**

**Vdd=1V Lc=20nm D=3nm**

Fig. 14. Impact of BSE on the inverter characteristic. Comparison between classical (i.e,

result is a reduction of the oscillation frequency when BSE are taken into account.

Regarding the ring oscillator, the results seem opposite to the inverter case. The better performances are for the classical case and introducing quantum confinement reduces the oscillation frequency. This is due to the fact that the ring-oscillator frequency is directly proportional to the ON-state current in strong inversion regime Vdd=1.5V (far from the threshold voltage). QE increases Vth and consequently reduces the current. The injection velocity also impacts directly the current and then the oscillation frequency is affected. The

**0 0.2 0.4 0.6 0.8 1**

**0.9 0.92 0.94 0.96 0.98 1**

**0.45 0.46 0.47 0.48 0.49**

**Vin (V)**

**4.4 Impact on performances of small circuits based on nanowire MOSFETs** 

decade when BSE is considered.

frequency of a ring oscillator.

Vdd

Vin Vout

confinement by smoothing the CMOS inverter switch.

**0**

without QE), quantum (QE) and low dimensions effects (QE+BSE).

**0.2**

**0.4**

**0.6**

**Vout (V)**

**0.8**

**1**

In this chapter, the potential of silicon nanowires for microelectronics applications was evaluated. Regarding the evolution of transistor architectures, they appear as the best configuration for the gate control of the device. The particular shape with a surrounding gate provides an ideal electrostatic control to immunize transistors against perturbations generated by the scaling down of dimensions. In this work, we have developed a complete model of the electrostatic of transistors based on nanowires. The physical phenomena affecting the electrostatics was considered: short-channel effects due to the channel length reduction or the quantum mechanical effects due to the diameter thinning. Moreover, ultimate mechanisms as the modification of the band shape of silicon material is studied based on advanced simulations (essentially tight-binding Schrodinger-Poisson solving). All this physics (thanks to analytical model development) is transposed to higher simulation levels as characteristics of transistor or small-circuit performances. Following this idea, we have seen the impact of short-channel, quantum or band structure effect on the threshold voltage. For ultra-thin nanowires, we highlighted the necessity to consider all these phenomena to be as close as possible to experimental data. Then, a study of their impact on transport was performed with the analysis of ballistic drain current of single transistor and performances of inverters or ring oscillators. In all cases, the evaluation of performances is inaccurate if quantum or band structure effects are not considered. For example, one decade and a half of difference on OFF-state current or a reduction of factor 2 or 3 on the oscillation frequency show the importance of the electrostatics (and so a realistic modeling) if we envisage nanowires as the future technological solution in microelectronics.

#### **6. References**

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Autran, J.L. (2011). Analytical model of ballistic current for GAA nanowire MOSFET including band structure effects: Application to ring oscillator, *Proceedings of 2011 Ultimate Integration of Silicon*, ISBN 978-1-4577-0089-7, Cork, Ireland, March

Y.C.; Hong, B.H. & Sung Woo H. (2007). Temperature-dependent characteristics of cylindrical gate-all-around twin silicon nanowire MOSFETs (TSNWFETs). *IEEE Electron Devices Letters*, Vol. 28, No. 12, (December 2007), pp. 1129-1131, 0741-3106


**Part 6** 

**Electrostatic Actuation** 


**Part 6** 

**Electrostatic Actuation** 

136 Electrostatics

Yu, B.; Lu, H.; Liu, M. & Taur, Y. (2007). Explicit continuous models for double-gate and

Zervos, M. & Feiner, L.F. (2004). Electronic structure of piezoelectric double-barrier

(October 2007), pp. 2715-2722, 0018-9383

(January 2004), pp. 281-291, 0021-8979

surrounding-gate MOSFETs, IEEE Trans. Electron Devices, Vol. 54, No. 10,

InAs/InP/InAs/InP/InAs (111) nanowires, *Journal of Appl. Phys.*, Vol. 95, No. 1,

*1,2Turkey* 

**New Approach to Pull-In Limit and** 

Ali Yildiz1, Cevher Ak1 and Hüseyin Canbolat2

*Electronics and Communication Engineering, Ankara,* 

**Within the Pull-In Limit** 

*2Yildirim Beyazit University, Department of* 

 **Position Control of Electrostatic Cantilever** 

*1Mersin University, Electrical and Electronics Engineering Department, Mersin,* 

Since electrostatic cantilevers are very easy to fabricate, have small dimension, and consume low power, they have been very popular as a sensor. They had been used as a capacitive pressure sensor for measuring blood pressure (Hin-Leung Chau & Wise, 1988), as a microwave switch (Dooyoung Hah et al., 2000), as an air flow sensor (Yu-Hsiang Wang et al., 2007), as a micro-actuator for probe-based data storage (Lu & Fedder, 2004), and in well known commercial applications like inkjet head (Kamusuki et al., 2000), and optical

