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## **Meet the editor**

Vakhrushev Alexander Vasilyevich received his PhD degree in Technical Sciences from Izhevsk Institute of Mechanical Engineering, Russia, in 1982, and degree of Doctor of Sciences in Physics and Mathematics from the Institute of Continuous Media Mechanics of Ural Branch of Russian Academy of Sciences, Russia, in 2003. Now, he is the head of the Department of "Nanotechnol-

ogy and Microsystems Technics" of Kalashnikov Izhevsk State Technical University and head of the Department of "Mechanics of Nanostructures" of the Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences. He has over 400 publications to his name, including monographs, articles, reports, reviews, and patents. He has received several awards, including an Academician A. F. Sidorov Prize from the Ural Division of the Russian Academy of Sciences for significant contribution to the creation of the theoretical fundamentals of physical processes taking place in multilevel nanosystems and Honorable Scientist of the Udmurt Republic. He is currently a member of the editorial board of several journals, including Computational Continuum Mechanics, Chemical Physics and Mesoscopia, and Nanobuild. His research interests include multiscale mathematical modeling of physical-chemical processes into the nanoheterosystems at nano-, micro-, and macrolevels; static and dynamic interaction of nanoelements; and basic laws relating the structure and macrocharacteristics of nanoheterostructures.

## Contents

#### **Preface XI**


## Preface

In recent years, nanotechnology is the basis for the development of modern production. This determined the urgency of the intensive development of the new direction of mechanics and nanomechanics, for the scientific description of nanotechnological processes and the solu‐ tion of a number of topical nanotechnology problems: reduction of time to prepare for the production, acceleration of technological processes, automation of technological processes, reduction in the cost of production, improvement in the quality of materials and products, preparation of high-performance materials, improvement of the ecology, etc. How are nano‐ technological processes different from traditional technologies and how does nanomechan‐ ics fundamentally differ from classical mechanics? The specific feature of the physical processes in nanoscale is that the key phenomena determining the behavior of a nanoscale system in real time at the macroscale take place at small space and time scales. The proper‐ ties of a nanoscale system depend not only on the properties of its constituent elements but also on the regularities of the spatial arrangement of the nanoscale system and the parame‐ ters of the nanoelement interaction. So, classical physical body mechanics, which is based on the hypothesis of continuum per that all the properties of a body are determined by the me‐ chanics of continuum, cannot be applied to nanoscale systems. In this connection, it is obvi‐ ous that describing of nanosystems requires the use of different physical models and mathematical methods and a detailed complex analysis of the "history" of the nanoscale ma‐ terial existence at different structural scales. Therefore, for nanosystems, we use currently nanomechanics, which allows to describe their behavior and properties at different structur‐ al levels. That is why, the development of nanomechanics is very necessary. Nanomechanics includes different physics and mathematics models: quantum mechanics, molecular dynam‐ ics, mesodynamics, and continuum mechanics. Because of the multilevel nature of nanotech‐ nological processes, the problems of nanomechanics are very complex.

This book presents a collection of eight scientific papers on theoretical and practical advan‐ ces in nanomechanics at different structural levels. Topics included in the book cover a wide range of research in the field of nanomechanics: thermomass theory of nanosystems, defor‐ mation of nanomaterials, interface mechanics of assembly carbon nanotube, nanomechanics on surface, molecular interactions and transformations, nanomechanical sensors, nano‐ beams and micromembranes, nanostructural organic and inorganic materials, green synthe‐ sis of metallic nanoparticles. The main goal of these works is the establishment of the nanosystem macroparameter dependence on its nanoparameters using nanomechanics.

Chapter 1 presents the complete material on the problem of interfacial mechanical behaviors in carbon nanotube assemblies. Chapter 2 presents the complete investigations on regulari‐ ties of deformation at high strain rates and damage and fracture of ultrafine-grained alloys in static and in dynamic loading conditions. Chapter 3 presents the complete material on the

problem of changes in the parameters of thermal conductivity when the size of the nanosys‐ tem changes. In Chapter 4, it is shown how to single out small ligands able to inhibit protein misfolding, to measure energetics of surface-confined ferritin during iron loading, and to realize a universal probe for ammine-based designer medication. Chapter 5 presents the complete experimental investigation, analytical calculation, and numerical simulation by the method of MFE of micromembranes supported by serial-parallel connected hinges. Chapter 6 presents the modified couple stress theory for static, vibration, and buckling analysis of nanobeams. Chapter 7 presents the eco-friendly methods of obtaining nanostructures using plant experimental results of metallic and carbon nanostructures formation *ficus-indica* using the extract from *Opuntia* cladodes. Chapter 8 presents the results of experimental and simu‐ lation study of structure and properties of materials, surface of which was modified by ori‐ ented carbon nanotubes and, additionally, was treated by surface electromagnetic waves.

This book will be useful for engineers, technologists, and researchers interested in methods of nanomechanics and in advanced nanomaterials with complex behavior and their applications.

I would like to express my appreciation to all the contributors of this book. My special thanks to the Publishing Process Manager, Ms. Martina Usljebrka, and other staff of InTech publishing for their kind support and great efforts in bringing the book to completion.

#### **Prof. Alexander Vakhrushev**

Head of the Department of "Mechanics of Nanostructures" Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia

Head of the Department of "Nanotechnology and Microsystems," Kalashnikov Izhevsk State Technical University, Izhevsk, Russia

## **Interfacial Mechanical Behaviors in Carbon Nanotube Assemblies**

Xiaohua Zhang

problem of changes in the parameters of thermal conductivity when the size of the nanosys‐ tem changes. In Chapter 4, it is shown how to single out small ligands able to inhibit protein misfolding, to measure energetics of surface-confined ferritin during iron loading, and to realize a universal probe for ammine-based designer medication. Chapter 5 presents the complete experimental investigation, analytical calculation, and numerical simulation by the method of MFE of micromembranes supported by serial-parallel connected hinges. Chapter 6 presents the modified couple stress theory for static, vibration, and buckling analysis of nanobeams. Chapter 7 presents the eco-friendly methods of obtaining nanostructures using plant experimental results of metallic and carbon nanostructures formation *ficus-indica* using the extract from *Opuntia* cladodes. Chapter 8 presents the results of experimental and simu‐ lation study of structure and properties of materials, surface of which was modified by ori‐ ented carbon nanotubes and, additionally, was treated by surface electromagnetic waves. This book will be useful for engineers, technologists, and researchers interested in methods of nanomechanics and in advanced nanomaterials with complex behavior and their applications. I would like to express my appreciation to all the contributors of this book. My special thanks to the Publishing Process Manager, Ms. Martina Usljebrka, and other staff of InTech publishing for their kind support and great efforts in bringing the book to completion.

VIII Preface

**Prof. Alexander Vakhrushev**

Ekaterinburg, Russia

Izhevsk, Russia

Institute of Continuous Media Mechanics

Head of the Department of "Mechanics of Nanostructures"

Head of the Department of "Nanotechnology and Microsystems,"

of the Ural Branch of the Russian Academy of Sciences,

Kalashnikov Izhevsk State Technical University,

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67764

#### **Abstract**

Interface widely exists in carbon nanotube (CNT) assembly materials, taking place at different length scales. It determines severely the mechanical properties of these assembly materials. In this chapter, I assess the mechanical properties of individual CNTs and CNT bundles, the interlayer or intershell mechanics in multiwalled CNTs, the shear properties between adjacent CNTs, and the assembly-dependent mechanical and multifunctional properties of macroscopic CNT fibers and films.

**Keywords:** carbon nanotube, assembly materials, mechanics, interface

### **1. Introduction**

Carbon nanotubes (CNTs) are hollow cylinders consisting of single or multiple sheets of graphite (graphene) wrapped into a cylinder. After the first observation in 1991 by Iijima [1], extensive work has been carried out to CNTs toward their broad mechanical, electronic, thermal, and optical applications [2, 3]. The fundamental mechanical properties, such as their stiffness, strength, and deformability, have been investigated with extensive theoretical and experimental researches, and been well reviewed, in the past two decades [4–8]. Nowadays, for industrial and engineering applications, CNTs have been assembled into macroscopic materials such as fibers, films, forests, and gels [9–15]. Due to the assembly feature of these materials, the interfacial structure between CNTs always plays a crucial role in determining the tensile or compressive behavior, dynamic response, and coupling phenomenon between

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

multiple physical properties that take place at the interfaces. Therefore, the utilization efficiency of the mechanical property from individual CNTs to their assembly is a sophisticated function of the assembly structure and types of interfacial interactions.

In this chapter, I will review the mechanics of individual CNTs, discuss the sliding friction between CNTs, and report our recent progresses on the fabrication technique and mechanical properties of macroscopic CNT assembly materials. The general features of the assembly design and interface design will be illustrated from the perspective of interface engineering. The chapter is organized as follows. Section 2 contains a brief description of the mechanical properties of individual CNTs, in terms of modulus, strength, compressibility, and deformability. Section 3 describes the interlayer (intershell) sliding of multiwalled CNTs (MWCNTs) and intertube friction between adjacent CNTs, and the strategies to enhance the interfacial frictions. Section 4 discusses the recent investigations on the interfacial mechanics in macroscopic CNT assembly materials, especially CNT fibers and films where the CNTs are highly aligned or highly entangled. Finally, the chapter concludes with a brief summary and gives an outlook on future developments in the field.

## **2. Mechanical properties of carbon nanotubes**

#### **2.1. Young's modulus and tensile strength**

The basic mechanical properties of CNT are strongly related to the basic properties of a graphene sheet. In these materials, carbon atoms are covalently bonded with three nearest carbon atoms in a hexagonal lattice by forming three σ bonds in the sheet plane and one delocalized π-bond over the basal plane. As the sp2 σ bond is shorter and stronger even than the sp<sup>3</sup> bond in diamond, the axial elastic modulus of CNT and graphene is super high, as expected to be up to several TPa [6, 7]. The first experimental study on the modulus was based on the analysis of thermal vibration of MWCNTs, which showed a wide range of 0.4–4.15 TPa [16]. Later, the modulus was also estimated by the compressive response using micro Raman spectroscopy and was 2.8–3.6 TPa for singlewalled CNTs (SWCNTs) and 1.7–2.4 GPa for MWCNTs [17]. Direct tensile tests showed the modulus ranged from 270 to 950 GPa for individual MWCNTs [18], and from 320 to 1470 GPa when several SWCNTs bundled together as a CNT rope [19]. Besides the experimental evaluations, theoretical estimations also show the similar results. For example, by using an empirical Keating Hamiltonian with parameters determined from first principles, a modulus ranging from 1.5 to 5.0 TPa was estimated [20]. A molecular dynamics (MD) approach showed a modulus of ~1 TPa and a shear modulus of ~0.5 TPa and predicted that chirality, radius, and number of walls have little effect on the value of Young's modulus [21].

Different from the modulus estimation, measuring the tensile strength of CNTs is a very challenging task. Compared with experiment, it is easier to compute the strength by considering the effects of defects [22, 23], loading rate, temperature [24], and number of walls [25]. Typically, the fraction of CNT involves bond breakage and/or rotation, usually resulting in the formation of specific types of dislocation, such as the pentagon-heptagon (5-7), (5-7-7-5), (5-7-5-8-5) defects [26]. MD simulations showed that CNT under tension behaves as a brittle material at high strain (15%) and low temperature (1300 K), or as a ductile material at low strain (3%) and high temperature (3000 K) [26–28]. Such tensile behaviors depend on the evolution of the dislocation. For example, the (5-7-7-5) dislocation can either separate into two (5-7) pairs to result in a ductile transformation or involve into a crack to result in a brittle fracture. An alternative pathway for the fracture of CNT was also proposed by the direct bond-breaking through the formation of a series of virtual defects at high tensions [23]. These studies were usually based on the consideration of the formation of energy for the dislocations. However, another MD study showed that the fracture behavior is almost independent of the separation energy and to depend primarily on the inflection point in the interatomic potential [22]. The fracture strength should be moderately dependent on tube chirality, and the strength and fracture strain were estimated to be 93.5–112 GPa and 15.8–18.7%, respectively. **Figure 1a** shows two different stress-strain curves for a (20,0) CNT, where the modified Morse potentials with interatomic force peaks at 19 and 13% strain were used. To show the effect of structural defect, a result for a missing atom is provided where the strength was reduced by about 25%.

multiple physical properties that take place at the interfaces. Therefore, the utilization efficiency of the mechanical property from individual CNTs to their assembly is a sophisticated

In this chapter, I will review the mechanics of individual CNTs, discuss the sliding friction between CNTs, and report our recent progresses on the fabrication technique and mechanical properties of macroscopic CNT assembly materials. The general features of the assembly design and interface design will be illustrated from the perspective of interface engineering. The chapter is organized as follows. Section 2 contains a brief description of the mechanical properties of individual CNTs, in terms of modulus, strength, compressibility, and deformability. Section 3 describes the interlayer (intershell) sliding of multiwalled CNTs (MWCNTs) and intertube friction between adjacent CNTs, and the strategies to enhance the interfacial frictions. Section 4 discusses the recent investigations on the interfacial mechanics in macroscopic CNT assembly materials, especially CNT fibers and films where the CNTs are highly aligned or highly entangled. Finally, the chapter concludes with a brief summary and gives

The basic mechanical properties of CNT are strongly related to the basic properties of a graphene sheet. In these materials, carbon atoms are covalently bonded with three nearest carbon atoms in a hexagonal lattice by forming three σ bonds in the sheet plane and one delocalized π-bond over

the axial elastic modulus of CNT and graphene is super high, as expected to be up to several TPa [6, 7]. The first experimental study on the modulus was based on the analysis of thermal vibration of MWCNTs, which showed a wide range of 0.4–4.15 TPa [16]. Later, the modulus was also estimated by the compressive response using micro Raman spectroscopy and was 2.8–3.6 TPa for singlewalled CNTs (SWCNTs) and 1.7–2.4 GPa for MWCNTs [17]. Direct tensile tests showed the modulus ranged from 270 to 950 GPa for individual MWCNTs [18], and from 320 to 1470 GPa when several SWCNTs bundled together as a CNT rope [19]. Besides the experimental evaluations, theoretical estimations also show the similar results. For example, by using an empirical Keating Hamiltonian with parameters determined from first principles, a modulus ranging from 1.5 to 5.0 TPa was estimated [20]. A molecular dynamics (MD) approach showed a modulus of ~1 TPa and a shear modulus of ~0.5 TPa and predicted that chirality, radius, and number of walls

Different from the modulus estimation, measuring the tensile strength of CNTs is a very challenging task. Compared with experiment, it is easier to compute the strength by considering the effects of defects [22, 23], loading rate, temperature [24], and number of walls [25]. Typically, the fraction of CNT involves bond breakage and/or rotation, usually resulting in the formation of specific types of dislocation, such as the pentagon-heptagon (5-7), (5-7-7-5), (5-7-5-8-5) defects [26]. MD simulations showed that CNT under tension behaves as a brittle

σ bond is shorter and stronger even than the sp<sup>3</sup>

bond in diamond,

function of the assembly structure and types of interfacial interactions.

an outlook on future developments in the field.

**2.1. Young's modulus and tensile strength**

the basal plane. As the sp2

2 Nanomechanics

**2. Mechanical properties of carbon nanotubes**

have little effect on the value of Young's modulus [21].

The tensile strength and the fracture of CNT were also measured directly with experiment [18, 19]. The measurement of 19 MWCNTs showed a strength of 11–63 GPa and a strain at break up to 12% [18]. The failure of MWCNTs was described in term of the "sword-insheath" type fracture, where the outermost shell broke first followed by the pull out of the rest of the shells from the outermost shell. When SWCNTs were bundled to be a rope, the tensile strength was measured to be 13–52 GPa, based on 15 measurements, and the strain at break was up to 5.3%, see **Figure 1b** [19]. The stretchability of CNT was also experimentally obtained by deforming freely suspended SWCNT ropes, where the maximum strain was up to 5.8 ± 0.9% [29]. By measuring the stress-induced fragmentation of MWCNTs in a polymer matrix, the tube strength was up to 55 GPa [30]. Another pulling and bending tests on individual CNTs in-situ in a transition electron microscope (TEM) showed a tensile strength of ~150 GPa, suggesting that the strength is a large fraction of the elastic modulus [31].

**Figure 1.** (a) A strong dependence on the inflection point in the interatomic potential can be seen from the stress-stain curves for a (20,0) CNT, and a missing atom has also a significant effect on strength, reducing it by about 25% [22]. Experimental measurements on MWCNTs are provided for comparison [18]. (b) Eight stress-strain curves obtained from the tensile-loading experiments on SWCNT ropes [19].

#### **2.2. Compressibility and deformability**

Different from the high rigidity and high strength along the axial direction, CNT is relatively compressible and deformable in the transverse direction. When individual SWCNTs are bundled together, significant deformation could occur by van der Waals (vdW) interaction between the adjacent tubes [32]. Fully collapsed MWCNTs were first observed with TEM [33]. Large diameter CNTs could also form the partial and full collapse on substrate [34–36]. The structural deformation can be induced by applying pressure. Various experiments on SWCNT bundles have shown clear evidence of structural deformation or transition [37–42]. MD simulations revealed that the compressibility and deformability are dependent on the tube diameter and the number of walls [43]. By applying a certain high pressure, there could be a circular-to-elliptical shape transition of the tube's cross section for either individual CNTs or their bundles [44–51]. These structural deformations are all reversible upon unloading the pressure. In a different way, a very large force exerted on CNT could still produce reversible and elastic deformation, and that radial mechanical forces might not be capable of cutting a CNT [43]. By examining the ballistic impact and bouncing-back processes, large diameter CNTs could withstand high bullet speeds [52].

Upon compressing, bending, or twisting, structural deformations of CNT become more complicated. With an increasing compression stress, different buckling patterns in CNT at the point of instabilities can be created [53]. As shown in **Figure 2**, each shape change corresponds

**Figure 2.** The strain energy (a) displays four singularities corresponding to shape changes (b–e) with increasing the axial compression [53].

to an abrupt release of energy and a singularity in the stress-strain curve. MD simulations showed that the presence of intertube vdW interactions tends to weaken the CNT bundles under compression [55]. On the other hand, with increasing the tube diameter, the bending modulus was found to decrease sharply [56], as a rippling mode becomes energetically favorable [57, 58]. The modeling on the wavelike distortion indicated that there is a critical diameter at a given load and a CNT length, for the emergence of the rippling mode [59]. It was also found that thick MWCNTs are very prone to develop rippling deformations in bending and twisting, governed by the interplay of strain energy relaxation and intertube interactions [58, 60]. **Figure 3** shows experimental observations of partially collapsed segments in freestanding and twisted MWCNTs with TEM [54].

**Figure 3.** TEM images showing a suspended, fully collapsed, and twisted MWCNT (a), and a freestanding, fully collapsed, and twisted MWCNT (b) on a lacy carbon TEM grid [7, 54].

### **3. Interlayer and intertube interactions**

#### **3.1. Sliding between MWCNT shells**

**2.2. Compressibility and deformability**

4 Nanomechanics

compression [53].

Different from the high rigidity and high strength along the axial direction, CNT is relatively compressible and deformable in the transverse direction. When individual SWCNTs are bundled together, significant deformation could occur by van der Waals (vdW) interaction between the adjacent tubes [32]. Fully collapsed MWCNTs were first observed with TEM [33]. Large diameter CNTs could also form the partial and full collapse on substrate [34–36]. The structural deformation can be induced by applying pressure. Various experiments on SWCNT bundles have shown clear evidence of structural deformation or transition [37–42]. MD simulations revealed that the compressibility and deformability are dependent on the tube diameter and the number of walls [43]. By applying a certain high pressure, there could be a circular-to-elliptical shape transition of the tube's cross section for either individual CNTs or their bundles [44–51]. These structural deformations are all reversible upon unloading the pressure. In a different way, a very large force exerted on CNT could still produce reversible and elastic deformation, and that radial mechanical forces might not be capable of cutting a CNT [43]. By examining the ballistic impact and bouncing-back processes, large diameter CNTs could withstand high bullet speeds [52].

Upon compressing, bending, or twisting, structural deformations of CNT become more complicated. With an increasing compression stress, different buckling patterns in CNT at the point of instabilities can be created [53]. As shown in **Figure 2**, each shape change corresponds

**Figure 2.** The strain energy (a) displays four singularities corresponding to shape changes (b–e) with increasing the axial

Different from SWCNTs, the interlayer interactions in MWCNTs exhibit interesting sliding phenomena during the tensile stretching and self-oscillation. By modeling of the interlayer interaction with different atomic potentials, such as the classic Lennard-Jones (LJ) potential and the registry-dependent Kolmogorov-Crespi (KC) potential (zeroth generation RDP0 [61] and first generation RPD1 [62]), could cause significantly different energy corrugation between graphitic layers [62–65]. Nevertheless, the interlayer sliding has been systematically studied despite the type of atomic potential. Experiments have also shown the interlayer sliding and superlubricity of CNTs.

Experimental measurements on the dependence of the sliding force against the contact length between the MWCNT shells suggested several responsible sources including the surface tension, the shear elastic force, and the edge (tube termination) effect force [66]. The shear strength was found to be 0.08–0.3 MPa depending on the interlayer commensurability. After the inner shells were pulled out from the outer shells, spontaneous retraction was observed with TEM due to the attractive vdW forces between these shells [67]. The interlayer static and dynamic shear strengths were estimated to be 0.43–0.66 MPa. The telescoping motion of an MWCNT also showed an ultralow friction, below 1.4 × 10−15 N per atom (8.7 × 10−4 meV Å−1) [68]. More direct observation of the superlubricity in doublewalled CNTs (DWCNTs) was conducted by extracting the inner shell out from a centimeters-long DWCNT, where the inner-shell friction (vdW force) was estimated to be 1.37–1.64 nN (the total number of atoms was unknown, yet could be over 10<sup>8</sup> –109 for a tube length of 1 cm) [69], corresponding to a friction force of about 10−6–10−5 meV Å−1 per atom. By following the low-friction induced telescoping, gigahertz oscillators were theoretically proposed based on the oscillatory extrusion and retraction of the inner shells [70–77], and the interlayer commensurability determines the rate of energy dissipation during the oscillation [71]. The friction force between oscillatory layers was estimated to range from 10−17 to 10−14 N per atom, about 6 × 10−6–0.006 meV Å−1 [73]. Similarly, the rotation between CNT shells and the rotational friction was observed or measured by experiments and simulations [78–81]. The rotational friction is a broadband phenomenon, as it does not depend strongly on a specific vibrational mode but rather appears to occur from the aggregate interactions of many modes at different frequencies [82].

At high sliding speeds, the friction between CNT layers was not hydrodynamical, in fact not even monotonic with the sliding speed [83]. A high speed can develop a sharp friction peak and onset parametric excitation of CNT's "breathing" phonon modes. **Figure 4a** shows that there is

**Figure 4.** (a) There is a massive increase in friction at ~1100 m s−1 for the (5,5)@(10,10) DWCNT [83]. (b) The center-ofmass angular momentum *J* CM (top) and the sliding frictional force per inner tube atom (bottom) as functions of the sliding speed [84]. A, B, and C are three resonance regions related the giant friction forces.

a massive increase in sliding friction at a speed of ~1100 m s−1 for a (5,5)@(10,10) DWCNT. The highest friction force could be up to 170 pN, corresponding to ~0.15 meV Å−1 per inner tube atom. The main source of the interlayer friction was believed to be the inner nanotube terminations [85], while the "bulk" friction was unfortunately considered to be irrelevant.

To understand the "bulk" frictional behavior, two groups of studies were carried out [65, 84, 86]; pulling simulations where a constant external force was applied on each atom of the inner tube along the tube axis, and rigid sliding simulations where the inner tube was constrained to slide as a whole at a fixed speed. The pulling simulation revealed a static friction (depinning) force of 0.02 meV Å−1 at 300 K and 0.1 meV Å−1 at 50 K for the (5,5)@(10,10) DWCNT. After the depinning, the speed of the inner tube jumps directly to plateaus. Three speed plateaus are seen at v ≈ 450, 720, and 780 m s−1. This phenomenon is ubiquitous as it is also observed for an incommensurate case where a chiral (11,2) tube slides inside a (12,12). The general occurrence of plateaus and jumps is a natural consequence of frictional peaks for growing the speed. **Figure 4b** shows sharp frictional peaks near *v* ≈ 450, 570, 720, and an important threshold onset near 780 m s−1. These peaks are known to generally arise out of parametric excitation of the "breathing" phonon modes, classified by an angular momentum index *n* (for tangential quantization around the tube axis). Such nonmonotonic force-speed characteristics imply a "negative differential friction," whereby an increasing applied force yields an inner tube sliding speed that grows by jumps and plateaus, rather than smoothly.

The surprise comes from analyzing the two parts of angular momentum *J* = *J* CM + *J*pseudo around the tube axis such as its center-of-mass (rigid body rotation with angular velocity *ω*) and shape-rotation ("pseudorotational") parts [65, 84]. Generally, zero at generic sliding speed due to lack of nanotube chirality, *J* pseudo = −*J*CM = 0, *J*pseudo jumps to nonzero values at the frictional peaks and past the threshold, where *J* pseudo = −*J*CM / ≠ 0, see **Figure 4b**. This is a spontaneous breaking of chiral symmetry occurring in nanoscale friction, whose mechanism is from the third- and fourth-order energy nonlinearities. The inner-tube sliding causes a washboard frequency due to the lattice periodicity *a*CC = 2.46 Å. At the critical frequencies, the '*n* = 5' or '2 + 3 = 5' excitations are observed, where the DWCNT's *n* = 5 mode is resonantly excited or the *n* = 2, 3, 5 modes are jointly excited by the simple matching condition 2 + 3 = 5. Besides these "normal" excitations (labeled A in **Figure 4b**), there are '*n* = 5s' and '2 + 3s = 5' excitations, where the 3s and 5s are the single outer tube's phonon modes. These excitations are labeled B in **Figure 4b**. The two C-peaks in **Figure 4b** are ascribed to the phenomenon of strong stickslip frictions, possibly related to the phonon modes along the tube axis.

#### **3.2. Intertube friction**

Experimental measurements on the dependence of the sliding force against the contact length between the MWCNT shells suggested several responsible sources including the surface tension, the shear elastic force, and the edge (tube termination) effect force [66]. The shear strength was found to be 0.08–0.3 MPa depending on the interlayer commensurability. After the inner shells were pulled out from the outer shells, spontaneous retraction was observed with TEM due to the attractive vdW forces between these shells [67]. The interlayer static and dynamic shear strengths were estimated to be 0.43–0.66 MPa. The telescoping motion of an MWCNT also showed an ultralow friction, below 1.4 × 10−15 N per atom (8.7 × 10−4 meV Å−1) [68]. More direct observation of the superlubricity in doublewalled CNTs (DWCNTs) was conducted by extracting the inner shell out from a centimeters-long DWCNT, where the inner-shell friction (vdW force) was estimated

length of 1 cm) [69], corresponding to a friction force of about 10−6–10−5 meV Å−1 per atom. By following the low-friction induced telescoping, gigahertz oscillators were theoretically proposed based on the oscillatory extrusion and retraction of the inner shells [70–77], and the interlayer commensurability determines the rate of energy dissipation during the oscillation [71]. The friction force between oscillatory layers was estimated to range from 10−17 to 10−14 N per atom, about 6 × 10−6–0.006 meV Å−1 [73]. Similarly, the rotation between CNT shells and the rotational friction was observed or measured by experiments and simulations [78–81]. The rotational friction is a broadband phenomenon, as it does not depend strongly on a specific vibrational mode but rather appears to occur from the aggregate interactions of many modes at different frequencies [82].

At high sliding speeds, the friction between CNT layers was not hydrodynamical, in fact not even monotonic with the sliding speed [83]. A high speed can develop a sharp friction peak and onset parametric excitation of CNT's "breathing" phonon modes. **Figure 4a** shows that there is

**Figure 4.** (a) There is a massive increase in friction at ~1100 m s−1 for the (5,5)@(10,10) DWCNT [83]. (b) The center-of-

speed [84]. A, B, and C are three resonance regions related the giant friction forces.

CM (top) and the sliding frictional force per inner tube atom (bottom) as functions of the sliding

mass angular momentum *J*

6 Nanomechanics

–109

for a tube

to be 1.37–1.64 nN (the total number of atoms was unknown, yet could be over 10<sup>8</sup>

In most of the cases, individual CNTs are bundled together as a CNT rope, owing to the strong aggregation tendency between adjacent CNTs. As mentioned above, experimental measurement showed a tensile strength just up to 13–52 GPa [19], far below the ideal strength for individual CNTs. The easy sliding between adjacent CNTs has become a severe problem hindering their engineering applications. For example, slip rather than breakage of individual CNTs was observed in the fracture test of polymers reinforced by CNT bundles [87]. As CNTs can contact with each other in parallel way or at certain cross angles, the interfacial shear strengths between CNTs should be discussed separately.

The static and kinetic frictions between two perpendicular CNTs were investigated experimentally. A coefficient of friction of 0.006 ± 0.003 was obtained by sliding an MWCNT perpendicularly on a SWCNT surface, and the shear strength was derived to be 4 ± 1 MPa [88]. The shear strength was one to two orders of magnitude larger than the intershell shear strength of 0.05 MPa for MWCNTs in vacuum, possibly due to the presence of water at the intertube interface in ambient. A following experiment, namely a vertical friction loop measurement, was proposed to characterize the adhesion and friction properties between CNTs [89]. An MWCNT tip was ramped in the vertical direction against a suspended SWCNT. During ramping, a stick-slip motion was found to dominate the sliding, attributed to the presence of defectinduced or amorphous-carbon-associated high energy points on the tip surface. Surprisingly, the coefficients of static friction and shear strength were finally evaluated to be about 0.2 and 1.4 GPa, respectively.

MD simulations showed that the sliding behaviors and interfacial shear strengths between two crossly contacting CNTs are much more complicated than the frictions between CNTs and graphite, graphite and graphite, and the inner and outer shells of MWCNTs [90]. The simulation was performed on two SWCNTs in contact at different cross angles. For the parallel contact, the axial interfacial shear strengths between the zigzag-zigzag (~0.25 GPa) and armchair-armchair pairs (~0.5 GPa) are two orders of magnitude larger than those of CNTs having different chiralities (0.5–1 MPa). CNTs with diameters larger than 1 nm will slide relatively from an AB stacking position to the next nearest AB stacking position while smaller ones do not slide through AB stacking positions. For two cross contacting CNTs, the magnitude of interfacial shear strength is much less dependent upon tube chirality. The highest value of shear strength was reported to be about 1 GPa in SWCNT bundles, however, still as quite low as to be improved for applications in high-performance composite materials [91].

#### **3.3. Strategies to improve the intertube load transfer**

A molecular mechanics modeling showed that in order to make the strength of a CNT bundle as high as that of the lowest value of individual CNTs (11 GPa), the contact length should be up to ~3800 nm, by assuming the CNTs are bundled together without changing their circular cross sections [63]. With considering the vdW-force-induced radial deformation [32], the intertube distance-of-closest changes from 3.34 to 3.25 Å for (10,10) CNT bundles, the required contact distance for 11 GPa can decrease to ~1300 nm. This implies that the key to improve the intertube load transfer is the interface design between CNTs.

Twist is an effective way to induce strong radial deformation for CNT bundles. **Figure 5a** shows the cross sections obtained from MD simulations with increasing the twist level and **Figure 5b** shows the corresponding transferred load due to the shape deformation at twist angles of 0, 30, 60, 90, 120, 150, 180, and 210° [63]. It was observed that the individual CNTs start to collapse in the cross section, and therefore, more atoms are in close contact. For the case of no twist, a force of only ~0.048 eV Å−1 is transferred to the center tube, while at 120° the transferred load increases to 1.63 eV Å−1. Clearly, the radial deformation strongly depends on the twist angle, which consequently changes the nature of the contact and contributes to a new interlayer tribology.

The static and kinetic frictions between two perpendicular CNTs were investigated experimentally. A coefficient of friction of 0.006 ± 0.003 was obtained by sliding an MWCNT perpendicularly on a SWCNT surface, and the shear strength was derived to be 4 ± 1 MPa [88]. The shear strength was one to two orders of magnitude larger than the intershell shear strength of 0.05 MPa for MWCNTs in vacuum, possibly due to the presence of water at the intertube interface in ambient. A following experiment, namely a vertical friction loop measurement, was proposed to characterize the adhesion and friction properties between CNTs [89]. An MWCNT tip was ramped in the vertical direction against a suspended SWCNT. During ramping, a stick-slip motion was found to dominate the sliding, attributed to the presence of defectinduced or amorphous-carbon-associated high energy points on the tip surface. Surprisingly, the coefficients of static friction and shear strength were finally evaluated to be about 0.2 and

MD simulations showed that the sliding behaviors and interfacial shear strengths between two crossly contacting CNTs are much more complicated than the frictions between CNTs and graphite, graphite and graphite, and the inner and outer shells of MWCNTs [90]. The simulation was performed on two SWCNTs in contact at different cross angles. For the parallel contact, the axial interfacial shear strengths between the zigzag-zigzag (~0.25 GPa) and armchair-armchair pairs (~0.5 GPa) are two orders of magnitude larger than those of CNTs having different chiralities (0.5–1 MPa). CNTs with diameters larger than 1 nm will slide relatively from an AB stacking position to the next nearest AB stacking position while smaller ones do not slide through AB stacking positions. For two cross contacting CNTs, the magnitude of interfacial shear strength is much less dependent upon tube chirality. The highest value of shear strength was reported to be about 1 GPa in SWCNT bundles, however, still as quite low as to be improved for applications in high-performance composite materials [91].

A molecular mechanics modeling showed that in order to make the strength of a CNT bundle as high as that of the lowest value of individual CNTs (11 GPa), the contact length should be up to ~3800 nm, by assuming the CNTs are bundled together without changing their circular cross sections [63]. With considering the vdW-force-induced radial deformation [32], the intertube distance-of-closest changes from 3.34 to 3.25 Å for (10,10) CNT bundles, the required contact distance for 11 GPa can decrease to ~1300 nm. This implies that the key to

Twist is an effective way to induce strong radial deformation for CNT bundles. **Figure 5a** shows the cross sections obtained from MD simulations with increasing the twist level and **Figure 5b** shows the corresponding transferred load due to the shape deformation at twist angles of 0, 30, 60, 90, 120, 150, 180, and 210° [63]. It was observed that the individual CNTs start to collapse in the cross section, and therefore, more atoms are in close contact. For the case of no twist, a force of only ~0.048 eV Å−1 is transferred to the center tube, while at 120° the transferred load increases to 1.63 eV Å−1. Clearly, the radial deformation strongly depends on the twist angle, which consequently changes the nature of the contact and contributes to a

improve the intertube load transfer is the interface design between CNTs.

1.4 GPa, respectively.

8 Nanomechanics

new interlayer tribology.

**3.3. Strategies to improve the intertube load transfer**

**Figure 5.** (a) Change in cross section at the mid-length of a SWCNT bundle with increasing the twist level and (b) the corresponding load transfer ability as a function of twist angle [63].

Even a slight structural deformation could cause an improved load transfer. For example, when a DWCNT was slightly bent into a curved one, the energy transfer between the two tubes becomes much faster than that between two straight tubes [81]. Slight displacement of the inner tube could add a large degree of disturbance to the DWCNT's original state of equilibrium and thus present strong inertia to oppose the extraction process, indicating the importance of the inner tube as structural support for the outer tube [92].

Pressure-induced structural phase transition could be a new strategy to improve the intertube load transfer [51]. For a CNT bundle containing 16 (23,0) SWCNTs, the intertube static friction can be improve from 0.1 eV Å−1 per atom to 0.4–0.45 eV Å−1 after all the tubes collapse under pressure higher than 0.5 GPa, see **Figure 6**. Due to the large tube diameter, the collapsed CNT structure

**Figure 6.** The intertube friction force can be increased by a factor of 1.5–4, depending on tube chirality and radius, when all tubes collapse above a critical pressure and when the bundle remains collapsed with unloading down to atmospheric pressure [51].

can be well maintained even by unloading the pressure down to 0.05 GPa. In order to realize a 10 GPa tensile strength, the minimum tube length should be just 167 nm, much smaller than the length of 1090 nm before the phase transition. For a CNT bundle containing totally different tube chiralities, the dynamic friction force can be improved from 0.0017 to 0.0026 eV Å−1 after the phase transition. Such study also indicated that fewwalled large-diameter CNTs could be very important for engineering applications due to the easy ability to induce the collapse phase transition.

Covalent bonding between the shells in MWCNTs or between adjacent CNTs can increase the interlayer and intertube shear strength by several orders of magnitude [93–95]. The covalent intershell bonds can be formed due to the on-shell vacancies or intershell interstitials. According to the types of defect, the force needed to initiate sliding of the shells varies from 0.08 to 0.4 nN for a single vacancy to 3.8–7.8 nN for two vacancies, an intershell interstitial, and an intershell dimer [93]. Therefore, small-dose electron or ion irradiations are suggested to partially transfer the load to the nanotube inner shells. MD simulations revealed that interwall sp3 bonds and interstitial carbon atoms can increase load transfer between DWCNT walls and that interwall sp3 bonds are the most effective [95]. Similarly, by using moderate electron-beam irradiation inside a TEM, stable links between adjacent CNTs within bundles were covalently formed [94]. At a high irradiation energy of 200 keV, there was a substantial increase in effective bending modulus and thus an increase in shear modulus for low doses, while a long irradiation time rather damaged the well-ordered structure of tubes (**Figure 7**). At 80 keV, electrons were shown not to damage isolated CNTs but had enough energy to lead to the formation

**Figure 7.** Behavior of bending modulus as a function of received dose for two incident electron energies [94]. For 200 keV, the modulus increases on short exposures, due to cross-linking and degrades at higher exposures because of structural damage. The bundle irradiated with 80-keV electrons shows a much stronger and sharper increase of the bending modulus.

of mobile interstitial atoms in the confined space between adjacent CNTs. As a result of the newly formed covalent bonds, a huge increase of the bending modulus by a factor of thirty was observed (**Figure 7**). However, a long-time dose at low energies could still cause the accumulation of damage to the structure, making the mechanical performance to decrease eventually.

Even without introducing covalent bonding between the sp2 layers, the existence of structural defects in either CNT layer could lead to a sharp increase in friction and energy dissipation rate [64].

However, besides the tube structure, defect level, and intertube contact, there are many other issues in macroscopic CNT assembly materials. For example, the CNT packing density, alignment, entanglement, aggregation size, and surface functionalization could be important structural parameters, as to be discussed below.

## **4. Carbon nanotube assembly materials**

So far, CNT fibers, films, forests, and gels are well-known and typical macroscopic assembly materials where the CNTs are the main constituents, especially at a mass fraction much more than 50% [10]. For example, CNT fiber is a one-dimensional assembly containing millions of individual tubes [9, 11–13, 15]. It has been found that CNT fibers could have much higher specific modulus and specific strength than those of commercial carbon and polymeric fibers. For the composite materials, there have been two different applications of CNTs such as a dispersion-based fabrication by using CNTs as reinforcing fillers in a polymer matrix [96, 97] and an assembly-based fabrication by directly assembling CNTs into composites [14]. In the former, the CNT content is strongly limited to be no larger than ~5 wt%, while in the latter, a small amount of polymer molecules (about 20–30 wt%) are used to strengthen the interfaces between CNTs. To describe these macroscopic CNT materials, different structural parameters are proposed and investigated in the past decade.

#### **4.1. CNT fibers**

can be well maintained even by unloading the pressure down to 0.05 GPa. In order to realize a 10 GPa tensile strength, the minimum tube length should be just 167 nm, much smaller than the length of 1090 nm before the phase transition. For a CNT bundle containing totally different tube chiralities, the dynamic friction force can be improved from 0.0017 to 0.0026 eV Å−1 after the phase transition. Such study also indicated that fewwalled large-diameter CNTs could be very important for engineering applications due to the easy ability to induce the collapse phase transition. Covalent bonding between the shells in MWCNTs or between adjacent CNTs can increase the interlayer and intertube shear strength by several orders of magnitude [93–95]. The covalent intershell bonds can be formed due to the on-shell vacancies or intershell interstitials. According to the types of defect, the force needed to initiate sliding of the shells varies from 0.08 to 0.4 nN for a single vacancy to 3.8–7.8 nN for two vacancies, an intershell interstitial, and an intershell dimer [93]. Therefore, small-dose electron or ion irradiations are suggested to partially transfer the load to the nanotube inner shells. MD simulations revealed that interwall

bonds and interstitial carbon atoms can increase load transfer between DWCNT walls and

irradiation inside a TEM, stable links between adjacent CNTs within bundles were covalently formed [94]. At a high irradiation energy of 200 keV, there was a substantial increase in effective bending modulus and thus an increase in shear modulus for low doses, while a long irradiation time rather damaged the well-ordered structure of tubes (**Figure 7**). At 80 keV, electrons were shown not to damage isolated CNTs but had enough energy to lead to the formation

**Figure 7.** Behavior of bending modulus as a function of received dose for two incident electron energies [94]. For 200 keV, the modulus increases on short exposures, due to cross-linking and degrades at higher exposures because of structural damage. The bundle irradiated with 80-keV electrons shows a much stronger and sharper increase of the

bonds are the most effective [95]. Similarly, by using moderate electron-beam

sp3

10 Nanomechanics

that interwall sp3

bending modulus.