An electrostatic MEMS cantilever is a simple capacitor consists of two parallel conductive plates. The bottom conductive plate is coated on a substrate and fixed on it, the top plate is suspended with a surface area *A*. The top electrode is separated by a gap spacing *d* above the bottom one and fixed from one end. The other end is free to move. When a potential difference (V) is applied between electrodes, free end will tilt downwards (δ) due to

Initial Position

d-

(displacement)

Free End

Bottom Electrode (fixed on substrate)

L

Top Electrode

**1. Introduction** 

scanners (Schenk et al., 2000).

electrostatic force (Fig.1.)

d

Fixed End

V

Fig. 1. Electrostatic actuator (Side View)

### **New Approach to Pull-In Limit and Position Control of Electrostatic Cantilever Within the Pull-In Limit**

Ali Yildiz1, Cevher Ak1 and Hüseyin Canbolat2

*1Mersin University, Electrical and Electronics Engineering Department, Mersin, 2Yildirim Beyazit University, Department of Electronics and Communication Engineering, Ankara, 1,2Turkey* 

#### **1. Introduction**

Since electrostatic cantilevers are very easy to fabricate, have small dimension, and consume low power, they have been very popular as a sensor. They had been used as a capacitive pressure sensor for measuring blood pressure (Hin-Leung Chau & Wise, 1988), as a microwave switch (Dooyoung Hah et al., 2000), as an air flow sensor (Yu-Hsiang Wang et al., 2007), as a micro-actuator for probe-based data storage (Lu & Fedder, 2004), and in well known commercial applications like inkjet head (Kamusuki et al., 2000), and optical scanners (Schenk et al., 2000).

An electrostatic MEMS cantilever is a simple capacitor consists of two parallel conductive plates. The bottom conductive plate is coated on a substrate and fixed on it, the top plate is suspended with a surface area *A*. The top electrode is separated by a gap spacing *d* above the bottom one and fixed from one end. The other end is free to move. When a potential difference (V) is applied between electrodes, free end will tilt downwards (δ) due to electrostatic force (Fig.1.)

Fig. 1. Electrostatic actuator (Side View)

New Approach to Pull-In Limit and

**2. Lumped model** 

Position Control of Electrostatic Cantilever Within the Pull-In Limit 141

As it can be seen from Fig. 2, only zero angle constrain is considered and zero displacement is ignored. Nevertheless, the model is very simple. Therefore, calculations are easy and pull-

<sup>1</sup> <sup>2</sup> U = CV

dU F =

d 1 1 dC 2 2 dV F = CV = <sup>V</sup> ( = 0 since <sup>V</sup> is constant) dx 2 2 dx dx

<sup>0</sup> <sup>ε</sup> <sup>A</sup> C = x

ε0 is permittivity of free space, A is area of one of the parallel plates, and x is the distance

(3)

2 (1)

dx (2)

(4)

in limit can be computed in few steps as one-third of the initial gap.

Fig. 2. Lumped model of a cantilever actuator.

Stored energy in a parallel plate capacitor is

Force due to this energy is

Value of a parallel plate capacitor is

Therefore,

between plates.

When potential difference is removed, top electrode will come back to its initial (original) position due to restoring force of the bended structure. It is also sometimes called as spring force. If applied potential is increased beyond the some limit, top electrode will collapse onto the bottom electrode. While spring force term is proportional with the displacement, electrostatic force is proportional to square of the displacement. Hence, after some point, spring force cannot balance the electrostatic force any more. Then, top electrode collapses onto the bottom electrode. This point is named as pull-in limit.

Because cantilever is an electromechanical coupled system, its behavior is non-linear. Thus, having an analytical formula for pull-in limit is impossible. So, there is no simple formula to calculate it. People have been using lumped model (Hyun-Ho Yang et al., 2010; Seeger & Boser, 1999; Nielson & Barbastathis, 2006; Faris et al., 2006; Mol et al., 2007; Chowdhury et al., 2006; Owusu & Lewis, 2007) for decades. Lump model estimates the pull-in limit as one-third of the initial gap (d/3). However, experimental part of a study showed that pull-in limit is at a different value (Hu et al., 2004). Hu et al. utilized a linearized governing equation of a cantilever to demonstrate their analytical approach. They obtained the total energy expressions including the kinetic energy, strain energy, and electric potential energy. The total energy expression was substituted into Hamilton's principle, and obtained partial differential equation with a nonlinear force term. This force term expanded by Taylor series about the equilibrium position and higher order terms neglected with an assumption of small displacement. At the end, structureelectrostatic coupling linear partial differential equation was obtained. Since small deflection was assumed and Taylor series expanded about equilibrium position, as the cantilever tip gets away from initial (original) position, error percentage also gets bigger (as high as 10%) when the tip is closer to pull-in limit(Hu et al., 2004). A later study has used Generalized Differential Quadrature Method which is accurate and efficient way to analyze a linear vector space by high-order polynomial approximation (Sadeghian et al., 2007). This approach gives smaller error in some measurements but not in all of them. Error gets as high as 5% when we close to pull-in limit.