CNT fiber can be produced by the coagulation-based "wet spinning" [98–101], the "direct spinning" from a CNT aerogel [102–104] or similarly from a pre-formed CNT film [105], and the "forest spinning" based on vertically aligned CNT forests [106–108]. A CNT fiber with a diameter of ~10 μm usually contains more than 10<sup>6</sup> individual CNTs. Such assembly feature is different from those well-known fibers for many years, which are usually a solid structure without internal interfaces. CNT fiber can also be considered a continuous length of interlocked "filaments" (CNT bundles), where the bundles are formed during the CNT growth rather than in the spinning process. As I have pointed out recently [109], although also being called as CNT yarn, CNT fiber is indeed not a yarn because the CNT bundles are not macroscopically processable. On the contrary, the basic components of a yarn, the long and parallel or interlocked filaments, are usually processable objects with a width larger than several micrometers [110].

Twist is the most fundamental treatment to make the CNTs assembled into a continuous fiber with a circular cross section. The as-spun fiber is relatively loose with noticeable spaces between CNTs or CNT bundles when the twist angle is small. Increasing the twist angle is an effective method to densify CNT fibers and thus improves the friction coefficient between the CNTs. **Figure 8** shows the strength-twist relationship for different CNT fibers [111–113]. All studies showed that the highest fiber strength was found at a twist angle of 15–20°. However, with further increasing the twist, the load (upon stretching) compresses the fiber rather than acts totally along the fiber axis due to the twist angle between the CNTs and the axis, making the utilization efficiency to decrease at high twist angles. The experimental observations that the strength initially increases with twist angle and then decreases are almost the same as those in traditional textile yarns. Nevertheless, the constituent CNTs are different from the filaments in a textile yarn; the CNTs are hollow cylinders and can deform the cross section under pressure due to a structural phase transition [44–49]. For those fibers spun from fewwalled CNTs (wall numbers smaller than 6), it can be seen that beyond the optimal twist angle, an additional strength peak is at large twist angle of 27–30°. This second peak arises from the collapse transformation of the CNT hollow structure, which renders the CNT bundles much stronger and reduces the bundle cross-sectional area.

Solvent densification is another interfacial treatment to CNT assemblies, due to a large capillary force [114, 115]. Therefore, liquid infiltration by using water, ethanol, acetone, or dimethyl sulfoxide (DMSO) is also often used in the method of array spinning to densify CNT fibers [116–119]. Although the densification process does not improve nanotube orientation, it enhances the load transfer between the nanotubes, thus ensuring that most of them are fully load-bearing. For example, the fiber diameter was shrunk from 11.5 to 9.7 μm after acetone shrinking [117]. However, the role of solvent is hardly known as the capillary force can be influenced by the solvents volatility (boiling point), surface tension, and interaction with CNT surfaces. By comparing various solvents including the nonpolar solvents of *n*-hexane, cyclohexane, cyclohexene, toluene, and styrene, polar protic solvents of glycerine, methanol, ethanol, water, ethylene glycol, and 1,3-propanediol, and polar aprotic solvents of

**Figure 8.** (a) Double-peak strength behavior of CNT fibers with increasing twist angles [111]. The surface twist angle (*θs* ) depends on the twisting and drawing speeds (*vt* and *vd* ) by *θ<sup>s</sup>* = arccos {(*vd* /*vt* )/[(π*d*<sup>f</sup> ) 2 + (*vd* /*vt* ) 2 ]1/2}, where *d*<sup>f</sup> is the fiber diameter and is >10.5 μm. (b) Dependence of yarn tensile strength on helix angle for forest-spun MWCNT fibers [112]. The fiber diameter was ~20 μm. (c) Relationship between the fiber's specific strength and tenacity with the surface twist angle [113].

ethyl acetate, acetone, acetonitrile, *N,N*-dimethylformamide (DMF), DMSO, and *N*-methyl-2-pyrrolidone (NMP), only few highly polar solvents show high densifying ability to CNTs, namely DMF, DMSO, NMP, and ethylene glycol, despite of their high boiling points and low evaporation rates [118]. These solvents increased the fiber strength from 864 MPa (dryspun) or 1.19 GPa (ethanol) to 1.14–1.35 GPa (DMF, DMSO, NMP) or 1.33–1.58 GPa (ethylene glycol). Clearly, the solvent polarity shows a key role in determining the capillary, as the polarity induced attractive binding energy with CNT (*E*ind) goes up quadratically with the solvent's local dipole moment *μ*, by *E*ind = −*μ*<sup>2</sup> *α*/(4π*ε*<sup>0</sup> )2 *r*6 , where α is the static polarizability of CNT, *ε*<sup>0</sup> is the vacuum permittivity, and *r* is the distance between the dipole moment and the CNT surface [120, 121].

Twist is the most fundamental treatment to make the CNTs assembled into a continuous fiber with a circular cross section. The as-spun fiber is relatively loose with noticeable spaces between CNTs or CNT bundles when the twist angle is small. Increasing the twist angle is an effective method to densify CNT fibers and thus improves the friction coefficient between the CNTs. **Figure 8** shows the strength-twist relationship for different CNT fibers [111–113]. All studies showed that the highest fiber strength was found at a twist angle of 15–20°. However, with further increasing the twist, the load (upon stretching) compresses the fiber rather than acts totally along the fiber axis due to the twist angle between the CNTs and the axis, making the utilization efficiency to decrease at high twist angles. The experimental observations that the strength initially increases with twist angle and then decreases are almost the same as those in traditional textile yarns. Nevertheless, the constituent CNTs are different from the filaments in a textile yarn; the CNTs are hollow cylinders and can deform the cross section under pressure due to a structural phase transition [44–49]. For those fibers spun from fewwalled CNTs (wall numbers smaller than 6), it can be seen that beyond the optimal twist angle, an additional strength peak is at large twist angle of 27–30°. This second peak arises from the collapse transformation of the CNT hollow structure, which renders the CNT bun-

Solvent densification is another interfacial treatment to CNT assemblies, due to a large capillary force [114, 115]. Therefore, liquid infiltration by using water, ethanol, acetone, or dimethyl sulfoxide (DMSO) is also often used in the method of array spinning to densify CNT fibers [116–119]. Although the densification process does not improve nanotube orientation, it enhances the load transfer between the nanotubes, thus ensuring that most of them are fully load-bearing. For example, the fiber diameter was shrunk from 11.5 to 9.7 μm after acetone shrinking [117]. However, the role of solvent is hardly known as the capillary force can be influenced by the solvents volatility (boiling point), surface tension, and interaction with CNT surfaces. By comparing various solvents including the nonpolar solvents of *n*-hexane, cyclohexane, cyclohexene, toluene, and styrene, polar protic solvents of glycerine, methanol, ethanol, water, ethylene glycol, and 1,3-propanediol, and polar aprotic solvents of

**Figure 8.** (a) Double-peak strength behavior of CNT fibers with increasing twist angles [111]. The surface twist angle

diameter and is >10.5 μm. (b) Dependence of yarn tensile strength on helix angle for forest-spun MWCNT fibers [112]. The fiber diameter was ~20 μm. (c) Relationship between the fiber's specific strength and tenacity with the surface twist

) by *θ<sup>s</sup>*

= arccos {(*vd*

/*vt* )/[(π*d*<sup>f</sup> ) 2 + (*vd* /*vt* )2

]1/2}, where *d*<sup>f</sup>

is the fiber

and *vd*

(*θs*

angle [113].

12 Nanomechanics

) depends on the twisting and drawing speeds (*vt*

dles much stronger and reduces the bundle cross-sectional area.

In addition to twisting and liquid densification, polymer impregnation is another effective treatment to enhance the mechanical properties of CNT fibers. The polymers are introduced to bridge nonneighboring CNTs, such as polyethyleneimine (PEI) [122], polydopamine (PDA) [123], polyvinyl alcohol (PVA) [105, 118, 124], polyacrylate [125], polyvinylidene fluoride (PVDF) [126], epoxy [105], polyimide (PI) [118, 127], and bismaleimide (BMI) [118, 128]. **Figure 9** shows several typical stress-strain curves for polymer-reinforced CNT fibers, such as PI, BMI, PEI, and PDA [118, 122, 123]. The enhanced mechanical properties of CNT fibers with polymer impregnation are attributed to couplings between the CNT network and polymer chains occurring at the molecular level. As a result, the sliding between CNTs can be remarkably hindered by the polymer network, especially that formed by the cured thermosetting polymers. Therefore, more external strain can be transferred to the CNTs as shown in **Figure 10**, where the Raman peaks of CNTs downshift to a lower wavenumber by different values under axial strains [105]. Both PVA and epoxy enhanced the downshifts, corresponding to the improved load transfer between CNTs. The cured thermosetting polymer network also shows advantages in improving the fiber modulus, as indicated by the largest change in G'-peak wavelength. Similarly, PI and BMI are other efficient thermosetting polymers for producing high-modulus CNT fibers [118, 127, 128]. However, other polymers might be good at improving the fiber's stretchability and toughness by preventing the sliding-induced fiber fracture, possibly due to the polymer wrapping around the CNT connections [126], cooperative deformation mechanisms of the soft and hard segments [125], iron-mediated covalent cross-linking [122], or cross-linked polymeric binders [123].

**Figure 9.** (a) Stress-strain curves of dry-spun, ethanol-densified, ethylene-glycol-densified, PI-reinforced, and BMIreinforced CNT fibers [118]. (b) Stress-strain curves of PEI-catechol-reinforced CNT fibers, including one after addition of Fe(III) [122]. (c) Stress-strain curves of as-drawn CNT fiber, CNT/PDA fiber, and CNT/(pyrolyzed PDA) fiber [123].

**Figure 10.** Variations of Raman spectra under strain in different systems [105]. (a) Typical G' band Raman spectra of epoxy- and PVA-infiltrated CNT fibers. (b) Downshifts of the peak position in different systems.

The structure of CNT fiber is hierarchical, with CNTs self-assembling into bundles and the bundles forming an aligned network in the fiber [129, 130]. Other assembly parameters also include the tube waviness or straightness [131], which can also result in different elasticity and modulus for CNT fibers. Notice that the above-mechanical improvements are irrelevant to the microstructure of individual CNTs but are relevant to the macroscopic assembly parameters. From the point of view of tube structure, fewwalled CNTs have shown advantages in producing high-performance fibers, rather than the ideal SWCNTs [131]. For the SWCNTs, usually 1–1.5 nm in diameter, simulations and experiments have shown that a certain high pressure (hundreds of MPa) is necessary to introduce even a slight deformation of their cross sections. That means, the intertube contact area is far limited compared with the overall circumference of the cross section. While for the fewwalled tubes, the tube diameter can be large up to ≥5 nm, making it much easy to deform the cross section and increase the contact area. Experiments even have shown, upon certain twisting and/or tensioning, collapse of fewwalled tubes were possibly introduced [111, 132].

Furthermore, covalent interface engineering could be an efficient posttreatment for the asproduced CNT fibers. Acid treatment could introduce rich hydroxyl, methyl/methylene, and carbonyl groups onto CNT surfaces, which also made CNT fibers shrunk slightly, or more densified. Such treatment enhanced the load transfer between CNTs, according to a theoretical study [94], and thus resulted in an increase in fiber strength and modulus [133]. Recently, by using an incandescent tension annealing process (ITAP), where the current-induced Joule heating increase the temperature of CNT fiber up to 2000°C in vacuum, sp<sup>3</sup> covalent bonding was observed in CNT fibers [134]. As a result, the fiber modulus could be improved from 37 to 170 GPa, once the ITAP was performed with maintaining a high tension on the fiber.

In order to further improve the utilization efficiency of the individual CNT's mechanical properties in CNT fibers, more effort should be paid to the delicate combining of the choice of CNT structure, controlling in CNT packing density, alignment, and straightness, and intertube mechanics engineering by polymers or covalent functionalization.

#### **4.2. CNT films**

The structure of CNT fiber is hierarchical, with CNTs self-assembling into bundles and the bundles forming an aligned network in the fiber [129, 130]. Other assembly parameters also include the tube waviness or straightness [131], which can also result in different elasticity and modulus for CNT fibers. Notice that the above-mechanical improvements are irrelevant to the microstructure of individual CNTs but are relevant to the macroscopic assembly parameters. From the point of view of tube structure, fewwalled CNTs have shown advantages in producing high-performance fibers, rather than the ideal SWCNTs [131]. For the SWCNTs, usually 1–1.5 nm in diameter, simulations and experiments have shown that a certain high pressure (hundreds of MPa) is necessary to introduce even a slight deformation of their cross sections. That means, the intertube contact area is far limited compared with the overall circumference of the cross section. While for the fewwalled tubes, the tube diameter can be large up to ≥5 nm, making it much easy to deform the cross section and increase the contact area. Experiments even have shown, upon certain twisting and/or tensioning, collapse of fewwalled tubes were

**Figure 10.** Variations of Raman spectra under strain in different systems [105]. (a) Typical G' band Raman spectra of

epoxy- and PVA-infiltrated CNT fibers. (b) Downshifts of the peak position in different systems.

Furthermore, covalent interface engineering could be an efficient posttreatment for the asproduced CNT fibers. Acid treatment could introduce rich hydroxyl, methyl/methylene, and carbonyl groups onto CNT surfaces, which also made CNT fibers shrunk slightly, or more densified. Such treatment enhanced the load transfer between CNTs, according to a theoretical study [94], and thus resulted in an increase in fiber strength and modulus [133]. Recently, by using an incandescent tension annealing process (ITAP), where the current-induced Joule

was observed in CNT fibers [134]. As a result, the fiber modulus could be improved from 37 to

In order to further improve the utilization efficiency of the individual CNT's mechanical properties in CNT fibers, more effort should be paid to the delicate combining of the choice

170 GPa, once the ITAP was performed with maintaining a high tension on the fiber.

covalent bonding

heating increase the temperature of CNT fiber up to 2000°C in vacuum, sp<sup>3</sup>

possibly introduced [111, 132].

14 Nanomechanics

CNT films are another important macroscopic material with high mechanical performances. Various processing methods have been developed to produce CNT films by using preformed CNT assemblies, among which these two methods are of great importance: the layer-by-layer stacking of aligned CNT sheets and the stretching on entangled CNT webs [14]. Despite of the fabrication methods, there are also many common problems concerning with the interface mechanics, as the CNT fibers do.

For instance, liquid densification is widely used to CNT films [135–137], and polymer impregnation can remarkably improve the interfacial load transfer [138–143]. Different from CNT fibers where the tubes are assembled with the aid of twisting, there is more freedom to tune the assembly structure. The CNTs can be got superaligned by stretching the films in a wet environment [135], or by microcombing before they are layer-by-layer stacked [136]. After improving the CNT alignment, the pure CNT films could be as strong as 3.2 GPa in tensile strength, with a Young's modulus of 124–172 GPa [135, 136]. **Figure 11** shows the stress-strain curves for the CNT films being both densified and superaligned. Once these aligned CNTs were impregnated with polymers, especially thermosetting polymers, the film's strength and modulus could be further improved. At a mass fraction of 50–55% for CNT, the CNT/BMI composite films exhibited a strength of 3.8 GPa and a modulus of 293 GPa [139]. Such superior properties are derived from the long length, high mass fraction, good alignment, and reduced waviness of the CNTs.

The stretching on entangled CNT webs is another way to obtain highly aligned CNT films. With the aid of BMI impregnation and curing, the strength and modulus of the CNT/BMI composite films could be improved from 620 MPa and 47 GPa (un-stretched) to 1600 MPa and

**Figure 11.** Stress-strain curves for CNT films obtained by a stretch-dip-drying method (a) [135] and a microcombing method (b) [136]. The films treated with acetone (ADD) and ethanol (EDD) were much stronger than the as-spun and spray-wound (SW) films. The films after microcombing had a larger Young's modulus than the ADD and EDD films.

122 GPa, 1800 MPa and 150 GPa, and 2088 MPa and 169 GPa after being stretched by 30, 35, and 40%, respectively [140]. By further introducing surface functionalization for CNTs, the strength and modulus were surprisingly improved up to 3081 MPa and 350 GPa, respectively [141].

However, the direct stacking of CNT sheets and stretching on CNT webs could not avoid the CNT aggregation due to the vdW interactions. As shown in **Figure 12a** and **b**, the aggregation size was up to hundreds of nanometer [143]. Such aggregation of nanometer-sized components has become a severe problem in paving the way to stronger materials [144]. In the aggregation phase, the intertube load transfer is not as efficient as at the CNT-polymer interface. Thus, such aggregation phase becomes the weak parts in the composites and hinders further reinforcement. In an ideal structure, the nanometer-sized components should be uniformly distributed in the matrix without forming any aggregation phases. Therefore, all the interfaces can play roles in shear load transfer. As inspired by the formation process of natural composite structures, the BMI resins were infiltrated into CNT webs before any stretching was performed. Since the liquid treatment just densifies the CNT network without inducing CNT aggregation, the impregnation before stretching causes each CNT or CNT bundle uniformly covered by a thin layer of polymers. Therefore, the following stretching could just align the CNTs but avoid the aggregation due to the thin layer of polymer, resulting in an optimized composite structure. By adapting a multistep stretching process, the final

**Figure 12.** Comparison of CNT assembly structure for different films [143]. (a,b) CNT aggregation in the layer-by-layer stacking of array-drawn CNT sheets and the stretched dry films composed by entangled CNTs, respectively. (c) The small-sized CNT bundles did not aggregate but were surrounded by BMI polymers and uniformly distributed. (d, e) 750°C treated CNT/BMI composite films where the stretching was performed before and after resin impregnation, respectively, showing the aggregation and unaggregation. (f) Cross section of the optimal CNT/BMI composite structure by using focused ion beam treatment, indicating an aggregation size of 20–50 nm for the CNTs.

composite film exhibited superior mechanical performances. The highest tensile strength and modulus were up to 5.77–6.94 and 212–351 GPa at a CNT-to-resin mass ratio of 7:3 and a stretching magnitude of 34% [143].

**Figure 12c** shows the morphology of the optimized CNT/BMI composites, where the CNT bundles were found to be uniformly distributed and aligned. To characterize the aggregation of CNTs, thermal treatment and focused ion beam treatment were used. After being heated to 750°C to decompose the polymers, the remaining CNTs were found to be aggregated/ unaggregated if the stretching was performed before/after the polymer impregnation, see **Figure 12d** and **e**. The direct observation of the cross section with focused ion beam treatment showed an aggregation size of just 20–50 nm (**Figure 12f**), a size for the bundling that was still difficult to avoid during the growth process.

On the whole, for the macroscopic CNT assemblies, the interface mechanics becomes much more complicated as the material exhibits a collective dynamics from the constituent CNTs. Assembly parameters, including CNT packing density, alignment and/or entanglement, twisting level, cross-linking, and aggregation size, are found to be critical for the mechanical properties. By controlling the interface mechanics in the CNT assemblies, we can not only optimize the mechanical properties, but also introduce multifunctionalities, as discussed below.

#### **4.3. Multifunctionalization by interface engineering**

122 GPa, 1800 MPa and 150 GPa, and 2088 MPa and 169 GPa after being stretched by 30, 35, and 40%, respectively [140]. By further introducing surface functionalization for CNTs, the strength and modulus were surprisingly improved up to 3081 MPa and 350 GPa, respectively

However, the direct stacking of CNT sheets and stretching on CNT webs could not avoid the CNT aggregation due to the vdW interactions. As shown in **Figure 12a** and **b**, the aggregation size was up to hundreds of nanometer [143]. Such aggregation of nanometer-sized components has become a severe problem in paving the way to stronger materials [144]. In the aggregation phase, the intertube load transfer is not as efficient as at the CNT-polymer interface. Thus, such aggregation phase becomes the weak parts in the composites and hinders further reinforcement. In an ideal structure, the nanometer-sized components should be uniformly distributed in the matrix without forming any aggregation phases. Therefore, all the interfaces can play roles in shear load transfer. As inspired by the formation process of natural composite structures, the BMI resins were infiltrated into CNT webs before any stretching was performed. Since the liquid treatment just densifies the CNT network without inducing CNT aggregation, the impregnation before stretching causes each CNT or CNT bundle uniformly covered by a thin layer of polymers. Therefore, the following stretching could just align the CNTs but avoid the aggregation due to the thin layer of polymer, resulting in an optimized composite structure. By adapting a multistep stretching process, the final

**Figure 12.** Comparison of CNT assembly structure for different films [143]. (a,b) CNT aggregation in the layer-by-layer stacking of array-drawn CNT sheets and the stretched dry films composed by entangled CNTs, respectively. (c) The small-sized CNT bundles did not aggregate but were surrounded by BMI polymers and uniformly distributed. (d, e) 750°C treated CNT/BMI composite films where the stretching was performed before and after resin impregnation, respectively, showing the aggregation and unaggregation. (f) Cross section of the optimal CNT/BMI composite structure

by using focused ion beam treatment, indicating an aggregation size of 20–50 nm for the CNTs.

[141].

16 Nanomechanics

Besides load transfer at the intertube interfaces, energy dissipation is also a common interfacial phenomenon. As CNT fiber is formed by assembling millions of individual tubes, the assembly features provide the fiber with rich interface structures and various ways of energy dissipation, including the internal viscosity and intertube friction [109]. Therefore, a modified Kelvin-Voigt model was adopted, where an elastic spring *K*, a viscous damping coefficient *η*, and an intertube friction *f* are connected in parallel [145]. Due to the friction, there exists a friction-dependent component in loss tangent, which is not dependent on the frequency. As shown in **Figure 13**, the loss tangent was nonzero (about 0.045) for the dry-spun CNT fibers, while that for the T300 carbon fiber was nearly zero due to the absence of interfaces. When the CNT fibers were densified

**Figure 13.** The frequency dependence of loss tangent for the dry-spun CNT fiber, ethylene-glycol-densified CNT fiber, and T300 carbon fiber [109].

by ethylene glycol, the CNTs became constrained, corresponding to reduced loss tangents, see **Figure 13**. Based on the friction-based Kelvin-Voigt model, the damping performance of CNT fibers can be tuned in a very wide range. The introduction of thermosetting polymers further decreases the loss tangent, while plying CNT fibers into a yarn increases the loss tangent [109].

The CNT entanglement provides another way to enhance the damping ability [146]. The as-produced CNT films based on an injection chemical vapor deposition exhibited a loss tangent of 0.37–0.42 at 200 Hz [146], or 0.2 at 50 Hz [143]. After the network was stretched and strengthened with BMI, the loss tangent could decrease remarkably down to about 0.05 at 50 Hz [143].

When an electric current is passing through CNT assemblies, there exist parallel currents along the aligned CNTs. The collective electromagnetic force between parallel current can cause a volume contraction for the assembly [147]. Based on such mechanism, CNT fibers can be developed as electromechanical actuator. Detailed study revealed that the electromechanical coupling is quite complicated beyond the actuation [128]. For example, when a current passes through a CNT fiber, the fiber's modulus could decrease remarkably due to the current-induced weakening in C−C σ bond. Besides the electromechanical coupling, the current could induce a strong Joule heating, making the fiber temperature to increase very sharply. Such electrothermal coupling can be used for fast curing to the impregnated thermosetting resins at a moderate temperature of 200–300°C [128], or for the formation of intertube sp3 covalent bonding at a high temperature of 2000°C in vacuum [134].

## **5. Conclusion**

The control of interface mechanics is a key strategy to develop high-performance CNT assembly structures. High tensile properties (strength up to 3–6 GPa and modulus up to 200–350 GPa) can be obtained for aligned CNT fibers and films, corresponding to an improved utilization efficiency of CNT's mechanical properties. Furthermore, the dynamical performance and multifunctionalities both depend strongly on the interface between CNTs, also in terms of CNT packing density, alignment, entanglement, twisting level, cross-linking, and aggregation size. For future development of high-performance CNT assembly materials, more delicate design on the multiscale interfaces is required. I hope this review could cast lights for the interface design ranging widely from the nanometer scale to the macroscopic scale.

### **Acknowledgements**

The author thanks Prof. Xingao Gong of Fudan University, Prof. Erio Tosatti of the International School for Advanced Studies, and Prof. Qingwen Li of Suzhou Institute of Nano-Tech and Nano-Bionics for their kind collaborations, which are the important sources for this chapter. Financial supports from the National Natural Science Foundation of China (11302241, 51561145008), Youth Innovation Promotion Association of the Chinese Academy of Sciences (2015256), and National Key Research and Development Program of China (2016YFA0203301) are acknowledged.

## **Author details**

by ethylene glycol, the CNTs became constrained, corresponding to reduced loss tangents, see **Figure 13**. Based on the friction-based Kelvin-Voigt model, the damping performance of CNT fibers can be tuned in a very wide range. The introduction of thermosetting polymers further decreases the loss tangent, while plying CNT fibers into a yarn increases the loss tangent [109]. The CNT entanglement provides another way to enhance the damping ability [146]. The as-produced CNT films based on an injection chemical vapor deposition exhibited a loss tangent of 0.37–0.42 at 200 Hz [146], or 0.2 at 50 Hz [143]. After the network was stretched and strengthened with BMI, the loss tangent could decrease remarkably down to about 0.05

When an electric current is passing through CNT assemblies, there exist parallel currents along the aligned CNTs. The collective electromagnetic force between parallel current can cause a volume contraction for the assembly [147]. Based on such mechanism, CNT fibers can be developed as electromechanical actuator. Detailed study revealed that the electromechanical coupling is quite complicated beyond the actuation [128]. For example, when a current passes through a CNT fiber, the fiber's modulus could decrease remarkably due to the current-induced weakening in C−C σ bond. Besides the electromechanical coupling, the current could induce a strong Joule heating, making the fiber temperature to increase very sharply. Such electrothermal coupling can be used for fast curing to the impregnated thermosetting resins at a moderate temperature of 200–300°C [128], or for the formation of intertube sp3

The control of interface mechanics is a key strategy to develop high-performance CNT assembly structures. High tensile properties (strength up to 3–6 GPa and modulus up to 200–350 GPa) can be obtained for aligned CNT fibers and films, corresponding to an improved utilization efficiency of CNT's mechanical properties. Furthermore, the dynamical performance and multifunctionalities both depend strongly on the interface between CNTs, also in terms of CNT packing density, alignment, entanglement, twisting level, cross-linking, and aggregation size. For future development of high-performance CNT assembly materials, more delicate design on the multiscale interfaces is required. I hope this review could cast lights for the interface design

The author thanks Prof. Xingao Gong of Fudan University, Prof. Erio Tosatti of the International School for Advanced Studies, and Prof. Qingwen Li of Suzhou Institute of Nano-Tech and Nano-Bionics for their kind collaborations, which are the important sources for this chapter. Financial supports from the National Natural Science Foundation of China (11302241, 51561145008), Youth Innovation Promotion Association of the Chinese Academy of Sciences (2015256), and National Key Research and Development Program of China (2016YFA0203301) are acknowledged.

covalent bonding at a high temperature of 2000°C in vacuum [134].

ranging widely from the nanometer scale to the macroscopic scale.

at 50 Hz [143].

18 Nanomechanics

**5. Conclusion**

**Acknowledgements**

#### Xiaohua Zhang

Address all correspondence to: zhangxhcm@gmail.com

1 Suzhou Institute of Nano-Tech and Nano-Bionics, Chinese Academy of Sciences, Suzhou, China

2 Jiangsu Key Laboratory of Engineering Mechanics and Department of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing, China

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## **Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading**

Vladimir A. Skripnyak and Evgeniya G. Skripnyak

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68291

#### Abstract

Researches of the last years have allowed to establish that the laws of deformation and fracture of bulk ultrafine-grained (UFG) and coarse-grained (CG) materials are various both in static and in dynamic loading conditions. The influence of average grain size on the yield stress, the tensile strength, and the compression strength was established for metal alloys with a face-centered cubic (FCC), a body-centered cubic (BCC), and a hexagonal close-packed (HCP) structures. The study of the microstructure of the alloys after severe plastic deformation (SPD) by the electron backscatter diffraction (EBSD) technique showed the presence of a bimodal grain size distribution in the UFG alloys. Metal alloys with a bimodal grain size distribution possess a negative strain rate sensitivity of the yield stress and higher ductility at quasi-static strain rates. In this chapter, we will discuss the regularities of deformation at high strain rates, damage, and fracture of ultrafine-grained alloys.

Keywords: nanostructured metals, ultrafine-grained alloys, severe plastic deformation, high strain rates, yield stress, spall strength

#### 1. Introduction

During the last decade, the interest in research of mechanical behavior of ultrafine-grained (UFG) and nanostructured (NS) metal alloys has been risen sharply [1–13]. Compared to their coarse-grained counterparts, UFG and NS metals exhibit significantly higher static yield strength, hardness, but lower tensile elongation. Ultrafine-grained materials produced by severe plastic deformation (SPD) methods exhibit a high strength, good damping properties, and superplastic properties over a limited range of strain rates at high temperatures. The increase of UFG material strength can be explained on the behavior of dislocation at submicrometer grain size. Laws of plastic deformation operating at room temperature for

coarse-grained metals usually are based on dislocation mechanisms [1–9]. When grain size is decreased in nanometer range, dislocation activity becomes difficult. Therefore, plastic deformation resulting to dislocation nucleation and motion limits ductility of NS and UFG materials. Strain rate sensitivity of flow stress also depends on grain size [9–11].

Similarity of mechanical behavior of materials in wide range of loading conditions takes place for metals combining into several groups [14]. Metal alloys with face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close-packed (HCP) crystalline lattices can be related to three groups. Mechanical behavior of these groups of metals has to be described by various constitutive models [2–4]. We can expect that UFG metals with FCC, BCC, and HCP lattices will have various laws of plastic deformation at high strain rates also.

Mechanical behavior of NS and UFG materials under shock wave loading requires understanding of fundamental laws of the deformation and failure mechanisms at high strain rates.

## 2. Mechanical behavior of nanostructured and ultrafine-grained metal alloys

Important information about mechanical behavior of nanostructured and ultrafine-grained materials under dynamic loading is contained in the experimental data on high-speed interaction of plates; results of tests on compression and tension by using a split-Hopkinson pressure bar (SHPB), also known as Kolsky bar test; and results of tests using high-rate servo-hydraulic machine Instron VHS.

Experimental results obtained from the tests of metals and alloys by these methods showed that the patterns of deformation in ultrafine-grained and coarse-grained materials are different not only under quasi-static but also intense dynamic loading.

The free surface profiles of shock wave in the UFG Ti-6Al-2Sn-2Zr-2Mo-Si titanium alloy are shown in Figure 1.

The metal plate was loaded with shock pulses, which were obtained by the impact of thin aluminum. The metal plate was loaded with shock pulses, which were obtained at impact thin aluminum plates at a speed of 650 m/s [13, 15].

The movement of the back surface of the plates when exposed to a shock pulse was recorded using a laser Doppler interferometer VISAR with subnanosecond time resolution [15].

Shock wave in UFG alloys has the two-wave structure as in CG alloys. The shock wave in the UFG alloys has a double structure in coarse-grained alloys. Elastic precursor propagates with the longitudinal sound speed. The bulk compression wave or the plastic wave propagates through compressed material with the bulk speed of sound.

Curves 1 and 2 correspond to specimens with grain size ~300 nm, and the Curve 3 corresponds to coarse-grained specimen with grain size of 20 μm [13, 15]. The thicknesses of UFG specimens 1 and 2 were 0.835 and 6.36 mm, respectively. The thickness of coarse-grained specimen was 10 mm.

Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading http://dx.doi.org/10.5772/intechopen.68291 33

Figure 1. Free surface velocity profiles.

coarse-grained metals usually are based on dislocation mechanisms [1–9]. When grain size is decreased in nanometer range, dislocation activity becomes difficult. Therefore, plastic deformation resulting to dislocation nucleation and motion limits ductility of NS and UFG

Similarity of mechanical behavior of materials in wide range of loading conditions takes place for metals combining into several groups [14]. Metal alloys with face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close-packed (HCP) crystalline lattices can be related to three groups. Mechanical behavior of these groups of metals has to be described by various constitutive models [2–4]. We can expect that UFG metals with FCC, BCC, and HCP

Mechanical behavior of NS and UFG materials under shock wave loading requires understanding of fundamental laws of the deformation and failure mechanisms at high strain rates.

Important information about mechanical behavior of nanostructured and ultrafine-grained materials under dynamic loading is contained in the experimental data on high-speed interaction of plates; results of tests on compression and tension by using a split-Hopkinson pressure bar (SHPB), also known as Kolsky bar test; and results of tests using high-rate servo-hydraulic

Experimental results obtained from the tests of metals and alloys by these methods showed that the patterns of deformation in ultrafine-grained and coarse-grained materials are different

The free surface profiles of shock wave in the UFG Ti-6Al-2Sn-2Zr-2Mo-Si titanium alloy are

The metal plate was loaded with shock pulses, which were obtained by the impact of thin aluminum. The metal plate was loaded with shock pulses, which were obtained at impact thin

The movement of the back surface of the plates when exposed to a shock pulse was recorded

Shock wave in UFG alloys has the two-wave structure as in CG alloys. The shock wave in the UFG alloys has a double structure in coarse-grained alloys. Elastic precursor propagates with the longitudinal sound speed. The bulk compression wave or the plastic wave propagates

Curves 1 and 2 correspond to specimens with grain size ~300 nm, and the Curve 3 corresponds to coarse-grained specimen with grain size of 20 μm [13, 15]. The thicknesses of UFG specimens 1 and 2 were 0.835 and 6.36 mm, respectively. The thickness of coarse-grained specimen

using a laser Doppler interferometer VISAR with subnanosecond time resolution [15].

not only under quasi-static but also intense dynamic loading.

through compressed material with the bulk speed of sound.

aluminum plates at a speed of 650 m/s [13, 15].

2. Mechanical behavior of nanostructured and ultrafine-grained metal

materials. Strain rate sensitivity of flow stress also depends on grain size [9–11].

lattices will have various laws of plastic deformation at high strain rates also.

alloys

32 Nanomechanics

machine Instron VHS.

shown in Figure 1.

was 10 mm.

The Hugoniot elastic limit (HEL) (dynamic elastic limit) for an elastic plastic material can be calculated by the formula [15, 16]

$$
\sigma\_{\rm HEL} = (1/2)\rho\_0 \mathbf{C}\_{\rm L} u\_{\rm HEL} \tag{1}
$$

where σHEL is the Hugoniot elastic limit, r<sup>0</sup> is the initial mass density, CL is the longitudinal speed of sound, and upHEL = ufsHEL/2 is the maximum particle velocity in the elastic precursor.

The average strain rate in the elastic precursor is measured by the ratio [16]

$$
\dot{\varepsilon}\_{eq}^{HEL} = \frac{1}{2} \frac{u\_{pHE}}{\mathbb{C}\_L \Delta t\_{HEL}},\tag{2}
$$

where ΔtHEL is a time, which is achieved by the amplitude of the elastic precursor.

The calculated ratio of the bulk modulus of nanostructured titanium to the bulk modulus of coarse-grained titanium agrees with experimental data. Sound velocities of materials can be calculated by relations [16]

$$\mathbf{c}\_{\mathbf{s}} = \sqrt{\mu}/\rho, \quad \mathbf{c}\_{\mathbf{b}} = \sqrt{\mathbf{B}/\rho}, \quad \mathbf{c}\_{\mathbf{L}} = \sqrt{\mathbf{c}\_{\mathbf{b}}^2 + (4/3)\mathbf{c}\_{\mathbf{s}}^2} \tag{3}$$

where Cs, Cb, CL are the shear, bulk, and longitudinal sound velocities; B is the bulk modulus; and μ is the shear modulus.

The effective bulk modulus B of nanostructured material can be determined by the equation

$$B = B^{(c)}\mathcal{C}^{(c)} + B^{(g)}\mathcal{C}^{(g)} + B^{(t\dagger)}\mathcal{C}^{(t\dagger)},\tag{4}$$

where B(c) is the isentropic bulk modulus of a crystalline phase; B(g) ≈ 0.45B(c) and B(tj) ≈ 0. 5B(c) are the bulk modulus of a grain boundary phase and the bulk modulus of material near the triple junctions correspondingly; and C(c), C(g), and C(tj) are the volume fractions of these phases correspondingly.

The volume fraction of the triple junction and the volume fraction of grain boundaries were described by the equations

$$\mathbb{C}^{(\mathcal{g})} = \beta^{(\mathcal{g})} \exp \left( -a^{(\mathcal{g})} d\_{\mathcal{S}} / b \right), \tag{5}$$

$$\mathbf{C}^{(\dagger)} = \beta^{(\dagger)} \exp\left(-\alpha^{(\dagger)} d\_{\S}/b\right),\tag{6}$$

where dg is the grain size; b is the modulus of the Burgers vector; and β(g), β(tj), α(g), and α(tj) are the constants of a material.

The values of C(g) and C(tj) versus grain size are shown in Figure 2.

As the grain size is decreased into the nanometer range (<100 nm), the volume of grain interfaces sufficiently increased. The calculated ratio of the bulk modulus of nanostructured materials to the bulk modulus of coarse-grained materials is demonstrated in Figure 2.

Figure 2. Volume fractions of grain boundary phases versus grain size for titanium. Filled square denotes the experimental data [17].

From Eqs. (3)–(6), it follows that the longitudinal and bulk sound velocities for the UFG and CG alloys differ insignificantly. Thus, in the absence of pores and microcracks, UFG and CG alloys have close values of mass density, elastic moduli, and sound velocities Cs, Cb, and CL.

The dynamic yield stress of the material σ<sup>y</sup> acting on the shock adiabat is determined by the formula [15, 16]

$$
\sigma\_Y = \sigma\_{\rm HEL} \frac{1 - 2\nu}{1 - \nu},
\tag{7}
$$

where ν is Poisson's ratio.

The effective bulk modulus B of nanostructured material can be determined by the equation

where B(c) is the isentropic bulk modulus of a crystalline phase; B(g) ≈ 0.45B(c) and B(tj) ≈ 0. 5B(c) are the bulk modulus of a grain boundary phase and the bulk modulus of material near the triple junctions correspondingly; and C(c), C(g), and C(tj) are the volume fractions of these

The volume fraction of the triple junction and the volume fraction of grain boundaries were

<sup>C</sup>ð Þ<sup>g</sup> <sup>¼</sup> <sup>β</sup>ð Þ<sup>g</sup> exp �αð Þ<sup>g</sup> dg=<sup>b</sup>

<sup>C</sup>ð Þ tj <sup>¼</sup> <sup>β</sup>ð Þ tj exp �αð Þ tj dg=<sup>b</sup>

where dg is the grain size; b is the modulus of the Burgers vector; and β(g), β(tj), α(g), and α(tj) are

As the grain size is decreased into the nanometer range (<100 nm), the volume of grain interfaces sufficiently increased. The calculated ratio of the bulk modulus of nanostructured materials to the bulk modulus of coarse-grained materials is demonstrated in Figure 2.

B/B0

The values of C(g) and C(tj) versus grain size are shown in Figure 2.

0 10 20 30 40 50 60 70 80

d g (nm)

Figure 2. Volume fractions of grain boundary phases versus grain size for titanium. Filled square denotes the experimen-

phases correspondingly.

34 Nanomechanics

described by the equations

the constants of a material.

0,0

<sup>C</sup>(tj) <sup>C</sup>(g)

C(gb)

0,2

0,4

B/B0 , C

tal data [17].

(g), C

(tj), C

(gb)

0,6

0,8

1,0

<sup>B</sup> <sup>¼</sup> <sup>B</sup>ð Þ<sup>c</sup> <sup>C</sup>ð Þ<sup>c</sup> <sup>þ</sup> <sup>B</sup>ð Þ<sup>g</sup> <sup>C</sup>ð Þ<sup>g</sup> <sup>þ</sup> <sup>B</sup>ð Þ tj <sup>C</sup>ð Þ tj ; <sup>ð</sup>4<sup>Þ</sup>

; ð5Þ

; ð6Þ

The experimental results in Figure 1 showed that the UFG and CG titanium alloys have similar amplitude values of the elastic precursors at the equivalent distances to the free surface of the samples. Thus, the Hugoniot elastic limits of UFG and CG titanium alloys can be the same. But the quasi-static yield strength of titanium alloys depends on grain size in accordance with the Hall-Petch relation, as shown in Figure 3.

Filled symbols show experimental data for titanium alloys [13, 18] and magnesium alloys [7, 8, 19].

Experimental investigations indicate that the prevailing deformation mechanisms are changed for the grain size below ~100 nm. The Hall-Petch relation (Eq. (8)) is satisfied for many polycrystalline materials, but the values of coefficients σ<sup>0</sup> and k are changed at the

Figure 3. The yield strength of HCP alloys versus the inverse square root of the grain size.

Figure 4. Free surface velocity versus time in UFG and CG copper.

grain size about several hundred nm (range UFG) and 30–20 nm (range NS) as shown in Figure 3. Filled symbols show experimental data for titanium alloys [13, 18] and magnesium alloys [7, 8, 19]:

$$
\sigma\_s = \sigma\_0 + k d\_g^{-1/2},
\tag{8}
$$

where σ<sup>0</sup> is the yield strength, dg is the grain size, σ0, and k are the material constants.

Figure 4 shows shock wave profiles in UFG and CG copper specimens [15, 20, 21]. Copper refers to isomechanical group of materials with the FCC lattice [15]. Curve 1 corresponds to UFG structure of specimen with grain size of ~500 nm. Curve 2 is received for a material in which fine grains occupied only 85% of volume. These experimental results testify that the Hugoniot elastic limit (HEL) and the spall strength (σsp) of copper sharply decrease with the decrease of the relative volume occupied with UFG structure.

The solid Curve 1 corresponds to CG specimen, and dashed Curve 2 corresponds to UFG specimen. The HEL of UFG copper M2 (~0.4 GPa) is about twice that of polycrystalline alloy.

Precipitate-strengthened aluminum, magnesium alloys, copper, and some other metals possess a high strength and ductility. The presence of nanoparticles and precipitate phases influences on the yield stress and the spall strength of UFG alloys under shock wave loadings.

In Figure 5, shock wave profiles in precipitate-strengthened aluminum alloy AA6063T6- AlMg0.5Si0.4 are shown [6].

Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading http://dx.doi.org/10.5772/intechopen.68291 37

Figure 5. Free surface velocities versus time in UFG and CG aluminum alloy AA6063 T6.

grain size about several hundred nm (range UFG) and 30–20 nm (range NS) as shown in Figure 3. Filled symbols show experimental data for titanium alloys [13, 18] and magnesium

Time (nanoseconds)

0 100 200 300 400 500 600

~ 2.95 GPa

~ 2.05 GPa Copper M2

d g

d g

~0.5 mm

~110 mm

<sup>σ</sup><sup>s</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup> <sup>þ</sup> kd�1=<sup>2</sup>

Figure 4 shows shock wave profiles in UFG and CG copper specimens [15, 20, 21]. Copper refers to isomechanical group of materials with the FCC lattice [15]. Curve 1 corresponds to UFG structure of specimen with grain size of ~500 nm. Curve 2 is received for a material in which fine grains occupied only 85% of volume. These experimental results testify that the Hugoniot elastic limit (HEL) and the spall strength (σsp) of copper sharply decrease with the

The solid Curve 1 corresponds to CG specimen, and dashed Curve 2 corresponds to UFG specimen. The HEL of UFG copper M2 (~0.4 GPa) is about twice that of polycrystalline alloy.

Precipitate-strengthened aluminum, magnesium alloys, copper, and some other metals possess a high strength and ductility. The presence of nanoparticles and precipitate phases influences on the yield stress and the spall strength of UFG alloys under shock wave loadings.

In Figure 5, shock wave profiles in precipitate-strengthened aluminum alloy AA6063T6-

where σ<sup>0</sup> is the yield strength, dg is the grain size, σ0, and k are the material constants.

decrease of the relative volume occupied with UFG structure.

<sup>g</sup> ; ð8Þ

alloys [7, 8, 19]:

0

Figure 4. Free surface velocity versus time in UFG and CG copper.

50

100

150

Free surface velocity (m/s)

36 Nanomechanics

200

250

300

σ sp

<sup>2</sup> <sup>1</sup> <sup>σ</sup>HEL < 0.2 GPa

σ sp

350

AlMg0.5Si0.4 are shown [6].

Curves 1, 2, and 3 correspond to AA6063T6 alloy after two and eight passes of equal-channel angular pressing (ECAP) and without ECAP, respectively. Coarse-grained alloy specimens had a grain size of ~100 μm. UFG structure was created by repeated ABC ECAP passes. Nanoparticles of Mg2Si were formed by means of thermal treatment of alloy before ECAP treatment. The HEL of coarse-grained AA6063T6 was equal to ~0.33 GPa, and the spall strength was equal to ~1.29 GPa, whereas the HEL values of UFG alloy were ~5.7% higher (~0.35 GPa), and the spall strength was equal to ~1.48 GPa. Thus, the spall strength and the yield strength of UFG alloys depend not only on grain size but also on concentration and sizes of phase precipitates.

The Hugoniot elastic limit of ultrafine-grained metal alloys and their coarse-grained counterparts is not significantly different. The rate of elastic precursor attenuation is above in UFG alloys, than in coarse-grained counterparts.

The tensile strength at high strain rate can be determined from the analysis of shock wave profiles shown in Figures 1, 4, and 5. These values correspond to the spall strength. The spall strength is determined by formulas [15, 16]

$$
\sigma\_{\rm sp} = (1/2)\rho\_0 \mathbb{C}\_b \{ \Delta u\_{\rm f} + \delta \}, \qquad \delta = (\hbar/\mathbb{C}\_b - \hbar/\mathbb{C}\_L) \cdot |\dot{u}\_1|, \tag{9}
$$

where σsp is the spall strength; r<sup>0</sup> is the initial mass density; Cb is the bulk speed of sound in the shock-compressed material; Δufs is the difference between the maximum speed of the free surface of the target and speed in front of the spall pulse; δ is the amendment, to account for the velocity gradients at the free surface in front of spall pulse and in the front; h is the thickness of the target; CL is the longitudinal sound velocity; and u\_ <sup>1</sup> is the change in the free surface velocity in time in front of the spall pulse.

The spall strength σsp of copper specimen with grain size of 500 nm is ~2.95 GPa, whereas the spall strength of specimen with grain size of 100 μm is only ~ 2.05 GPa. The spall strength of UFG and CG copper is shown in Figure 6.

The spall strength of UFG and CG and single crystal aluminum alloys is shown in Figure 7.

Filled symbols show experimental data for Al single crystal [15], AD1 [15], AMg6 [15], and AA6063T6 [6].

These results correlate with data on the increasing of the quasi-static yield stress of FCC metals at a decrease of grain sizes (Figure 8). Experimental data [4–6, 18, 20, 21] for copper and aluminum alloys are shown by the filled symbols.

Changes of curve slope both FCC and HCP metal alloys (Figures 3 and 8) testify to peculiarities of the mechanical behavior of UFG alloys when the grain size varied from 1000 to ~10 nm.

Bulk UFG alloys usually have a dispersion of grain size appearing at their manufacturing by severe plastic deformation (SPD) methods. A specific volume, which is occupied with UFG structure, influences on the HEL and the spall strength of alloys.

Figure 6. The spall strength of UFG and CG copper.

Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading http://dx.doi.org/10.5772/intechopen.68291 39

Figure 7. The spall strength of UFG and CG aluminum alloy AA6063 T6.

the velocity gradients at the free surface in front of spall pulse and in the front; h is the thickness of the target; CL is the longitudinal sound velocity; and u\_ <sup>1</sup> is the change in the free

The spall strength σsp of copper specimen with grain size of 500 nm is ~2.95 GPa, whereas the spall strength of specimen with grain size of 100 μm is only ~ 2.05 GPa. The spall strength of

The spall strength of UFG and CG and single crystal aluminum alloys is shown in Figure 7.

Filled symbols show experimental data for Al single crystal [15], AD1 [15], AMg6 [15], and

These results correlate with data on the increasing of the quasi-static yield stress of FCC metals at a decrease of grain sizes (Figure 8). Experimental data [4–6, 18, 20, 21] for copper and

Changes of curve slope both FCC and HCP metal alloys (Figures 3 and 8) testify to peculiarities of the mechanical behavior of UFG alloys when the grain size varied from 1000 to

Bulk UFG alloys usually have a dispersion of grain size appearing at their manufacturing by severe plastic deformation (SPD) methods. A specific volume, which is occupied with UFG

surface velocity in time in front of the spall pulse.

aluminum alloys are shown by the filled symbols.

Figure 6. The spall strength of UFG and CG copper.

structure, influences on the HEL and the spall strength of alloys.

UFG and CG copper is shown in Figure 6.

AA6063T6 [6].

38 Nanomechanics

~10 nm.

Figure 8. The yield strength versus the inverse square root of the grain size.

## 3. Dissipation of mechanical energy during deformation of ultrafinegrained and coarse-grained titanium and zirconium alloys

Research of deformation localization and fracture mechanisms of NS and UFG alloys was investigated by a combination of structure-sensitive methods [1, 2, 22–27]. The high sensitivity of modern infrared (IR) thermal vision systems in combination with the opportunity of noncontact continuous temperature and strain measurements gives us important information on processes of damage and fracture. Recently, it has been shown that UFG titanium has a peak of an internal friction at temperatures of 450–500C, which results from the processes of recovery and recrystallization [22]. Experimental data were obtained about the deformation and thermal characteristics of the UFG and CG VT1-0 alpha titanium alloy and E110 zirconium alloy [26–28]. The mechanical tests were performed by means of servo-hydraulic machine Instron VHS 40/50–20. These specimens were nanostructured by means of a combined method of severe plastic deformation (SPD).

The residual stresses in the specimens were relieved by means of annealing. Samples from UFG titanium and zirconium were converted to coarse grain using a high-temperature annealing. The forces were registered by the sensor DYNACELL with accuracy up to 0.2%.

The obtained curves of true stress—true strain were used for determination of the yield strength, the strength, and the strain to failure. The temperature distributions on the specimen surface under tension with constant strain rate were registered by using the FLIR system SC7700M series infrared camera.

IR thermograms were registered with maximum recording frequency of 115 Hz. The resolution was ~20 micron and the image format is of 640 512 mm.

The thermal camera was at a distance of 0.4 m from the specimen. The software of the FLIR system SC7700M was used for interpretation of the registered data [24]. The reflection surface of the samples was close to the "absolutely black body" through the use of carbon blackening.

The thermal vision system allowed one to record not only the specimen's surface temperature distribution but also the current dimensions of specimen during the deformation. The specimen size, the loading force, and the loading times were recorded in the process of quasi-static tensile tests. Engineering true stress, strain, and true strain were calculated using recorded data.

True stress-strain curves of coarse-grained VT1-0 titanium and E110 zirconium specimens under tension are shown in Figures 9 and 10, respectively. True stress-true strain curves are marked as Curve 1. Curve 2 corresponds to the engineering stress and strain diagrams. Figures 9 and 10 also showed the temperature distribution on the surface of the samples for several increasing strains.

Temperature change is caused by dissipation of mechanical energy during plastic deformation.

The stress-strain curve has a parabolic ascending branch, which passes into the descending portion having a negative coefficient of strain hardening. True stress-strain diagrams of UFG

Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading http://dx.doi.org/10.5772/intechopen.68291 41

Figure 9. Stress-strain curves of coarse-grained VT1-0 titanium.

3. Dissipation of mechanical energy during deformation of ultrafine-

Research of deformation localization and fracture mechanisms of NS and UFG alloys was investigated by a combination of structure-sensitive methods [1, 2, 22–27]. The high sensitivity of modern infrared (IR) thermal vision systems in combination with the opportunity of noncontact continuous temperature and strain measurements gives us important information on processes of damage and fracture. Recently, it has been shown that UFG titanium has a peak of an internal friction at temperatures of 450–500C, which results from the processes of recovery and recrystallization [22]. Experimental data were obtained about the deformation and thermal characteristics of the UFG and CG VT1-0 alpha titanium alloy and E110 zirconium alloy [26–28]. The mechanical tests were performed by means of servo-hydraulic machine Instron VHS 40/50–20. These specimens were nanostructured by means of a combined method

The residual stresses in the specimens were relieved by means of annealing. Samples from UFG titanium and zirconium were converted to coarse grain using a high-temperature annealing. The forces were registered by the sensor DYNACELL with accuracy up to 0.2%.

The obtained curves of true stress—true strain were used for determination of the yield strength, the strength, and the strain to failure. The temperature distributions on the specimen surface under tension with constant strain rate were registered by using the FLIR system

IR thermograms were registered with maximum recording frequency of 115 Hz. The resolu-

The thermal camera was at a distance of 0.4 m from the specimen. The software of the FLIR system SC7700M was used for interpretation of the registered data [24]. The reflection surface of the samples was close to the "absolutely black body" through the use of carbon blackening. The thermal vision system allowed one to record not only the specimen's surface temperature distribution but also the current dimensions of specimen during the deformation. The specimen size, the loading force, and the loading times were recorded in the process of quasi-static tensile tests. Engineering true stress, strain, and true strain were calculated using recorded

True stress-strain curves of coarse-grained VT1-0 titanium and E110 zirconium specimens under tension are shown in Figures 9 and 10, respectively. True stress-true strain curves are marked as Curve 1. Curve 2 corresponds to the engineering stress and strain diagrams. Figures 9 and 10 also showed the temperature distribution on the surface of the samples for

Temperature change is caused by dissipation of mechanical energy during plastic deformation. The stress-strain curve has a parabolic ascending branch, which passes into the descending portion having a negative coefficient of strain hardening. True stress-strain diagrams of UFG

tion was ~20 micron and the image format is of 640 512 mm.

grained and coarse-grained titanium and zirconium alloys

of severe plastic deformation (SPD).

40 Nanomechanics

SC7700M series infrared camera.

data.

several increasing strains.

Figure 10. Stress-strain curves of coarse-grained E110 zirconium.

Figure 11. Stress-strain diagrams of ultrafine-grained VT1-0 titanium alloy.

VT1-0 titanium and E110 zirconium specimens under tension are shown in Figures 11 and 12, respectively.

Curves of true stress-true strain are shown as Curve 1. Curves of engineering stress and strain are marked as Curve 2. The insets show the temperature distribution on the specimen surface under certain deformations.

Dimensions of specimens at thermograms for UFG titanium and zirconium alloys showed that the fractional decrease in the area at fracture of specimens is twice less ones in comparison with values of CG alloys. Deformation localization in the zone of necking caused a significant growth of true flow stress and increase of the hardening coefficient. Deformation localization is accompanied by local temperature rise.

The maximum temperature on the surface of CG titanium specimen raised proportionally to the strain rate up to a certain level (in the range from 50 to 55C) and then increased sharply on ~510C (up to the temperature 58.4560.54C) due to the localization of the plastic deformation near damages or crack.

The analyses of IR thermograms showed that character of temperature changing under deformation of UFG and CG titanium specimens is different. These results indicate that macroscopic regularities of strain localization in the UFG and CG titanium and zirconium alloys differ.

The maximum temperature of the UFG sample is directly proportional to strain rate and reaches a temperature of 50–58 C, after that the temperature did not change much [28].

Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading http://dx.doi.org/10.5772/intechopen.68291 43

Figure 12. The diagram of ultrafine-grained E110 zirconium alloy.

VT1-0 titanium and E110 zirconium specimens under tension are shown in Figures 11 and 12,

Curves of true stress-true strain are shown as Curve 1. Curves of engineering stress and strain are marked as Curve 2. The insets show the temperature distribution on the specimen surface

Dimensions of specimens at thermograms for UFG titanium and zirconium alloys showed that the fractional decrease in the area at fracture of specimens is twice less ones in comparison with values of CG alloys. Deformation localization in the zone of necking caused a significant growth of true flow stress and increase of the hardening coefficient. Deformation localization is

The maximum temperature on the surface of CG titanium specimen raised proportionally to the strain rate up to a certain level (in the range from 50 to 55C) and then increased sharply on ~510C (up to the temperature 58.4560.54C) due to the localization of the plastic deforma-

The analyses of IR thermograms showed that character of temperature changing under deformation of UFG and CG titanium specimens is different. These results indicate that macroscopic regularities of strain localization in the UFG and CG titanium and zirconium alloys differ.

The maximum temperature of the UFG sample is directly proportional to strain rate and reaches a temperature of 50–58 C, after that the temperature did not change much [28].

respectively.

42 Nanomechanics

under certain deformations.

tion near damages or crack.

accompanied by local temperature rise.

Figure 11. Stress-strain diagrams of ultrafine-grained VT1-0 titanium alloy.

Figure 13 shows the SEM images of fracture surfaces of UFG titanium alloy and UFG zirconium alloy. Chains of micropores and microcracks are present on the fracture surface.

The maximum temperature at the surface of UFG zirconium alloy specimen increased from 40 to 68C; then, there was an abrupt temperature decrease. This effect is caused by rapid damage accumulation and formation of macrocrack.

The grain size distributions of UFG zirconium and titanium specimens differ due to imperfect process of the intensive plastic deformation [29].

The results of the SEM study of fracture surface confirmed the difference of damage in the samples of UFG titanium and zirconium alloys.

The results of fractographic studies of the fracture surfaces and the results of quasi-static and dynamic tests prove that the fracture of UFG FCC and HCP alloys has the ductile character.

The experimental and theoretical results can be used to develop in defining equations for the UFG alloys over the last decade.

Figure 13. SEM image of the fracture surface. (a) Specimen of UFG VT1-0 titanium and (b) specimen of UFG E110 zirconium.

## 4. The constitutive equation for ultrafine-grained alloys covering a wide range of strain rates

The computational model is required to predict the mechanical behavior of ultrafine-grained FCC, HCP, and BCC alloys under dynamic loading. Therefore, constitutive equations taking into account the grain size of metal alloys were offered by Meyers et al. [4, 18], Armstrong and Zerilli [30], and Khan et al. [31, 32]. Mishra et al. modified the model of Johnson-Cook [33]. A satisfactory quantitative description of the mechanical behavior of FCC and HCP alloys with coarse-grained and ultrafine-grained structure can be achieved by changing parameters of the materials. This limits opportunities to use models for the prediction of mechanical behavior of ultrafine-grained titanium alloys under dynamic loadings.

Obtaining realistic predictions of yield and strength in a wide range of strain rates remains an important problem.

We have developed model of mechanical behavior of the alloys for studying of laws of deformation and fracture of UFG alloys in a wide range of strain rates [12, 34–36].

The approach of damaged medium with stress relaxation for numerical simulation of solid response under shock wave loading was used. The stress of material particle is considered as a result of two competitive processes. Increasing of stresses and internal energy is caused by external impacts. Relaxation of the shear stress and energy dissipation are caused by the increments of inelastic deformation.

The model uses the effective parameters of mechanical state and averaged parameters of mechanical properties (mass density, elastic constants, Grüneisen parameter, etc.).

We consider the possibility of damage accumulation during inelastic deformation of NS and UFG metals in consequence of limited ductility.

The local damage parameter D was used in the scalar form

$$D = \Delta \varepsilon^p / \varepsilon\_f,\tag{10}$$

where Δε<sup>p</sup> <sup>¼</sup> ðt t0 ε\_ p eq dt, ε\_ p eq is the equivalent of inelastic strain rate tensor, and ε<sup>f</sup> is the average

value of an inelastic strain of a material particle at fracture.

4. The constitutive equation for ultrafine-grained alloys covering a wide

Figure 13. SEM image of the fracture surface. (a) Specimen of UFG VT1-0 titanium and (b) specimen of UFG E110

The computational model is required to predict the mechanical behavior of ultrafine-grained FCC, HCP, and BCC alloys under dynamic loading. Therefore, constitutive equations taking into account the grain size of metal alloys were offered by Meyers et al. [4, 18], Armstrong and Zerilli [30], and Khan et al. [31, 32]. Mishra et al. modified the model of Johnson-Cook [33]. A satisfactory quantitative description of the mechanical behavior of FCC and HCP alloys with coarse-grained and ultrafine-grained structure can be achieved by changing parameters of the materials. This limits opportunities to use models for the prediction of mechanical behavior of

Obtaining realistic predictions of yield and strength in a wide range of strain rates remains an

We have developed model of mechanical behavior of the alloys for studying of laws of

The approach of damaged medium with stress relaxation for numerical simulation of solid response under shock wave loading was used. The stress of material particle is considered as a result of two competitive processes. Increasing of stresses and internal energy is caused by external impacts. Relaxation of the shear stress and energy dissipation are caused by the

The model uses the effective parameters of mechanical state and averaged parameters of

We consider the possibility of damage accumulation during inelastic deformation of NS and

deformation and fracture of UFG alloys in a wide range of strain rates [12, 34–36].

mechanical properties (mass density, elastic constants, Grüneisen parameter, etc.).

ultrafine-grained titanium alloys under dynamic loadings.

range of strain rates

zirconium.

44 Nanomechanics

important problem.

increments of inelastic deformation.

UFG metals in consequence of limited ductility.

The local failure criterion of a material particle was used in the form

$$\mathbf{D} = \mathbf{1} \tag{11}$$

The constitutive equation of UFG alloy was determined for material particle in the form

$$
\sigma\_{\vec{\eta}} = \sigma\_{\vec{\eta}}^{(m)} \varphi(D), \qquad \sigma\_{\vec{\eta}}^{(m)} = -p^{(m)} \delta\_{\vec{\eta}} + \mathcal{S}\_{\vec{\eta}}^{(m)}.\tag{12}
$$

where σij is components of the stress tensor, p is the pressure, Sij is components of the deviatoric part of the stress tensor, δij is the Kronecker delta, ϕ (D) is the function of damage, and the parameters with index m correspond to the condensed phase of the damaged material.

The equation of state, Mie-Grüneisen, was used to calculate the pressure [16]. The relaxation equation was used to calculate the deviator of the stress tensor in the undamaged phase material:

$$dS\_{ij}^{(m)}/dt = \mathfrak{D}\mu^{(m)} \left( \dot{\mathfrak{e}}\_{ij}^{(m)} - \dot{\mathfrak{e}}\_{ij}^{p(m)} \right),\tag{13}$$

where d/dt is the Jaumann derivative, μ(m) is the shear modulus of the undamaged phase material, e\_ ð Þ m ij is components of the strain rate tensor, and e\_ p mð Þ ij is components of the inelastic strain rate tensor.

The deviator of the inelastic strain rate tensor was calculated in the framework of the theory of Hencky-Ilyushin:

$$
\dot{\varepsilon}\_{ij}^{p(m)} = (\mathfrak{Z}/2) \left[ \mathbf{S}\_{i\dot{\jmath}}{}^{(m)}{}\_{i\dot{\jmath}} \dot{\varepsilon}\_{eq}^{p}(m) / \sigma\_{eq} \right], \tag{14}
$$

$$
\sigma\_{\mathbf{eq}} = \left[ (\mathbf{3}/\mathbf{2}) \mathbf{S}\_{\mathbf{ij}}^{(m)} \mathbf{S}\_{\mathbf{ij}}^{(m)} \right]^{1/2}. \tag{15}
$$

The scalar function e\_ p mð Þ eq is defined by the sum of the components basing on physical modes of inelastic deformation by relations:

$$\begin{aligned} \dot{\mathbf{e}}\_{\text{eq}}^{\text{p}(\text{m})} &= \sum\_{\mathbf{k}} \left[ \dot{\mathbf{e}}\_{\text{eq}}^{\text{p}} \right]\_{\mathbf{k}} \Delta \mathbf{F}\_{\text{k}},\\ \left[ \dot{\mathbf{e}}\_{\text{eq}}^{\text{p}} \right]\_{\mathbf{k}} &= \left[ \dot{\mathbf{e}}\_{\text{eq}}^{\text{p}} \right]\_{\text{disl}} + \left[ \dot{\mathbf{e}}\_{\text{eq}}^{\text{p}} \right]\_{\text{disl \, nuclei}} + \left[ \dot{\mathbf{e}}\_{\text{eq}}^{\text{p}} \right]\_{\text{t} \,\text{t} \boldsymbol{v}},\\ \Delta F\_{\text{k}} &= \int\_{d\_{\mathcal{S}} - 1}^{d\_{\mathcal{S}} + 1} f(d\_{\mathcal{S}}) d(d\_{\mathcal{S}}), \end{aligned} \tag{16}$$

where f(dg)=1 � exp(�[(dg � dg min)/δ] � n ) and δ, dg min, and n are parameters of grain size distribution function, e\_ p eq is the equivalent rate of plastic strain rate, e\_ p eq h i disl is the equivalent rate of plastic strain rate due to dislocation movement, e\_ p eq h i dislnucl is the equivalent rate plastic strain rate due to heterogeneous dislocation nucleation, and e\_ p eq h i tw is the equivalent rate of plastic strain rate due to twinning.

We used experimental function f(dg) of grain size distribution in UFG metal alloys treated by the equal channel angular pressing (ECAP). The grain size distribution and the average grain size are varied after several passes of ECAP [4]. Therefore, the total increments of inelastic deformation will depend on specific volume of small grains.

Peculiarity of deformation laws of FCC and HCP groups of metals has to be described by various kinetics (Eq. (17)). Equations can be used in phenomenological relation forms [12, 14]:

$$\begin{aligned} \left[\dot{\mathcal{e}}\_{eq}^{p}\right]\_{\text{slickov}} &= b\upsilon\rho\_{m}\exp(-\Delta G\_{1}/RT), \\ \Delta G\_{1} &= \Delta G\_{0} \left[1 - \left(\sigma\_{eq}/\sigma^{\*}\right)^{m}\right]^{\eta}, \\ \left[\dot{\mathcal{e}}\_{eq}^{p}\right]\_{\text{slickov}} &= b\dot{\rho}(\ell)\exp(-\Delta G\_{2}/kT), \\ \left[\dot{\mathcal{e}}\_{eq}^{p}\right]\_{\text{liv}} &= A\_{\text{lve}}\exp(-\Delta G\_{3}/RT) \times \left[1 - \sigma\_{\text{eff}}/\sigma^{\*\ast}\left(d\_{\mathcal{g}^{k}},T,P\right)\right]^{m\_{\text{l}}}, \\ \sigma\_{eq}^{\ast} &= \sigma\_{eq}^{\ast}0 + kd\_{\mathcal{g}^{k}}^{\ast}^{-1/2}, \sigma\_{\text{eff}} = \sigma\_{\text{eq}} - \Delta\sigma\_{\text{lvs}} \\ \sigma^{\ast\ast} &= \sigma\_{0}^{\ast} + k\_{1}d\_{\mathcal{g}^{k}}^{-1/2} + k\_{2}\left(\varepsilon\_{\text{eq}}^{p}\right)^{1/2} + k\_{3}p^{1/2}, \\ \Delta\sigma\_{\text{lvs}} &= k\_{1}d\_{\mathcal{g}^{k}}^{-1/2} + k\_{2}\left(\varepsilon\_{\text{eq}}^{p}\right)^{1/2} + \Delta\sigma\_{\text{bvs}}, \end{aligned} \tag{17}$$

where R is the gas constant; k is the Boltzmann constant; T is temperature; b is the modulus of Burgers vector; g ≈ 0.5 is the orientation coefficient; G0 and G2 are specific activation energy of dislocation movement and nucleation, respectively; v is the average dislocation velocity; G3 is the specific activation energy of twinning; dg is the grain size; r<sup>m</sup> is a density of dislocation movement; r\_ is the rate of dislocation nucleation; and Atw, n1, m1,q are the material constants.

Constants n, q, ΔG0, and ΔG2 are varied for FCC and HCP metals [14].

The precipitate hardening of metals can be described by Orowan's equation [37]:

$$
\Delta \sigma\_{\text{bs}, pr} = 0.4 \mu bM \ln(2r/b) / \left( \pi \lambda \sqrt{1 - \nu} \right), \quad \lambda = \left( \sqrt{3\pi/4} \xi - 1.64 \right) r, \tag{18}
$$

where μ is the shear modulus, b is the modulus of Burgers vector, M is the material constant, r is the radius of particles, f is the concentration of particles, and ν is Poisson's ratio.

In Refs. [12–14], numerical values of model parameters have been discussed for UFG titanium alloys. Used numerical method has been discussed in [12]. Chemical composition of alloys is shown in Table 1.


Table 1. Chemical composition of Ti-6-22-22S and VT1-0 titanium alloys.

where f(dg)=1 � exp(�[(dg � dg min)/δ]

plastic strain rate due to twinning.

e\_ p eq h i

e\_ p eq h i

e\_ p eq h i

σ� eq ¼ σ�

shown in Table 1.

σ�� ¼ σ�

Δσbs ¼ k1d

dislmov

dislnucl

p

rate of plastic strain rate due to dislocation movement, e\_

strain rate due to heterogeneous dislocation nucleation, and e\_

deformation will depend on specific volume of small grains.

<sup>Δ</sup>G<sup>1</sup> <sup>¼</sup> <sup>Δ</sup>G<sup>0</sup> <sup>1</sup> � <sup>σ</sup>eq=σ� � �<sup>n</sup><sup>1</sup> � �<sup>q</sup>

eq<sup>0</sup> <sup>þ</sup> kdg k�1=<sup>2</sup>

�1=2 gk <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>ε</sup><sup>P</sup>

Constants n, q, ΔG0, and ΔG2 are varied for FCC and HCP metals [14].

Δσbs pr <sup>¼</sup> <sup>0</sup>:4μbM ln 2ð Þ <sup>r</sup>=<sup>b</sup> <sup>=</sup> πλ ffiffiffiffiffiffiffiffiffiffiffi

The precipitate hardening of metals can be described by Orowan's equation [37]:

is the radius of particles, f is the concentration of particles, and ν is Poisson's ratio.

<sup>0</sup> þ k1d

�1=2 gk <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>ε</sup><sup>P</sup>

distribution function, e\_

46 Nanomechanics

� n

eq is the equivalent rate of plastic strain rate, e\_

We used experimental function f(dg) of grain size distribution in UFG metal alloys treated by the equal channel angular pressing (ECAP). The grain size distribution and the average grain size are varied after several passes of ECAP [4]. Therefore, the total increments of inelastic

Peculiarity of deformation laws of FCC and HCP groups of metals has to be described by various kinetics (Eq. (17)). Equations can be used in phenomenological relation forms [12, 14]:

,

tw <sup>¼</sup> Atwexpð Þ� �ΔG3=RT <sup>1</sup> � <sup>σ</sup>eff <sup>=</sup>σ�� dg k; <sup>T</sup>; <sup>P</sup> � � � � <sup>m</sup><sup>1</sup>

þ Δσbspr;

<sup>þ</sup> <sup>k</sup>3p<sup>1</sup>=<sup>2</sup>,

, σeff ¼ σeq � Δσbs,

where R is the gas constant; k is the Boltzmann constant; T is temperature; b is the modulus of Burgers vector; g ≈ 0.5 is the orientation coefficient; G0 and G2 are specific activation energy of dislocation movement and nucleation, respectively; v is the average dislocation velocity; G3 is the specific activation energy of twinning; dg is the grain size; r<sup>m</sup> is a density of dislocation movement; r\_ is the rate of dislocation nucleation; and Atw, n1, m1,q are the material constants.

> 1 � ν � � <sup>p</sup> , <sup>λ</sup> <sup>¼</sup>

where μ is the shear modulus, b is the modulus of Burgers vector, M is the material constant, r

In Refs. [12–14], numerical values of model parameters have been discussed for UFG titanium alloys. Used numerical method has been discussed in [12]. Chemical composition of alloys is

eq � �<sup>1</sup>=<sup>2</sup>

eq � �<sup>1</sup>=<sup>2</sup>

¼ gbvrmexpð Þ �ΔG1=RT ,

<sup>¼</sup> gbr\_ð Þ<sup>ℓ</sup> expð Þ �ΔG2=kT ,

p eq h i

dislnucl

p eq h i

) and δ, dg min, and n are parameters of grain size

p eq h i

,

ffiffiffiffiffiffiffiffiffiffiffiffiffi 3π=4f q

� �

� 1:64

r; ð18Þ

ð17Þ

disl is the equivalent

is the equivalent rate plastic

tw is the equivalent rate of

Eqs. (10)–(17) were used for numerical simulation of high velocity impacts of plates. Calculated profiles of shock pulses in UFG titanium alloy are shown in Figure 14.

In simulated loading condition, UFG Ti-6-22-22S of 0.835 and 6.36 mm thicknesses was impacted by aluminum plates with 0.12 and 0.835 mm thicknesses at impact velocities of about 1250 and 680 m/s [15]. Dashed and solid curves indicate calculated and experimental profiles, respectively.

Results of computer simulation of shock loading of UFG alloys testify that fast relaxation of elastic precursor amplitude takes place at the first millimeter. Fast attenuation of elastic precursor can be explained by high rate of dislocation nucleation at grain sizes of several hundreds of nanometers.

Unlike coarse-grained alloys, strain hardening of fine-grained alloys at deformation under shock loading of several GPa is not enough. Satisfactory description of spall fracture was

Figure 14. Measured (solid lines) and simulated (dashed lines) free surface velocity histories for UFG Ti-6-22-22S [12].

achieved in calculation too, in spite of a quasi-elastic unloading. We have not found considerable Bauschinger effect.

Figure 15 showed calculated stress-strain curves for Ti-6-22-22S. Solid curves correspond to experimental data. Dashed curves correspond to results of calculation.

Stress-strain curves for UFG Ti-6-22-22S have a good agreement with experimental data at strain rates within the range from 10<sup>3</sup> to 103 s 1 . Eqs. (15)–(16) allow us to describe the work hardening of titanium alloys in the range of deformation before necking.

Figure 16 showed calculated stress-strain curves for VT1-0 titanium alloy.

Eqs. (15)–(16) describe the flow stress dependence on the strain rates and the plastic strains. Results of simulation of deformation of VT1-0 titanium alloy are shown in Figure 16.

Experimental stress-strain curves are shown as solid curves. The dashed lines 1 were obtained under assumption that k in the relation (Eq. (8)) is the constant for UFG and CG alloys. The dashed line 2 was obtained under assumption that k decreased when grain size became less 1 μm. These results are consistent with the experimental data for the UFG alloy, shown in Figure 3.

Figure 15. Stress versus strain of Ti-6-22-22S titanium alloy.

Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading http://dx.doi.org/10.5772/intechopen.68291 49

Figure 16. Stress versus strain of titanium alloy VT1-0 at strain rate 10�<sup>3</sup> s �1 .

Figure 17 shows the logarithm of the shear stress versus the logarithm of normalized strain rate for coarse grain and UFG Ti-6-22-22S alloys. Experimental data are shown by symbols [12, 13, 38]. Curves 1 and 2 correspond to UFG and CG alloy, respectively. For normalization of the equivalent, strain rate was used ε\_ p eq=1 s�<sup>1</sup> . Analysis of the stress-strain rate data shows that the parameter <sup>m</sup> <sup>¼</sup> <sup>d</sup> lgσeq <sup>=</sup><sup>d</sup> lgε\_ p eq T, ε p eq of strain rate sensitivity is less than that

of CG Ti-6-22-22S alloy.

achieved in calculation too, in spite of a quasi-elastic unloading. We have not found consider-

Figure 15 showed calculated stress-strain curves for Ti-6-22-22S. Solid curves correspond to

Stress-strain curves for UFG Ti-6-22-22S have a good agreement with experimental data at

Eqs. (15)–(16) describe the flow stress dependence on the strain rates and the plastic strains.

Experimental stress-strain curves are shown as solid curves. The dashed lines 1 were obtained under assumption that k in the relation (Eq. (8)) is the constant for UFG and CG alloys. The dashed line 2 was obtained under assumption that k decreased when grain size became less 1 μm. These results are consistent with the experimental data for the UFG alloy, shown in Figure 3.

Results of simulation of deformation of VT1-0 titanium alloy are shown in Figure 16.

1

. Eqs. (15)–(16) allow us to describe the work

experimental data. Dashed curves correspond to results of calculation.

hardening of titanium alloys in the range of deformation before necking. Figure 16 showed calculated stress-strain curves for VT1-0 titanium alloy.

strain rates within the range from 10<sup>3</sup> to 103 s

Figure 15. Stress versus strain of Ti-6-22-22S titanium alloy.

able Bauschinger effect.

48 Nanomechanics

Change of curve slope at strain rates above ~103 s �<sup>1</sup> testifies the change of dominant physical mechanisms of shear stress relaxation in both CG and UFG alloys. Thus, the constitutive equations for ultrafine-grained FCC and HCP metal alloys should have various forms likewise for coarse-grained counterparts.

The strain rate sensitivity index of the flow stress is determined by the relation

$$m = \frac{d(\ln \sigma\_s)}{d \ln(\dot{\varepsilon}\_{eq}/\dot{\varepsilon}\_0)} \bigg|\_{\Upsilon, \varepsilon\_{eq}^p} = m\_0 + \mathbb{C}\_m d\_g^{-1/2},\tag{19}$$

where <sup>σ</sup><sup>s</sup> is the flow stress; <sup>ε</sup>\_eq <sup>¼</sup> ð Þ <sup>2</sup>=<sup>3</sup> <sup>ε</sup>\_ijε\_ij <sup>1</sup>=<sup>2</sup> is the equivalent strain rate; and <sup>ε</sup>\_ <sup>0</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> <sup>s</sup>�1, m0, and Cm are the material parameters.

The strain rate sensitivity index of the yield strength is a private parameter value m at a predetermined value of ε p eq ¼ 0:002.

Figure 17. Strain rate sensitivity of UFG and CG Ti-6-22-22S titanium alloy.

Figure 18 shows the dependence of the strain rate sensitivity index m of the yield strength versus the inverse square root of the grain size. Experimental data are shown by symbols [39, 40].

Curve 1 corresponds to HCP alloys; Curve 2 corresponds to FCC alloys. Filled triangles are experimental data [39]; filled squares are experimental data [40].

The tensile deformation to fracture under static loading increases with the decrease of grain size, as shown in Figure 19 [40]:

$$
\varepsilon\_{f\text{ static}} = \varepsilon\_{f0} \left[ 1 + D\_{\epsilon f} d\_{\mathcal{g}}^{-1/2} \right] / (1 - \Theta),
\tag{20}
$$

where εf static is the limit of plastic deformation to fracture under tension, dg is the average grain size, εf0,Dε<sup>f</sup> are structure-sensitive parameters of the material,θ = (T � Tr)/(Tm � Tr) T is the temperature in absolute scale, Tr =295 К is the room temperature, and Tm is the melting temperature.

The decrease of the strain to fracture under tension of UFG alloys is proportional to the logarithm of the strain rate in the range from 10�<sup>3</sup> to 10<sup>3</sup> s �<sup>1</sup> [5, 41]. Experimental data on the normalized ultimate tensile strain versus the logarithm of normalized strain rate for the UFG and NC aluminum and magnesium alloys are shown in Figure 19.

Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading http://dx.doi.org/10.5772/intechopen.68291 51

Figure 18. The strain rate sensitivity index of the yield strength versus the inverse square root of the grain size.

Figure 18 shows the dependence of the strain rate sensitivity index m of the yield strength versus the inverse square root of the grain size. Experimental data are shown by sym-

Curve 1 corresponds to HCP alloys; Curve 2 corresponds to FCC alloys. Filled triangles are

The tensile deformation to fracture under static loading increases with the decrease of grain

where εf static is the limit of plastic deformation to fracture under tension, dg is the average grain size, εf0,Dε<sup>f</sup> are structure-sensitive parameters of the material,θ = (T � Tr)/(Tm � Tr) T is the temperature in absolute scale, Tr =295 К is the room temperature, and Tm is the melting

The decrease of the strain to fracture under tension of UFG alloys is proportional to the

normalized ultimate tensile strain versus the logarithm of normalized strain rate for the UFG

g h i

=ð Þ 1 � θ ; ð20Þ

�<sup>1</sup> [5, 41]. Experimental data on the

<sup>ε</sup>f static <sup>¼</sup> <sup>ε</sup><sup>f</sup> <sup>0</sup> <sup>1</sup> <sup>þ</sup> <sup>D</sup>ε<sup>f</sup> <sup>d</sup>�1=<sup>2</sup>

experimental data [39]; filled squares are experimental data [40].

Figure 17. Strain rate sensitivity of UFG and CG Ti-6-22-22S titanium alloy.

logarithm of the strain rate in the range from 10�<sup>3</sup> to 10<sup>3</sup> s

and NC aluminum and magnesium alloys are shown in Figure 19.

bols [39, 40].

50 Nanomechanics

temperature.

size, as shown in Figure 19 [40]:

Figure 19. Normalized ultimate tensile strain versus the logarithm of normalized strain rate for the UFG and NC aluminum and magnesium alloys.

The dashed curve shows data obtained according to the formula

$$
\varepsilon\_{f\text{ static}} / \varepsilon\_{f\text{ dynamic}} = \left[1 + \mathbb{C}\_{\varepsilon f} \ln \left(\dot{\varepsilon}\_{eq} / \dot{\varepsilon}\_0\right)\right] / (1 - \theta), \tag{21}
$$

where <sup>ε</sup><sup>f</sup> is the strain to fracture, <sup>ε</sup>\_eq <sup>¼</sup> ð Þ <sup>2</sup>=<sup>3</sup> <sup>ε</sup>\_ijε\_ij <sup>1</sup>=<sup>2</sup> is the equivalent strain rate, <sup>ε</sup>\_ <sup>0</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> <sup>s</sup>�1, and Cε<sup>f</sup> is material parameter.