When it is also checked by a software (ANSYS) which utilizes finite element method, pull-in limit seems to be at around 44% of initial gap which is consistent with experimental results (Hu et al., 2004; Sadeghian et al., 2007). Table 1 shows some simulation results for different initial gaps.


Table 1. Ansys simulation Pull-in results of a cantilever with L = 200 µm.

These results show that the lumped model is not a very good approximation of the system. In fact, the cantilever beam system has two constrains: fixed end of the top electrode has zero displacement and zero angle even when voltage applied between electrodes. However, lumped model considers only second constrain for the sake of simplicity.

#### **2. Lumped model**

140 Electrostatics

When potential difference is removed, top electrode will come back to its initial (original) position due to restoring force of the bended structure. It is also sometimes called as spring force. If applied potential is increased beyond the some limit, top electrode will collapse onto the bottom electrode. While spring force term is proportional with the displacement, electrostatic force is proportional to square of the displacement. Hence, after some point, spring force cannot balance the electrostatic force any more. Then, top electrode collapses

Because cantilever is an electromechanical coupled system, its behavior is non-linear. Thus, having an analytical formula for pull-in limit is impossible. So, there is no simple formula to calculate it. People have been using lumped model (Hyun-Ho Yang et al., 2010; Seeger & Boser, 1999; Nielson & Barbastathis, 2006; Faris et al., 2006; Mol et al., 2007; Chowdhury et al., 2006; Owusu & Lewis, 2007) for decades. Lump model estimates the pull-in limit as one-third of the initial gap (d/3). However, experimental part of a study showed that pull-in limit is at a different value (Hu et al., 2004). Hu et al. utilized a linearized governing equation of a cantilever to demonstrate their analytical approach. They obtained the total energy expressions including the kinetic energy, strain energy, and electric potential energy. The total energy expression was substituted into Hamilton's principle, and obtained partial differential equation with a nonlinear force term. This force term expanded by Taylor series about the equilibrium position and higher order terms neglected with an assumption of small displacement. At the end, structureelectrostatic coupling linear partial differential equation was obtained. Since small deflection was assumed and Taylor series expanded about equilibrium position, as the cantilever tip gets away from initial (original) position, error percentage also gets bigger (as high as 10%) when the tip is closer to pull-in limit(Hu et al., 2004). A later study has used Generalized Differential Quadrature Method which is accurate and efficient way to analyze a linear vector space by high-order polynomial approximation (Sadeghian et al., 2007). This approach gives smaller error in some measurements but not in all of them.

When it is also checked by a software (ANSYS) which utilizes finite element method, pull-in limit seems to be at around 44% of initial gap which is consistent with experimental results (Hu et al., 2004; Sadeghian et al., 2007). Table 1 shows some simulation results for different

These results show that the lumped model is not a very good approximation of the system. In fact, the cantilever beam system has two constrains: fixed end of the top electrode has zero displacement and zero angle even when voltage applied between electrodes. However,

**Initial Gap(µm) Pull-in Gap(µm) Pull-in Gap/Initial Gap** 

2 0.881 0.4405 5 2.203 0.4406 10 4.403 0.4403

Table 1. Ansys simulation Pull-in results of a cantilever with L = 200 µm.

lumped model considers only second constrain for the sake of simplicity.

onto the bottom electrode. This point is named as pull-in limit.

Error gets as high as 5% when we close to pull-in limit.

initial gaps.

As it can be seen from Fig. 2, only zero angle constrain is considered and zero displacement is ignored. Nevertheless, the model is very simple. Therefore, calculations are easy and pullin limit can be computed in few steps as one-third of the initial gap.

Fig. 2. Lumped model of a cantilever actuator.

Stored energy in a parallel plate capacitor is

$$\text{U} = \frac{1}{2}\text{CV}^2\tag{1}$$

Force due to this energy is

$$\mathbf{F} = \frac{\mathbf{d} \mathbf{U}}{\mathbf{d} \mathbf{x}} \tag{2}$$

Therefore,

$$\mathbf{F} = \frac{\mathbf{d}}{\mathrm{d}\mathbf{x}} \left(\frac{1}{2}\mathbf{C}\mathbf{V}^2\right) = \frac{1}{2}\frac{\mathrm{d}\mathbf{C}}{\mathrm{d}\mathbf{x}}\mathbf{V}^2 \qquad\qquad\qquad\left(\frac{\mathrm{d}\mathbf{V}}{\mathrm{d}\mathbf{x}} = \mathbf{0} \text{ since } \mathbf{V} \text{ is constant}\right) \tag{3}$$

Value of a parallel plate capacitor is

$$\mathbf{C} = \frac{\mathbf{c}\_0 \mathbf{A}}{\mathbf{x}} \tag{4}$$

ε0 is permittivity of free space, A is area of one of the parallel plates, and x is the distance between plates.

New Approach to Pull-In Limit and

**3. Bisection model** 

*k* can be obtained for a cantilever as (Saha et al., 2006)

and insert this in Eq. (16), and replace *A* with w*L*. We can get

one third of the movable part since it has to be close to pivot.

Fig. 3. New Approach (Bisection Model) to cantilever actuator.

Position Control of Electrostatic Cantilever Within the Pull-In Limit 143

8 kd V =

<sup>3</sup> 2 t k = Ew 3 L

Since model has two different sections, it is named as *Bisection Model* and can be seen in Fig.3. Bisection model considers two constrains of fixed end of the top electrode. Fixed end has both zero displacement and zero angle while the other end is free to tilt linearly around pivot. Pivot point is placed 1/3 of the cantilever length from fixed end of top electrode. There is no specific reason for 1/3 ratio exactly. Since, the left side of the cantilever is fixed, there is very small movement at the left side. So, we model the structure in a that way 1/3 of the cantilever is not moving at all and rest is moving linearly around the break point (pivot). By doing this, we still have very simple model as Lumped model and also consider both constraint of fixed end. Therefore, we can have a simple formula for the structure. Electrostatic force is placed at the free end of the top electrode since this end is close to bottom electrode and force gets its biggest value over there. Restoring force is placed at the

critical

critical

3

<sup>27</sup> <sup>ε</sup> <sup>A</sup> (16)

(17)

<sup>ε</sup> <sup>L</sup> = (18)

0

3

3 4 0 16 t 81 Ed <sup>V</sup>

So,

$$\frac{\text{dC}}{\text{dx}} = -\frac{\varepsilon\_0 \text{A}}{\text{x}^2} \tag{5}$$

Then, electrostatic force term is

$$\mathbf{F\_e = -\frac{\varepsilon\_0 \mathbf{A} \mathbf{V}^2}{2\chi^2}} \qquad \text{( - sign shows direction of the force)} \tag{6}$$

Electrostatic force term and spring force term will be equal to each other for equilibrium,

$$\mathbf{F\_e = F\_s} \tag{7}$$

$$\frac{\varepsilon\_0 \mathbf{A} \mathbf{V}^2}{2\mathbf{x}^2} = \mathbf{k} \boldsymbol{\delta} \; \; \; \; \; \; \; \delta = (\mathbf{d} - \mathbf{x}) \tag{8}$$

$$2\mathbf{k}\left(\mathbf{x}^2\mathbf{d} - \mathbf{x}^3\right) = \varepsilon\_0 \mathbf{A} \mathbf{V}^2 \tag{9}$$

Potential can be get as

$$\mathbf{V} = \sqrt{\frac{2\mathbf{k}}{\varepsilon\_0 \mathbf{A}} \left(\mathbf{x}^2 \mathbf{d} - \mathbf{x}^3\right)}\tag{10}$$

Top electrode will collapse when

$$\frac{\text{dV}}{\text{dx}} = 0\tag{11}$$

So, if derivative is taken of Eq. (10) we have

$$\mathbf{12xd - 3x^2 = 0}\tag{12}$$

Then critical x value can be get as

$$\mathbf{x}\_{\text{critical}} = \frac{\mathbf{2}}{\mathbf{3}} \mathbf{d} \tag{13}$$

Displacement of top electrode at the limit condition can be found as

$$\mathbf{G}\_{\text{critical}} = \frac{1}{3}\mathbf{d} \tag{14}$$

Thus, critical value of the potential difference can be calculated as

$$\mathbf{V}\_{\text{critical}} = \sqrt{\frac{2\mathbf{k}}{\varepsilon\_0 \mathbf{A}} \left(\frac{2}{3}\mathbf{d}\right)^2 \mathbf{d} - \left(\frac{2}{3}\mathbf{d}\right)^3} \tag{15}$$

Therefore, it can be simplified as

$$\mathbf{V}\_{\text{critical}} = \sqrt{\frac{8}{27}} \frac{\text{kd}^3}{\varepsilon\_0 \mathbf{A}} \tag{16}$$

*k* can be obtained for a cantilever as (Saha et al., 2006)

$$\mathbf{k} = \frac{2}{3} \mathbf{Ew} \left(\frac{\mathbf{t}}{\mathbf{L}}\right)^3 \tag{17}$$

and insert this in Eq. (16), and replace *A* with w*L*. We can get

$$\mathbf{V}\_{\text{critical}} = \sqrt{\frac{16}{81} \frac{\text{Ed}^3 \text{t}^3}{\varepsilon\_0 \text{L}^4}}\tag{18}$$

#### **3. Bisection model**

142 Electrostatics

Electrostatic force term and spring force term will be equal to each other for equilibrium,

23 2

0 2k V= x d x

> critical <sup>2</sup> x =d

> critical <sup>1</sup> <sup>δ</sup> = d

0 2k 2 2 V = dd d

ε A3 3 

2 3

Displacement of top electrode at the limit condition can be found as

Thus, critical value of the potential difference can be calculated as

critical

dV = 0

2 3

2 0 2

0 2

<sup>ε</sup> AV F = ( - sign shows direction of the force) <sup>e</sup> 2x (6)

dC <sup>ε</sup> <sup>A</sup> <sup>=</sup> dx <sup>x</sup> (5)