In the aluminum, magnesium, and titanium alloys, the formation of mesoscopic cracks is preceded by the stage of nucleation and growth of micro-voids. In this regard, the width of the plastic zone in the considered alloys will be determined by the critical size of voids.

Valiev et al. reported the log-normal grain size distributions for fine-grained polycrystalline metals processed by equal channel angular extrusion (ECAE) [42]. The distribution effect of grain size on the flow stress, ultimate strength at compression and tension, and elongation to failure has been shown to CG and UFG alloys [43, 44].

Starting with the pioneering work of Tellkamp et al. [45], the formation of bimodal distribution of grain size is used to produce ultrafine-grained and nanostructured alloys with high strength and satisfactory ductility. A bimodal distribution of grain size in UFG aluminum and magnesium alloys is the cause of several anomalies of the mechanical behavior. Negative strain rate sensitivity of plastic flow stress and increase of the ductility in a wide range of strain rates (from 10�<sup>4</sup> to 1 s�<sup>1</sup> ) were revealed for UFG Al–7.5 wt.% Mg, Al 5083, and magnesium alloys Mg-7wt.%Y [46–52].

The threshold value for the strain to fracture can be calculated by the ratio

$$
\varepsilon\_f^n \left( d\_\S \right) / \varepsilon\_{f0}^{\rm CG} = A\_2 + (A\_1 + A\_2) / \left( 1 + \exp[(\mathbf{x} - \mathbf{x}\_0)/\Xi] \right), \qquad \varepsilon\_{f0}^{\rm CG} = D\_1 (P^\* + T^\*)^{D2}, \tag{22}
$$

where εCG <sup>f</sup> <sup>0</sup> is the strain to fracture of CG alloy; x is equal to dg � 1/2; A1, A2, x0, and x are material constants; T\* = σsp/PHEL, P\* = p/PHEL, and PHEL are the pressure in elastic precursor; and D1 and D2 are material constants.

Increased crack growth resistance in Al-Mg alloys with a bimodal grain size distribution is a result of deflection of microcracks on borders between UFG and coarse-grained (CG) zones [53]. As a result of it, resistance to crack growth at mesoscale level increases with the increase of the plastic deformation.

Features of mechanical behavior of UFG alloys with bimodal grain size distribution are defined by shear band formation, nucleation and growth of damages, coalescence damages under formation of mesoscale, and macroscale cracks.

Thus, processes of shear banding in UFG alloys with bimodal and unimodal grain size distributions depend on strain rates.

## 5. Multiscale model of mechanical behavior of UFG alloys under dynamic loadings

The mechanisms of dynamic fracture in alloys with bimodal grain size distributions are poorly investigated. We present computational model and results of numerical simulation of damaging and fracture of aluminum and magnesium alloys under dynamic loading.

The multilevel computer simulation method was used for numerical research on damage and formation of cracks within structured representative volume element (RVE) of the alloy [34–36, 54]. Several types of grain structure that take place in alloys after severe plastic deformation were simulated.

Grain size distributions of aluminum and magnesium alloys after various numbers of passes of equal channel angular pressing (ECAP) are reported in Refs. [41–49]. The analysis of grain size distributions has shown that there are several types of grain structures. The model-structured RVE of alloys was created using the experimental data on grain structures.

Models of RVE have the volume fracture of coarse grains: 0, 5, 10, 15, 30, 50, 75, and 100%. When volume fracture of coarse grains exceeds the percolation limit (~0.25), the cell grain structures were taken into consideration. Mechanical behavior of alloy is described by means of averaging the mechanical response of structured RVE at the mesoscale level under loading of high strain rates. 3D models of representative volume element (RVE) of alloys with a bimodal grain size distribution have a dimension of 100 � <sup>100</sup> � 5 (μm)<sup>3</sup> .

Dynamics of RVE is described within approach of a continuum mechanics [14]. Smooth particle hydrodynamic (SPH) method was used for the simulation [54]. The kinematic boundary condition (Eq. (23)) is convenient when using SPH method. The boundary conditions (Eq. (23)) can be used for load combination of compression, shear, and tension. The scheme of boundary conditions is shown in Figure 20:

$$
\mu\_k(\mathbf{x}\_k, t) = \mu\_k(\mathbf{x}\_k, t), \qquad \mathbf{x}\_k \in \Gamma\_{4\prime}, \mathbf{x}\_k \in \Gamma\_{2\prime}, \mathbf{x}\_k \in \Gamma\_3, \tag{23}
$$

where xk is Cartesian coordinates, t is time, and uk is components of particle velocity vector.

Figure 20. Scheme of boundary conditions.

The dashed curve shows data obtained according to the formula

where ε<sup>f</sup> is the strain to fracture, ε\_eq ¼ ð Þ 2=3 ε\_ijε\_ij

failure has been shown to CG and UFG alloys [43, 44].

and Cε<sup>f</sup> is material parameter.

52 Nanomechanics

(from 10�<sup>4</sup> to 1 s�<sup>1</sup>

Mg-7wt.%Y [46–52].

εn <sup>f</sup> dg =εCG

and D2 are material constants.

increase of the plastic deformation.

butions depend on strain rates.

under formation of mesoscale, and macroscale cracks.

where εCG

εf static=εf dynamic ¼ 1 þ Cε<sup>f</sup> ln ε\_eq=ε\_<sup>0</sup>

<sup>1</sup>=<sup>2</sup>

In the aluminum, magnesium, and titanium alloys, the formation of mesoscopic cracks is preceded by the stage of nucleation and growth of micro-voids. In this regard, the width of the plastic zone in the considered alloys will be determined by the critical size of voids.

Valiev et al. reported the log-normal grain size distributions for fine-grained polycrystalline metals processed by equal channel angular extrusion (ECAE) [42]. The distribution effect of grain size on the flow stress, ultimate strength at compression and tension, and elongation to

Starting with the pioneering work of Tellkamp et al. [45], the formation of bimodal distribution of grain size is used to produce ultrafine-grained and nanostructured alloys with high strength and satisfactory ductility. A bimodal distribution of grain size in UFG aluminum and magnesium alloys is the cause of several anomalies of the mechanical behavior. Negative strain rate sensitivity of plastic flow stress and increase of the ductility in a wide range of strain rates

The threshold value for the strain to fracture can be calculated by the ratio

<sup>f</sup> <sup>0</sup> is the strain to fracture of CG alloy; x is equal to dg

<sup>f</sup> <sup>0</sup> <sup>¼</sup> <sup>A</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>A</sup><sup>1</sup> <sup>þ</sup> <sup>A</sup><sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>þ</sup> exp½ � ð Þ <sup>x</sup> � <sup>x</sup><sup>0</sup> <sup>=</sup><sup>x</sup> , <sup>ε</sup>CG

constants; T\* = σsp/PHEL, P\* = p/PHEL, and PHEL are the pressure in elastic precursor; and D1

Increased crack growth resistance in Al-Mg alloys with a bimodal grain size distribution is a result of deflection of microcracks on borders between UFG and coarse-grained (CG) zones [53]. As a result of it, resistance to crack growth at mesoscale level increases with the

Features of mechanical behavior of UFG alloys with bimodal grain size distribution are defined by shear band formation, nucleation and growth of damages, coalescence damages

Thus, processes of shear banding in UFG alloys with bimodal and unimodal grain size distri-

) were revealed for UFG Al–7.5 wt.% Mg, Al 5083, and magnesium alloys

<sup>=</sup>ð Þ <sup>1</sup> � <sup>θ</sup> ; <sup>ð</sup>21<sup>Þ</sup>

is the equivalent strain rate, <sup>ε</sup>\_ <sup>0</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> <sup>s</sup>�1,

<sup>f</sup> <sup>0</sup> <sup>¼</sup> <sup>D</sup><sup>1</sup> <sup>P</sup>� <sup>þ</sup> <sup>T</sup>� ð Þ<sup>D</sup><sup>2</sup>

� 1/2; A1, A2, x0, and x are material

; ð22Þ

The system of equations of continuum mechanics is applied to describe the deformation of the structured representative volume under loading conditions. Components of the strain rate tensor are described by the relation

$$\dot{\varepsilon}\_{ij} = \frac{1}{2} \left( \nabla\_i u\_j + \nabla\_j u\_i \right), \tag{24}$$

where ε\_ij is the strain rate tensor component, ui is the particle velocity vector component, and ∇<sup>i</sup> is the Hamilton operator.

Components of strain rate tensor are expressed by the sum of the elastic and the inelastic parts:

$$
\dot{\varepsilon}\_{\dot{\imath}\dot{\jmath}} = \dot{\varepsilon}\_{\dot{\imath}\dot{\jmath}}^{\epsilon} + \dot{\varepsilon}\_{\dot{\imath}\dot{\jmath}}^{\eta},\tag{25}
$$

where ε\_<sup>e</sup> ij is the elastic strain rate tensor component and ε\_<sup>n</sup> ij is the inelastic strain rate tensor component.

The bulk inelastic strain rate is described by the relation

$$
\dot{\varepsilon}\_{kk}^{n} = \frac{1}{3} \frac{\dot{D}}{(1 - D)},
\tag{26}
$$

where D\_ is the rate of change of the damage parameter.

The bulk inelastic strain rate will be equal to zero at <sup>D</sup>\_ <sup>¼</sup> 0 as follows from Eq. (26). In this case, to describe the plastic flow can be used in the theory of plasticity, which assumes the implementation of the postulate of constant volume for plastic deformation. However, the plastic deformation of NS or UFG alloys is accompanied by the damage (microcrack) nucleation and accumulation due to the limited ductility of the grains.

However, plastic deformation of NS or UFG alloys is accompanied by the nucleation and accumulation of microdamage (microcracks or pores) due to the limited ductility of mansized grains.

The damage parameter D was introduced in the form

$$D = \int\_0^{t\_f} \frac{\dot{\varepsilon}\_{eq}^n}{\varepsilon\_f^n} dt. \tag{27}$$

A local fracture criterion (Eq. (11)) was used for the UFG and the NS alloy with unimodal and bimodal distribution of grain size. The model uses Eq. (28) and phenomenological relation for the threshold inelastic strain ε<sup>n</sup> f .

Damage parameter is defined by the relation

Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading http://dx.doi.org/10.5772/intechopen.68291 55

$$D = \frac{1}{\mathbb{C}\_1} \int\_0^{\varepsilon\_{f0}^n + kl\_{\varepsilon}^{-1/2}} \exp\left(-\frac{p}{\sigma\_{eq}} \mathbb{C}\_2\right) d\varepsilon\_{eq}^n,\tag{28}$$

where dg is the grain size of mesoscale region with the unimodal distribution of grain sizes; p is the pressure; and ε<sup>n</sup> <sup>f</sup> <sup>0</sup>, k, C1, and C2 are the parameters for the local volumes of the material

The system of equations of continuum mechanics is applied to describe the deformation of the structured representative volume under loading conditions. Components of the strain rate

where ε\_ij is the strain rate tensor component, ui is the particle velocity vector component, and

Components of strain rate tensor are expressed by the sum of the elastic and the inelastic parts:

D\_

The bulk inelastic strain rate will be equal to zero at <sup>D</sup>\_ <sup>¼</sup> 0 as follows from Eq. (26). In this case, to describe the plastic flow can be used in the theory of plasticity, which assumes the implementation of the postulate of constant volume for plastic deformation. However, the plastic deformation of NS or UFG alloys is accompanied by the damage (microcrack) nucleation and

However, plastic deformation of NS or UFG alloys is accompanied by the nucleation and accumulation of microdamage (microcracks or pores) due to the limited ductility of man-

A local fracture criterion (Eq. (11)) was used for the UFG and the NS alloy with unimodal and bimodal distribution of grain size. The model uses Eq. (28) and phenomenological relation for

D ¼ ðtf 0 ε\_ n eq εn f

ε\_ij ¼ ε\_ e ij þ ε\_ n

ij is the elastic strain rate tensor component and ε\_<sup>n</sup>

ε\_ n kk <sup>¼</sup> <sup>1</sup> 3

The bulk inelastic strain rate is described by the relation

where D\_ is the rate of change of the damage parameter.

accumulation due to the limited ductility of the grains.

The damage parameter D was introduced in the form

f .

; ð24Þ

ij; ð25Þ

ð Þ <sup>1</sup> � <sup>D</sup> ; <sup>ð</sup>26<sup>Þ</sup>

dt: ð27Þ

ij is the inelastic strain rate tensor

<sup>ε</sup>\_ij <sup>¼</sup> <sup>1</sup> <sup>2</sup> <sup>∇</sup> i uj þ ∇ j ui � �

tensor are described by the relation

∇<sup>i</sup> is the Hamilton operator.

where ε\_<sup>e</sup>

54 Nanomechanics

component.

sized grains.

the threshold inelastic strain ε<sup>n</sup>

Damage parameter is defined by the relation

with unimodal distribution of grain size: e p ij <sup>¼</sup> ð Þ <sup>2</sup>=<sup>3</sup> <sup>e</sup><sup>n</sup> ijen ij h i<sup>1</sup>=<sup>2</sup> , ε<sup>n</sup> ij <sup>¼</sup> ð Þ <sup>1</sup>=<sup>3</sup> <sup>ε</sup><sup>n</sup> kkδij <sup>þ</sup> <sup>e</sup><sup>n</sup> ij, σeq = [3/ 2SijSij] 1/2, <sup>σ</sup>ij <sup>=</sup> � <sup>p</sup>δij <sup>+</sup> Sij, and <sup>δ</sup>ij is the Kronecker delta.

Eq. (12) was used as a constitutive equation of material particles. The pressure was calculated by polynomial equation of state [12]. The stress tensor deviator is calculated by Eq. (13). The deviator of the inelastic strain rate tensor is calculated by Eq. (14).

The scalar function e\_ p mð Þ eq is defined by the sum of components based on physical mechanisms of inelastic deformation by Eq. (16).

The total plastic work per volume unit under high strain rates was calculated by the relation

$$\mathcal{W}^p = (1/2) \int\_0^{t\_f} \sigma\_{i\bar{j}} \dot{\varepsilon}\_{i\bar{j}}^n dt,\tag{29}$$

where σij is the stress tensor component, ε<sup>n</sup> ij is the inelastic strain rate tensor component, and tf is the time of inelastic deformation before local fracture.

The local increment of temperature due to dissipation of plastic work at high strain rate was calculated by the equation

$$
\Delta T(t) = \left(\beta/\rho \mathbb{C}\_p\right) \int\_0^t \sigma\_{\vec{\imath}\vec{\jmath}} \dot{\varepsilon}\_{\vec{\imath}\vec{\jmath}}^n dt,\tag{30}
$$

where ΔT is the temperature increment associated with plastic deformation, β is a phenomenological parameter, Cp is the specific heat of material, and r is the mass density:

$$\beta(\varepsilon\_{\mathbf{eq}}^{\mathbb{P}}) = 1 - \ln \left( \frac{\varepsilon\_{\mathbf{eq}}^{\mathbb{P}}}{\varepsilon\_0} \right)^{n-1},\tag{31}$$

where ε<sup>0</sup> is the yield strain and n is the hardening exponent of the material.

Eqs. (10)–(15) and (24)–(31) were used for describing mechanical response of structured RVE of FCC and HCP alloys under dynamic loadings.

The distribution of grain sizes is specified in the model of a representative volume of the alloys.

Model RVE of alloys with a bimodal grain size distribution includes separate large grains and the volumes consisting of fine grains. RVEs taking into account the distribution of grain size similar to those identified in samples of UFG aluminum alloy Al 1560 (Al-Mg) and magnesium alloy MA2-1 (Mg-Al-Zn) were used for model calibration. Parameters of materials in constitutive equations were discussed in Refs. [12, 34, 53]. Numerical method was discussed in Refs. [12, 34–36].

The equivalent stresses in UFG magnesium alloy with quasi-regular grains structure at the time ~15.7 ns under shock wave loading with peak stress of 3.8 GPa are shown in Figure 21.

The width of shock wave front is less than the average grain size ~300 nm in the model of NS aluminum alloy with regular grain structure. Mechanical behavior of aluminum alloys with similar grain structure under quasi-static loading was discussed in Ref. [55].

A distribution of material particle velocities was found on the mesoscale level. The distribution of particle velocities arises from the wave interaction with grain boundaries and triple joints of grains.

The distribution of the parameter of damage behind the front of a shock wave with a peak stress of 3.2 GPa is shown in Figure 22. Two large grains with size of 20 μm are contained in the model volume of the alloy. These large grains were surrounded by UFG phase with average grain size 1 μm.

Deformation of RVE occurred nonuniformly under shock compression. Shear bands have been formed in the UFG phase at high strain rates. Damage to the mesoscale shear band was formed in 40 ns. The damage parameter increased rapidly when the local tensile stresses acted in areas filled with small grains and on the border between large grains and fine grains. Fracture of RVE is a result of the formation of flat clusters of damaged particles.

The distribution of grains size affects on the formation of damage and the distribution of the specific plastic work per unit volume over the RVE. The distribution of specific plastic work in RVE is shown in Figure 23. The size of large grains and its specific volume was ~20 μm and ~38%, respectively.

The damage parameter increased in local areas where the maximum values of plastic work have been achieved. Fracture mechanisms of alloys with bimodal distribution of grain size are similar in compression and tension. Damage also accumulates in the zones of plastic strain

Figure 21. Stress in UFG magnesium alloy with quasi-regular grain structure.

Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading http://dx.doi.org/10.5772/intechopen.68291 57

Figure 22. Damage in Al 1560 alloy under shock wave loading with peak stress of 3.2 GPa.

alloy MA2-1 (Mg-Al-Zn) were used for model calibration. Parameters of materials in constitutive equations were discussed in Refs. [12, 34, 53]. Numerical method was discussed in Refs.

The equivalent stresses in UFG magnesium alloy with quasi-regular grains structure at the time ~15.7 ns under shock wave loading with peak stress of 3.8 GPa are shown in Figure 21. The width of shock wave front is less than the average grain size ~300 nm in the model of NS aluminum alloy with regular grain structure. Mechanical behavior of aluminum alloys with

A distribution of material particle velocities was found on the mesoscale level. The distribution of particle velocities arises from the wave interaction with grain boundaries and triple joints

The distribution of the parameter of damage behind the front of a shock wave with a peak stress of 3.2 GPa is shown in Figure 22. Two large grains with size of 20 μm are contained in the model volume of the alloy. These large grains were surrounded by UFG phase with

Deformation of RVE occurred nonuniformly under shock compression. Shear bands have been formed in the UFG phase at high strain rates. Damage to the mesoscale shear band was formed in 40 ns. The damage parameter increased rapidly when the local tensile stresses acted in areas filled with small grains and on the border between large grains and fine grains. Fracture of RVE is a result of the formation of flat clusters of damaged particles.

The distribution of grains size affects on the formation of damage and the distribution of the specific plastic work per unit volume over the RVE. The distribution of specific plastic work in RVE is shown in Figure 23. The size of large grains and its specific volume was ~20 μm and

The damage parameter increased in local areas where the maximum values of plastic work have been achieved. Fracture mechanisms of alloys with bimodal distribution of grain size are similar in compression and tension. Damage also accumulates in the zones of plastic strain

Figure 21. Stress in UFG magnesium alloy with quasi-regular grain structure.

similar grain structure under quasi-static loading was discussed in Ref. [55].

[12, 34–36].

56 Nanomechanics

of grains.

average grain size 1 μm.

~38%, respectively.

localization near borders of large grains of alloys with a bimodal distribution of grains. Specific plastic work per unit volume and equivalent plastic strain are shown in Figures 23 and 24, respectively.

Therefore, kinetics of fracture depends on the specific volume of coarse grains.

The equivalent plastic strain in RVE of Al 1560 alloy under shock compression with peak stress of 3.2 GPa is shown in Figure 24. The average intensity of plastic deformation in coarse grains is higher than in the volume filled with fine grains. The maximum equivalent plastic strain is localized in a fine-grained phase near the boundaries of large grains.

Figure 25 shows the calculated plastic work under the tension of RVE at high strain rates. The tensile loading at high strain rates is realized in the spall zone where interaction of unloading waves takes place.

Figure 23. Specific plastic work per unit volume in Al 1560 alloy under shock wave loading with peak stress of 3.2 GPa.

Figure 24. The equivalent plastic strain in RVE under shock wave loading with peak stress of 3.2 GPa.

Figure 25. Specific work per unit volume under tension of RVE of Al 1560 aluminum alloy.

Mesoscale cracks in fine-grained volume formed under a tensile strain rates above ~10<sup>3</sup> s 1 . The presence of large grains prevents the rapid growth of these cracks and leads to an increase in plastic work per unit volume. Thus, UFG alloys with bimodal distribution of grain sizes under dynamic loading have a higher ductility compared to UFG counterparts with unimodal distribution of grain sizes.

Tensile stress in the spall zone is shown in Figure 26. The representative volume of Al 1560 aluminum alloy consisted from ~75% of large grains with a size ~20 μm. The average size of fine grains was assumed equal to 1 μm. The relaxations of stresses in a spall zone have been accelerated by the damage accumulation.

The use of multi-pass ECAP leads to stochastic distribution of large grains in the volume of alloys. As a result, fracture of UFG alloys with bimodal grain size distribution under dynamic loading has probabilistic character. Large grains can be formed in the NS and UFG alloys as a result of dynamic recrystallization [56, 57].

In the used model, Eq. (18) takes into account the influence of nanostructured precipitates on dislocation kinetics. Wang et al. showed that in the UFG Al the deformation mechanism is operated by the dislocation interactions at high strain rates, while in the lower strain rate range, the deformation mechanism may be related to grain boundary sliding [58].

The increase in the concentration of nano-size precipitates leads to the growth of the yield stress of both large and small grains of alloys. The concentration of precipitates in UFG alloys affects the grain size distribution and the distance between the shear bands. These results agree

Figure 26. Tensile stress in RVE of Al 1560 aluminum alloy.

Figure 24. The equivalent plastic strain in RVE under shock wave loading with peak stress of 3.2 GPa.

58 Nanomechanics

Figure 25. Specific work per unit volume under tension of RVE of Al 1560 aluminum alloy.

with experimental data [59]. Chrominski et al. showed that the grain size, grain boundary character, and dislocation substructure are strongly depended on the precipitation in 6082 aluminum alloy. It was found that needlelike precipitates are homogeneously distributed in the grains of micron-size [59].

The ductility of UFG alloys increases when a specific volume of coarse grains is decreased. The ductility versus specific volume of coarse grains in Al-Mg alloy with a bimodal grain size distribution is shown in Figure 27. Experimental data [45, 48, 49] are shown by filled symbols. Decreasing of the specific volume of large grains in RVE of UFG alloy is accompanied by a decrease of average strain to failure under quasi-static and dynamic loading.

Figure 28 shows the dependence of strain to fracture of aluminum and magnesium alloys on logarithm of strain rates.

Experimental data reported by Ulacia for coarse-grained magnesium alloy AZ31 are marked by filled symbols [60]. Data for aluminum alloys are shown in circular and triangular symbols [48, 49]. UFG Al-Mg alloys have fine-grained size of 1 μm and coarse-grained size of 20 μm.

Figure 27. The strain to fracture of Al-Mg alloy with a bimodal grain size distribution versus specific volume of coarse grains.

Mechanical Behavior of Nanostructured and Ultrafine-Grained Metal Alloy under Intensive Dynamic Loading http://dx.doi.org/10.5772/intechopen.68291 61

Figure 28. Strain to fracture of aluminum and magnesium alloys versus logarithm of strain rate.

with experimental data [59]. Chrominski et al. showed that the grain size, grain boundary character, and dislocation substructure are strongly depended on the precipitation in 6082 aluminum alloy. It was found that needlelike precipitates are homogeneously distributed in

The ductility of UFG alloys increases when a specific volume of coarse grains is decreased. The ductility versus specific volume of coarse grains in Al-Mg alloy with a bimodal grain size distribution is shown in Figure 27. Experimental data [45, 48, 49] are shown by filled symbols. Decreasing of the specific volume of large grains in RVE of UFG alloy is accompanied by a

Figure 28 shows the dependence of strain to fracture of aluminum and magnesium alloys on

Experimental data reported by Ulacia for coarse-grained magnesium alloy AZ31 are marked by filled symbols [60]. Data for aluminum alloys are shown in circular and triangular symbols [48, 49]. UFG Al-Mg alloys have fine-grained size of 1 μm and coarse-grained size of

0,0 0,5 1,0 0,00

Figure 27. The strain to fracture of Al-Mg alloy with a bimodal grain size distribution versus specific volume of coarse

Specific volume of coarse grains

decrease of average strain to failure under quasi-static and dynamic loading.

the grains of micron-size [59].

logarithm of strain rates.

0,06

0,12

0,18

20 μm.

60 Nanomechanics

ε f

grains.

The calculated values of total strain to failure of coarse-grained magnesium alloy AZ31 versus the logarithm of strain rate are depicted by dashed Curve 1 in Figure 28. Rodriguez [61] later received that the experimental values of strain to failure of AZ31 alloy at quasi-static loading conditions confirmed the adequacy of the prediction.

Curve 2 shows the calculated strain to fracture of Ma2-1 magnesium alloy with volume concentration of large grains about ~70%. Curve 3 shows strain to fracture for magnesium alloy with specific volume of large grains about ~10%.

Curve 4 shows the ductility of Al-Mg alloys with a bimodal grain structure in a wide range of strain rates. Curve 5 displays decreasing dynamic ductility of Al-Mg alloys with a bimodal grain structure and specific volume of large grains ~10%.

The plasticity of the alloys under quasi-static tension increases when specific volume of large grains in UFG Al-Mg alloy is more than 30%.

The calculated strain to failure for Al-M alloys can be described by the equation

$$
\varepsilon\_f^\eta = 0.01 \exp\left(\mathsf{C}\_{\mathsf{G}\mathfrak{g}}/0.363\right),
\tag{32}
$$

where ε<sup>n</sup> <sup>f</sup> is the strain to failure under quasi-static tension and Ccg is the specific volume of coarse-grained size.

Eq. (32) describes ductility of UFG Al-Mg alloys with a bimodal grain distribution versus the specific volume of large grains. Results of simulation agree with the available experimental data [48, 49, 52].

## 6. Conclusions

The study of the mechanical behavior of NS and UFG alloys under intensive dynamic loadings allows you to deepen understanding of the basic laws of deformation and failure of alloys in extreme states.

Experimental studies have shown that the difference between the Hugoniot elastic limits of alloys with UFG and CG structure can be negligible.

The spall strength and the yield strength of UFG alloys are depended not only on grain size but also on distribution of grain sizes, a concentration of a nano-sized precipitates.

Fine precipitates in alloys not only affect the hardening but also lead to change the influence of the grain size distribution on volume concentration of shear bands.

Multiscale computer modeling can be used to study the regularities of deformation and fracture of advanced structural alloys at high strain rates.

It was shown that the use of models taking into account the unimodal and bimodal distribution of grain size allows predicting the strength and ductility of a UFG alloys at high strain rates under compression and tension.

The results of computer simulation showed that the dynamic strength and ductility depend on the distribution of grain size in the HCP and FCC alloys processed by severe plastic deformation.

Experimental and numerical studies have shown that strain to failure of UFG alloys with a bimodal grain size distribution nonlinearly decreases with the increase of specific volume of submicrometer grains.

The dynamic ductility of UFG alloys is decreased when specific volume of submicrometer grains is above ~70%.

It should be noted that researches on the influence of grain size distribution of UFG alloys on dynamic strength and ductility are at an early stage. The peculiarities of the mechanical behavior of UFG alloys under dynamic loading at elevated temperatures are insufficiently studied.

## Author details

Vladimir A. Skripnyak\* and Evgeniya G. Skripnyak

\*Address all correspondence to: skrp2006@yandex.ru

National Research Tomsk State University, Russia

## References

6. Conclusions

62 Nanomechanics

extreme states.

deformation.

submicrometer grains.

grains is above ~70%.

insufficiently studied.

Author details

alloys with UFG and CG structure can be negligible.

The study of the mechanical behavior of NS and UFG alloys under intensive dynamic loadings allows you to deepen understanding of the basic laws of deformation and failure of alloys in

Experimental studies have shown that the difference between the Hugoniot elastic limits of

The spall strength and the yield strength of UFG alloys are depended not only on grain size but

Fine precipitates in alloys not only affect the hardening but also lead to change the influence of

Multiscale computer modeling can be used to study the regularities of deformation and

It was shown that the use of models taking into account the unimodal and bimodal distribution of grain size allows predicting the strength and ductility of a UFG alloys at high strain

The results of computer simulation showed that the dynamic strength and ductility depend on the distribution of grain size in the HCP and FCC alloys processed by severe plastic

Experimental and numerical studies have shown that strain to failure of UFG alloys with a bimodal grain size distribution nonlinearly decreases with the increase of specific volume of

The dynamic ductility of UFG alloys is decreased when specific volume of submicrometer

It should be noted that researches on the influence of grain size distribution of UFG alloys on dynamic strength and ductility are at an early stage. The peculiarities of the mechanical behavior of UFG alloys under dynamic loading at elevated temperatures are

also on distribution of grain sizes, a concentration of a nano-sized precipitates.

the grain size distribution on volume concentration of shear bands.

fracture of advanced structural alloys at high strain rates.

Vladimir A. Skripnyak\* and Evgeniya G. Skripnyak

\*Address all correspondence to: skrp2006@yandex.ru

National Research Tomsk State University, Russia

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## **Thermomass Theory: A Mechanical Pathway to Analyze Anomalous Heat Conduction in Nanomaterials**

Yuan Dong, Bingyang Cao and Zengyuan Guo

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67780

#### Abstract

The synthesis and measurements of nanomaterials have yielded significant advances in the past decades. In the area of thermal conduction, the nanomaterials exhibit anomalous behavior such as size-dependent thermal conductivity, thermal rectification, and ultra-high thermoelectric properties. The theoretical understanding and modeling on these behaviors are much desired. In this chapter, we study the thermal conduction in nanomaterials through the thermomass theory, which models the heat transfer from a fluid mechanics viewpoint. The control equations of the equivalent mass of the thermal energy are formulated following the continuum mechanics principles, which give the general heat conduction law. It incorporates nonlinear effects such as spatial acceleration and boundary resistance, which can overcome the drawbacks of the traditional Fourier's law in nanoscale systems. By the thermomass theory, we successfully model the sizedependent effective thermal conductivity in nanosystems. Furthermore, the thermal rectification as well as the thermoelectric enhancement in nanosystems is also discussed with the present framework.

Keywords: thermomass theory, nanomaterial, thermal conductivity, thermal rectification, thermoelectric

#### 1. Introduction

The Fourier law proposed in 1822 [1] is the fundamental of thermal conduction. It indicates that the heat flux passing through a material is proportional to the local gradient of temperature

$$q = -\kappa \nabla T \tag{1}$$

where q is the heat flux, ∇T is the local temperature gradient, and κ is the thermal conductivity, which represents the material capability of transferring heat. In a long term, the Fourier law can accurately model the heat conduction. In the middle of twentieth century, theoretical

physicists started to question the Fourier law because of its contradiction to the second law of thermodynamics [2]. After that, the heat waves were observed in low-temperature experiments [3] and aroused people's interest as well as controversy. In 1980s, the short pulse laser experiment stimulated a lot of research and led to several relaxational [4], hyperbolic [5], or lagging types [6] of models, which can be regarded as the generalization of Fourier law. The above research focused on the distortion of ordinary heat transfer in short time scales. On the other hand, the shrink of space scales caused another type of distortion and began to be realized in the early 1990s, when sign of failure of Fourier law was perceived in thin dielectric films [7]. The phenomena of anomalous heat transfer in small scale materials can be fundamentally understood through the kinetic theory of phonons, that is, the thermal conductivity of dielectric materials can be formulated as [8, 9]

$$
\kappa = \frac{1}{3} \mathbf{C} \nu \lambda \tag{2}
$$

where C is the specific heat per unit volume, v is the average group velocity of phonon, and λ is the phonon mean free path (MFP). When the material size is much larger than MFP, the MFP can be regarded as a constant and is dominated by the intrinsic phonon-phonon scattering and phonon-defect scattering rates. Therefore, the thermal conductivity is independent on the system size. In contrast, when the material size reduces to comparable value with the MFP, the phonon-boundary scattering becomes considerable. In this condition, the smaller system size induces higher boundary scattering rates and consequently shorter effective phonon MFP. By using Eq. (2) one figures out the reduction of thermal conductivity of nanomaterials.

The reduced thermal conductivity of nanofilms is a disadvantage for the heat dissipation in IC chips or semiconductor lasers. Nevertheless, it is an advantage for the thermoelectric devices. Experiments showed that the silicon nanowires have very high figure of merit (ZT) [10, 11]. The nanocomposites also demonstrate considerable ZT benefiting from the nano-sized superlattice or grains significantly scatter the phonons and reduce the effective thermal conductivity [12, 13]. Therefore, a lot of effort has been made to fabricate materials with ultra-low thermal conductivity through nanotechnology with the target at high ZT for the applications in advanced heating and cooling, waste heat recovery [14], as well as solar thermoelectric generators [15].

Due to the fast growth of energy-related nanomaterial synthesis and its transition from laboratory to industrial applications, modeling the thermal conducting behavior in nanosystems is in urgent need. Ideally, it should rise from a perspective of characterizing the fundamental physics and approach to simply structured theory which can be conveniently used by engineers. Nevertheless, this goal has not been satisfactorily achieved and current research is paving toward it. The gray model proposed by Majumdar is a pioneer work in this path. It predicts the effective thermal conductivity as [7]

$$\frac{\kappa\_{\rm eff}}{\kappa\_0} = \frac{1}{1 + \beta \frac{\lambda}{L}} = \frac{1}{1 + \beta \text{Kn}}\tag{3}$$

where κeff is the effective thermal conductivity, κ<sup>0</sup> is the thermal conductivity of the bulk material, L is the characteristic size of system, and β is a dimensionless parameter. Except that the temperature is much lower than the Debye temperature, the phonon scattering at most engineering surfaces can be regarded as diffusive. In this case, it was derived that for the inplane heat conductivity of nanofilms, β = 3/8. For the cross-plane heat conductivity of nanofilms, β = 4/3. For the longitudinal heat conductivity of nanowires, β can be selected as 4/3 [16]. Kn is the Knudsen number, which is the ratio of MFP over L. Kn is actually a concept in gas dynamics, and it is well known that rarefaction effects should be considered in high Kn situations [17]. Eq. (3) was derived from an analogy between photons and phonons as wave packets of energy. Therefore, radiative transfer was assumed for phonons. It is easily found that Eq. (3) retreats to the Fourier law when the system size is much larger than MFP, that is, at the bulk limit. When the system size is comparable with the MFP, Eq. (3) delineates the size dependency of thermal conductivity. However, along with the progress in measuring the thermal conductivity of thin silicon films [18–21], the accuracy of Eq. (3) was questioned. It was claimed that the MFP of monocrystalline silicon should be around 300 nm to match the experiment results [19], while the value based on Eq. (2) is around 42 nm. Chen et al. [22–24] proposed that the phonon MFPs of single-crystal Si at room temperature should be 210–260 nm considering that the phonons of different frequencies contribute differently to the heat conduction. This amendment partly resolves the inaccuracy of gray model. However, it still exhibits considerable deviations to predict the experiment value of nanowires [25]. McGaughey et al. [16] developed a model which accounts the full dispersion relation and the directional dependent scattering chances with surfaces. This model matches well with experiments for nanofilms, while still overestimating the experiments for nanowires.

physicists started to question the Fourier law because of its contradiction to the second law of thermodynamics [2]. After that, the heat waves were observed in low-temperature experiments [3] and aroused people's interest as well as controversy. In 1980s, the short pulse laser experiment stimulated a lot of research and led to several relaxational [4], hyperbolic [5], or lagging types [6] of models, which can be regarded as the generalization of Fourier law. The above research focused on the distortion of ordinary heat transfer in short time scales. On the other hand, the shrink of space scales caused another type of distortion and began to be realized in the early 1990s, when sign of failure of Fourier law was perceived in thin dielectric films [7]. The phenomena of anomalous heat transfer in small scale materials can be fundamentally understood through the kinetic theory of phonons, that is, the thermal conductivity

> <sup>κ</sup> <sup>¼</sup> <sup>1</sup> 3

where C is the specific heat per unit volume, v is the average group velocity of phonon, and λ is the phonon mean free path (MFP). When the material size is much larger than MFP, the MFP can be regarded as a constant and is dominated by the intrinsic phonon-phonon scattering and phonon-defect scattering rates. Therefore, the thermal conductivity is independent on the system size. In contrast, when the material size reduces to comparable value with the MFP, the phonon-boundary scattering becomes considerable. In this condition, the smaller system size induces higher boundary scattering rates and consequently shorter effective phonon MFP. By using Eq. (2) one figures out the reduction of thermal conductivity of nanomaterials.

The reduced thermal conductivity of nanofilms is a disadvantage for the heat dissipation in IC chips or semiconductor lasers. Nevertheless, it is an advantage for the thermoelectric devices. Experiments showed that the silicon nanowires have very high figure of merit (ZT) [10, 11]. The nanocomposites also demonstrate considerable ZT benefiting from the nano-sized superlattice or grains significantly scatter the phonons and reduce the effective thermal conductivity [12, 13]. Therefore, a lot of effort has been made to fabricate materials with ultra-low thermal conductivity through nanotechnology with the target at high ZT for the applications in advanced heating

Due to the fast growth of energy-related nanomaterial synthesis and its transition from laboratory to industrial applications, modeling the thermal conducting behavior in nanosystems is in urgent need. Ideally, it should rise from a perspective of characterizing the fundamental physics and approach to simply structured theory which can be conveniently used by engineers. Nevertheless, this goal has not been satisfactorily achieved and current research is paving toward it. The gray model proposed by Majumdar is a pioneer work in this path. It

where κeff is the effective thermal conductivity, κ<sup>0</sup> is the thermal conductivity of the bulk material, L is the characteristic size of system, and β is a dimensionless parameter. Except that

<sup>¼</sup> <sup>1</sup>

and cooling, waste heat recovery [14], as well as solar thermoelectric generators [15].

κeff κ0

<sup>¼</sup> <sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>β</sup> <sup>λ</sup> L

Cvλ (2)

<sup>1</sup> <sup>þ</sup> <sup>β</sup>Kn (3)

of dielectric materials can be formulated as [8, 9]

70 Nanomechanics

predicts the effective thermal conductivity as [7]

The phonon hydrodynamics [26–31] is another pathway to model the nanoscale heat conduction. It originates from the solving of linearized Boltzmann equation. An additional term representing the second order spatial derivative of heat flux, ∇<sup>2</sup> q, is involved in the governing equation of heat conduction. Since the heat flux is similar to a fluid flow flux, ∇2 q is in analogy with the viscous dissipation term in Navier-Stokes equation for fluid mechanics. Therefore, the heat flux could be nonuniform in the heat transfer cross-section due to the drag from the boundary, forming a Poiseuille flow of heat. This behavior induces the terminology of "phonon hydrodynamics." The analysis based on phonon hydrodynamics indicated the effective thermal conductivity of nanosystems should be inversely proportional to the square of Kn due to the nonuniform distribution of heat flux profile. However, the experiments indicated that the effective thermal conductivity is approximately linear to the characteristic size rather than the square of size. It is thereby further elucidated that the boundary velocity slip would happen in case of large Kn [29, 31]. By introducing the slip boundary condition into the governing equation, the linear size-dependent effective thermal conductivity can be achieved. The drawbacks of present phonon hydrodynamics analysis are: 1. The arbitrary in choosing the style and parameters of slip boundary condition. 2. The deviation from the physical picture of original derivation of Boltzmann equation, where it was the normal (N) scattering processes that induced the second order spatial derivative of heat flux. The present phonon hydrodynamic models just simply use the MFP of resistive (R) scattering processes as the parameter of ∇<sup>2</sup> q.

Upon the abovementioned progresses and their defects, the development of better models characterizing heat conducting in nanomaterials should base on capturing the essential feature of its physics. In recent years, the thermomass theory has been developed in our group, which proposes a mechanical analysis framework for heat transfer [32–35]. The generalized heat conduction governing equations are established based on such analysis. In the following sections, we will present the application of thermomass theory in nanomaterial heat conduction. The size dependency of thermal conductivity, thermal rectification, and thermoelectric effects will be addressed.

#### 2. Thermomass theory

In history, the nature of heat was regarded as either a fluid, that is, caloric theory. The caloric theory regards heat as a weightless, self-repulsive fluid. In the eighteenth and the first half of nineteenth centuries, the caloric theory was the mainstream theory. It was extinct after the midnineteenth century and replaced by the dynamic theory that the nature of heat is the random motion of particles in a body. In twentieth century, Einstein's relativity theory introduced the well-known mass-energy equivalence relation, E = mc<sup>2</sup> , where c is the speed of light. According to this theory, the thermal energy should correspond to a certain amount of mass. To illustrate his theory, Einstein elucidated "a piece of iron weighs more when red hot than when cool" [36], which means the adding of the thermal energy into material, that is, raise its temperature and at the same time increase the mass. The mass increase induced by heat was defined as "thermomass," which is very small in ordinary conditions. For example, the thermomass of Si at room temperature is 10�<sup>12</sup> of the total mass. Such small amount of mass is negligible when dealing with the dynamic problem, like movement and balance of the body. However, the heat conduction is the movement of thermomass itself relative to molecular or the lattice. It is driven by the pressure gradient induced by the concentration difference of thermomass among the materials. The forces and inertia of thermomass are comparable and lead to the limited acceleration and drift velocity of it. The advantage to bring in the concept of thermomass is that the analysis of heat conduction can follow a mechanical framework. The corresponding forces, velocities, accelerations, and momentums can be properly defined.

Consider the dielectric solids, the phonons are the main heat carriers. In this case, the internal energy per unit volume, e, is the summation of all phonon energies [9]

$$\begin{split} e = \mathcal{C}T &= \left(2\pi\right)^{-3} \sum\_{n} \int [\hbar\omega^{n}f^{n}(k,\mathbf{x},t)]d^{3}k\\ &= \sum\_{n} \int\_{k} \hbar\omega^{n}f^{n}(k,\mathbf{x},t) \end{split} \tag{4}$$

where ħ is the reduced Planck constant (Dirac constant), ω is the phonon frequency, k is the wave vector, and n denotes the index of phonon branches. f is the phonon distribution function. In equilibrium state, f obeys the Bose-Einstein distribution

$$f\_0 = \left[\exp(\hbar\omega/k\_\mathrm{B}T) - 1\right]^{-1} \tag{5}$$

where k<sup>B</sup> is the Boltzmann constant. The density of the thermomass, that is, the equivalent mass of the phonon gas, is obtained by using the Einstein's mass-energy equivalence relation

Thermomass Theory: A Mechanical Pathway to Analyze Anomalous Heat Conduction in Nanomaterials http://dx.doi.org/10.5772/67780 73

$$
\rho\_{\rm TM} = \frac{\rm CT}{c^2} \tag{6}
$$

It should be reminded that the frequently used expression for thermal conductivity of phonon systems, Eq. (2), is from the analogy between gas and heat carriers. The scattering of phonons induces resistance on heat transport. Generally, the scattering accounted for thermal resistance is the R processes, including the Umklapp scattering, defect scattering, and boundary scattering. These scattering events eliminate the quasi-momentum of phonons. The MFP defined in Eq. (2) refers to the traveled distance of a phonon between succeeding R scatterings. However, in ideal gas systems, the collision among gas molecules does not perish the momentums. Therefore, the R processes of phonons are more resemble to the collision of gas molecules to residential barriers. It is the case when a gas flows through a porous medium. The collision frequency between gas molecules and material skeleton determines the resistance experienced by the gas flow. In the porous flow, the Darcy's law describes the effective flow velocity is proportional to the pressure gradient

physics. In recent years, the thermomass theory has been developed in our group, which proposes a mechanical analysis framework for heat transfer [32–35]. The generalized heat conduction governing equations are established based on such analysis. In the following sections, we will present the application of thermomass theory in nanomaterial heat conduction. The size dependency of thermal conductivity, thermal rectification, and thermoelectric effects will be addressed.

In history, the nature of heat was regarded as either a fluid, that is, caloric theory. The caloric theory regards heat as a weightless, self-repulsive fluid. In the eighteenth and the first half of nineteenth centuries, the caloric theory was the mainstream theory. It was extinct after the midnineteenth century and replaced by the dynamic theory that the nature of heat is the random motion of particles in a body. In twentieth century, Einstein's relativity theory introduced the

to this theory, the thermal energy should correspond to a certain amount of mass. To illustrate his theory, Einstein elucidated "a piece of iron weighs more when red hot than when cool" [36], which means the adding of the thermal energy into material, that is, raise its temperature and at the same time increase the mass. The mass increase induced by heat was defined as "thermomass," which is very small in ordinary conditions. For example, the thermomass of Si at room temperature is 10�<sup>12</sup> of the total mass. Such small amount of mass is negligible when dealing with the dynamic problem, like movement and balance of the body. However, the heat conduction is the movement of thermomass itself relative to molecular or the lattice. It is driven by the pressure gradient induced by the concentration difference of thermomass among the materials. The forces and inertia of thermomass are comparable and lead to the limited acceleration and drift velocity of it. The advantage to bring in the concept of thermomass is that the analysis of heat conduction can follow a mechanical framework. The corresponding

Consider the dielectric solids, the phonons are the main heat carriers. In this case, the internal

X n

where ħ is the reduced Planck constant (Dirac constant), ω is the phonon frequency, k is the wave vector, and n denotes the index of phonon branches. f is the phonon distribution func-

where k<sup>B</sup> is the Boltzmann constant. The density of the thermomass, that is, the equivalent mass of the phonon gas, is obtained by using the Einstein's mass-energy equivalence relation

ð ħωnf

<sup>n</sup> ½ � ð Þ <sup>k</sup>; <sup>x</sup>; <sup>t</sup> <sup>d</sup><sup>3</sup>

k

<sup>f</sup> <sup>0</sup> <sup>¼</sup> expð Þ� <sup>ħ</sup>ω=kB<sup>T</sup> <sup>1</sup> � ��<sup>1</sup> (5)

(4)

forces, velocities, accelerations, and momentums can be properly defined.

energy per unit volume, e, is the summation of all phonon energies [9]

<sup>e</sup> <sup>¼</sup> CT <sup>¼</sup> ð Þ <sup>2</sup><sup>π</sup> �<sup>3</sup>

tion. In equilibrium state, f obeys the Bose-Einstein distribution

<sup>¼</sup> <sup>X</sup> n

ð k ħωnf n ð Þ k; x; t

, where c is the speed of light. According

2. Thermomass theory

72 Nanomechanics

well-known mass-energy equivalence relation, E = mc<sup>2</sup>

$$
u \propto -\nabla p\tag{7}$$

The pressure gradient can be regarded as the driving force of flow. From a viewpoint of force balance, the driving force is actually balanced by the friction force. Thereby Eq. (7) essentially depicts that the friction force is proportional to the flow velocity. It is a general case in laminar flow.

In analogy to the gas flow in porous medium, the velocity of thermomass is defined as

$$
\mu\_{\rm TM} = \frac{q}{CT} \tag{8}
$$

The mass and momentum balance equations of thermomass can be derived as [32–34]

$$\frac{\partial \rho\_{\rm TM}}{\partial t} + \nabla \cdot \left(\rho\_{\rm TM} \mu\_{\rm TM}\right) = 0 \tag{9}$$

$$
\rho\_{\rm TM} \frac{\partial \mu\_{\rm TM}}{\partial t} + \left(\rho\_{\rm TM} \mu\_{\rm TM} \cdot \nabla\right) \mu\_{\rm TM} + \nabla p\_{\rm TM} = f\_{\rm TM} \tag{10}
$$

where pTM is the phonon gas pressure, and fTM is the friction force impeding the phonon gas. Eq. (9) gives the energy conservation equation by applying Eqs. (6) and (8). Eq. (10) characterizes the heat transport, which is the motion of thermomass through the materials. To obtain the explicit heat transport governing equation, the pressure and friction terms need to be determined. If the phonons are viewed as moving particles with finite mass, the pressure of them can be derived by accounting the momentum change when these particles hit and rebound from a unit area of the container surface, in analogy to the kinetic theory of gas. In a result, the pressure of phonon gas can be expressed as

$$p\_{\rm TM} = \frac{1}{3} \frac{\nu\_{\rm g}^2}{c^2} CT = \frac{1}{3} \nu\_{\rm g}^2 \rho\_{\rm TM} \tag{11}$$

where v<sup>g</sup> is the group velocity of phonons. For bulk material, the friction experienced by thermomass can be extracted from Eq. (7). When discussing the nanosystems, the boundary effect needs to be considered. The Darcy's law for porous flow was extended to Darcy-Brinkman relation when the boundary effect is nonnegligible [37, 38]

$$-\nabla p = \frac{\mu}{K}u - \mu \nabla^2 u \tag{12}$$

where μ is the viscosity,Kis the permeability with a unit of m2 . Eq. (12) indicates that the boundary slip velocity attenuates from the boundary with a characteristic length of K1/2 to the uniform velocity in the porous medium. The introducing of a second-order spatial derivative term also makes Eq. (12) the same order as the governing equations for free flow. In the steady flow, the driving force is balanced with the friction force. Following the form of Eq. (12), when the boundary effect is considered, the friction of thermomass can be formulated as

$$f\_{\rm TM} = -\chi \rho\_{\rm TM} \mu\_{\rm TM} + \mu\_{\rm TM} \nabla^2 \mu\_{\rm TM} \tag{13}$$

where χ is the friction factor. The permeability of the thermomass in heat conducting medium is

$$K\_{\rm TM} = \frac{\mu\_{\rm TM}}{\chi \rho\_{\rm TM}} \tag{14}$$

In large systems, the boundary effect is negligible. Then, Eq. (13) reduces to the Darcy's law with the first term much more important than the second term on the right hand side.

When the spatial gradient and changing rate of physical quantities are not significant, the first and second terms in Eq. (10) can be neglected. In this case, Eq. (10) exhibits the balance between driving force and friction force. The heat conduction is steady in such a nonequilibrium system. Combining Eqs. (13) and (10) leads to

$$
\lambda \chi \eta = -\nabla \left(\frac{1}{3} \nu\_{\text{g}}^2 \text{CT}\right) \tag{15}
$$

For the simplest case, v<sup>g</sup> and C are assumed to be temperature independent. Then, Eq. (15) actually gives the Fourier law with

$$
\chi = \nu\_\mathrm{g}^2 \mathbb{C} / 3\kappa \tag{16}
$$

When the boundary effect is considerable, the second term in Eq. (13) needs to be accounted. In this case, the combination of Eqs. (13) and (10) gives

$$-\kappa \nabla T = q - l\_\text{B}^2 \nabla^2 q \tag{17}$$

where l<sup>B</sup> equaling to the square root of KTM is a characteristic length.

Eq. (17) is a generalization of Fourier law when boundary effect needs to be considered. It predicts the reduction of effective thermal conductivity in nanosystems by the additional resistance term. When the system size is bigger, the spatial gradient of q is smaller. Thus, κeff increases with the system size growing larger. Nevertheless, to quantitatively predict the size dependency of κeff and compare it with experiments, the exact value of l<sup>B</sup> needs to be determined for certain material. The thermal conductivity is a macroscopic physical quantity, which is usually obtained by experiments. Similarly, with plenty of experimental data, the value of thermomass permeability and l<sup>B</sup> could be evaluated. However, nowadays the experiments in nanosystems are still expensive and have large uncertainty. Therefore, in the following, a bottom-up strategy, namely, raising from microscopic phonon properties, is used to extract the value of lB.

#### 3. Phonon Boltzmann derivation

effect needs to be considered. The Darcy's law for porous flow was extended to Darcy-Brinkman

slip velocity attenuates from the boundary with a characteristic length of K1/2 to the uniform velocity in the porous medium. The introducing of a second-order spatial derivative term also makes Eq. (12) the same order as the governing equations for free flow. In the steady flow, the driving force is balanced with the friction force. Following the form of Eq. (12), when the boundary

<sup>f</sup> TM ¼ �χρTMuTM <sup>þ</sup> <sup>μ</sup>TM∇<sup>2</sup>

where χ is the friction factor. The permeability of the thermomass in heat conducting medium is

<sup>K</sup>TM <sup>¼</sup> <sup>μ</sup>TM χρTM

In large systems, the boundary effect is negligible. Then, Eq. (13) reduces to the Darcy's law

When the spatial gradient and changing rate of physical quantities are not significant, the first and second terms in Eq. (10) can be neglected. In this case, Eq. (10) exhibits the balance between driving force and friction force. The heat conduction is steady in such a nonequilibrium system.

> 1 3 v2 <sup>g</sup>CT

For the simplest case, v<sup>g</sup> and C are assumed to be temperature independent. Then, Eq. (15)

When the boundary effect is considerable, the second term in Eq. (13) needs to be accounted. In

Eq. (17) is a generalization of Fourier law when boundary effect needs to be considered. It predicts the reduction of effective thermal conductivity in nanosystems by the additional resistance term. When the system size is bigger, the spatial gradient of q is smaller. Thus, κeff increases with the system size growing larger. Nevertheless, to quantitatively predict the size dependency

2 B∇<sup>2</sup>

<sup>χ</sup> <sup>¼</sup> <sup>v</sup><sup>2</sup>

�κ∇T ¼ q � l

with the first term much more important than the second term on the right hand side.

χq ¼ �∇

<sup>K</sup> <sup>u</sup> � <sup>μ</sup>∇<sup>2</sup>

u (12)

. Eq. (12) indicates that the boundary

uTM (13)

<sup>g</sup>C=3κ (16)

q (17)

(14)

(15)

�∇<sup>p</sup> <sup>¼</sup> <sup>μ</sup>

relation when the boundary effect is nonnegligible [37, 38]

74 Nanomechanics

where μ is the viscosity,Kis the permeability with a unit of m2

effect is considered, the friction of thermomass can be formulated as

Combining Eqs. (13) and (10) leads to

actually gives the Fourier law with

this case, the combination of Eqs. (13) and (10) gives

where l<sup>B</sup> equaling to the square root of KTM is a characteristic length.

For dielectric solids, the Boltzmann equation describes the evolution of phonon density of state as in Ref. [26, 27]

$$Df(k, \mathbf{x}, t) = \mathbb{C}f(k, \mathbf{x}, t) \tag{18}$$

where D is the drift operator and C is the collision operator. Eq. (18) indicates that the phonon gas can freely drift without the disturbance of collision. The drift operator is

$$Df(\mathbf{k}, \mathbf{x}, t) = \left(\frac{\partial}{\partial t} + \nu\_k^n \cdot \nabla\right) f^n \tag{19}$$

where v<sup>k</sup> is the phonon velocity in one Cartesian direction. The collision, such as the phononphonon scattering, reshapes the phonon distribution function. In phonon theory, the collisions can be sorted to R and N processes. The R processes break the phonon quasi-momentum, while the N processes conserve it. In this sense, the collision operator can be simply formulated as

$$\mathbf{C}f(k, \mathbf{x}, t) = \frac{f\_0^n - f^n}{\tau\_\mathbb{R}} + \frac{f\_\text{D}^n - f^n}{\tau\_\mathbb{N}} \tag{20}$$

where τ<sup>R</sup> and τ<sup>N</sup> are the characteristic relaxation time between succeeding R and N events. f<sup>0</sup> is the equilibrium distribution given by Eq. (5), f<sup>D</sup> is the displaced distribution

$$f\_{\rm D} = \frac{1}{\exp[(\hbar\omega - \hbar k \cdot \mu\_{\rm D})/k\_{\rm B}T] - 1} \tag{21}$$

where u<sup>D</sup> is the drift velocity of phonon gas. Eq. (20) illustrates that the R processes tend to bring f back to f0, while N processes tend to bring f to fD.

If f can be approximated with fD, a solution of Eq. (20) can be obtained with a second-order Taylor expansion of f<sup>D</sup> around f<sup>0</sup> and then integrating [33]

$$\frac{\partial q\_i}{\partial t} + \frac{15}{16} \nabla\_j \frac{q\_i q\_j}{e} + \frac{1}{3} \nu\_8^2 \mathbf{C} \nabla\_i T = -\frac{q\_i}{\tau\_\mathbf{R}} \tag{22}$$

If the friction force in Eq. (10) only has the first term, which is linear to the thermomass velocity, Eq. (22) is identical to Eq. (10) except the coefficient 15/16 in ahead of the second term on the left hand side. This difference is caused by the Doppler Effect during the drift motion of phonon gas. From this perspective, the phonon gas is slightly different from the real gas. The phonon energy varies due to the dispersion, causing the "eclipse" of the convection term. In a nondispersive medium, the frequency is independent of k. Then, Eq. (22) is consistent with Eq. (10). Nevertheless, the second-order spatial derivative term, like in Eq. (17), is dismissed. In nanosystems, the boundary condition should be considered in solving Eq. (18). For example, if the boundary is completely diffusive, the drift velocity in Eq. (21) is dragged to zero. In this case, the phonon distribution function is assumed to have the following form.

$$f = f\_{\rm D} + \nu\_{\rm g} \tau\_{\rm N} \nabla f\_{\rm D} \tag{23}$$

It indicates that with the diffusive boundary, the N processes induce a deviation from fD, with the relaxation length λ<sup>N</sup> = vgτN, i.e., the MFP of N processes. The additional term in Eq. (23) gives a second-order spatial derivative term. By the integration of Eq. (18), one gets

$$\frac{\partial q\_i}{\partial t} + \frac{15}{16} \nabla\_{\rangle} \frac{q\_i q\_j}{e} + \frac{1}{3} \nu\_{\neq}^2 \mathbf{C} \nabla\_i T = -\frac{q\_i}{\tau\_{\mathbf{R}}} + \frac{\tau\_{\mathbf{N}} \nu\_{\neq}^2}{5} \nabla^2 q\_i \tag{24}$$

Keep in mind that the thermal conductivity in bulk limit is expressed by Eq. (2), in steady state, one-dimensional heat conduction case, Eq. (24) can be simplified to

$$-\kappa \nabla T = q - \frac{\lambda \mathbf{R} \lambda \mathbf{N}}{5} \nabla^2 q \tag{25}$$

Eq. (25) can be regarded as the first order Chapman-Enskog expansion [17] of the phonon distribution function. In fluid mechanics, the viscous term in Navier-Stokes equation can be derived from the first order Chapman-Enskog expansion of the state distribution function of fluid molecular. Without the Chapman-Enskog expansion, the solution of Boltzmann equation gives the Euler equation, which is the dynamic equation without the viscous dissipation. This case happens when the interested region is far away from the boundary, or the boundary layer thickness is negligible compared with the flow region, like the large Reynolds number flow around the aircrafts. The difference between the thermomass flow and ordinary gas flow is that the R processes causes residential friction forces to the flow, which makes the transfer diffusive. In low temperature crystals, or low dimensional materials, such as graphene, the R processes can be rare. Then the heat conduction will exhibit obvious hydrodynamic behaviors. Therefore, based on the phonon Boltzmann derivation, the value of l<sup>B</sup> in Eq. (17) can be determined as lB <sup>2</sup> = λRλN/5.

#### 4. Phonon gas flow in Si nanosystems

Based on Eq. (25) we can calculate the effective thermal conductivity of nanosystems. The silicon nanofilms and nanowires are investigated here because the experimental results are available for comparison. The geometries of nanofilms and nanowires are shown in Figure 1. The direction of heat conduction is in-plane for nanofilms and longitudinal for nanowires.

Figure 1. Heat conduction in nanosystems. (a) In-plane nanofilm; (b) nanowire.

left hand side. This difference is caused by the Doppler Effect during the drift motion of phonon gas. From this perspective, the phonon gas is slightly different from the real gas. The phonon energy varies due to the dispersion, causing the "eclipse" of the convection term. In a nondispersive medium, the frequency is independent of k. Then, Eq. (22) is consistent with Eq. (10). Nevertheless, the second-order spatial derivative term, like in Eq. (17), is dismissed. In nanosystems, the boundary condition should be considered in solving Eq. (18). For example, if the boundary is completely diffusive, the drift velocity in Eq. (21) is dragged to zero. In this

It indicates that with the diffusive boundary, the N processes induce a deviation from fD, with the relaxation length λ<sup>N</sup> = vgτN, i.e., the MFP of N processes. The additional term in Eq. (23)

Keep in mind that the thermal conductivity in bulk limit is expressed by Eq. (2), in steady state,

Eq. (25) can be regarded as the first order Chapman-Enskog expansion [17] of the phonon distribution function. In fluid mechanics, the viscous term in Navier-Stokes equation can be derived from the first order Chapman-Enskog expansion of the state distribution function of fluid molecular. Without the Chapman-Enskog expansion, the solution of Boltzmann equation gives the Euler equation, which is the dynamic equation without the viscous dissipation. This case happens when the interested region is far away from the boundary, or the boundary layer thickness is negligible compared with the flow region, like the large Reynolds number flow around the aircrafts. The difference between the thermomass flow and ordinary gas flow is that the R processes causes residential friction forces to the flow, which makes the transfer diffusive. In low temperature crystals, or low dimensional materials, such as graphene, the R processes can be rare. Then the heat conduction will exhibit obvious hydrodynamic behaviors. Therefore, based on the phonon Boltzmann derivation, the value of l<sup>B</sup> in Eq. (17) can be determined as

Based on Eq. (25) we can calculate the effective thermal conductivity of nanosystems. The silicon nanofilms and nanowires are investigated here because the experimental results are available for comparison. The geometries of nanofilms and nanowires are shown in Figure 1. The direction of heat conduction is in-plane for nanofilms and longitudinal for nanowires.

�κ∇<sup>T</sup> <sup>¼</sup> <sup>q</sup> � <sup>λ</sup>Rλ<sup>N</sup>

<sup>g</sup>C∇iT ¼ � qi

τR þ τNv<sup>2</sup> g <sup>5</sup> <sup>∇</sup><sup>2</sup>

<sup>5</sup> <sup>∇</sup><sup>2</sup>

f ¼ f <sup>D</sup> þ vgτN∇f <sup>D</sup> (23)

qi (24)

q (25)

case, the phonon distribution function is assumed to have the following form.

gives a second-order spatial derivative term. By the integration of Eq. (18), one gets

∂qi ∂t þ 15 <sup>16</sup> <sup>∇</sup><sup>j</sup> qi qj e þ 1 3 v2

4. Phonon gas flow in Si nanosystems

lB

<sup>2</sup> = λRλN/5.

76 Nanomechanics

one-dimensional heat conduction case, Eq. (24) can be simplified to

Assume the boundary is completely diffusive, i.e., the phonon gas drift velocity is zero on the boundary. The solution of Eq. (25) for a nanofilm is [39]

$$q(\mathbf{y}) = -\kappa\_0 \nabla T \left[ 1 - \frac{\cosh(\mathbf{y}/l\_\mathrm{B})}{\cosh(l/2l\_\mathrm{B})} \right] \tag{26}$$

$$\kappa\_{\rm eff}^{\rm nf} = \frac{\int\_l qdy}{-\nabla Tl} = \kappa\_0 [1 - 2Br \cdot \tanh(1/2Br)]\tag{27}$$

where l is the thickness of film and Br = lB/l is the Brinkman number. The solution for nanowire is

$$q(\mathbf{y}) = -\kappa\_0 \nabla T \left[ 1 - \frac{J\_0(\text{i}\mathbf{y}/l\_\text{B})}{J\_0(\text{i}l/2l\_\text{B})} \right] \tag{28}$$

$$\kappa\_{\rm eff}^{\rm new} = \kappa\_0 \left[ 1 - (4Br) \cdot \frac{J\_1(i/2Br)}{i l\_0 (i/2Br)} \right] = \kappa\_0 \left[ 1 - \sum\_{t=0}^{\infty} \frac{(4Br)^{-2t}}{t!(t+1)!} \right] \tag{29}$$

where l is the diameter (thickness) of the wire and J is the cylindrical Bessel function

$$f\_n(\mathbf{x}) = \left(\frac{\mathbf{x}}{2}\right)^n \sum\_{t=0}^{\infty} \frac{(-1)^t (\mathbf{x}/2)^{2t}}{t!(t+n)!} \tag{30}$$

Eqs. (26) and (28) show the heat flux is nonuniform at the cross-section. If the system size is much larger than lB, q(y) tends to be constant. Then, the effective thermal conductivity renders the bulk limit, κ0. If the system size is comparable with lB, q(y) is significantly affected by the boundary. Thereby, κeff is strongly reduced.

The analytical derivation of Eqs. (26)–(30) is based on the assumption that the l<sup>B</sup> is constant. However, in nanosystems, the phonon would scatter with boundary, which shortens the MFPs. For the pure diffusive boundary, the scattering on boundary will terminate the MFPs. It can be seen as the additional collision event into the ordinary scatterings. If the boundary is located at r away from the originating point, the effective MFP of phonons can be expressed as

$$\frac{\lambda\_{\rm eff}}{\lambda\_0} = 1 - \exp\left(-\frac{r}{\lambda\_0}\right) \tag{31}$$

In this way, the effective MFPs in nanosystems can be obtained by integrating over the sphere angle. For nanofilms, the local value of MFPs is [40]

$$\frac{\lambda\_{\text{eff}}(\mathbf{y})}{\lambda\_0} = 1 + \frac{1}{2} \left[ (\alpha - 1)e^{-\alpha} + (\beta - 1)e^{-\beta} - \alpha^2 E\_i(\alpha) - \beta^2 E\_i(\beta) \right] \tag{32}$$

where <sup>α</sup> = (l/2�y)/λ0, <sup>β</sup> = (l/2+y)/λ0, and Eið Þ¼ <sup>x</sup> <sup>Ð</sup> <sup>∞</sup> 1 t �<sup>1</sup>e�txdt. For nanowires, we have

$$\frac{\lambda\_{\rm eff}(\mathbf{y})}{\lambda\_0} = \frac{1}{\pi} \int\_0^{\pi} \int\_0^{\frac{\pi}{2}} \left[ 1 - \exp\left( -\frac{\mathbf{y}\cos\theta + \sqrt{l^2/4 - \mathbf{y}^2\sin^2\theta}}{\lambda\_0 \sin\varphi} \right) \right] \sin\varphi d\varphi d\theta \tag{33}$$

Therefore, the MFPs are significantly shortened in nanosystems. It reveals that the boundary has dual effects on heat conduction in nanosystems. First, the second spatial derivative of heat flux, which represents the viscous effect of phonon gas, imposes additional resistance on heat transfer due to the nonslip boundary condition. Second, the collision on boundary changes the effective MFPs. This effect is similar to the rarefaction of gas flow in high Kn case. By accounting both the dual effects, the thermal conduction in nanosystems is described as

$$-\frac{\lambda\_{\text{R,eff}}(r)}{\lambda\_{\text{R,0}}}\kappa\_0 \nabla T = q(r) - \frac{\lambda\_{\text{R,eff}}(r)\lambda\_{\text{N,eff}}(r)}{5}\nabla^2 q(r) \tag{34}$$

It is worth noting that in fluid mechanics, the rarefaction is not necessarily happened at the same time of viscous effect based on the Darcy-Brinkman relation. Consider the water flow in porous material. The permeability of porous flow is determined by the size of pores, which typically is in the order of micrometers. The MFP among water molecule is typically subnanometer. Therefore, the square root of permeability differs much from the MFP. The effects of Darcy-Brinkman boundary layer and rarefaction can be unconjugated. On the other hand, if the fluid is replaced by gas, the MFP of fluid could be comparable to the square root of permeability. In this case, the Darcy-Brinkman boundary layer and the rarefaction should be considered simultaneously. For the phonon gas flow, the relative magnitude of λR, λN, and l decides the conjugation of boundary layer and rarefaction. λ<sup>R</sup> represents the "size of pores" while λ<sup>N</sup> represents the viscosity of phonon gas. The bulk limit is achieved when l >> λ<sup>R</sup> and l >> λN. If λ<sup>R</sup> >> l >> λN, the first term on the right hand side of Eq. (34) is less important than the second term. The flow mimics a dense fluid passing through a sparse medium. The boundary transmits momentum efficiently across the flow region. The phonon hydrodynamics can be observed. If λ<sup>N</sup> >> l >> λR, the flow mimics a dilute fluid passing through a dense medium. The velocity profile will be close to linear. In this case, only the rarefaction effect needs to be considered. If λ<sup>N</sup> >> l and λ<sup>R</sup> >> l, both the rarefaction and boundary drag affect the resistance on flow and need to be modeled simultaneously.

The numerical solution of Eq. (34) gives the effective thermal conductivity for Si nanofilms and nanowires at room temperature, as shown in Figure 2. The physical properties are adopted as κ<sup>0</sup> = 148W/(m K) (standard experiment value for monocrystalline Si), λR,0 = 42 nm (according to the direct calculation based on Eq. (2)), λN,0 = 360 nm. The predictions based on the gray model [7], McGaughey model [16], and Ma model [31] are also presented in Figure 2. It shows the gray model and McGaughey model overestimate the thermal conductivities. Ma model gives close results to experiments. However, Ma model assumes a MFP of 210 nm, which is lack of physical support. It also shows an unreasonable drop at D = 1000–2000 nm for nanowires. According to Figure 2, our model achieves the best agreement with current available experiment and numerical results.

Figure 2. Effective in-plane thermal conductivities of single crystal Si nanosystems at room temperature. Bottom solid line: present model (Eq. (34)); top solid line: gray model [7]; solid and circle: McGaughey model [16]; dash line: Ma model [31]. (a) Nanofilm. Symbols E1–E4 are experimental results from Refs. [18–21]. (b) Nanowire. Square is experimental results from Ref. [25]. Triangle is numerical results from Refs. [41, 42].

#### 5. Thermal rectification in nanosystems

λeff λ0

<sup>2</sup> ð Þ <sup>α</sup> � <sup>1</sup> <sup>e</sup>

<sup>1</sup> � exp � <sup>y</sup> cos <sup>θ</sup> <sup>þ</sup>

ing both the dual effects, the thermal conduction in nanosystems is described as

0 @

angle. For nanofilms, the local value of MFPs is [40]

¼ 1 þ 1

> ðπ 2 0

2 4

� <sup>λ</sup>R, effð Þ<sup>r</sup> λR,<sup>0</sup>

where <sup>α</sup> = (l/2�y)/λ0, <sup>β</sup> = (l/2+y)/λ0, and Eið Þ¼ <sup>x</sup> <sup>Ð</sup> <sup>∞</sup>

λeffð Þy λ0

> ¼ 1 π ðπ 0

λeffð Þy λ0

78 Nanomechanics

simultaneously.

<sup>¼</sup> <sup>1</sup> � exp � <sup>r</sup>

In this way, the effective MFPs in nanosystems can be obtained by integrating over the sphere

�<sup>α</sup> <sup>þ</sup> <sup>β</sup> � <sup>1</sup> � �<sup>e</sup>

Therefore, the MFPs are significantly shortened in nanosystems. It reveals that the boundary has dual effects on heat conduction in nanosystems. First, the second spatial derivative of heat flux, which represents the viscous effect of phonon gas, imposes additional resistance on heat transfer due to the nonslip boundary condition. Second, the collision on boundary changes the effective MFPs. This effect is similar to the rarefaction of gas flow in high Kn case. By account-

<sup>κ</sup>0∇<sup>T</sup> <sup>¼</sup> q rð Þ� <sup>λ</sup>R, effð Þ<sup>r</sup> <sup>λ</sup>N, effð Þ<sup>r</sup>

It is worth noting that in fluid mechanics, the rarefaction is not necessarily happened at the same time of viscous effect based on the Darcy-Brinkman relation. Consider the water flow in porous material. The permeability of porous flow is determined by the size of pores, which typically is in the order of micrometers. The MFP among water molecule is typically subnanometer. Therefore, the square root of permeability differs much from the MFP. The effects of Darcy-Brinkman boundary layer and rarefaction can be unconjugated. On the other hand, if the fluid is replaced by gas, the MFP of fluid could be comparable to the square root of permeability. In this case, the Darcy-Brinkman boundary layer and the rarefaction should be considered simultaneously. For the phonon gas flow, the relative magnitude of λR, λN, and l decides the conjugation of boundary layer and rarefaction. λ<sup>R</sup> represents the "size of pores" while λ<sup>N</sup> represents the viscosity of phonon gas. The bulk limit is achieved when l >> λ<sup>R</sup> and l >> λN. If λ<sup>R</sup> >> l >> λN, the first term on the right hand side of Eq. (34) is less important than the second term. The flow mimics a dense fluid passing through a sparse medium. The boundary transmits momentum efficiently across the flow region. The phonon hydrodynamics can be observed. If λ<sup>N</sup> >> l >> λR, the flow mimics a dilute fluid passing through a dense medium. The velocity profile will be close to linear. In this case, only the rarefaction effect needs to be considered. If λ<sup>N</sup> >> l and λ<sup>R</sup> >> l, both the rarefaction and boundary drag affect the resistance on flow and need to be modeled

1 t

l 2

λ<sup>0</sup> sin ϕ

q

λ0 � �

�<sup>β</sup> � <sup>α</sup><sup>2</sup>

Eið Þ� <sup>α</sup> <sup>β</sup><sup>2</sup>Ei <sup>β</sup> � � � � (32)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

=4 � y<sup>2</sup> sin <sup>2</sup>θ

�<sup>1</sup>e�txdt. For nanowires, we have

1 A

3

<sup>5</sup> <sup>∇</sup><sup>2</sup>q rð Þ (34)

5 sin ϕdϕdθ (33)

(31)

Thermal rectification refers that the heat conduction in one direction of the device leads to higher heat flux than following the opposite direction, even though the same temperature difference is applied. It currently raises much interest since the first experimental report by carbon nanotubes [43]. The thermal rectification effect is anticipated to realize thermal diode [44], thermal logic gate [45], or thermal transistors [46, 47]. Though much effort has been paid for searching useful mechanisms and realizing considerable rectification ratio, the ambitious goal that controlling heat as electricity is still far away [48].

The mechanism of thermal rectification has been widely studied. It is found that various effects can induce rectification, such as the different temperature dependences of the thermal conductivity at the different parts of the device [49], the asymmetric transmission rates of phonons across the interfaces [50], and the temperature dependence of electromagnetic resonances [51]. Here, another rectification mechanism is proposed through the thermomass theory, following an analogy to fluid mechanics. In Navier-Stokes equations, the convective acceleration term indicates when the fluid experiences speed up or slow down. Therefore, if the cross-section area of a flow channel is changing (e.g. the trapezoidal channel), the flow rate under the same pressure difference is different in the convergent direction or in the divergent direction. In the convergent direction, the channel serves as a nozzle, which accelerates the fluid and converts part of its potential energy to the kinetic energy. In the divergent direction, the channel serves as a diffuser, which decelerates the fluid and converts part of its kinetic energy to the potential energy. The acceleration of fluid increases the velocity head and consumes the dynamic head of flow. Therefore, the total fluid flux in the convergent direction will be less than that in the divergent direction. In terms of thermal conduction, it means that with the same temperature difference between the heat source and sink, the total heat flux in the wide-to-narrow direction is smaller than that in the narrow-towide direction, which is the thermal rectification. Nevertheless, it should be stressed that for a flow channel with large angle of divergence, the flow separation could happen when the fluid velocity is high. In case of flow separation, the effective resistance of the diffuser will be much increased. It may cause the total heat flux in the wide-to-narrow direction larger than that in the narrow-to-wide direction, that is, the reverse of rectification.

In steady state, the generalized conduction law, Eq. (10), can be reformulated as

$$-\pi\_{\rm R}\nabla\_{\dagger}\frac{q\_{i}q\_{j}}{\varepsilon} - \kappa\nabla T = q\_{i} - l\_{\rm B}^{2}\nabla^{2}q\tag{35}$$

The difference between Eqs. (35) and (25) is the additional convective term, �τR∇jqiqj/e. The first term on the left hand side mimics to the spatial inertia term in fluid mechanics. It induces rectification effect. Consider a trapezoidal material with heat conducting through the symmetry axis, as shown in Figure 3. The thickness of the material is H; the widths at the narrow and the wide ends are L<sup>W</sup> and LN, respectively, and the separation between these ends is L. If L is much larger than L<sup>N</sup> and LW, the heat conduction can be assumed as quasione-dimensional. The mainstream of heat flux is in the x direction, qx >> qy. The total heat flux (Q) at each cross-section perpendicular to x direction is constant. Due to the boundary friction, the Laplacian of qx in the y direction is much larger than in the x direction. Then, the x component of Eq. (35) is

$$-\kappa \frac{\partial T}{\partial x} = q\_x \left( 1 + \tau\_\mathsf{R} \frac{\partial}{\partial x} \frac{q\_x}{\mathcal{C} T} \right) - l\_\mathsf{B}^2 \frac{\partial^2}{\partial \mathbf{y}^2} q\_x = q\_x (1 + \mathsf{C}\_\mathsf{R}) - l\_\mathsf{B}^2 \frac{\partial^2}{\partial \mathbf{y}^2} q\_x \tag{36}$$

where C<sup>R</sup> consists of two terms

$$\mathbf{C}\_{\text{R}} = \frac{\tau\_{\text{R}}}{\underline{\text{CT}}} \frac{\partial q\_{\text{x}}}{\partial \mathbf{x}} - \frac{\tau\_{\text{R}} q\_{\text{x}}}{\underline{\text{CT}}^2} \frac{\partial T}{\partial \mathbf{x}} \tag{37}$$

Thermomass Theory: A Mechanical Pathway to Analyze Anomalous Heat Conduction in Nanomaterials http://dx.doi.org/10.5772/67780 81

Figure 3. Trapezoidal nanoribbon for demonstrating thermal rectification.

phonons across the interfaces [50], and the temperature dependence of electromagnetic resonances [51]. Here, another rectification mechanism is proposed through the thermomass theory, following an analogy to fluid mechanics. In Navier-Stokes equations, the convective acceleration term indicates when the fluid experiences speed up or slow down. Therefore, if the cross-section area of a flow channel is changing (e.g. the trapezoidal channel), the flow rate under the same pressure difference is different in the convergent direction or in the divergent direction. In the convergent direction, the channel serves as a nozzle, which accelerates the fluid and converts part of its potential energy to the kinetic energy. In the divergent direction, the channel serves as a diffuser, which decelerates the fluid and converts part of its kinetic energy to the potential energy. The acceleration of fluid increases the velocity head and consumes the dynamic head of flow. Therefore, the total fluid flux in the convergent direction will be less than that in the divergent direction. In terms of thermal conduction, it means that with the same temperature difference between the heat source and sink, the total heat flux in the wide-to-narrow direction is smaller than that in the narrow-towide direction, which is the thermal rectification. Nevertheless, it should be stressed that for a flow channel with large angle of divergence, the flow separation could happen when the fluid velocity is high. In case of flow separation, the effective resistance of the diffuser will be much increased. It may cause the total heat flux in the wide-to-narrow direction larger than

that in the narrow-to-wide direction, that is, the reverse of rectification.

�τR∇<sup>j</sup>

x component of Eq. (35) is

80 Nanomechanics

�κ ∂T

where C<sup>R</sup> consists of two terms

<sup>∂</sup><sup>x</sup> <sup>¼</sup> qx <sup>1</sup> <sup>þ</sup> <sup>τ</sup><sup>R</sup>

∂ ∂x qx CT

<sup>C</sup><sup>R</sup> <sup>¼</sup> <sup>τ</sup><sup>R</sup> CT ∂qx <sup>∂</sup><sup>x</sup> � <sup>τ</sup>Rqx CT<sup>2</sup>

� l 2 B ∂2

In steady state, the generalized conduction law, Eq. (10), can be reformulated as

qi qj

<sup>e</sup> � <sup>κ</sup>∇<sup>T</sup> <sup>¼</sup> qi � <sup>l</sup>

The difference between Eqs. (35) and (25) is the additional convective term, �τR∇jqiqj/e. The first term on the left hand side mimics to the spatial inertia term in fluid mechanics. It induces rectification effect. Consider a trapezoidal material with heat conducting through the symmetry axis, as shown in Figure 3. The thickness of the material is H; the widths at the narrow and the wide ends are L<sup>W</sup> and LN, respectively, and the separation between these ends is L. If L is much larger than L<sup>N</sup> and LW, the heat conduction can be assumed as quasione-dimensional. The mainstream of heat flux is in the x direction, qx >> qy. The total heat flux (Q) at each cross-section perpendicular to x direction is constant. Due to the boundary friction, the Laplacian of qx in the y direction is much larger than in the x direction. Then, the

2 B∇<sup>2</sup>

<sup>∂</sup>y<sup>2</sup> qx <sup>¼</sup> qxð Þ� <sup>1</sup> <sup>þ</sup> <sup>C</sup><sup>R</sup> <sup>l</sup>

∂T

2 B ∂2

<sup>∂</sup><sup>x</sup> (37)

<sup>∂</sup>y<sup>2</sup> qx (36)

q (35)

The sign of the first term of C<sup>R</sup> will be positive for the heat conduction in a convergent channel, which means the acceleration of heat flux creates additional effective resistance, and reduces the total heat flux. Oppositely, heat conduction in a divergent channel will increase the total heat flux. The second term of C<sup>R</sup> will not change sign with the direction of heat transport. It characterizes the acceleration due to density variation since thermomass is compressible. It is insignificant except for the case of ultra-high heat flux [52].