F =F e s (7)

<sup>ε</sup> AV = kδ , <sup>δ</sup> = (d - x) 2x (8)

<sup>0</sup> 2k x d x = ε AV (9)

<sup>ε</sup> <sup>A</sup> (10)

dx (11)

<sup>2</sup> 2xd - 3x = 0 (12)

3 (13)

3 (14)

(15)

So,

Then, electrostatic force term is

Potential can be get as

Top electrode will collapse when

Then critical x value can be get as

Therefore, it can be simplified as

So, if derivative is taken of Eq. (10) we have

2 0 2

> Since model has two different sections, it is named as *Bisection Model* and can be seen in Fig.3. Bisection model considers two constrains of fixed end of the top electrode. Fixed end has both zero displacement and zero angle while the other end is free to tilt linearly around pivot. Pivot point is placed 1/3 of the cantilever length from fixed end of top electrode. There is no specific reason for 1/3 ratio exactly. Since, the left side of the cantilever is fixed, there is very small movement at the left side. So, we model the structure in a that way 1/3 of the cantilever is not moving at all and rest is moving linearly around the break point (pivot). By doing this, we still have very simple model as Lumped model and also consider both constraint of fixed end. Therefore, we can have a simple formula for the structure. Electrostatic force is placed at the free end of the top electrode since this end is close to bottom electrode and force gets its biggest value over there. Restoring force is placed at the one third of the movable part since it has to be close to pivot.

Fig. 3. New Approach (Bisection Model) to cantilever actuator.

New Approach to Pull-In Limit and

The electrostatic force term can be obtained as

The storing force term can be written as

substituting 2*L*/3 to *L*, *k* can be calculated as

can write

limit.

and *k* can be obtained for a cantilever as (Saha et al., 2006)

Where E and t are Young's modulus and thickness of top electrode.

Position Control of Electrostatic Cantilever Within the Pull-In Limit 145

d d ln d+ln <sup>δ</sup>+<sup>δ</sup> <sup>1</sup> d-<sup>δ</sup> d-<sup>δ</sup> F = <sup>ε</sup> wL <sup>V</sup>

e 0 2

3 δ d-δ

s <sup>δ</sup> F =k

<sup>3</sup> 2 t k = Ew 3 L

In the Bisection Model only 2*L*/3 part of the upper electrode is free to move. So, by

<sup>3</sup> 9 t k = Ew 4 L

for Bisection Model. Since moments of the electrostatic and restoring forces are equal, we

e s 6 2 F= F

3 3

This equation is valid not only for pull-in limit, but also for all values within the pull-in

<sup>d</sup> <sup>δ</sup>Et d <sup>δ</sup> <sup>L</sup> <sup>ε</sup> ln <sup>δ</sup> d +<sup>δ</sup>

0

2 22

3Et <sup>d</sup> <sup>4</sup><sup>δ</sup> 3dδ+ 3<sup>δ</sup> -6δd+3d ln dV 4 d <sup>δ</sup> = = <sup>0</sup>

d δ

4 d d ε L ( ln d ln δ δ d δ d δ

Therefore, the relation between applied voltage and displacement is given by

3 Et <sup>δ</sup> (d <sup>δ</sup>) V =

4 0

To find the pull-in limit, we have to take derivative of *V* with respect to *δ*. So;

3 4

3

dδ

2

3 (25)

(26)

(27)

9 9 (28)

1 2

1 3 2 (29)

(30)

(24)

In this model, capacitance of the system has two parts (see Fig. 4.)

Fig. 4. Bisection Model capacitor calculation.

C1 and C2 can be found as

$$\mathbf{C}\_{1} = \varepsilon\_{0} \frac{\mathbf{w} \mathbf{L}}{\Re \mathbf{d}} \quad \text{and} \quad \mathbf{C}\_{2} = \frac{2 \varepsilon\_{0} \mathbf{w} \mathbf{L}}{3 \delta} \ln \left( \frac{\mathbf{d}}{\mathbf{d} \cdot \boldsymbol{\delta}} \right) \tag{19}$$

Where w and L are width and length of the cantilever respectively.