To enhance the thermal rectification, the directional sensitive part in Eq. (36) should be amplified over the directional non-sensitive part. If the diffusive boundary condition is replaced with slip boundary condition, or the system size is large compared with the boundary layer, the Laplacian term of heat flux can be neglected. In room temperature, the second term of C<sup>R</sup> is usually much less than the first term. In this case, Eq. (36) can be simplified to

$$-\kappa \frac{\partial T}{\partial \mathbf{x}} = q\_{\mathbf{x}} \left( 1 + \frac{\tau\_{\mathbf{R}}}{CT} \frac{\partial q\_{\mathbf{x}}}{\partial \mathbf{x}} \right) \tag{38}$$

Consider a silicon ribbon with the average temperature 300 K. Assume that H = 1000 nm, L = 300 nm, L<sup>N</sup> = 300 nm, L<sup>w</sup> varies from 300 to 2000 nm. The relaxation time τ<sup>R</sup> is set as 1.5e�10s based on experimental results [53]. The temperatures on both ends are 330 and 270 K, respectively. By numerically solving Eq. (38), we can get the rectification ratio (defined as the thermal conductance in narrow-to-wide direction over that in the opposite direction), as shown in Figure 4. It shows that the rectification ratio grows with Lw, from zero to a considerable value of 32.3%. This value is large enough to construct thermal diode or thermal logic gate.

Figure 4. Thermal rectification ratio depending on L<sup>w</sup> of the trapezoidal silicon ribbon.

#### 6. Thermoelectricity of nanosystems

The ZT for nanomaterials could be much enhanced [10–13]. The mechanism of such enhancement can be that the nanostructures reduce the thermal conductivity by strong phononboundary scattering while maintaining the electrical conductivity. Although a lot of work has been done in searching high ZT materials through nanotechnology, the thermodynamic analysis and the role of nonlocal and nonlinear transports, which are highly possible to happen in nanosystems, are not fully discussed [54, 55]. In recent years, the nonlocal effects raised by the MFP reduction due to geometry constraint [56], the electron and phonon temperature [57], and the breakdown of Onsager reciprocal relation (ORR) [58, 59] in nanosystems have been investigated from the framework of extended irreversible thermodynamics (EIT). These works showed that the nonlinear and nonlocal effects influence the efficiency of devices. The breakdown of ORR not only possesses theoretical importance but also shed light on approaches to further increase efficiency.

Here, we analyze the thermoelectric effect from the thermomass theory perspective. There could be various effects when the individual motion of phonon gas and electron gas is separately considered. The most apparent one is the energy exchange between phonons and electrons [60]. In a one-dimensional thermoelectric medium, the conservation of energy gives

$$
\nabla q - I\mathbf{E} = \mathbf{0} \tag{39}
$$

where I is the electrical current and E is the electrical field. IE equals the adding or subtracting rate of thermal energy. Dividing Eq. (39) by c <sup>2</sup> illustrates that the electrical current performs as the mass source or sink of thermomass. The nonconservation of mass brings additional term in Eq. (10). In steady state, we obtain the governing equation of thermomass momentum as

$$\left(\rho\_{\rm TM}\mu\_{\rm TM}\cdot\nabla\right)\mu\_{\rm TM} + \mu\_{\rm TM}\nabla\cdot\left(\rho\_{\rm TM}\mu\_{\rm TM}\right) + \nabla p\_{\rm TM} = f\_{\rm TM} \tag{40}$$

The second term on the left hand side is nonzero because of the energy conversion. It increases the spatial inertia of thermomass. For simplicity, we do not consider the Brinkman extension of the friction force and assume the material cross-section is constant, and then Eq. (40) turns to

$$-\kappa \frac{\partial T}{\partial \mathbf{x}} = q\_x (\mathbf{1} + \mathbf{C}\_{\mathbb{R}}) \tag{41}$$

where

6. Thermoelectricity of nanosystems

0

Figure 4. Thermal rectification ratio depending on L<sup>w</sup> of the trapezoidal silicon ribbon.

10

20

Rectification Ratio %

82 Nanomechanics

30

40

rate of thermal energy. Dividing Eq. (39) by c

further increase efficiency.

The ZT for nanomaterials could be much enhanced [10–13]. The mechanism of such enhancement can be that the nanostructures reduce the thermal conductivity by strong phononboundary scattering while maintaining the electrical conductivity. Although a lot of work has been done in searching high ZT materials through nanotechnology, the thermodynamic analysis and the role of nonlocal and nonlinear transports, which are highly possible to happen in nanosystems, are not fully discussed [54, 55]. In recent years, the nonlocal effects raised by the MFP reduction due to geometry constraint [56], the electron and phonon temperature [57], and the breakdown of Onsager reciprocal relation (ORR) [58, 59] in nanosystems have been investigated from the framework of extended irreversible thermodynamics (EIT). These works showed that the nonlinear and nonlocal effects influence the efficiency of devices. The breakdown of ORR not only possesses theoretical importance but also shed light on approaches to

0 500 1000 1500 2000

*<sup>L</sup>*W (nm)

Here, we analyze the thermoelectric effect from the thermomass theory perspective. There could be various effects when the individual motion of phonon gas and electron gas is separately considered. The most apparent one is the energy exchange between phonons and electrons [60]. In a one-dimensional thermoelectric medium, the conservation of energy gives

where I is the electrical current and E is the electrical field. IE equals the adding or subtracting

∇q � IE ¼ 0 (39)

<sup>2</sup> illustrates that the electrical current performs as

$$\mathbf{C}\_{\rm R} = \frac{2\tau\_{\rm R}}{CT}IE - \frac{\tau\_{\rm R}q\_{\rm x}}{CT^2}\frac{\partial T}{\partial \mathbf{x}} \tag{42}$$

Compared with Eq. (37), the first term of C<sup>R</sup> has a coefficient 2 because of the energy exchange between phonons and electrons. The electrical current couples with the heat flux and induces additional spatial acceleration force on the thermomass flow. This inertia increase is insignificant in ordinary conditions due to the small value of τR. It could be considerable in case of strong power thermoelectric convertor with large electrical current and intense electrical field. Neglecting the second term of CR, it can be derived that the effective thermal conductivity and Seebeck coefficient change to

$$\kappa' = (1 + \mathbb{C}\_{\mathbb{R}})^{-1} \kappa \tag{43}$$

$$\mathbf{S}' = (\mathbf{1} + \mathbf{C}\_{\mathbb{R}})^{-1}\mathbf{S} \tag{44}$$

Since ZT is S<sup>2</sup> σ/κT, the effective ZT is (1 + CR) �<sup>1</sup> of the original one without considering the inertia effect of thermomass. Therefore, when IE > 0, the electrical energy converts to thermal energy. It is typically the case of thermoelectric cooler. The heat flux is additionally impeded. The ZT is decreased. When IE < 0, the temperature gradient drives electric current. It is typically a thermoelectric generator. The heat flux is further pumped, and the effective ZT is enhanced. The inertia effect could be beneficial for a higher ZT of the device in this case.

#### 7. Conclusion

In this chapter, we present a mechanical analysis on the thermal conduction in nanosystems with the thermomass theory. Firstly, the boundary resistance in nanosystems on heat flow is modeled with the Darcy-Brinkman analogy. The permeability of thermomass in materials is derived based on the phonon Boltzmann equation. The size-dependent effective thermal conductivity of Si nanosystems thereby is accurately predicted with the present model. Then, the spatial inertia effect of thermomass is shown to induce the thermal rectification in asymmetry nanosystems. The predicted rectification ratio can be as high as 32.3% in a trapezoidal Si nanoribbon. Finally, the energy conversion in thermoelectric devices can be coupled with the spatial inertia of thermomass flow. The ZT tends to be increased in case of a thermoelectric generator.

## Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (nos. 51136001, 51356001) and the Tsinghua University Initiative Scientific Research Program.

## Author details

Yuan Dong1 \*, Bingyang Cao<sup>2</sup> and Zengyuan Guo2

\*Address all correspondence to: dony327@163.com

1 Department of Mechanical & Aerospace Engineering, University of Missouri, Columbia, Missouri, USA

2 Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing, China

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asymmetry nanosystems. The predicted rectification ratio can be as high as 32.3% in a trapezoidal Si nanoribbon. Finally, the energy conversion in thermoelectric devices can be coupled with the spatial inertia of thermomass flow. The ZT tends to be increased in case of a

This work was financially supported by the National Natural Science Foundation of China (nos. 51136001, 51356001) and the Tsinghua University Initiative Scientific Research Program.

1 Department of Mechanical & Aerospace Engineering, University of Missouri, Columbia,

2 Key Laboratory for Thermal Science and Power Engineering of Ministry of Education,

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\*Address all correspondence to: dony327@163.com

thermoelectric generator.

Acknowledgements

Author details

Yuan Dong1

84 Nanomechanics

Missouri, USA

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## **Surface Nanomechanics of Biomolecules and Supramolecular Systems**

Paolo Bergese and Stefania Federici

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68293

#### **Abstract**

Surface nanomechanics of biomolecules and supramolecular systems is an interdisci‐ plinary and vital area of current research, with implications/applications spanning from synthetic biology to regenerative medicine, from smart surfaces to molecular machines. Biomolecule surface transformations and nanomachinery arise upon "wiring" them onto surfaces and interfaces. Surface confinement of biomolecules is a common feature of biological systems (e.g., cell membranes) and often a mandatory step for translating their properties into real‐world applications (e.g., biosensors). On surfaces biomolecules undergo peculiar transformations and interactions which they do not experience in solu‐ tion. Such unedited effects open challenges in synthetic systems, for example, by altering or hindering the designed/expected property, but also disclose a wealth of opportunities and surprises. Based on our latest research, this chapter will bring fresh excerpts from the field. It will start with an accessible description of thermodynamics of surface nano‐ mechanics of biomolecules and supramolecular systems and then will show how it can be implemented to gain understanding of grow factor cell signaling, to single out small ligands able to inhibit protein misfolding, to measure energetics of surface confined fer‐ ritin during iron loading, and to realize a universal probe for ammine‐based designer drugs.

**Keywords:** molecular transformations, surface, nanomechanics, nanomechanical sensors, grow factors, ferritin, abiotic supramolecular receptors, designer drugs

## **1. Describing and probing molecule collective surface nanomechanics**

The section introduces the description of surface molecule transformations by classical inter‐ facial thermodynamics. This will be helpful to better grasp the working principle of nanome‐ chanical sensors, which will be presented in the next subsection. Nanomechanical sensors are

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

the basic technology used to probe and quantify molecule collective surface nanomechanics. The level of the treatment is kept concise and accessible to a wide readership; those interested in the throughout treatment are redirected to Refs. [1–3].

For simplicity, let us restrict to the case of a solid supported monolayer of proteins that can only switch between two conformational states, A and B. The switch from A to B can be directed by changing the electrolyte (salt) molar concentration of the solution in which the system is immersed. This changes the amount of ions bound to the protein and the Debye‐Hückel screen‐ ing of the charge interactions on the protein, which in turn trigger the exposition to the solution of peptide groups that were buried in conformation A, driving the protein into conformation B.

With the visual help of **Figure 1**, it can be intuitively seen that the switch from state A to state B involves a change of the in‐plane interactions between the proteins, because the switch is intertwined with several nanoscale and subnanoscale changes, such as intermolecular dis‐ tances, surface charge and monolayer thickness. This can be thermodynamically described by an additional surface work that accompanies the surface transformation with respect to the same transformation occurring in "free" solution. In particular, it can be shown [2, 3] that the surface standard molar Gibbs energy, ∆*<sup>r</sup> G*0 *σ* , of the surface switch from state A to state B is composed by the sum of an excess surface work, *W<sup>σ</sup>* , and the molar Gibbs energy of the same transformation occurring in bulk solution, ∆*<sup>r</sup> G*0 *σ* :

**Figure 1.** Working principle of nanomechanical sensors in the representative case of a solid supported monolayer of proteins switching between state A (left images) to state B (right images) upon binding to cations. The switch drives the overall variation of surface tension Δγ, triggered at the nanoscale by the change of in‐plane interactions between proteins.

Surface Nanomechanics of Biomolecules and Supramolecular Systems http://dx.doi.org/10.5772/intechopen.68293 91

$$
\Delta\_r G\_0^\sigma = \quad \Delta\_r G\_0^b + \mathbf{W}^\sigma \tag{1}
$$

In addition, *W<sup>σ</sup>* can be further detailed as [2, 3]

the basic technology used to probe and quantify molecule collective surface nanomechanics. The level of the treatment is kept concise and accessible to a wide readership; those interested

For simplicity, let us restrict to the case of a solid supported monolayer of proteins that can only switch between two conformational states, A and B. The switch from A to B can be directed by changing the electrolyte (salt) molar concentration of the solution in which the system is immersed. This changes the amount of ions bound to the protein and the Debye‐Hückel screen‐ ing of the charge interactions on the protein, which in turn trigger the exposition to the solution of peptide groups that were buried in conformation A, driving the protein into conformation B. With the visual help of **Figure 1**, it can be intuitively seen that the switch from state A to state B involves a change of the in‐plane interactions between the proteins, because the switch is intertwined with several nanoscale and subnanoscale changes, such as intermolecular dis‐ tances, surface charge and monolayer thickness. This can be thermodynamically described by an additional surface work that accompanies the surface transformation with respect to the same transformation occurring in "free" solution. In particular, it can be shown [2, 3] that

> *G*0 *σ*

> > *G*0 *σ* :

**Figure 1.** Working principle of nanomechanical sensors in the representative case of a solid supported monolayer of proteins switching between state A (left images) to state B (right images) upon binding to cations. The switch drives the overall variation of surface tension Δγ, triggered at the nanoscale by the change of in‐plane interactions

, of the surface switch from state A to state B is

, and the molar Gibbs energy of the same

in the throughout treatment are redirected to Refs. [1–3].

90 Nanomechanics

the surface standard molar Gibbs energy, ∆*<sup>r</sup>*

transformation occurring in bulk solution, ∆*<sup>r</sup>*

between proteins.

composed by the sum of an excess surface work, *W<sup>σ</sup>*

$$\mathbf{W}^o = \frac{\Delta \mathbf{y}}{\Gamma\_o} \tag{2}$$

where ∆*γ* is the thermodynamic surface tension, for simplicity hereafter referred as surface tension, that gathers both the mechanical, ∆*σ*, and electrostatic, *ω<sup>e</sup> σ* , contributions, and *Γ<sup>B</sup>* is the surface density of proteins in state B. From a (nano)mechanical view point, one can say that ∆*γ* exerts a surface stress on the surface that supports the molecular monolayer or thin film.

The above considerations and equations have general validity. Molecular transforma‐ tions, recognition, binding and nanomachinery at a solid‐liquid interface involve nano‐ mechanical work with point of application at the surface. This arises from a very complex co‐operative action of electrostatic, steric (hydration) and thermal fluctuation (entropic) forces triggered by the transformation that macroscopically appear as a variation of the surface tension, or the applied surface stress. The nature of the forces is determined by the specific solid‐solution interfacial environment, that is, by the molecules and their bind‐ ing partners, the molecule surface density, the solution composition and ionic strength, the solid surface and modification chemistry, the solid geometry and nanostructure, and so on. This phenomenon is leveraged by static nanomechanical sensors [4]. In particular, the molecular transformations confined on the sensor surface cumulate and perform an overall surface tension change in the order of mN/m [2] that can be probed and translated by tensiometric techniques such as contact angle [5] or microcantilever (MC) beams [6], as sketched in **Figure 1**.

The working principle of MC biosensor is quite simple: the MC surface is functionalized with a receptor that can selectively bind the target species. Adsorption and binding site interactions of the targets change the mechanical response of the MC system (because of the surface stress generated by changes in Gibbs free energy), providing the transduction/ sensing mechanism. CONAMORE (COntact Angle MOlecularREcognition) technique is based on the sessile drop contact angle principle. When a droplet containing the target species is placed onto a solid surface functionalized with a receptor, it reaches equilibrium with the surface and the surroundings under the action of the interfacial tensions at the contact line at which drop, surface, and surroundings meet, forming a definite contact angle [5].

The variation of the overall surface tension *Δγ* can be directly determined by CONAMORE measurement, where the molecular transformations can be univocally associated with the differential of the solid‐solution interfacial tensions of the systems represented in **Figure 1a** and **b**. In case of MC experiments, *Δγ* can be easily calculated through the relation between *Δγ* and MC deflection, *Δz*, given by the Stoney equation [7]:

$$
\Delta \gamma = -\frac{\Delta z \to t^2}{4} \tag{3}
$$

where *Δz* is the cantilever deflection (with the z axis individuated by the unitary vector nor‐ mal to the top surface of the cantilever base, i.e., *Δz*< 0 for a downward cantilever bending), *E* is the cantilever Young's modulus, *t* is the cantilever thickness, *L* is the cantilever length, and *ν* is the cantilever Poisson's ratio.

Application of nanomechanical sensors to biosensing has become in the last 15 years a break‐ through in biochemistry, life science and medicine, depicting how nanomechanics and biology can grow together [6, 8, 9]. Research in this direction is growing substantially after the milestone work of Fritz and coworkers in 2000, in which they report the specific transduction driven by the surface stress change of DNA hybridization without reported labels [6]. Several experiments have been successfully performed afterwards, revealing DNA hybridization and DNA switch [10–12], detecting proteins and antibodies [5, 13, 14], single virus particles [15] or bacteria [16].

## **2. Surface nanomechanics of biomolecules**

#### **2.1. Role of nanomechanics in the activation of cell membrane growth factors**

Ligand‐receptor protein interactions are a fundamental mechanism for every biological system, in both physiological and pathological conditions. In particular, the interaction between cell membrane growth factor receptors and their key ligands plays an important role in different processes, including cancer [17]. The growth, survival, and metastatic spreading of solid tumors strongly depend on the formation of a novel vessel network (tumor angiogenesis), making the pro‐angiogenic molecular machinery a target for new strategies in cancer therapy [18, 19]. This approach requires to disentangle the complex array of transduction signals activated by the interaction of pro‐angiogenic growth factors with their cognate cell membrane tyrosine kinase receptors [20].

In view of this, a nanoliter CONAMORE assay was assessed to investigate the interactions of the vascular endothelial growth factor receptor‐2 (VEGFR2), which is the major pro‐angio‐ genic receptor expressed by endothelial cells [21], with the canonical ligand VEGF‐A, which is the major pro‐angiogenic factor of the VEGF family. The activated complex has a crucial role in physiological and pathological angiogenesis through distinct signal transduction pathways regulating endothelial cell survival, proliferation, migration, vascular permeability, tubulo‐ genesis, and gene expression [22].

The CONAMORE assay scheme and working principle are sketched in **Figure 2**. The rec‐ ognition between the VEGF‐A ligand at nanomolar concentration and the surface confined VEGFR2 was characterized in different scenarios: in the presence of noninteracting proteins, in competitive binding experiments and testing the detection of binding of small peptide ligands to VEGFR2 [23]. In particular, the VEGFR2/VEGFA recognition was clearly detectable in the presence of a ten‐fold molar excess of an unrelated protein (1.0 μМ BSA) and com‐ bined with irrelevant immunoglobulins. It was also distinguished from the nonspecific inter‐ actions occurring after denaturation of the receptors. The specificity and robustness of the

where *Δz* is the cantilever deflection (with the z axis individuated by the unitary vector nor‐ mal to the top surface of the cantilever base, i.e., *Δz*< 0 for a downward cantilever bending), *E* is the cantilever Young's modulus, *t* is the cantilever thickness, *L* is the cantilever length, and

Application of nanomechanical sensors to biosensing has become in the last 15 years a break‐ through in biochemistry, life science and medicine, depicting how nanomechanics and biology can grow together [6, 8, 9]. Research in this direction is growing substantially after the milestone work of Fritz and coworkers in 2000, in which they report the specific transduction driven by the surface stress change of DNA hybridization without reported labels [6]. Several experiments have been successfully performed afterwards, revealing DNA hybridization and DNA switch [10–12], detecting proteins and antibodies [5, 13, 14], single virus particles [15] or bacteria [16].

**2.1. Role of nanomechanics in the activation of cell membrane growth factors**

Ligand‐receptor protein interactions are a fundamental mechanism for every biological system, in both physiological and pathological conditions. In particular, the interaction between cell membrane growth factor receptors and their key ligands plays an important role in different processes, including cancer [17]. The growth, survival, and metastatic spreading of solid tumors strongly depend on the formation of a novel vessel network (tumor angiogenesis), making the pro‐angiogenic molecular machinery a target for new strategies in cancer therapy [18, 19]. This approach requires to disentangle the complex array of transduction signals activated by the interaction of pro‐angiogenic growth factors with their cognate cell membrane tyrosine kinase

In view of this, a nanoliter CONAMORE assay was assessed to investigate the interactions of the vascular endothelial growth factor receptor‐2 (VEGFR2), which is the major pro‐angio‐ genic receptor expressed by endothelial cells [21], with the canonical ligand VEGF‐A, which is the major pro‐angiogenic factor of the VEGF family. The activated complex has a crucial role in physiological and pathological angiogenesis through distinct signal transduction pathways regulating endothelial cell survival, proliferation, migration, vascular permeability, tubulo‐

The CONAMORE assay scheme and working principle are sketched in **Figure 2**. The rec‐ ognition between the VEGF‐A ligand at nanomolar concentration and the surface confined VEGFR2 was characterized in different scenarios: in the presence of noninteracting proteins, in competitive binding experiments and testing the detection of binding of small peptide ligands to VEGFR2 [23]. In particular, the VEGFR2/VEGFA recognition was clearly detectable in the presence of a ten‐fold molar excess of an unrelated protein (1.0 μМ BSA) and com‐ bined with irrelevant immunoglobulins. It was also distinguished from the nonspecific inter‐ actions occurring after denaturation of the receptors. The specificity and robustness of the

*ν* is the cantilever Poisson's ratio.

92 Nanomechanics

receptors [20].

genesis, and gene expression [22].

**2. Surface nanomechanics of biomolecules**

**Figure 2.** Scheme of the CONAMORE assay in sessile drop configuration for quantification of the nanomachinery of VEGF‐A/VEGFR2 surface recognition. (a) Initial state with ligands free in solution. (b) Ligand‐receptor recognition equilibrium.

technique were confirmed also by a competition experiment, where the interaction VEGF‐A/ VEGFR2 was suppressed by a neutralizing anti–VEGF‐A antibody, and by a successful detec‐ tion of an interaction between VEGFR2 and a low molecular weight (LMW) molecule (2 k Dacyclo‐peptide).

These preliminary studies set the basis to investigate the role of nanomechanics in the activa‐ tion of cell membrane growth factors [24]. We started from observations that identified the bone morphogenic protein‐antagonist gremlin as a novel pro‐angiogenic ligand of VEGFR2, distinct from canonical VEGFs, increasing the complexity of extracellular interactions involv‐ ing this receptor [24].

VEGF‐A/VEGFR2 and gremlin/VEGFR2 surface recognition were first characterized by Surface Plasmon Resonance (SPR) spectroscopy, which is a gold‐standard mass‐based bio‐ sensor [25]. The SPR isotherms of VEGF‐A and gremlin overlapped, demonstrating that a similar number of VEGF‐A and gremlin molecules interacting with VEGFR2 (VEGF‐A and gremlin have very close masses). But, to the contrary, we found the interactions significantly differentiate in terms of binding kinetics and in‐plane intermolecular forces, suggesting that the binding of VEGF‐A or gremlin induces different VEGFR2 conformational changes and/or clustering in respect to gremlin. Such nanomechanical differences resulted exactly mirrored and supported by the in‐vitro experiments. In fact, we showed that VEGF‐A trig‐ gers a more rapid receptor clustering and a more potent biological response in endothelial cells with respect to gremlin. The key nanomechanical experimental and results are sum‐ marized below.

The SPR dose‐response experiments were repeated with CONAMORE, by exploiting the fact that with CONAMORE technique is possible to perform the nanomechanical sensing on the same chips used for SPR experiments. **Figure 3a** shows the typical binding curves obtained by plotting *Δγ*SB as a function of ligand concentration. In view of the SPR data, the extent of binding of VEGF‐A and gremlin matches at any concentration. Therefore, the isotherms indicate that for the same extent of binding to surface‐immobilized VEGFR2, VEGF‐A exerts a *Δγ*SB (blue dots) that is two‐ to fivefold higher than the *Δγ*SB exerted by gremlin (red reverse triangle). At 100 nM, *Δγ*SB is (8.3 ± 2.1) mN/m and (3.7 ± 0.4) mN/m for VEGF‐A and gremlin, respectively. Remarkably, these values are consistent with MC measurements of cooperative surface mechanical work performed by protein conformational changes [26].

**Figure 3.** Results of VEGF‐A/VEGFR2 and gremlin/VEGFR2 surface recognition. (a) CONAMORE binding isotherms of VEGF‐A (blue circles) and gremlin (red reverse triangles) with VEGFR2 immobilized on surface. (b) Normalized *Δγ*SB binding isotherms.

**Figure 3b** shows the normalized *Δγ*SB isotherms and the resulting fitting curves with a Langmuir‐like equation for monovalent binding. It is possible to estimate an apparent equilib‐ rium constant that explains the nanomechanical aspects of ligand/VEGFR2 surface recognition, named surface nanomechanical affinity, *K<sup>σ</sup> mech*, and its reciprocal, *Kd <sup>σ</sup> mech* = 1/*K<sup>σ</sup> mech*, named sur‐ face nanomechanical dissociation constant. The VEGF‐A isotherm features a sharply steeper rise with respect to the gremlin one, indicating a significant difference in terms of *Kd <sup>σ</sup> mech*. This is supported by the fitting results, which give *Kd <sup>σ</sup> mech* = (2.0 ± 0.7) nM and *Kd <sup>σ</sup> mech* = (32 ± 9) nM, for VEGF‐A and gremlin, respectively. Thus, VEGF‐A has about 16‐fold higher surface nanome‐ chanical affinity for VEGFR2 with respect to gremlin.

#### **2.2. Nanomechanics and protein folding disorders**

Protein conformational changes are a key event in protein's activity, and their characteriza‐ tion is a central goal of biology. Several diseases arise from protein misfolding, in which the misfolded protein self‐associates and becomes deposited in amyloid‐like aggregates in diverse organs, inducing tissue damage and organ dysfunction.

Beta2‐microglobulin (β2‐m) is a key protein acting in the onset of the dialysis related amyloi‐ dosis (DRA), that is a severe complication occurring in patients subjected to chronic hemodi‐ alysis, where insoluble and toxic β2‐m amyloid deposits (fibrils) localize in the skeletal tissues [27]. The fibrils formation follows a complex and still unclear mechanism, where protein con‐ formational changes, among other factors, play a crucial role.

MCs biosensors are suited to probe protein conformational changes, as the biomolecular transformation confined on MC surface can be directly translate in MC bending [6]. In par‐ ticular, silicon MCs with *ad hoc* copolymer coating were employed to probe the effect of an unfolded intermediate of β2‐m, driven by pH changes, and in turn to single out within a pilot set of LMW ligands the ones able to influence or even suppress such effect. The working con‐ cept is presented in the cartoon in **Figure 4**.

The set of small ligands were selected in order to cover the most relevant scenarios: congo red, a dye that probes fibril formation, speeds up the protein refolding kinetics and can abolish in

**Figure 4.** Cartoon representing the effect of small ligands on β2‐m conformational stability probed by MC sensors during pH switch. In the top scheme, the pH switch between 8.0 and 1.5 induces a β2‐m conformational change that drives MC deflection. In the bottom scheme, the presence of small molecule ligands prevents the protein conformational transformation and in turn MC deflection.

**Figure 3b** shows the normalized *Δγ*SB isotherms and the resulting fitting curves with a Langmuir‐like equation for monovalent binding. It is possible to estimate an apparent equilib‐ rium constant that explains the nanomechanical aspects of ligand/VEGFR2 surface recognition,

**Figure 3.** Results of VEGF‐A/VEGFR2 and gremlin/VEGFR2 surface recognition. (a) CONAMORE binding isotherms of VEGF‐A (blue circles) and gremlin (red reverse triangles) with VEGFR2 immobilized on surface. (b) Normalized *Δγ*SB

face nanomechanical dissociation constant. The VEGF‐A isotherm features a sharply steeper

VEGF‐A and gremlin, respectively. Thus, VEGF‐A has about 16‐fold higher surface nanome‐

Protein conformational changes are a key event in protein's activity, and their characteriza‐ tion is a central goal of biology. Several diseases arise from protein misfolding, in which the misfolded protein self‐associates and becomes deposited in amyloid‐like aggregates in

Beta2‐microglobulin (β2‐m) is a key protein acting in the onset of the dialysis related amyloi‐ dosis (DRA), that is a severe complication occurring in patients subjected to chronic hemodi‐ alysis, where insoluble and toxic β2‐m amyloid deposits (fibrils) localize in the skeletal tissues [27]. The fibrils formation follows a complex and still unclear mechanism, where protein con‐

MCs biosensors are suited to probe protein conformational changes, as the biomolecular transformation confined on MC surface can be directly translate in MC bending [6]. In par‐ ticular, silicon MCs with *ad hoc* copolymer coating were employed to probe the effect of an unfolded intermediate of β2‐m, driven by pH changes, and in turn to single out within a pilot set of LMW ligands the ones able to influence or even suppress such effect. The working con‐

The set of small ligands were selected in order to cover the most relevant scenarios: congo red, a dye that probes fibril formation, speeds up the protein refolding kinetics and can abolish in

*<sup>σ</sup> mech* = (2.0 ± 0.7) nM and *Kd*

rise with respect to the gremlin one, indicating a significant difference in terms of *Kd*

*<sup>σ</sup> mech* = 1/*K<sup>σ</sup> mech*, named sur‐

*<sup>σ</sup> mech* = (32 ± 9) nM, for

*<sup>σ</sup> mech*. This is

named surface nanomechanical affinity, *K<sup>σ</sup> mech*, and its reciprocal, *Kd*

diverse organs, inducing tissue damage and organ dysfunction.

formational changes, among other factors, play a crucial role.

cept is presented in the cartoon in **Figure 4**.

supported by the fitting results, which give *Kd*

binding isotherms.

94 Nanomechanics

chanical affinity for VEGFR2 with respect to gremlin.

**2.2. Nanomechanics and protein folding disorders**

vitro fibril deposition [28], suramin, a urea derivative which also binds the protein but does not interfere with its refolding and without antiamyloid activity [28], and a reference sulfo‐ nated molecule that does not bind the protein, hereafter referred as nonbinder.

The pH switch, set between 8.0 and 1.5, drives the MCs functionalized with the native form of β2‐m to a mean differential deflection of *Δz* = (−8 ± 2) nm (**Figure 5** "no ligand"). To test the

**Figure 5.** Experimental MC results and related binding‐conformational scenarios of incubation of β2‐m with the selected small molecule ligands. See the main text for the explanation.

activity of the selected set of small ligands, the β2‐m MCs were incubated *ex situ* in a 6 μM solution of congo red, suramin or nonbinder (see Ref. [29] for the ligand concentration). Then, the β2‐m MCs system was exposed again to the pH switch. **Figure 5** reports the results and cartoons to explain the different binding‐conformation scenarios. The β2‐m MCs incubated with congo red show no differential deflection (*Δz* = 0) after the pH switch. To the contrary, β2‐m MCs incubated with suramin and nonbinder drive a deflection of about −10 nm, that is compatible with the deflection of the native β2‐m MCs. MC deflection therefore indicates that only congo red is able to stabilize the β2‐m in its native conformation and prevent the confor‐ mational change upon the applied pH shift, while suramin and the nonbinder compound fail. Remarkably, it is known that suramin binds to β2‐m with the same binding affinity of congo red (in order of 10−5 M) [29, 30], but only congo red exerts an in vitro antiamyloid activity [28], matching the nanomechanical results.

These findings demonstrate how nanomechanical sensors can be an extraordinary platform for the screening small ligands of proteins involved in pathological processes.

#### **2.3. The nanomechanical side of ferritin iron loading**

Sections 2.1 and 2.2 refer to nanomechanical biomolecular recognition driven by ligand‐recep‐ tor interactions and molecular switches, where deflection is determined mostly by biomolecu‐ lar conformational changes. Instead, in the following section, the study of a class of proteins with negligible conformational rearrangements is presented. In the particular case herein dis‐ cussed, the surface energy change does not take origin from conformational changes, but is related to the electrostatic interaction between the inorganic new‐born nanocores in ferritin cage proteins and other nonspecific short range forces, such as steric, bridging and depletion forces. These findings give the first observation on the in‐plane forces arising upon ferritin iron loading confined at a solid‐liquid interface [31].

Ferritin is a mineralization protein dedicated to the storage of intracellular free iron and per‐ oxides, protecting the cell from oxidative damage [32]. Mammalian ferritin is made of 24 subunits that self‐assemble in a 12 nm shell structure with an inner cavity of 8 nm in diameter able to accommodate up to about 4500 iron atoms [33]. The potential application of ferritin in the field of nanotechnology and nanomedicine [34], together with the rapid development of novel nanomaterials [35, 36], increases the need to understand and control the properties, interactions and iron loading activity of surface confined ferritin.

The rationale depicted in **Figure 6** shows a ferritin‐MC assay based on recombinant human ferritin H chain (FTH) and a mutant without ferroxidase activity (Mutant). A thin film of active FTH (green circles) is deposited onto a MC, which balances by bending the variation of surface energy triggered by iron loading. A control MC is prepared by the surface function‐ alization with the Mutant that is not able to take up iron in the experimental conditions (light blue circles).

**Figure 7a** reports in dark gray line the absolute deflection of the MC modified with FTH (FTH‐MC), in light gray line the absolute deflection of the reference MC modified with the

activity of the selected set of small ligands, the β2‐m MCs were incubated *ex situ* in a 6 μM solution of congo red, suramin or nonbinder (see Ref. [29] for the ligand concentration). Then, the β2‐m MCs system was exposed again to the pH switch. **Figure 5** reports the results and cartoons to explain the different binding‐conformation scenarios. The β2‐m MCs incubated with congo red show no differential deflection (*Δz* = 0) after the pH switch. To the contrary, β2‐m MCs incubated with suramin and nonbinder drive a deflection of about −10 nm, that is compatible with the deflection of the native β2‐m MCs. MC deflection therefore indicates that only congo red is able to stabilize the β2‐m in its native conformation and prevent the confor‐ mational change upon the applied pH shift, while suramin and the nonbinder compound fail. Remarkably, it is known that suramin binds to β2‐m with the same binding affinity of congo red (in order of 10−5 M) [29, 30], but only congo red exerts an in vitro antiamyloid activity [28],

These findings demonstrate how nanomechanical sensors can be an extraordinary platform

Sections 2.1 and 2.2 refer to nanomechanical biomolecular recognition driven by ligand‐recep‐ tor interactions and molecular switches, where deflection is determined mostly by biomolecu‐ lar conformational changes. Instead, in the following section, the study of a class of proteins with negligible conformational rearrangements is presented. In the particular case herein dis‐ cussed, the surface energy change does not take origin from conformational changes, but is related to the electrostatic interaction between the inorganic new‐born nanocores in ferritin cage proteins and other nonspecific short range forces, such as steric, bridging and depletion forces. These findings give the first observation on the in‐plane forces arising upon ferritin

Ferritin is a mineralization protein dedicated to the storage of intracellular free iron and per‐ oxides, protecting the cell from oxidative damage [32]. Mammalian ferritin is made of 24 subunits that self‐assemble in a 12 nm shell structure with an inner cavity of 8 nm in diameter able to accommodate up to about 4500 iron atoms [33]. The potential application of ferritin in the field of nanotechnology and nanomedicine [34], together with the rapid development of novel nanomaterials [35, 36], increases the need to understand and control the properties,

The rationale depicted in **Figure 6** shows a ferritin‐MC assay based on recombinant human ferritin H chain (FTH) and a mutant without ferroxidase activity (Mutant). A thin film of active FTH (green circles) is deposited onto a MC, which balances by bending the variation of surface energy triggered by iron loading. A control MC is prepared by the surface function‐ alization with the Mutant that is not able to take up iron in the experimental conditions (light

**Figure 7a** reports in dark gray line the absolute deflection of the MC modified with FTH (FTH‐MC), in light gray line the absolute deflection of the reference MC modified with the

for the screening small ligands of proteins involved in pathological processes.

matching the nanomechanical results.

96 Nanomechanics

**2.3. The nanomechanical side of ferritin iron loading**

iron loading confined at a solid‐liquid interface [31].

blue circles).

interactions and iron loading activity of surface confined ferritin.

**Figure 6.** Cartoon depicting the ferritin‐MC assay based on recombinant human ferritin H chain (FTH, green circles) and a mutant without ferroxidase activity (Mutant, light blue circles). The variation of surface energy triggered by the iron loading in FTH drives the FTH MC deflection, Δz.

Mutant (ref‐MC), and in red line the differential deflection between the two signals, *Δz*. Vertical arrows indicate a sequential injections of Fe(II) solution necessary to reach a pro‐ gressive iron loading and to limit the oxidation of nonloaded Fe(II) on the outer structure of the protein leading to Fe(III) precipitation. The presentation of *Δz* signal reduces any contribution to MCs deflection due to unspecific adsorption of Fe(II) and/or other compo‐ nents of the buffer solution. The trend depicted in **Figure 7a** is confirmed by the bar chart plot in **Figure 7b** that shows the mean equilibrium *Δz* over 4 replicate MC (the error bars represent the SD of the mean).The *Δz* value reaches a plateau value at *Δz* = (28.1 ± 9.6) nm, corresponding to a net tensile surface stress change, *Δσ* = (6.0 ± 1.5) mJ/m<sup>2</sup> . By further cal‐ culation (see Ref. [31] for details) is possible to convert the *Δz* value at saturation in in‐ plane interferritin interactions force of about 40 k<sup>b</sup> T at room temperature, coming along the formation of iron cores. This value indicates long‐lived van der Waals and electrostatic interactions [37], consistent with reports on nanomechanical biomolecular recognition. Finally, by building up a purely attractive model between the iron cores, described by van der Waals (VDW) interaction potential, it comes out that the 40% of the measured energy variation of the interactions can be ascribed to VDW attractive forces between the newborn ferritin iron cores, and the remaining part ascribed to nonspecific short‐range forces typi‐ cal of biomolecule surface confined systems, such as hydration, steric, bridging and deple‐ tion forces [37, 38].

**Figure 7.** MC responses upon sequential injection of Fe(II). (a) Single MC deflection signals of: FTH‐MC (dark gray line), ref‐MC (light gray line) and differential deflection (red line). (b) Mean differential deflection, Δ*z*, signals at equilibrium at each injection step.

## **3. Breaking good. Probing designer drug family with a unique supramolecular nanomechanical sensor**

The systems presented in previous sections relate in general to biological systems and, in particular, to interactions of biomolecules confined at solid‐solution interface, neglecting the large pool of available synthetic receptors. Nanomechanical sensors are limited, in fact, by the availability of coatings that interact selectively with the target analyte. By introducing the phosphonate cavitands as a versatile class of synthetic receptors [39] that are capable of binding inorganic and organic cations [40, 41] as well as neutral molecules [42] is possible to extend the surface nanomechanics to supramolecular systems.

Phosphonate cavitands are synthetic abiotic receptors (hosts) [39, 40, 43] with molecular recognition properties that have been exploited in gas sensing [44], supramolecular poly‐ mers [45, 46], surface self‐assembly [47], and product protection [48]. They are specifically designed to target small molecules bearing amino‐functionalities via a synergistic com‐ bination of weak interactions such as H‐bonding, dipole−dipole, and CH−π interactions. Probing small molecules bearing amino‐functionalities is a key issue from both the funda‐ mental and the applied sides. N‐Methylated moieties, in particular, are present in a broad range of biologically active compounds, from drugs [49] to cancer biomarkers [50] and neurotransmitters [51].

In the Section 3.1, we will present the preliminary study toward the viability of cavitand‐ MC nanomechanical sensors for probing small molecules bearing amino‐functionalities. In Section 3.2, we show how we implemented these results to realize a nanomechanical device for label‐free detection of amine‐based illicit and designer drug in water.

#### **3.1. Alkyl ammonium series**

**3. Breaking good. Probing designer drug family with a unique** 

The systems presented in previous sections relate in general to biological systems and, in particular, to interactions of biomolecules confined at solid‐solution interface, neglecting the large pool of available synthetic receptors. Nanomechanical sensors are limited, in fact, by the availability of coatings that interact selectively with the target analyte. By introducing the phosphonate cavitands as a versatile class of synthetic receptors [39] that are capable of binding inorganic and organic cations [40, 41] as well as neutral molecules [42] is possible to

**Figure 7.** MC responses upon sequential injection of Fe(II). (a) Single MC deflection signals of: FTH‐MC (dark gray line), ref‐MC (light gray line) and differential deflection (red line). (b) Mean differential deflection, Δ*z*, signals at equilibrium

Phosphonate cavitands are synthetic abiotic receptors (hosts) [39, 40, 43] with molecular recognition properties that have been exploited in gas sensing [44], supramolecular poly‐ mers [45, 46], surface self‐assembly [47], and product protection [48]. They are specifically designed to target small molecules bearing amino‐functionalities via a synergistic com‐ bination of weak interactions such as H‐bonding, dipole−dipole, and CH−π interactions. Probing small molecules bearing amino‐functionalities is a key issue from both the funda‐ mental and the applied sides. N‐Methylated moieties, in particular, are present in a broad range of biologically active compounds, from drugs [49] to cancer biomarkers [50] and

In the Section 3.1, we will present the preliminary study toward the viability of cavitand‐ MC nanomechanical sensors for probing small molecules bearing amino‐functionalities. In Section 3.2, we show how we implemented these results to realize a nanomechanical device

for label‐free detection of amine‐based illicit and designer drug in water.

**supramolecular nanomechanical sensor**

neurotransmitters [51].

at each injection step.

98 Nanomechanics

extend the surface nanomechanics to supramolecular systems.

**Figure 8** sketches a label‐free selective detection of N‐methyl‐ammonium salts in methanol attained thanks to the use of MCs functionalized with tetraphosphonate cavitands. These molecules are LMW species of a mass equal to or lower than 150 Da, differencing only by a methyl group, which is 15 Da.

The bar chart reported in **Figure 9a** shows the mean value of the deflection peaks of MCs func‐ tionalized with the cavitand. The highest interaction intensity is obtained when methylbutyl ammonium chloride is injected, *Δz* = (−80 ± 10) nm. A comparable deflection is measured with dimethylbutyl ammonium chloride and butyl ammonium chloride, with deflection values of *Δz* = (−55 ± 6) nm, and the weakest response comes from trimethyl butyl ammonium chlo‐ ride, *Δz* = (−10 ± 2) nm. The trend is confirmed also by independent ITC experiments shown in **Figure 9b**. Deflection results can be read in terms of the applied surface stress driven by the host‐guest complexation, determined to be *Δσ* = (−17 ± 2) mJ/m<sup>2</sup> for methylbutyl ammonium chloride, and *Δσ* = (−12 ± 1) mJ/m<sup>2</sup> for the interaction with dimethylbutyl ammonium chloride and butyl ammonium chloride. The stress was much lower for the interaction between the cavitand and the last guest, *Δσ* = (−2 ± 0.4) mJ/m<sup>2</sup> . At the molecular level, the overall surface stress is triggered by the interplay between host‐guest complexation, cavitand desolvation, and interface adsorption, as also suggested by the order of magnitude of *Δσ*, which falls in the range of intermolecular forces [52].

**Figure 8.** Scheme of the detection of N‐methylammonium salts by cavitand‐functionalized MCs. (a) Cavitand‐MC architecture and (b) MC deflections driven by the different binding energies of N‐methyl‐ammonium salts and cavitands complexation.

**Figure 9.** Bar chart of (a) the mean value of the deflection peaks of MCs functionalized with the cavitand and of (b) the mean value of *K* evaluated by ITC experiments.

#### **3.2. Illicit and designer drugs**

The next level application of the supramolecular nanomechanical device based on the com‐ bination of tetraphosphonate cavitands and silicon MCs depicted in previous section can be enrolled to the frontier of designer drugs identification.

Designer drugs pose serious challenges when it comes to recognizing them with the current assays, which are tailored for identification of currently illicit substances but poorly effective, or even useless, for novel designer drugs. Actually, designer variants, featuring minor modi‐ fications with respect to an existing drug, show a different chemical composition and are cur‐ rently not illegal in many jurisdictions. The cavitand‐MC system overcomes this shortcoming for methamphetamines, being capable to recognize the methyl‐amine portion that is common to the entire drug family [53].

Data reported in **Figure 10** show the implementation of the cavitand‐MC system on the detection of ecstasy (MDMA), cocaine and amphetamine in the presence of an interferent,

**Figure 10.** (a) Absolute deflection signal of cavitand‐MC system upon injection of MDMA, Cocaine, Amphetamine and Caffeine and (b) mean deflection signal referring to different replicates.

such as caffeine. The different drugs drive MC responses with the same kinetics but dif‐ ferent final equilibrium values (**Figure 10a**). Namely, the latter range from an average of 55 nm for MDMA and cocaine to 22 nm for amphetamine (with a 10% of uncertainty). The surface stress generated by the interaction gives (in modulus) *Δσ* = 12 mN m−1 for MDMA and cocaine and *Δσ* = 5 mN m−1 for amphetamine, in fully agreement with several previ‐ ously reported studies, all related to biomolecules, where it is reported that recognition triggers 1 mN m−1 <*Δσ*< 50 mN m−1 [26, 54–56]. Other common excipients used in drug for‐ mulation, such as lactose and glucose, were investigated with the cavitand‐MC system, and both MDMA and cocaine are recognized with high fidelity in the presence of the excipients [53].

Finally, the device was tested against a real "street" sample, containing 45% of 3‐fluorometh‐ amphetamine (3‐FMA) and glucose as excipient. Signals reported in **Figure 11** (as absolute deflection curves in **Figure 11a** and related bar chart in **Figure 11b**) show the successful detec‐ tion of the drug also directly in a real sample.

This research, that moves from advanced understanding of molecular recognition at the solid‐liquid interface to complement the analytical toolbox for small molecules bearing amino functionalities, has broader horizons, including neurotransmitters and cancer biomarkers.

**Figure 11.** (a) Absolute deflection signal of cavitand‐MC system upon injection of the street sample at two different concentrations and (b) mean deflection signal referring to different replicates.

### **Acknowledgements**

**3.2. Illicit and designer drugs**

100 Nanomechanics

mean value of *K* evaluated by ITC experiments.

to the entire drug family [53].

enrolled to the frontier of designer drugs identification.

Caffeine and (b) mean deflection signal referring to different replicates.

The next level application of the supramolecular nanomechanical device based on the com‐ bination of tetraphosphonate cavitands and silicon MCs depicted in previous section can be

**Figure 9.** Bar chart of (a) the mean value of the deflection peaks of MCs functionalized with the cavitand and of (b) the

Designer drugs pose serious challenges when it comes to recognizing them with the current assays, which are tailored for identification of currently illicit substances but poorly effective, or even useless, for novel designer drugs. Actually, designer variants, featuring minor modi‐ fications with respect to an existing drug, show a different chemical composition and are cur‐ rently not illegal in many jurisdictions. The cavitand‐MC system overcomes this shortcoming for methamphetamines, being capable to recognize the methyl‐amine portion that is common

Data reported in **Figure 10** show the implementation of the cavitand‐MC system on the detection of ecstasy (MDMA), cocaine and amphetamine in the presence of an interferent,

**Figure 10.** (a) Absolute deflection signal of cavitand‐MC system upon injection of MDMA, Cocaine, Amphetamine and

The authors wish to thank all the colleagues and collaborators who participated to the pre‐ sented researches; without their contribution, they would have been not just impossible, but unimaginable. Warm acknowledgments first of all to Laura E.Depero, Giulio Oliviero and Daniele Maiolo and then to Marcella Chiari, Ersilia De Lorenzi, Marco Presta, PaoloArosio, Enrico Dalcanale Guido Condorelli and their research groups.

## **Author details**

Paolo Bergese1 \* and Stefania Federici2

\*Address all correspondence to: paolo.bergese@unibs.it

1 Department of Molecular and Translational Medicine, INSTM, CSGI, University of Brescia, Brescia, Italy

2 Department of Mechanical and Industrial Engineering, INSTM, University of Brescia, Brescia, Italy

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This chapter presents the experimental investigation and numerical simulation of micromembranes supported by serial-parallel connected hinges. The micromembranes can be used in optical applications or as the flexible mechanical element in radio frequency microelectromechanical system switches. A method to determine the micromembrane stiffness is presented. Experimentally, the out-of-plane micromembranes deflection is performed using an atomic force microscope. The dependence between deflection and the applied force gives the sample stiffness. The flexible plate of micromembranes is directly deflected to substrate, and the adhesion force is measured. The micromembranes are electroplated with gold, and two series of the serial connected hinges are investigated. Each of them has different parallel connected hinges. The experimental results of stiffness and adhesion force are compared with analytical and numerical results. The presented method is also applied to determine the stiffness of micromembranes supported by other types of hinges.

Keywords: micromembrane, stiffness, modeling, experimental investigation

### 1. Introduction

The micromembranes are microelectromechanical system (MEMS) components that accomplish one double role of supporting other components, which are regularly rigid and of providing the necessary flexibility in a microdevice that has moving parts [1–3]. A micromembrane has three significant parts: the mobile plate that is moved in different planes in response of an acting signal, the anchors that connect the flexible structures to substrate, and the hinges that connect the mobile

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

plate to anchors. Micromembranes have their thickness much smaller than in plane dimensions and can be implemented in RF-MEMS switches, MEMS accelerometer, or in optical applications [3]. The micromembranes are mechanically characterized here by means of their stiffness. Tribological investigations include the adhesion force measurement between micromembranes and their substrate.

The reliability issue concerning on MEMS has been developed in recent years. First step to understand MEMS reliability is to know the failure modes. The failure of a microsystem depends on the behavior of integrated microcomponents. For different MEMS applications, the mechanical flexible microcomponents are integrating on the same structure with electrical, optical, thermal, or magnetic components. In such systems, a multiphysics interaction occurs and the failure of one microcomponent means the failure of all system [4]. The flexible mechanical microcomponents from a MEMS device, such as sensor or actuator, can be a microcantilever, a microbridge, or micromembranes with different geometrical configurations. These structure MEMS are sensitive from mechanical structure point of view to phenomena like creep, fatigue, delamination, wear or adhesion [5].

Two of the main failure causes of micromembranes under large deflections are excessive stress in hinges and the stiction. Stiction is one of the most important and unavoidable failure problems of micromembranes under large deflection. Stiction is the adhesion of contacting surfaces due to surface forces (van der Waals, capillary forces, Casmir forces, hydrogen bridging, and electrostatic forces) [6]. The restorative force (pull-off force) of a micromembrane from substrate, opposite to the adhesion force, depends on the micromembrane stiffness.

There are several applications which include micromembranes as mechanical flexible component. For example, in a pressure sensor, a capacitive micromembrane deflects when the pressure is applied, changing the distance between electrodes and the capacitance. In optical MEMS, the micromembranes are used as micromirrors supported by hinges with high mobility. In an interferometer, where the laser light brought into the sensor by optical fiber and the light beam crosses a micromembrane, the deformation of micromembrane changes the light properties, and different propagation speed can result in phase shift. Micromembrane surface stress sensors from chemical and biological applications are fabricated from thin gold layer. The molecular interaction between probe and target molecules generates a surface stress on micromembrane. This surface stress causes the structural deflection of micromembrane which generates the capacitance change in electrical sensing [4].

One important parameter characteristic to micromembranes is their mechanical stiffness. The micromembrane stiffness is related to the geometry and material properties. Of micromembranes that are connected by hinges to the anchors, their mechanical stiffness is given by the geometrical and structural characteristics of hinges. Microhinges are utilized as joints in MEMS that provide relative motion between two adjacent rigid links through elastic deformation. The geometrical configuration of hinges has influence on the micromembrane response related to the mechanical stiffness. Hinges are deformed in bending or torsion as a function of the applied force.

This chapter presents a study case of micromembranes supported by serial-parallel connected hinges. The analysis includes theoretical approach, and numerical analysis of the out-of-plane stiffness of micromembranes is followed by experimental tests performed using an atomic force microscope (AFM). Moreover, the adhesion effect between flexible part of micromembranes and substrate is experimentally determined and compared with analytical results. In order to compute the analytical adhesion force, the roughness of the contact surfaces is measured using the AFM taping mode. Discussions and comparison with the other micromembranes supported by other type of hinges with different geometry is included at the end of chapter.

## 2. Geometry and dimensions of micromembrane with serial-parallel connected hinges

The samples for experimental tests are electroplated gold micromembranes with two different serial-parallel connected hinges (Figure 1). The flexible part of investigated micromembranes is suspended at 2 μm above a silicone substrate. Gold is the most used material from optical and electrical MEMS applications.

The geometrical dimensions of investigated micromembranes according to Figure 2 are the following:


plate to anchors. Micromembranes have their thickness much smaller than in plane dimensions and can be implemented in RF-MEMS switches, MEMS accelerometer, or in optical applications [3]. The micromembranes are mechanically characterized here by means of their stiffness. Tribological investigations include the adhesion force measurement between micromembranes

The reliability issue concerning on MEMS has been developed in recent years. First step to understand MEMS reliability is to know the failure modes. The failure of a microsystem depends on the behavior of integrated microcomponents. For different MEMS applications, the mechanical flexible microcomponents are integrating on the same structure with electrical, optical, thermal, or magnetic components. In such systems, a multiphysics interaction occurs and the failure of one microcomponent means the failure of all system [4]. The flexible mechanical microcomponents from a MEMS device, such as sensor or actuator, can be a microcantilever, a microbridge, or micromembranes with different geometrical configurations. These structure MEMS are sensitive from mechanical structure point of view to phenomena

Two of the main failure causes of micromembranes under large deflections are excessive stress in hinges and the stiction. Stiction is one of the most important and unavoidable failure problems of micromembranes under large deflection. Stiction is the adhesion of contacting surfaces due to surface forces (van der Waals, capillary forces, Casmir forces, hydrogen bridging, and electrostatic forces) [6]. The restorative force (pull-off force) of a micromembrane from

There are several applications which include micromembranes as mechanical flexible component. For example, in a pressure sensor, a capacitive micromembrane deflects when the pressure is applied, changing the distance between electrodes and the capacitance. In optical MEMS, the micromembranes are used as micromirrors supported by hinges with high mobility. In an interferometer, where the laser light brought into the sensor by optical fiber and the light beam crosses a micromembrane, the deformation of micromembrane changes the light properties, and different propagation speed can result in phase shift. Micromembrane surface stress sensors from chemical and biological applications are fabricated from thin gold layer. The molecular interaction between probe and target molecules generates a surface stress on micromembrane. This surface stress causes the structural deflection of micromembrane which

One important parameter characteristic to micromembranes is their mechanical stiffness. The micromembrane stiffness is related to the geometry and material properties. Of micromembranes that are connected by hinges to the anchors, their mechanical stiffness is given by the geometrical and structural characteristics of hinges. Microhinges are utilized as joints in MEMS that provide relative motion between two adjacent rigid links through elastic deformation. The geometrical configuration of hinges has influence on the micromembrane response related to the mechanical

This chapter presents a study case of micromembranes supported by serial-parallel connected hinges. The analysis includes theoretical approach, and numerical analysis of the out-of-plane stiffness of micromembranes is followed by experimental tests performed using an atomic force

stiffness. Hinges are deformed in bending or torsion as a function of the applied force.

substrate, opposite to the adhesion force, depends on the micromembrane stiffness.

and their substrate.

108 Nanomechanics

like creep, fatigue, delamination, wear or adhesion [5].

generates the capacitance change in electrical sensing [4].


Figure 1. Micromembranes supported by serial-parallel connected hinges: (a) micromembrane supported by 2 � 4 connected hinges; (b) micromembrane supported by 4 � 6 connected hinges.

Figure 2. Dimensions of the micromembrane supported by connected hinges (half geometry).

### 3. Analytical and numerical analysis of stiffness of micromembranes with serial-parallel connected hinges

Castigliano's second theorem, known as the displacement theorem, is utilized herein to derive the stiffness of investigated micromembranes and to compute the dependence between force and the sample-bending deflection. The force is considered to be applied in the mid position of the mobile plate as in experimental investigations [3]. A general formulation is derived here enabling stiffness computation for any combination of serial-parallel connected hinges. The analytical results of micromembranes with 2 � 4 and 4 � 6 serial-parallel connected hinges are compared in Section 6 of this chapter, with numerical and experimental results. As a consequence, a bending stiffness expression of micromembranes supported by serial-parallel connected hinges is obtained as following:

$$k\_b = \frac{2}{(\mathbf{S\_1} + \mathbf{S\_2} + \mathbf{S\_3} + \mathbf{S\_4})} \tag{1}$$

where

$$S\_{1} = \frac{\left(\frac{(l\_{4} + l\_{3} + l\_{2} + l\_{1})^{3} - (l\_{4} + l\_{3} + l\_{2})^{3}}{3} - A\left(\left(l\_{4} + l\_{3} + l\_{2} + l\_{1}\right)^{2} - \left(l\_{4} + l\_{3} + l\_{2}\right)^{2}\right) + A^{2}l\_{1}\right)}{EI\_{1}},$$

$$S\_{2} = \frac{\left(\frac{(l\_{4} + l\_{3} + l\_{2})^{3} - (l\_{4} + l\_{3})^{3}}{3} - A\left(\left(l\_{4} + l\_{3} + l\_{2}\right)^{2} - \left(l\_{4} + l\_{3}\right)^{2}\right) + A^{2}l\_{2}\right)}{EI\_{2}},$$

$$S\_{3} = \frac{\left(\frac{(l\_{4} + l\_{3})^{3} - l\_{4}^{2}}{3} - A\left(\left(l\_{4} + l\_{3}\right)^{2} - l\_{4}^{2}\right) + A^{2}l\_{3}\right)}{EI\_{3}}, S\_{4} = \frac{\left(\frac{l\_{4}^{2}}{3} - Al\_{4}^{2} + A^{2}l\_{4}\right)}{EI\_{4}}, \text{and}$$

Nanocharacterization of the Mechanical and Tribological Behavior of MEMS Micromembranes http://dx.doi.org/10.5772/intechopen.68292 111

$$A = \frac{\left(\frac{\left(l\_4 + l\_3 + l\_2 + l\_1\right)^2 - \left(l\_4 + l\_3 + l\_2\right)^2}{I\_1} + \frac{\left(l\_4 + l\_3 + l\_2\right)^2 - \left(l\_4 + l\_3\right)^2}{I\_2} + \frac{\left(l\_4 + l\_3\right)^2 - l\_4^2}{I\_3} + \frac{l\_4^2}{I\_4}\right)}{2 \cdot \left(\frac{l\_1}{I\_1} + \frac{l\_2}{I\_2} + \frac{l\_3}{I\_3} + \frac{l\_4}{I\_4}\right)} \cdot 2$$

In these expressions, l1, l2, l3, and l<sup>4</sup> are the characteristic lengths of hinges and mobile plates (Figure 2), and I1, I2, I3, and I<sup>4</sup> are the bending moments of inertia given by the micromembrane thickness and widths and influenced by the number of connected hinges n<sup>1</sup> and n<sup>2</sup> as:

$$I\_1 = \eta\_1 \cdot \frac{w\_1 t^3}{12}, \qquad I\_2 = I\_4 = \frac{w\_2 t^3}{12}, \qquad I\_3 = \eta\_2 \cdot \frac{w\_1 t^3}{12}$$

Using the stiffness expression given by Eq. (1), a theoretical analysis of the number of connected hinges influence on the membrane stiffness is performed (Figure 3). As the number of hinges increases, the stiffness increases respectively.

A numerical analysis of micromembranes stiffness was performed by Finite Element Analysis (FEA) using the static structural module in ANSYS Workbench 13 software. The mesh of the FEA model for the micromembrane with 2 � 4 serial-parallel connected hinges (Figure 1a) consists of 318,200 nodes and 58,786 hexahedral elements with a size of 2 μm. For the micromembrane with 4 � 6 serial-parallel connected hinges (Figure 1b), the created mesh has 534,835 nodes and 102,601 hexahedral elements with the same size. Boundary conditions applied on the bottom surface of the anchors correspond to a fixed support. A unitary force (1 μN) is applied in the mid position of the mobile plate, and the out-of-plane displacement is simulated. Considering the applied force and the resulting displacement, the bending stiffness is computed. The simulation is performed considering a value of modulus of elasticity equal by 72 GPa [7]. The own weight of the mobile plate upon the hinge deformation is very small and has been neglected.

3. Analytical and numerical analysis of stiffness of micromembranes with

Figure 2. Dimensions of the micromembrane supported by connected hinges (half geometry).

Castigliano's second theorem, known as the displacement theorem, is utilized herein to derive the stiffness of investigated micromembranes and to compute the dependence between force and the sample-bending deflection. The force is considered to be applied in the mid position of the mobile plate as in experimental investigations [3]. A general formulation is derived here enabling stiffness computation for any combination of serial-parallel connected hinges. The analytical results of micromembranes with 2 � 4 and 4 � 6 serial-parallel connected hinges are compared in Section 6 of this chapter, with numerical and experimental results. As a consequence, a bending stiffness expression of micromembranes supported by serial-parallel

kb <sup>¼</sup> <sup>2</sup>

<sup>2</sup> � <sup>l</sup> 2 4 <sup>þ</sup> <sup>A</sup><sup>2</sup> l3

ðl<sup>4</sup> þ l<sup>3</sup> þ l<sup>2</sup> þ l1Þ

EI<sup>1</sup>

ðl<sup>4</sup> þ l<sup>3</sup> þ l2Þ

EI<sup>2</sup>

<sup>ð</sup>S<sup>1</sup> <sup>þ</sup> <sup>S</sup><sup>2</sup> <sup>þ</sup> <sup>S</sup><sup>3</sup> <sup>þ</sup> <sup>S</sup>4<sup>Þ</sup> <sup>ð</sup>1<sup>Þ</sup>

<sup>2</sup> � ðl<sup>4</sup> <sup>þ</sup> <sup>l</sup><sup>3</sup> <sup>þ</sup> <sup>l</sup>2<sup>Þ</sup>

2 <sup>þ</sup> <sup>A</sup><sup>2</sup> l2

<sup>2</sup> � ðl<sup>4</sup> <sup>þ</sup> <sup>l</sup>3<sup>Þ</sup>

l 3 4 <sup>3</sup> � Al<sup>2</sup>

, S<sup>4</sup> ¼

2 <sup>þ</sup> <sup>A</sup><sup>2</sup> l1

<sup>4</sup> <sup>þ</sup> <sup>A</sup><sup>2</sup> l4

EI<sup>4</sup>

,

, and

,

serial-parallel connected hinges

connected hinges is obtained as following:

ðl4þl3þl2þl1Þ

S<sup>2</sup> ¼

ðl4þl3Þ 3 �l 3 4 <sup>3</sup> � A

3 �ðl4þl3þl2Þ 3 <sup>3</sup> � A

ðl4þl3þl2Þ 3 �ðl4þl3Þ 3 <sup>3</sup> � A

> ðl<sup>4</sup> þ l3Þ

EI<sup>3</sup>

where

110 Nanomechanics

S<sup>1</sup> ¼

S<sup>3</sup> ¼

Figure 3. Analytical stiffness variation as a function of the number of connected hinges n1�n<sup>2</sup> (n<sup>1</sup> represents the number of hinges that connect the membrane to anchor and n<sup>2</sup> represents the number of hinges between mobile plates).

Figure 4. Finite element analysis of micromembrane supported by: (a) 2 4 connected hinges; (b) 4 6 connected hinges. The force is applied in the mid position of the mobile plate.

A maximum displacement of 103.2 nm is simulated of the investigated micromembrane supported by 2 4 connected hinges (Figure 4a) that give a numerical stiffness equal by 9.69 N/m. For the investigated micromembrane supported by 4 6 connected hinges, an out-of-plane displacement of 60.19 nm is obtained (Figure 4b) for a unitary force applied in the mid position of the central mobile plate that gives a bending stiffness equal to 16.61 N/m.

## 4. Experimental investigations on stiffness of micromembranes with serial-parallel connected hinges

The aim of experimental investigations is to estimate the mechanical stiffness of micromembranes using an atomic force microscope (AFM) type XE-70 fabricated by Park System Co. The AFM probe used to deflect the micromembranes is TD21562 with a nominal value of the spring constant equal to 144 Nm<sup>1</sup> , the radius of tip smaller than 25 nm, the tip height of 109 μm, the length of cantilever 782 μm, and the thickness 24 μm. The tests were performed at room temperature (22C) and a relative humidity (RH) of 40%. During experimental tests, a mechanical force given by the bending deflection of AFM probe and its stiffness is applied in the mid position of the central plate and deflect it directly to substrate (Figure 5).

During experimental tests, the vertical displacement of the scanning head (Dz) is controlled, and the deflection of AFM probe (DAFM) is optically monitored (Figure 6).

The experimental AFM curve gives the dependence between vertical displacement of scanning head that is controlled by software and the bending deflection of AFM probe detected by a photodetector. The experimental AFM curve has two different slopes (Figure 7). First part of the curve (A and B) from the loading curve (lower curve) corresponds to the bending of AFM probe and samples (position b from Figure 6), and the second part (B and C) is given by the bending only of AFM probe (position c from Figure 6) [3]. In the point B (Figure 7), the sample reaches the substrate. After the point C, the scanning head and the AFM probe are coming to the initial position (the blue curve).

Nanocharacterization of the Mechanical and Tribological Behavior of MEMS Micromembranes http://dx.doi.org/10.5772/intechopen.68292 113

Figure 5. AFM probe in contact with the micromembrane supported by n<sup>1</sup> � n<sup>2</sup> ¼ 2 � 4 connected hinges.

A maximum displacement of 103.2 nm is simulated of the investigated micromembrane supported by 2 4 connected hinges (Figure 4a) that give a numerical stiffness equal by 9.69 N/m. For the investigated micromembrane supported by 4 6 connected hinges, an out-of-plane displacement of 60.19 nm is obtained (Figure 4b) for a unitary force applied in the mid position

Figure 4. Finite element analysis of micromembrane supported by: (a) 2 4 connected hinges; (b) 4 6 connected hinges.

of the central mobile plate that gives a bending stiffness equal to 16.61 N/m.

serial-parallel connected hinges

The force is applied in the mid position of the mobile plate.

112 Nanomechanics

the central plate and deflect it directly to substrate (Figure 5).

and the deflection of AFM probe (DAFM) is optically monitored (Figure 6).

constant equal to 144 Nm<sup>1</sup>

the initial position (the blue curve).

4. Experimental investigations on stiffness of micromembranes with

The aim of experimental investigations is to estimate the mechanical stiffness of micromembranes using an atomic force microscope (AFM) type XE-70 fabricated by Park System Co. The AFM probe used to deflect the micromembranes is TD21562 with a nominal value of the spring

length of cantilever 782 μm, and the thickness 24 μm. The tests were performed at room temperature (22C) and a relative humidity (RH) of 40%. During experimental tests, a mechanical force given by the bending deflection of AFM probe and its stiffness is applied in the mid position of

During experimental tests, the vertical displacement of the scanning head (Dz) is controlled,

The experimental AFM curve gives the dependence between vertical displacement of scanning head that is controlled by software and the bending deflection of AFM probe detected by a photodetector. The experimental AFM curve has two different slopes (Figure 7). First part of the curve (A and B) from the loading curve (lower curve) corresponds to the bending of AFM probe and samples (position b from Figure 6), and the second part (B and C) is given by the bending only of AFM probe (position c from Figure 6) [3]. In the point B (Figure 7), the sample reaches the substrate. After the point C, the scanning head and the AFM probe are coming to

, the radius of tip smaller than 25 nm, the tip height of 109 μm, the

Figure 6. Bending deflection of AFM probe and investigated micromembranes: (a) the initial contact between AFM probe and sample; (b) bending of AFM probe together with sample; (c) bending only of AFM probe (the sample touch the substrate).

The AFM data from the first part of curve (A and B) are used to estimate the sample deflection (Dz) as the difference between the controlled displacement of the scanning head (Dz) and the detected deflection of AFM probe (DAFM).

$$D\_s = D\_z - D\_{\rm{AFM}} \tag{2}$$

Figure 7. AFM experimental dependence between vertical displacements of the scanning head [μm] versus deflection of AFM probe [nm] for the investigated micromembrane with 2 � 4 connected hinges.

The force used in experiment can be determined based on the known stiffness of AFM probe and its bending deflection. This force and the determined deflection of sample given by Eq. (2) are used to compute the sample stiffness as:

$$k = \frac{F}{D\_s} \tag{3}$$

The experimental dependence between the sample deflection and the applied AFM force is presented in Figure 8. The slope of this curve gives the micromembrane bending stiffness.

Figure 8. Force [μN] versus deflection [μm] of the micromembrane with 2 � 4 connected hinges. The slope of experimental curve represents the bending stiffness.

The stiffness of the investigated micromembrane supported by 2 � 4 serial-parallel connected hinges (Figure 1a) is 8.930 7N/m (Figure 8). The same experiment is performed in order to estimate the stiffness of the micromembrane supported by 4 � 6 serial-parallel connected hinges (Figure 1b). For this micromembrane, a bending stiffness equal to 18.18 N/m is determined if the force is applied in the mid position of the mobile plate (Figure 5). The tests were repeated several times, and the results of micromembrane stiffness have closed values.

## 5. Analysis of adhesion force between micromembrane with serial-parallel connected hinges and the silicon substrate

Stiction is one of the main failure causes in MEMS, and it occurs when surface forces are too large and, as a consequence, the surfaces brought in contact cannot be separated again. Intensive experimental and theoretical research has been conducted in order to determine the effect of the main factors on the surface forces especially on the adhesion force. For example, the influence of material properties, surface characteristics, and environmental conditions on the adhesion force has been experimentally determined and mathematically validated for several MEMS structures [8].

The force used in experiment can be determined based on the known stiffness of AFM probe and its bending deflection. This force and the determined deflection of sample given by Eq. (2)

Figure 7. AFM experimental dependence between vertical displacements of the scanning head [μm] versus deflection of

<sup>k</sup> <sup>¼</sup> <sup>F</sup> Ds

The experimental dependence between the sample deflection and the applied AFM force is presented in Figure 8. The slope of this curve gives the micromembrane bending stiffness.

Figure 8. Force [μN] versus deflection [μm] of the micromembrane with 2 � 4 connected hinges. The slope of experimen-

ð3Þ

are used to compute the sample stiffness as:

114 Nanomechanics

tal curve represents the bending stiffness.

AFM probe [nm] for the investigated micromembrane with 2 � 4 connected hinges.

Researchers have developed several mathematical models for computing the adhesion force by taking into consideration the possible contact types between surfaces. For example, if two surfaces are characterized by high roughness, their contact was approximated by the contact between two elastic spheres (see DMT and JKR models [9]). Another example is the case of the elastic contact between a surface with high roughness and a surface with low roughness. The already mentioned mathematical models can be easily customized for this case [10]. However, the ideal case of elastic contact is rare at micro and nano scale, determining in some cases large differences between experimental and theoretical values. As mentioned in [8], when two surfaces are bought in contact, the highest asperities suffer a plastic deformation. Therefore, in order to correctly estimate the adhesion force between two rough surfaces, a plastic adhesion index has to be computed [8]:

$$
\lambda = \frac{\pi^2 H^4 r\_d \sigma}{8 \Delta \gamma^2 E^{\*2}} \tag{4}
$$

where H is the material hardness, r<sup>a</sup> is the mean radius of curvature of asperities, σ is the standard deviation of peak heights, Δγ is the work of adhesion, E\* is the equivalent Young's module given by:

$$E^\* = \left(\frac{1-\nu\_1^2}{E\_1} + \frac{1-\nu\_2^2}{E\_2}\right) \tag{5}$$

E1,2 are Young's moduli and ν1,2 are Poisson's ratios of the two surfaces. The work of adhesion is computed according to [10]:

$$
\Delta \mathcal{Y} = \mathcal{Y}\_1 + \mathcal{Y}\_2 - \mathcal{Y}\_{12} \tag{6}
$$

where γ<sup>1</sup> and γ<sup>2</sup> are the surface energies of the two materials brought in contact, and γ1,2 is the interface energy of contacting materials and was computed according to [11].

Considering the plastic deformation and assuming an exponential distribution of asperity heights, the adhesion force is expressed as a function of the compression force F as follows:

$$F\_{ad} = F\left(-1 + \frac{1+\lambda}{\lambda} \exp\left(-\frac{1}{\lambda}\right)\right) \tag{7}$$

while the total real contact area per unit area is given by:

$$A\_r = 2\pi nr\_d \sigma \tag{8}$$

and n is the number of asperities in contact per unit area.

The experimental determination of the adhesion force between the flexible part of the gold micromembrane and the silicon substrate was conducted using the spectroscopy in point mode of the AFM for the micromembrane with 2 � 4 connected hinges (Figure 9).

On the unloading AFM curve (Figure 7), the detachment of the micromembrane from substrate is delayed based on the adhesion effect that occurs. This effect gives a jump in the bending deflection of AFM probe that is directly proportional to the adhesion force. A zoom of the experimental AFM curve obtained when performing the spectroscopy in point is illustrated in Figure 9, and the blue part corresponds to the unloading test. In the same figure, the experimental value of the corresponding adhesion force given by the AFM software can be observed, and it is equal to 1.278 μm.

Figure 9. Adhesion force between the central part of micromembrane and the substrate in the case of micromembrane with 2 � 4 connected hinges.

In order to compute the adhesion force, the roughness of the contact surfaces is needed. To perform this step, a micromembrane is cut close to anchor and moved with the backside up to scan the contact surfaces [12]. The topography of the contact surfaces is taken using the tapping mode of AFM. The 3D images for each of the two surfaces that come in contact are provided in Figures 10 and 11, respectively, together with the statistical parameters provided for roughness by the software used for interpreting the experimental data (XEI Image Processing Tool for SPM).

Δγ ¼ γ<sup>1</sup> þ γ<sup>2</sup> � γ<sup>12</sup> ð6Þ

Ar ¼ 2πnraσ ð8Þ

ð7Þ

where γ<sup>1</sup> and γ<sup>2</sup> are the surface energies of the two materials brought in contact, and γ1,2 is the

Considering the plastic deformation and assuming an exponential distribution of asperity heights, the adhesion force is expressed as a function of the compression force F as follows:

1 þ λ

The experimental determination of the adhesion force between the flexible part of the gold micromembrane and the silicon substrate was conducted using the spectroscopy in point

On the unloading AFM curve (Figure 7), the detachment of the micromembrane from substrate is delayed based on the adhesion effect that occurs. This effect gives a jump in the bending deflection of AFM probe that is directly proportional to the adhesion force. A zoom of the experimental AFM curve obtained when performing the spectroscopy in point is illustrated in Figure 9, and the blue part corresponds to the unloading test. In the same figure, the experimental value of the corresponding adhesion force given by the AFM software can be

Figure 9. Adhesion force between the central part of micromembrane and the substrate in the case of micromembrane

mode of the AFM for the micromembrane with 2 � 4 connected hinges (Figure 9).

<sup>λ</sup> exp � <sup>1</sup>

λ

interface energy of contacting materials and was computed according to [11].

Fad ¼ F �1 þ

while the total real contact area per unit area is given by:

and n is the number of asperities in contact per unit area.

observed, and it is equal to 1.278 μm.

116 Nanomechanics

with 2 � 4 connected hinges.

Figure 10. AFM images and the surface parameters of the backside of flexible plate (gold).

Figure 11. AFM images and the surface parameters of the substrate (silicon).

The theoretical value for the adhesion force was obtained using the mathematical model presented above. First, the work of adhesion was computed using a surface energy of 1.51 J/ m2 [13] for the silicon substrate and of 1 J/m<sup>2</sup> [14] for the gold micromembrane, while the equivalent Young's module was computed using a value of 150 GPa for Young's module and 0.17 for Poisson's ratio for the silicon substrate and a value of 72 GPa for Young's module and 0.42 for Poisson's ratio for the gold membrane. Then, the adhesion force was computed for the micromembrane with 2 � 4 connected hinges considering the real contact area between the surfaces. Knowing that the compression force was 17.28 μN, an adhesion force of 1.34 μN was obtained in good agreement with the experimental ones.

## 6. Results and discussions

First, using the AFM technique, the effect of geometrical configuration of hinges on the bending stiffness of gold micromembranes was investigated. A method to estimate the out-of-plane stiffness by AFM is provided in this chapter, including its validation by analytical and numerical approaches. The mechanical force used to bend the flexible part of micromembrane is given by bending deflection of AFM probe and its stiffness. Second, by considering Castigliano's deformation theorem, an expression to compute the bending stiffness of micromembranes supported by serial-parallel connected hinges is provided. Moreover, simulation and finite element analysis is developed using a unitary force applied in the mid position of the mobile plate as in the experiments. The results obtained for out-of-plane stiffness in investigated samples are in good agreement as observed in Table 1.

The differences between results are influenced by the accuracy of the experimental tests and the differences of Young's modulus. In analytical and numerical analysis, a value of Young's modulus taken from literature is used, and it can differ by its experimental value. Moreover, the analytical model does not consider the influence of the holes from the plates. In any way, the holes have a negligible effect on the samples stiffness as confirmed by numerical analysis. An important influence of the holes can be observed if the micromembranes are dynamically actuated, because the damping given by surrounding medium decreases. Using different serial-parallel connected hinges to support the flexible part of micromembrane, different mechanical responses can be obtained. The stiffness of samples depends on the number of hinges and their connection.

The adhesion force for the contact between the flexible part and the silicon substrate of the gold micromembrane with 2 � 4 connected hinges was determined both experimentally and theoretically, and the results are in good agreement. The pull-off force of micromembranes from


Table 1. Bending stiffness of investigated micromembranes.

substrate depends on the adhesion force and the samples stiffness. Adhesion effect is influenced by the roughness of contact surfaces and the operating conditions. The roughness is the same for investigated micromembranes.

The theoretical value for the adhesion force was obtained using the mathematical model presented above. First, the work of adhesion was computed using a surface energy of 1.51 J/ m2 [13] for the silicon substrate and of 1 J/m<sup>2</sup> [14] for the gold micromembrane, while the equivalent Young's module was computed using a value of 150 GPa for Young's module and 0.17 for Poisson's ratio for the silicon substrate and a value of 72 GPa for Young's module and 0.42 for Poisson's ratio for the gold membrane. Then, the adhesion force was computed for the micromembrane with 2 � 4 connected hinges considering the real contact area between the surfaces. Knowing that the compression force was 17.28 μN, an adhesion force of 1.34 μN was

First, using the AFM technique, the effect of geometrical configuration of hinges on the bending stiffness of gold micromembranes was investigated. A method to estimate the out-of-plane stiffness by AFM is provided in this chapter, including its validation by analytical and numerical approaches. The mechanical force used to bend the flexible part of micromembrane is given by bending deflection of AFM probe and its stiffness. Second, by considering Castigliano's deformation theorem, an expression to compute the bending stiffness of micromembranes supported by serial-parallel connected hinges is provided. Moreover, simulation and finite element analysis is developed using a unitary force applied in the mid position of the mobile plate as in the experiments. The results obtained for out-of-plane stiffness in investigated samples are in good

The differences between results are influenced by the accuracy of the experimental tests and the differences of Young's modulus. In analytical and numerical analysis, a value of Young's modulus taken from literature is used, and it can differ by its experimental value. Moreover, the analytical model does not consider the influence of the holes from the plates. In any way, the holes have a negligible effect on the samples stiffness as confirmed by numerical analysis. An important influence of the holes can be observed if the micromembranes are dynamically actuated, because the damping given by surrounding medium decreases. Using different serial-parallel connected hinges to support the flexible part of micromembrane, different mechanical responses can be obtained. The stiffness of samples depends on the number of hinges and their connection.

The adhesion force for the contact between the flexible part and the silicon substrate of the gold micromembrane with 2 � 4 connected hinges was determined both experimentally and theoretically, and the results are in good agreement. The pull-off force of micromembranes from

Micromembrane 2 � 4 8.93 10.33 9.69 Micromembrane 4 � 6 18.18 17.86 16.61

Experimental Analytical Simulation

obtained in good agreement with the experimental ones.

6. Results and discussions

118 Nanomechanics

agreement as observed in Table 1.

Samples Stiffness [N/m]

Table 1. Bending stiffness of investigated micromembranes.

Depending on the micromembrane application, different types of hinges can be chosen in order to obtain different stiffness for in-plane and/or out-of-plane movement. The same methodology as presented in Section 3 of this chapter was applied to determine the stiffness of micromembrane with the other types of hinges.

Micromembranes supported by serpentine hinges are characterized by the possibility of in-plane and out-of-plane movements of the mobile plate [15]. Micromembranes supported by two and four serpentine hinges are presented in Figure 12. The number of hinges as well as their length have influence the micromembranes deflection under a mechanical force. Using the AFM tests, the displacement of the central plate is monitored under a mechanical force, and the micromembrane stiffness was determined. As the number of hinges and their length are modified, the stiffness is changed. The micromembrane 1 with two short hinges (Figure 12a) has an out-of-plane stiffness equal to 64.4 N/m. If the length of hinges increases for the micromembrane 2 (Figure 12b), the bending stiffness decreases 4.3 times. In the case of the same micromembrane, but supported by four hinges, the stiffness is modified. The micromembrane 3 with four short hinges (Figure 12c) is characterized by a stiffness of 127.5 N/m, and its value decreases five times if the length of hinges increases as in the case of the micromembrane 4 (Figure 12d). It can be noticed that longer hinges ensure more flexibility and the number of hinges multiply the stiffness.

Figure 12. Micromembranes supported by serpentine hinges: (a) Micromembrane-1; (b) Micromembrane-2; (c) Micromembrane-3; (d) Micromembrane-4.