Therefore, total capacitance of the system is

$$\mathbf{C}\_{\Gamma} = \mathbf{C}\_{1} + \mathbf{C}\_{2} = \frac{1}{3} \varepsilon\_{0} \mathbf{w} \mathbf{L} \left( \frac{1}{\mathbf{d}} + \frac{2}{\delta} \ln \left( \frac{\mathbf{d}}{\mathbf{d} \cdot \boldsymbol{\delta}} \right) \right) \tag{20}$$

And electrostatic force term can be obtained as

$$\mathbf{F}\_{\rm e} = \frac{1}{2} \left[ \left( \frac{\rm d}{d\delta} (\mathbf{C}\_1 + \mathbf{C}\_2) \right) \mathbf{V}^2 + \left( \frac{\rm d}{d\delta} \mathbf{V}^2 \right) (\mathbf{C}\_1 + \mathbf{C}\_2) \right] \tag{21}$$

Since *V* (potential difference between electrodes) is constant between plates and C1 is also constant, equation can be written shortly as

$$\mathbf{F}\_{\rm e} = \frac{1}{2} \frac{\mathbf{d} \mathbf{C}\_2}{\mathbf{d} \boldsymbol{\delta}} \mathbf{V}^2 \tag{22}$$

Since

$$\frac{\text{dC}\_2}{\text{d(d-\delta)}} = \frac{2}{3} \left( \frac{\varepsilon\_0 \text{wLLn}\left(\frac{\text{Ld}}{\text{Ld-L}\,\text{\textdegree\textdegree\&}}\right)}{\text{6}^2} + \frac{\varepsilon\_0 \text{w}\,\text{L}^2}{\text{6L}\,\text{\textdegree\textdegree\textdegree\textdegree\text&}} \right) \tag{23}$$

The electrostatic force term can be obtained as

144 Electrostatics

0

3 d δ d-δ

2 2

2 2

0 2

Ld

 

(19)

(21)

(23)

(20)

2 dδ (22)

3d 3δ d-δ

1 0 2

T 12 0

Where w and L are width and length of the cantilever respectively.

wL <sup>2</sup><sup>ε</sup> wL d C = <sup>ε</sup>and C = ln

1 12 d C =C +C = <sup>ε</sup> wL + ln

e 1 2 1 2 1 d <sup>d</sup> F = C +C V + V C +C 2 dδ dδ

Since *V* (potential difference between electrodes) is constant between plates and C1 is also

1 dC F= V

2 0 2

<sup>ε</sup> wLln dC 2 Ld-L<sup>δ</sup> <sup>ε</sup> wL = + d(d-δ) 3 δ δL d-δ

e

In this model, capacitance of the system has two parts (see Fig. 4.)

Fig. 4. Bisection Model capacitor calculation.

Therefore, total capacitance of the system is

And electrostatic force term can be obtained as

constant, equation can be written shortly as

Since

C1 and C2 can be found as

$$\mathbf{F}\_{\mathbf{e}} = \frac{1}{3} \varepsilon\_0 \mathbf{w} \mathbf{L} \left( \frac{-\ln\left(\frac{\mathbf{d}}{\mathbf{d} \cdot \boldsymbol{\delta}}\right) \mathbf{d} + \ln\left(\frac{\mathbf{d}}{\mathbf{d} \cdot \boldsymbol{\delta}}\right) \mathbf{\hat{\delta}} + \boldsymbol{\delta}}{\boldsymbol{\delta}^2 (\mathbf{d} \cdot \boldsymbol{\delta})} \right) \mathbf{V}^2 \tag{24}$$

The storing force term can be written as

$$\mathbf{F}\_s = \mathbf{k} \frac{\boldsymbol{\delta}}{\boldsymbol{\delta}} \tag{25}$$

and *k* can be obtained for a cantilever as (Saha et al., 2006)

$$\mathbf{k} = \frac{2}{3} \mathbf{Ew} \left(\frac{\mathbf{t}}{\mathbf{L}}\right)^3 \tag{26}$$

Where E and t are Young's modulus and thickness of top electrode.

In the Bisection Model only 2*L*/3 part of the upper electrode is free to move. So, by substituting 2*L*/3 to *L*, *k* can be calculated as

$$\mathbf{k} = \frac{9}{4} \mathbf{Ew} \left(\frac{\mathbf{t}}{\mathbf{L}}\right)^3 \tag{27}$$

for Bisection Model. Since moments of the electrostatic and restoring forces are equal, we can write

$$\frac{6}{9}\mathbf{F}\_{\mathbf{e}} = \frac{2}{9}\mathbf{F}\_{\mathbf{s}}\tag{28}$$

Therefore, the relation between applied voltage and displacement is given by

$$\mathbf{V} = \left( \frac{3}{4} \frac{\text{Er}^3 \text{\ $}^3 \text{\$ }^3 \text{(d} - \text{\ $)}}{\text{\$ }\_0 \text{\ $}^4 \text{\$ }^4 \text{(-\ln\left(\frac{\text{d}}{\text{d} - \text{\ $}}\right) \text{d} + \ln\left(\frac{\text{d}}{\text{d} - \text{\$ }}\right) \text{\ $} + \text{\$ }} \right)^{\frac{1}{2}} \tag{29}$$

This equation is valid not only for pull-in limit, but also for all values within the pull-in limit.

To find the pull-in limit, we have to take derivative of *V* with respect to *δ*. So;

$$\frac{\mathrm{d}\mathbf{V}}{\mathrm{d}\boldsymbol{\delta}} = \frac{-\frac{\sqrt{3}\mathrm{E}\mathrm{t}^{3}}{4} \Big( 4\boldsymbol{\delta}^{2} - 3\mathrm{d}\boldsymbol{\delta} + \Big( 3\boldsymbol{\delta}^{2} - 6\mathrm{d}\boldsymbol{\delta} + 3\mathrm{d}^{2} \Big) \ln\Big( \frac{\mathrm{d}}{\mathrm{d}-\mathrm{d}} \Big) \Big)}{\left( \Big\delta\mathrm{E}\mathrm{t}^{3} (\mathrm{d}-\mathrm{\boldsymbol{\delta}})\mathrm{L}^{4} \boldsymbol{\varepsilon}\_{0} \Big( \ln\Big( \frac{\mathrm{d}}{\mathrm{d}-\mathrm{\boldsymbol{\delta}}} \Big) (\mathrm{\boldsymbol{\delta}}-\mathrm{d}) + \mathrm{\boldsymbol{\delta}} \Big)^{3} \right)^{\frac{1}{2}}} = 0$$

New Approach to Pull-In Limit and

better results when compared with Lumped Model.

**(Initial gap=2µm) Vmax(V)** 

**(L=150µm, d=2µm) Voltage (V)** 

Position Control of Electrostatic Cantilever Within the Pull-In Limit 147

Eq. (33) is very similar with Lumped Model's Eq. (18) except coefficient which helps to get

Table 2 and 3 show comparison of Ansys simulation results and values obtained from

**Vmax(V)** 

**Voltage (V)** 

**From ANSYS % Error** 

**From ANSYS % Error** 

**From Bisection Model** 

L=150 27.4707 27.3408 0.475

L=200 15.4523 15.4179 0.223

L=250 9.8894 9.8985 0.092

L=300 6.8677 6.8284 0.575

L=400 3.8631 3.8604 0.069

L=500 2.4724 2.4715 0.036

Table 2. Comparison of Vmax (Pull-in Voltage) values for cantilevers with different lengths

**From Bisection Model** 

δ= 0.01415 (0.71 %) 5.0549 5.0 1.098

δ= 0.05829 (2.91%) 10.1072 10.0 1.072

δ= 0.1386 (6.93%) 15.1544 15.0 1.030

δ= 0.2714 (13.57%) 20.1942 20.0 0.971

δ= 0.5165 (25.83%) 25.2056 25.0 0.822

δ= 0.6028 (30.14%) 26.1918 26.0 0.737

δ= 0.7419 (37.10%) 27.1588 27.0 0.588

δ= 0.7654 (38.27%) 27.2542 27.1 0.569

δ= 0.7963 (39.82%) 27.3518 27.2 0.558

δ= 0.8146 (40.73%) 27.3949 27.25 0.531

δ= 0.8808 (44.04%) 27.4628 27.34 0.449

Table 3. Comparison of Voltage values for arbitrary δ displacements.

Bisection Model and percentage error for cantilever with different lengths.

There is no analytical solution to this equation. Hence, computational result has been obtained as:

$$\frac{\delta\_{\text{max}}}{\text{d}} = 0.440423 \,\text{tag} \tag{31}$$

In addition, when a graph of the equation is drawn, it shows stable and unstable regions (Fig. 5). It can be seen that the Pull-in Limit is at the 44% of the initial gap.

Thus, critical value of the potential difference can be calculated as

$$\mathbf{V}\_{\text{critical}} = \sqrt{\frac{69 \text{ kd}^3}{500 \text{ }\varepsilon\_0 \text{A}}} \tag{32}$$

and insert Eq. (27) into Eq. (32), and replace *A* with w*L*. We can get

$$\mathbf{V}\_{\text{critical}} = \sqrt{\frac{621 \text{ Ed}^3 \text{t}^3}{2000 \text{ } \varepsilon\_0 \text{L}^4}} \tag{33}$$

146 Electrostatics

There is no analytical solution to this equation. Hence, computational result has been

<sup>δ</sup>max = 0.440423

In addition, when a graph of the equation is drawn, it shows stable and unstable regions

(Fig. 5). It can be seen that the Pull-in Limit is at the 44% of the initial gap.

Fig. 5. Stable and Unstable Regions of Bisection Model.

Thus, critical value of the potential difference can be calculated as

and insert Eq. (27) into Eq. (32), and replace *A* with w*L*. We can get

criti

V

cal

critical 4

Ed <sup>V</sup>

2 621 000 3

3

3 t

0

<sup>ε</sup> <sup>=</sup> <sup>A</sup> (32)

<sup>ε</sup> <sup>L</sup> = (33)

0 69 kd 500

d (31)

obtained as:

Eq. (33) is very similar with Lumped Model's Eq. (18) except coefficient which helps to get better results when compared with Lumped Model.

Table 2 and 3 show comparison of Ansys simulation results and values obtained from Bisection Model and percentage error for cantilever with different lengths.


Table 2. Comparison of Vmax (Pull-in Voltage) values for cantilevers with different lengths


Table 3. Comparison of Voltage values for arbitrary δ displacements.

New Approach to Pull-In Limit and

Italie (2006)

ISSN 0018-9383

ISSN 1084-6999

1057-7157

821, ISSN 1057-7157

1340 ISSN 1057-7157

Jan., pp. 473-478.

pp.155-161 ISSN 0924-4247

Miyazaki, Japan, 23-27 Jan., pp. 793-798.

*on,* 16-19 Jan, Bangkok, pp.1190-1195

*and Simulation Workshop,* s.56-60,

Position Control of Electrostatic Cantilever Within the Pull-In Limit 149

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Mol, Lukas.; Rocha, Luis A.; Cretu, Edmond. & Wolffenbuttel, Reinoud F. (2007). Full-Gap

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cardiovascular catheter. *IEEE Trans. Electronic Devices*, vol.35, No.12, pp. 2355-2362,

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Table 4 shows comparison of previous experimental, analytical results (Hu et al., 2004; Sadeghian et al., 2007), Bisection Model Result, and percentage error with respect to experimental results. Ansys simulation results also added for comparison.

Table 4. Comparison of displacements for different voltage values. Errors are respect to experimental results.

#### **4. Conclusion**

Values calculated from Bisection Model are very close to those obtained from ANSYS. Especially, when the displacement is larger than 10% of the initial gap, all the errors are within 1% (see Table 3). Therefore, Bisection Model not only gives a better pull-in limit when compared with previous lumped model, but also has simpler analytical result when compared with previous discrete models. At the same time, it gives satisfactory results for applied voltages for given displacements of top electrode's free end. Bisection Model is also very successful when compared to experimental studies. Percentage error level of Bisection Model is comparable when displacement is small, and gets better when displacement is close to pull-in limit (see Table 4). Bisection Model also gives a simple formula to use instead of using numerical methods which is time consuming and requires computation capacity.

#### **5. References**


148 Electrostatics

Table 4 shows comparison of previous experimental, analytical results (Hu et al., 2004; Sadeghian et al., 2007), Bisection Model Result, and percentage error with respect to

> **Analytical (Sadeghian et al., 2007) / (Error)**

**Bisection Model / (%Error)** 

**ANSYS** 

experimental results. Ansys simulation results also added for comparison.

**Analytical (Hu et al., 2004) / (Error)** 

20 90.5 90.2 / (0.3%) 90.2 / (0.3%) 90.4 / (0.1%) 90.4

40 84.6 84.3 / (0.4%) 84.1 / (0.6%) 85.1 / (0.6%) 85.1

60 70.0 71.5 / (2.1%) 69.1 / (1.2%) 73.2 / (4.5%) 73.2

65 64.0 67.2 / (5.0%) 59.6 / (6.9%) 67.4 / (5.3%) 67.6

67 59.0 65.0 / (10.2%) - 64.1 / (8.7%) 64.5

Table 4. Comparison of displacements for different voltage values. Errors are respect to

Values calculated from Bisection Model are very close to those obtained from ANSYS. Especially, when the displacement is larger than 10% of the initial gap, all the errors are within 1% (see Table 3). Therefore, Bisection Model not only gives a better pull-in limit when compared with previous lumped model, but also has simpler analytical result when compared with previous discrete models. At the same time, it gives satisfactory results for applied voltages for given displacements of top electrode's free end. Bisection Model is also very successful when compared to experimental studies. Percentage error level of Bisection Model is comparable when displacement is small, and gets better when displacement is close to pull-in limit (see Table 4). Bisection Model also gives a simple formula to use instead of using numerical methods which is time consuming and requires

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**Experimental (Hu et al., 2004)** 

**Voltage(V)** 

experimental results.

computation capacity.

7157

**5. References** 

**4. Conclusion** 

*Integration and Packaging of MEMS/MOEMS*, DTIP 2006, Stresa, Lago Maggiore : Italie (2006)


Yu-Hsiang Wang.; Chia-Yen Lee. & Che-Ming Chiang. (2007). A Mems-based Air Flow Sensor with a Free-Standing Microcantilever Structure. *Sensors*, vol.7, no.10, pp.2389-2401

150 Electrostatics

Yu-Hsiang Wang.; Chia-Yen Lee. & Che-Ming Chiang. (2007). A Mems-based Air Flow

pp.2389-2401

Sensor with a Free-Standing Microcantilever Structure. *Sensors*, vol.7, no.10,

### *Edited by Hüseyin Canbolat*

In this book, the authors provide state-of-the-art research studies on electrostatic principles or include the electrostatic phenomena as an important factor. The chapters cover diverse subjects, such as biotechnology, bioengineering, actuation of MEMS, measurement and nanoelectronics. Hopefully, the interested readers will benefit from the book in their studies. It is probable that the presented studies will lead the researchers to develop new ideas to conduct their research.

Electrostatics

Electrostatics

*Edited by Hüseyin Canbolat*

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