The other investigations using the same AFM method were performed on micromembranes supported by bent beam hinges electroplated with gold in different geometrical dimensions are presented in Figure 13 [16]. The structures are characterized by different widths of hinges and different lengths of the central plate. Stiffness analysis considers fixed anchors and a central force perpendicular to the central plate of micromembranes. Different stiffness of micromembranes can be obtained if the geometrical dimensions are modified. The stiffness of micromembrane increases with the dimensions of hinges and decreases for a longer mobile plate.

Other example, presented in Figure 14, is a micromembrane supported by folded hinges [3]. The same AFM tests were performed to determine the micromembrane behavior under a

Figure 13. Micromembranes supported by bent beam hinges.

Figure 14. Micromembranes supported by folded hinges.

mechanical force. The stiffness was experimentally determined and validated by numerical and analytical results. As in the case of investigated micromembrane with serial-parallel connected hinges, the holes have a small influence upon stiffness under static deflection fact demonstrated by the numerical simulation. The holes have significant effect on the dynamic modulation because the mass of micromembrane is changed and the quality factor given by the air damping is improved [17].

## Acknowledgements

The other investigations using the same AFM method were performed on micromembranes supported by bent beam hinges electroplated with gold in different geometrical dimensions are presented in Figure 13 [16]. The structures are characterized by different widths of hinges and different lengths of the central plate. Stiffness analysis considers fixed anchors and a central force perpendicular to the central plate of micromembranes. Different stiffness of micromembranes can be obtained if the geometrical dimensions are modified. The stiffness of micromembrane

Other example, presented in Figure 14, is a micromembrane supported by folded hinges [3]. The same AFM tests were performed to determine the micromembrane behavior under a

increases with the dimensions of hinges and decreases for a longer mobile plate.

Figure 13. Micromembranes supported by bent beam hinges.

120 Nanomechanics

Figure 14. Micromembranes supported by folded hinges.

This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-1271.

## Author details

Marius Pustan\*, Cristian Dudescu, Corina Birleanu and Florina Rusu

\*Address all correspondence to: Marius.Pustan@omt.utcluj.ro

Technical University of Cluj-Napoca, Cluj-Napoca, Romania

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[10] Zhang LX, Zhao YP. Adhesion of rough surfaces with plastic deformation. Journal of

[11] Gutierrez-Miravete E. Adhesion between Contacting Surfaces, Chapter 4 [Internet], 2012. Available from: http://www.ewp.rpi.edu/hartford/ernesto/F2012/FWM/Notes/ch04.pdf

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[16] Pustan M, Dudescu C, Birleanu C, Rusu F, Chiorean R, Craciun S. Effect of geometrical dimensions on the tribomechanical response of a gold micromembrane with bent beam hinges. IOP Conference Series: Materials Science and Engineering. 2016;147(1):012022.

[17] Pustan M, Birleanu C, Dudescu C, Golinval JC. Dynamical behavior of smart MEMS in industrial applications. In: Nihtianov S, Estepa AL, editors. Smart Sensors and MEMS: Intelligent Devices and Microsystems for Industrial Applications. Woodhead Publishing;

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2013. pp. 349–365. DOI:10.1533/9780857099297.2.349

DOI: 10.1088/1757-899X/147/1/012022

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67973

#### Abstract

Static, vibration, and buckling analysis of nanobeams is studied based on modified couple stress theory (MCST) in this chapter. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new nonclassical beam model reduces the classical beam model when the length scale parameter is set to zero. The finite element formulations are derived for static, free vibration, and buckling problems of nanobeams within MCST and the Euler-Bernoulli beam theory. The effect of the material length scale parameter and geometry parameters on the static, vibration, and buckling responses of the nanobeam is investigated in both the classical beam theory (CBT) and MCST by using finite element method. Also, the difference between the classical beam theory (CBT) and modified couple stress theory is investigated.

Keywords: nanobeams, couple stress theory, finite element method, static, vibration, buckling

#### 1. Introduction

With the great advances in technology in recent years, micro and nanostructures have found many applications. In these structures, micro beams and micro tubes are widely used in nanoscale electromechanical systems such as sensors (Zook et al. [1], Pei et al. [2]), actuators (Senturia [3], Rezazadeh et al. [4]). In investigation of micro and nanostructures, the classical continuum mechanics are not effort of describing of the size-dependent mechanics. Nonclassical continuum theories such as higher-order gradient theories and the couple stress theory are capable of explanation of the size-dependent behaviors, which occur in micro/ nanoscale structures.

At the present time, the experimental investigations of the micro/nano materials are still a challenge because of difficulties confronted in the micro/nanoscale. Therefore, mechanical

theories and atomistic simulations have been used for nanostructural analysis. The process of the atomistic simulations is very difficult and takes much time. So, continuum theory is the most preferred method for the analysis of the micro and nanostructures. Classical continuum mechanics does not contain the size effect, because of its scale-free character. The nonlocal continuum theory initiated by Eringen [5] has been widely used to mechanical behavior of micro/nanostructures.

The size effect is very effective in the mechanical behavior of nanostructures at nanometer scale that the classic theory has failed to consider when the size reduces from macro to nano (Toupin [6], Mindlin [7, 8], Fleck and Hutchinson [9], Yang et al. [10], Lam et al. [11]). Therefore, higher-order theories of modified couple stress theory (MCST) and modified strain gradient are used in the mechanical model of the nano/microstructures (Yang et al. [10], Lam et al. [11]).

The determination of the micro/nanostructural material length scale parameters is very difficult experimentally. So, Yang et al. [10] studied the strain energy of the MCST with one length scale parameter. After this, the MCST and the strain gradient elasticity theories have been widely applied to static and dynamic analysis of beams (Park and Gao [12], Ma et al. [13], Kong et al. [14], Wang et. al. [15], Asghari et al. [16], Wang [17], Simsek [18], Kahrobaiyan et al. [19], Xia et al. [20], Ke et al. [21], Li et al. [22], Akgöz and Civalek [23, 24], Ansari et al. [25], Dos Santos and Reddy [26], Simsek et al. [27], Wang et al. [28], Kocatürk and Akbas [29], Kong [30], Daneshmehr et al. [31], Akgöz and Civalek [32], Ziaee [33], Islam et. al. [34], Miandoab et al. [35], Liu et al. [36], Tang et al. [37], Hosseini and Rahmani [38], Akbas [39, 40]).

The objective of this paper is to investigate static, vibration, and buckling solutions of nanobeams based on modified couple stress theory (MCST). The finite element formulations are derived for static, free vibration, and buckling problems of nanobeams within MCST and the Euler-Bernoulli beam theory. The effect of the material length scale parameter and geometry parameters on the static, vibration, and buckling responses of the nanobeam are investigated in both the classical beam theory (CBT) and MCST. Also, the difference between the classical beam theory and modified couple stress theory is investigated.

#### 2. Theory and formulations

A simple supported nanobeam of length L, width b, and height h, with X,Y,Z cartesian coordinate system is shown in Figure 1.

Figure 1. A simple supported nanobeam with X,Y,Z cartesian coordinate system and cross section.

The modified couple stress theory was proposed by Yang et al. [10]. Based on this theory, the strain energy density for a linear elastic material which is a function of both strain tensor and curvature tensor is introduced for the modified couple stress theory;

theories and atomistic simulations have been used for nanostructural analysis. The process of the atomistic simulations is very difficult and takes much time. So, continuum theory is the most preferred method for the analysis of the micro and nanostructures. Classical continuum mechanics does not contain the size effect, because of its scale-free character. The nonlocal continuum theory initiated by Eringen [5] has been widely used to mechanical behavior of

The size effect is very effective in the mechanical behavior of nanostructures at nanometer scale that the classic theory has failed to consider when the size reduces from macro to nano (Toupin [6], Mindlin [7, 8], Fleck and Hutchinson [9], Yang et al. [10], Lam et al. [11]). Therefore, higher-order theories of modified couple stress theory (MCST) and modified strain gradient are used in the mechanical model of the nano/microstructures (Yang et al.

The determination of the micro/nanostructural material length scale parameters is very difficult experimentally. So, Yang et al. [10] studied the strain energy of the MCST with one length scale parameter. After this, the MCST and the strain gradient elasticity theories have been widely applied to static and dynamic analysis of beams (Park and Gao [12], Ma et al. [13], Kong et al. [14], Wang et. al. [15], Asghari et al. [16], Wang [17], Simsek [18], Kahrobaiyan et al. [19], Xia et al. [20], Ke et al. [21], Li et al. [22], Akgöz and Civalek [23, 24], Ansari et al. [25], Dos Santos and Reddy [26], Simsek et al. [27], Wang et al. [28], Kocatürk and Akbas [29], Kong [30], Daneshmehr et al. [31], Akgöz and Civalek [32], Ziaee [33], Islam et. al. [34], Miandoab et al.

The objective of this paper is to investigate static, vibration, and buckling solutions of nanobeams based on modified couple stress theory (MCST). The finite element formulations are derived for static, free vibration, and buckling problems of nanobeams within MCST and the Euler-Bernoulli beam theory. The effect of the material length scale parameter and geometry parameters on the static, vibration, and buckling responses of the nanobeam are investigated in both the classical beam theory (CBT) and MCST. Also, the difference between the

A simple supported nanobeam of length L, width b, and height h, with X,Y,Z cartesian

[35], Liu et al. [36], Tang et al. [37], Hosseini and Rahmani [38], Akbas [39, 40]).

classical beam theory and modified couple stress theory is investigated.

Figure 1. A simple supported nanobeam with X,Y,Z cartesian coordinate system and cross section.

micro/nanostructures.

124 Nanomechanics

[10], Lam et al. [11]).

2. Theory and formulations

coordinate system is shown in Figure 1.

$$\mathcal{U} = \bigcap\_{V} (\sigma : \mathfrak{e} + \mathfrak{m} : \mathfrak{x}) \, dV \tag{1}$$

where σ is the stress tensor, ε is the strain tensor, m is the deviatoric part of the couple stress tensor, χ is the symmetric curvature tensor, defined by

$$
\sigma = \lambda \operatorname{tr}(\mathfrak{e}) I + 2\mu \mathfrak{e} \tag{2}
$$

$$\boldsymbol{\varepsilon} = \frac{1}{2} [\nabla \boldsymbol{\mu} + (\nabla \boldsymbol{\mu})^T] \tag{3}$$

$$
\mathfrak{m} = 2l^2 \mu \,\, \mathfrak{x} \tag{4}
$$

$$\mathbf{x} = \frac{1}{2} [\nabla \theta + (\nabla \theta)^T] \tag{5}$$

where λ and μ are Lame's constants, l is a material length scale parameter which is regarded as a material property characterizing the effect of couple stress, u is the displacement vector and θ is the rotation vector, given by

$$
\theta = \frac{1}{2} \operatorname{curl} \,\mathfrak{u} \tag{6}
$$

The parameters λ and μ in the constitutive equation are given by

$$
\lambda = \frac{E \text{ } \nu}{(1+\nu)(1-2\nu)}, \text{ } \mu = \frac{E}{2(1+\nu)}\tag{7}
$$

where E is the modulus of elasticity and ν is the Poisson's ratio. According to the coordinate system (X,Y,Z) shown in Figure 1, based on Euler-Bernoulli beam theory, the axial and the transverse displacement field are expressed as

$$u(X,Y,t) = u\_0(X,t) - Y \frac{\partial v\_0(X,t)}{\partial X} \tag{8}$$

$$
\upsilon(\mathbf{X}, \mathbf{Y}, t) = \upsilon\_0(\mathbf{X}, t) \tag{9}
$$

$$w(\mathbf{X}, \mathbf{Y}, \mathbf{t}) = \mathbf{0} \tag{10}$$

where u, v, w are the x, y, and z components of the displacements, respectively; u<sup>0</sup> and v<sup>0</sup> are the axial and the transverse displacements in the mid-plane; and t indicates time. Because the transversal surfaces of the beam are free of stress,

$$
\sigma\_{zz} = \sigma\_{yy} = 0 \tag{11}
$$

By using Eqs. (3) and (8)–(10), the strain-displacement relation can be obtained:

$$
\varepsilon\_{\rm xx} = \frac{\partial u}{\partial X} = \frac{\partial u\_0(X,t)}{\partial X} - Y \frac{\partial^2 v\_0(X,t)}{\partial X^2} \tag{12a}
$$

$$
\varepsilon\_{yy} = \varepsilon\_{zz} = Y \frac{\nu \,\mathrm{\partial^2 v\_0(X,t)}}{\partial X^2} \tag{12b}
$$

$$
\varepsilon\_{xz} = \varepsilon\_{yz} = \varepsilon\_{xy} = \mathbf{0} \tag{12c}
$$

By using Eqs. (6) and (8)–(10),

$$
\partial\_z = \frac{\partial v\_0(X, t)}{\partial X} \theta\_x = \theta\_y = 0 \tag{13}
$$

Substituting Eq. (13) into Eq. (5), the curvature tensor χ can be obtained as follows:

$$\chi\_{xz} = \frac{1}{2} \frac{\partial^2 v\_0(X, t)}{\partial X^2} \\ \chi\_{xx} = \chi\_{xy} = \chi\_{yy} = \chi\_{yz} = \chi\_{zz} = 0 \tag{14}$$

According to Hooke's law, the constitutive equations of the nanobeam are as follows:

$$
\sigma\_{\rm xx} = E \varepsilon\_{\rm xx} = E \left[ \frac{\partial u\_0(X,t)}{\partial X} - Y \frac{\partial^2 v\_0(X,t)}{\partial X^2} \right] \tag{15}
$$

where σxx and εxx are the normal stresses and normal strains in the X direction, respectively. Substituting Eq. (14) into Eq. (4), the couple stress tensor can be obtained as follows:

$$m\_{xz} = l^2 \mu \frac{1}{2} \frac{\partial^2 v\_0(\mathbf{X}, t)}{\partial \mathbf{X}^2} \tag{16a}$$

$$m\_{xx} = m\_{xy} = m\_{yy} = m\_{yz} = m\_{zz} = 0\tag{16b}$$

where μ is the shear modulus defined by Eq. (7). The elastic strain energy (Ui) of the nanobeam is expressed as

$$
\delta L I\_i = \frac{1}{2} \int\_0^L (\sigma\_{i\bar{\eta}} \,\, \varepsilon\_{i\bar{\eta}} + m\_{i\bar{\eta}} \,\, \chi\_{i\bar{\eta}}) dA \,\, dX \,\tag{17}
$$

By substituting Eqs. (12) and (14)–(16) into Eq. (17), elastic strain energy (Ui) can be rewritten as follows:

$$LI\_i = \frac{1}{2} \left[ \left[ EA \left( \frac{\partial u\_0(\mathbf{X}, t)}{\partial \mathbf{X}} \right)^2 + EI \left( \frac{\partial^2 v(\mathbf{X}, t)}{\partial \mathbf{X}^2} \right)^2 + \frac{1}{4} I^2 \mu A \left( \frac{\partial^2 v(\mathbf{X}, t)}{\partial \mathbf{X}^2} \right)^2 \right] d\mathbf{X} \tag{18}$$

where A is the area of the cross section, and I is the moment of inertia.

Static, Vibration, and Buckling Analysis of Nanobeams http://dx.doi.org/10.5772/67973 127

$$T = \frac{1}{2} \left[ \left[ \rho A \left( \frac{\partial u\_0}{\partial t} \right)^2 + \rho A \left( \frac{\partial v\_0}{\partial t} \right)^2 + \rho I \left( \frac{\partial^2 v\_0}{\partial X \ \partial t} \right)^2 \right] dX \tag{19}$$

The potential energy of the external load can be written as

<sup>ε</sup>xx <sup>¼</sup> <sup>∂</sup><sup>u</sup>

By using Eqs. (6) and (8)–(10),

126 Nanomechanics

is expressed as

as follows:

Ui <sup>¼</sup> <sup>1</sup> 2 ð L

0

EA <sup>∂</sup>u0ðX,t<sup>Þ</sup> ∂X � �<sup>2</sup>

<sup>χ</sup>xz <sup>¼</sup> <sup>1</sup> 2 <sup>∂</sup><sup>X</sup> <sup>¼</sup> <sup>∂</sup>u0ðX, t<sup>Þ</sup>

<sup>θ</sup><sup>z</sup> <sup>¼</sup> <sup>∂</sup>v0ðX, t<sup>Þ</sup>

Substituting Eq. (13) into Eq. (5), the curvature tensor χ can be obtained as follows:

According to Hooke's law, the constitutive equations of the nanobeam are as follows:

Substituting Eq. (14) into Eq. (4), the couple stress tensor can be obtained as follows:

mxz ¼ l 2 μ 1 2

Ui <sup>¼</sup> <sup>1</sup> 2 ð L

where A is the area of the cross section, and I is the moment of inertia.

0

ð

A

where σxx and εxx are the normal stresses and normal strains in the X direction, respectively.

where μ is the shear modulus defined by Eq. (7). The elastic strain energy (Ui) of the nanobeam

By substituting Eqs. (12) and (14)–(16) into Eq. (17), elastic strain energy (Ui) can be rewritten

<sup>þ</sup> EI <sup>∂</sup><sup>2</sup>vðX,t<sup>Þ</sup> ∂X<sup>2</sup> � �<sup>2</sup>

� �<sup>2</sup> " #

þ 1 4 l 2

<sup>σ</sup>xx <sup>¼</sup> <sup>E</sup>εxx <sup>¼</sup> <sup>E</sup> <sup>∂</sup>u0ðX, t<sup>Þ</sup>

<sup>∂</sup><sup>2</sup>v0ðX, t<sup>Þ</sup>

<sup>ε</sup>yy <sup>¼</sup> <sup>ε</sup>zz <sup>¼</sup> <sup>Y</sup> <sup>ν</sup> <sup>∂</sup><sup>2</sup> <sup>v</sup>0ðX, t<sup>Þ</sup>

<sup>∂</sup><sup>X</sup> � <sup>Y</sup> <sup>∂</sup><sup>2</sup>v0ðX, t<sup>Þ</sup>

<sup>∂</sup>X<sup>2</sup> <sup>ð</sup>12a<sup>Þ</sup>

ð15Þ

<sup>∂</sup>X<sup>2</sup> <sup>ð</sup>12b<sup>Þ</sup>

εxz ¼ εyz ¼ εxy ¼ 0 ð12cÞ

<sup>∂</sup><sup>X</sup> <sup>θ</sup><sup>x</sup> <sup>¼</sup> <sup>θ</sup><sup>y</sup> <sup>¼</sup> <sup>0</sup> <sup>ð</sup>13<sup>Þ</sup>

<sup>∂</sup>X<sup>2</sup> <sup>ð</sup>16a<sup>Þ</sup>

<sup>∂</sup>X<sup>2</sup> <sup>χ</sup>xx <sup>¼</sup> <sup>χ</sup>xy <sup>¼</sup> <sup>χ</sup>yy <sup>¼</sup> <sup>χ</sup>yz <sup>¼</sup> <sup>χ</sup>zz <sup>¼</sup> <sup>0</sup> <sup>ð</sup>14<sup>Þ</sup>

<sup>∂</sup><sup>X</sup> � <sup>Y</sup> <sup>∂</sup><sup>2</sup>v0ðX, t<sup>Þ</sup>

� �

<sup>∂</sup><sup>2</sup>v0ðX, t<sup>Þ</sup>

∂X<sup>2</sup>

mxx ¼ mxy ¼ myy ¼ myz ¼ mzz ¼ 0 ð16bÞ

ðσij εij þ mij χijÞdA dX ð17Þ

<sup>μ</sup><sup>A</sup> <sup>∂</sup><sup>2</sup>vðX,t<sup>Þ</sup> ∂X<sup>2</sup>

dX ð18Þ

$$dL\_t = \int\_0^L \left[ f\leftv(\mathbf{x}) + P\left(\frac{\partial v}{\partial \mathbf{x}}\right) \right] dX + Q\_i v\_i \tag{20}$$

where f is load function, Qi is point loads which contains point forces and moments, P is axial compressive load for buckling case. The nodal displacements q for a two-node beam element contain three degrees of freedom at each node, as shown in Figure 2, namely,

$$\{\boldsymbol{\varrho}(t)\}\_{\mathbf{e}} = [\boldsymbol{u}\_i^{(\boldsymbol{e})}(t), \,\boldsymbol{v}\_i^{(\boldsymbol{e})}(t), \,\,\,\theta\_i^{(\boldsymbol{e})}(t), \,\,\boldsymbol{u}\_j^{(\boldsymbol{e})}(t), \,\,\boldsymbol{v}\_j^{(\boldsymbol{e})}(t), \,\,\,\theta\_j^{(\boldsymbol{e})}(t) \,\,\big]^T \tag{21}$$

The displacement field of the finite element is expressed in terms of the nodal displacements as follows:

$$\mu^{(\mathbf{e})}(\mathbf{X}, \mathbf{t}) = \mathfrak{q}\_1^{(\mathrm{II})}(\mathbf{X})\,\mathfrak{u}\_{\mathbf{i}}(\mathbf{t}) + \mathfrak{q}\_2^{(\mathrm{II})}(\mathbf{X})\,\mathfrak{u}\_{\mathbf{j}}(\mathbf{t}) \,\ = [\mathfrak{q}^{(\mathrm{II})}] \left\{ \begin{array}{c} \mathfrak{u}\_{\mathbf{i}} \\ \mathfrak{u}\_{\mathbf{j}} \end{array} \right\} = [\mathfrak{q}^{(\mathrm{II})}] \{\mathbf{q}\}\_{\mathbf{i}\mathbf{l}} \tag{22}$$

$$\mathbf{v}^{(\mathbf{e})}(\mathbf{X},\mathbf{t}) = \mathbf{q}\_1^{(V)}(\mathbf{X})\,\mathbf{v}\_{\mathbf{i}}(\mathbf{t}) + \mathbf{q}\_2^{(V)}(\mathbf{X})\,\boldsymbol{\Theta}\_{\mathbf{i}}(\mathbf{t}) + \mathbf{q}\_3^{(V)}(\mathbf{X})\,\mathbf{v}\_{\mathbf{j}}(\mathbf{t}) + \mathbf{q}\_4^{(V)}(\mathbf{X})\,\boldsymbol{\Theta}\_{\mathbf{j}}(\mathbf{t})\tag{23a}$$

$$\begin{Bmatrix} \mathbf{q}^{(V)} \end{Bmatrix} \begin{Bmatrix} \mathbf{v}\_{\mathbf{i}} \\ \theta\_{\mathbf{i}} \\ \upsilon\_{\mathbf{j}} \\ \theta\_{\mathbf{j}} \end{Bmatrix} = \begin{bmatrix} \mathbf{q}^{(V)} \end{bmatrix} \{\mathbf{q}\}\_{V} \tag{23b}$$

where ui, vi, and θ<sup>i</sup> are the axial displacements, transverse displacements, and slopes at the two end nodes of the beam element, respectively, and ϕ<sup>ð</sup>U<sup>Þ</sup> <sup>i</sup> and <sup>ϕ</sup><sup>ð</sup>V<sup>Þ</sup> <sup>i</sup> are the Hermite shape functions for the axial and transverse displacements, respectively. The interpolation functions for the axial displacement are,

Figure 2. Two-node beam element.

$$\boldsymbol{\phi}^{(\mathrm{II})}(\mathbf{X}) = \left[ \boldsymbol{\phi}\_1^{(\mathrm{II})}(\mathbf{X}) \, \boldsymbol{\phi}\_2^{(\mathrm{II})}(\mathbf{X}) \right]^T \tag{24}$$

where

$$
\rho\_1^{(L)}(\mathbf{X}) = \left(-\frac{\mathbf{X}}{L\_\mathbf{e}} + \mathbf{1}\right) \tag{25a}
$$

$$
\rho\_2^{(\mathrm{II})}(\mathbf{X}) = \left(\frac{\mathbf{X}}{L\_\mathbf{e}}\right) \tag{25b}
$$

The interpolation functions for the transverse displacement are

$$\boldsymbol{\varrho}^{(V)}(\mathbf{X}) = \begin{bmatrix} \boldsymbol{\varrho}\_1^{(V)}(\mathbf{X}) \ \boldsymbol{\varrho}\_2^{(V)}(\mathbf{X}) \ \boldsymbol{\varrho}\_3^{(V)}(\mathbf{X}) \ \boldsymbol{\varrho}\_4^{(V)}(\mathbf{X}) \end{bmatrix}^T \tag{26}$$

where

$$\varphi\_1^{(V)}(\mathbf{X}) = \left(1 - \frac{3X^2}{L\_\mathbf{e}^2} + \frac{2X^3}{L\_\mathbf{e}^3}\right) \tag{27a}$$

$$\rho\_2^{(V)}(\mathbf{X}) = \left(-\mathbf{X} + \frac{2\mathbf{X}^2}{L\_\mathbf{e}} - \frac{\mathbf{X}^3}{L\_\mathbf{e}^2}\right) \tag{27b}$$

$$
\rho\_3^{(V)}(\mathbf{X}) = \left(\frac{3X^2}{L\_\mathbf{e}^2} - \frac{2X^3}{L\_\mathbf{e}^3}\right) \tag{27c}
$$

$$
\rho\_4^{(V)}(\mathbf{X}) = \left(\frac{\mathbf{X}^2}{L\_\mathbf{e}} - \frac{\mathbf{X}^3}{L\_\mathbf{e}^2}\right) \tag{27d}
$$

with L<sup>e</sup> indicating the length of the beam element. The Lagrangian functional of the problem is given as follows:

$$I = T - \left(\mathcal{U}\_{\text{i}} + \mathcal{U}\_{\text{e}}\right) \tag{28}$$

After substituting Eqs. (22) and (23) into Eq. (28) and then using the Lagrange's equations, one obtains the following equation:

$$\frac{\partial I}{\partial q\_k^{(e)}} - \frac{\partial}{\partial t} \frac{\partial I}{\partial \dot{q}\_k^{(e)}} = 0, k = 1, 2, 3, 4, 5, 6,\tag{29}$$

where q\_ ðeÞ <sup>k</sup> indicates the time derivative of the nodal displacements q.

The Lagrange's equations can be employed to yield the system of equations of motion for the finite element. By the usual assemblage procedure, the equations of motion can be obtained for the entire structure. For the free vibration problem, the finite element equation is as follows:

$$([K] - \omega^2 [M]) \{ \dot{q} \} = 0 \tag{30}$$

For the static problem, the finite element equation is as follows:

$$[K]\{q\} = \{F\} \tag{31}$$

For the buckling problem, the finite element equation is as follows:

$$([\mathbf{K}] - P\_{cr}[\mathbf{K}\_{\mathbb{G}}]) \{ \hat{q} \} = \mathbf{0} \tag{32}$$

where ω is the natural frequency, fq ^g is a vector of displacement amplitudes of the vibration, fFg is the global load vector, [K] is the stiffness matrix, [M] is the mass matrix, and [KG] is the stability matrix. The stiffness matrix [K] can be given as:

$$[K] = \begin{bmatrix} [K^A] & [0] \\ [0] & [K^D] \end{bmatrix} \tag{33}$$

where

ϕðU<sup>Þ</sup>

The interpolation functions for the transverse displacement are

<sup>ð</sup>XÞ ¼ <sup>ϕ</sup><sup>ð</sup>V<sup>Þ</sup>

ϕ<sup>ð</sup>V<sup>Þ</sup>

ϕ<sup>ð</sup>V<sup>Þ</sup>

ϕ<sup>ð</sup>V<sup>Þ</sup>

ϕ<sup>ð</sup>V<sup>Þ</sup>

∂I ∂q ðeÞ k

� ∂ ∂t ∂I ∂q\_ ðeÞ k

<sup>k</sup> indicates the time derivative of the nodal displacements q.

ϕðV<sup>Þ</sup>

ϕ<sup>ð</sup>U<sup>Þ</sup>

ϕ<sup>ð</sup>U<sup>Þ</sup>

<sup>1</sup> <sup>ð</sup>X<sup>Þ</sup> <sup>ϕ</sup><sup>ð</sup>V<sup>Þ</sup>

<sup>1</sup> <sup>ð</sup>XÞ ¼ <sup>1</sup> � <sup>3</sup>X<sup>2</sup>

<sup>2</sup> ðXÞ¼ �X þ

<sup>3</sup> <sup>ð</sup>XÞ ¼ <sup>3</sup>X<sup>2</sup>

<sup>4</sup> <sup>ð</sup>XÞ ¼ <sup>X</sup><sup>2</sup>

L2 e

Le

with L<sup>e</sup> indicating the length of the beam element. The Lagrangian functional of the problem is

After substituting Eqs. (22) and (23) into Eq. (28) and then using the Lagrange's equations, one

The Lagrange's equations can be employed to yield the system of equations of motion for the finite element. By the usual assemblage procedure, the equations of motion can be obtained for the entire structure. For the free vibration problem, the finite element equation is as follows:

where

128 Nanomechanics

where

given as follows:

where q\_

ðeÞ

obtains the following equation:

<sup>ð</sup>XÞ ¼ <sup>ϕ</sup><sup>ð</sup>U<sup>Þ</sup>

<sup>1</sup> <sup>ð</sup>XÞ¼ � <sup>X</sup>

<sup>2</sup> <sup>ð</sup>XÞ ¼ <sup>X</sup>

<sup>1</sup> <sup>ð</sup>X<sup>Þ</sup> <sup>ϕ</sup><sup>ð</sup>U<sup>Þ</sup>

Le þ 1 � �

> Le � �

<sup>2</sup> <sup>ð</sup>X<sup>Þ</sup> <sup>ϕ</sup><sup>ð</sup>V<sup>Þ</sup>

L2 e þ 2X<sup>3</sup> L3 e

!

2X<sup>2</sup> Le

> � <sup>2</sup>X<sup>3</sup> L3 e

� X3 L2 e

!

!

!

� X3 L2 e

h i<sup>T</sup>

<sup>3</sup> <sup>ð</sup>X<sup>Þ</sup> <sup>ϕ</sup><sup>ð</sup>V<sup>Þ</sup>

<sup>4</sup> ðXÞ

I ¼ T � ðU<sup>i</sup> þ UeÞ ð28Þ

¼ 0, k ¼ 1, 2, 3, 4, 5, 6, ð29Þ

<sup>2</sup> ðXÞ h i<sup>T</sup>

ð24Þ

ð25aÞ

ð25bÞ

ð26Þ

ð27aÞ

ð27bÞ

ð27cÞ

ð27dÞ

$$\left[K^{A}\right] = \int\_{0}^{L\_{q}} EA \left[\frac{\partial \varphi^{(L)}}{\partial X}\right]^{T} \left[\frac{\partial \varphi^{(L)}}{\partial X}\right] dX \tag{34a}$$

$$[K^D] = \int\_0^{L\_q} \left(\mathcal{E}I + \frac{1}{4}l^2\mu A\right) \left[\frac{\partial^2 \rho^{(V)}}{\partial X^2}\right]^T \left[\frac{\partial^2 \rho^{(V)}}{\partial X^2}\right] dX \tag{34b}$$

The mass matrix [M] can be expressed as the sum of four submatrices as follows:

$$[M] = [M\_{II}] + [M\_V] + [M\_{\mathcal{O}}] \tag{35}$$

where

$$\begin{aligned} \left[M\_{\rm II}\right] = \int\limits\_{0} \rho A \left[\rho^{(\rm II)}\right]^{T} \left[\rho^{(\rm II)}\right] dX \end{aligned} \tag{36a}$$

$$\mathbb{E}\left[M\_V\right] = \bigcap\_{0}^{L\_q} \rho A [\varphi^{(V)}]^T \, [\varphi^{(V)}] dX \tag{36b}$$

$$[M\_{\mathcal{Q}}] = \bigcap\_{\circ \in \mathcal{I}}^{L\_{\bullet}} \rho I[\varphi^{(\mathcal{Q})}]^T \ [\varphi^{(\mathcal{Q})}] dX \tag{36c}$$

The stability matrix [KG] can be given as:

$$\begin{bmatrix} \mathbf{K}\_G \end{bmatrix} = \begin{bmatrix} \begin{bmatrix} \mathbf{0} \end{bmatrix} & \begin{bmatrix} \mathbf{0} \end{bmatrix} \\ \begin{bmatrix} \mathbf{0} \end{bmatrix} & \begin{bmatrix} \mathbf{K}\_G^D \end{bmatrix} \end{bmatrix} \tag{37}$$

where

$$\begin{aligned} \left[K\_G^D\right] = \int\_0^{L\_q} \left[\frac{\partial \varphi^{(V)}}{\partial X}\right]^T \left[\frac{\partial \varphi^{(V)}}{\partial X}\right] dX \end{aligned} \tag{38}$$

The load vector {F} is expressed as

$$\{F\} = \bigcap\_{\mathbf{x}=\mathbf{0}}^{L\_{\mathbf{q}}} \{\boldsymbol{\varphi}(\mathbf{X})\}^{\mathrm{T}} \boldsymbol{f} \, d\mathbf{X} + \mathbf{Q}\_{i} \tag{39}$$

In the solution of the free vibration and buckling problems, the eigenvalue procedure is performed in Eqs. (30) and (32). When the material length scale parameter (l) is equal to zero, the finite element formulations reduce to classical beam theory.

After integration processing, the finite element matrixes can be expressed as follows:

½ �¼ K EA Le 0 0 � EA Le 0 0 <sup>12</sup>ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup> μAÞ Le �6ðEI þ 0:25l μAÞ Le <sup>0</sup> �12ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup> μAÞ Le �6ðEI þ 0:25l μAÞ Le �6ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup> μAÞ Le ðEI þ 0:25l μAÞ Le <sup>6</sup>ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup> μAÞ Le ðEI þ 0:25l μAÞ Le � EA Le 0 0 EA Le 0 0 �12ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup> μAÞ Le ðEI þ 0:25l μAÞ Le <sup>0</sup> <sup>12</sup>ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup> μAÞ Le ðEI þ 0:25l μAÞ Le �6ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup> μAÞ Le ðEI þ 0:25l μAÞ Le <sup>6</sup>ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup> μAÞ Le ðEI þ 0:25l μAÞ Le ð40aÞ

M� ¼ ρAL<sup>e</sup> 0 0 <sup>ρ</sup>AL<sup>e</sup> 0 0 ð ρAL<sup>e</sup> þ ρI Le Þ ð�11ρAL<sup>e</sup> � <sup>ρ</sup><sup>I</sup> <sup>Þ</sup> <sup>0</sup> <sup>ð</sup> ρAL<sup>e</sup> � <sup>6</sup>ρ<sup>I</sup> Le Þ ð13ρAL<sup>e</sup> � <sup>ρ</sup><sup>I</sup> Þ ð �11ρAL<sup>e</sup> � <sup>ρ</sup><sup>I</sup> Þ ðρAL<sup>e</sup> þ ρILe <sup>Þ</sup> <sup>0</sup> <sup>ð</sup> �13ρAL<sup>e</sup> <sup>þ</sup> <sup>ρ</sup><sup>I</sup> <sup>Þ</sup> �ρAL<sup>e</sup> � <sup>ρ</sup>ILe ρAL<sup>e</sup> 0 0 <sup>ρ</sup>AL<sup>e</sup> 0 0 ð ρAL<sup>e</sup> � <sup>6</sup>ρ<sup>I</sup> Le Þ ð�<sup>13</sup>ρAL<sup>e</sup> <sup>þ</sup> <sup>ρ</sup><sup>I</sup> <sup>Þ</sup> <sup>0</sup> <sup>ð</sup> ρAL<sup>e</sup> þ ρI Le Þ ð11ρAL<sup>e</sup> <sup>þ</sup> <sup>ρ</sup><sup>I</sup> Þ ð ρAL<sup>e</sup> � <sup>ρ</sup><sup>I</sup> Þ ð�<sup>ρ</sup>AL<sup>e</sup> � <sup>ρ</sup>ILe <sup>Þ</sup> <sup>0</sup> <sup>ð</sup> ρAL<sup>e</sup> <sup>þ</sup> <sup>ρ</sup><sup>I</sup> Þ ð <sup>ρ</sup>AL<sup>e</sup> þ ρILe Þ ð40bÞ

#### Static, Vibration, and Buckling Analysis of Nanobeams http://dx.doi.org/10.5772/67973

$$[K\_G] = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{6}{5L\_e} & \frac{-1}{10} & 0 & \frac{-6}{5L\_e} & \frac{-1}{10} \\ 0 & \frac{-1}{10} & \frac{2L\_e}{15} & 0 & \frac{1}{10} & \frac{-L\_e}{30} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{-6}{5L\_e} & \frac{1}{10} & 0 & \frac{6}{5L\_e} & \frac{1}{10} \\ 0 & \frac{-1}{10} & \frac{-L\_e}{30} & 0 & \frac{1}{10} & \frac{2L\_e}{15} \end{bmatrix} \tag{40c}$$

#### 3. Numerical results

KG� ¼ <sup>½</sup>0� ½0�

∂ϕðV<sup>Þ</sup> ∂X

KD <sup>G</sup>� ¼

fFg ¼

the finite element formulations reduce to classical beam theory.

 μAÞ

 μAÞ

> μAÞ

 μAÞ

ρI Le L ðe

L ðe

x¼0

After integration processing, the finite element matrixes can be expressed as follows:

 μAÞ

 μAÞ

 μAÞ

 μAÞ

 � <sup>ρ</sup><sup>I</sup>

 <sup>þ</sup> <sup>ρ</sup><sup>I</sup>

 � <sup>ρ</sup>ILe

ρILe

 þ

0 0 � EA

�6ðEI þ 0:25l

ðEI þ 0:25l

0 0 EA

ðEI þ 0:25l

ðEI þ 0:25l

0 0 <sup>ρ</sup>AL<sup>e</sup>

Þ ðρAL<sup>e</sup>

0 0 <sup>ρ</sup>AL<sup>e</sup>

Þ ð�11ρAL<sup>e</sup>

Þ ð�<sup>13</sup>ρAL<sup>e</sup>

Þ ð�<sup>ρ</sup>AL<sup>e</sup>

Le

Le

Le

Le

In the solution of the free vibration and buckling problems, the eigenvalue procedure is performed in Eqs. (30) and (32). When the material length scale parameter (l) is equal to zero,

Le

Le

<sup>Þ</sup> <sup>0</sup> <sup>ð</sup>

<sup>Þ</sup> <sup>0</sup> <sup>ð</sup>

<sup>Þ</sup> <sup>0</sup> <sup>ð</sup>

<sup>Þ</sup> <sup>0</sup> <sup>ð</sup>

<sup>0</sup> <sup>12</sup>ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup>

<sup>0</sup> �12ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup>

where

Nanomechanics

½ �¼ K

M� ¼

EA Le

� EA Le

ρAL<sup>e</sup>

ρAL<sup>e</sup>

ð

ð

ð

ð

<sup>12</sup>ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup>

�6ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup>

�12ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup>

�6ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup>

Le 

Le 

Le 

Le 

ρAL<sup>e</sup> þ

�11ρAL<sup>e</sup> � <sup>ρ</sup><sup>I</sup>

ρAL<sup>e</sup> � <sup>6</sup>ρ<sup>I</sup> Le

ρAL<sup>e</sup> � <sup>ρ</sup><sup>I</sup>

The load vector {F} is expressed as

0� ½K<sup>D</sup> G�

� �<sup>T</sup> ∂ϕðV<sup>Þ</sup>

∂X � �

� �

ð37Þ

dX ð38Þ

<sup>f</sup>ϕðXÞg<sup>T</sup> f dX <sup>þ</sup> Qi <sup>ð</sup>39<sup>Þ</sup>

0 0

0 0

�6ðEI þ 0:25l

ðEI þ 0:25l

ðEI þ 0:25l

ðEI þ 0:25l

Þ ð13ρAL<sup>e</sup>

<sup>Þ</sup> �ρAL<sup>e</sup>

Þ ð11ρAL<sup>e</sup>

Þ ð <sup>ρ</sup>AL<sup>e</sup>

Le 

Le

 � <sup>ρ</sup><sup>I</sup> Þ

 <sup>þ</sup> <sup>ρ</sup><sup>I</sup> Þ

> ρILe Þ

> > ð40bÞ

 þ

 � <sup>ρ</sup>ILe 

Le 

Le

 μAÞ

 μAÞ

 μAÞ

 μAÞ

ð40aÞ

 μAÞ

 μAÞ

> μAÞ

 μAÞ

0 0

0 0

ρI Le

Le 

Le 

Le 

Le 

ρAL<sup>e</sup> � <sup>6</sup>ρ<sup>I</sup> Le

�13ρAL<sup>e</sup> <sup>þ</sup> <sup>ρ</sup><sup>I</sup>

ρAL<sup>e</sup> þ

ρAL<sup>e</sup> <sup>þ</sup> <sup>ρ</sup><sup>I</sup>

<sup>6</sup>ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup>

<sup>6</sup>ðEI <sup>þ</sup> <sup>0</sup>:25<sup>l</sup>

In the numerical examples, the effects of the material length scale parameter and geometry parameters on the static, vibration, and buckling responses of the nanobeam are presented in both the classical beam theory (CBT) and MCST. Using the conventional assembly procedure for the finite elements, the system stiffness, mass, stability matrices, and the load vector are obtained from the element stiffness, mass, stability matrices, and load vectors. After that, the solution process outlined in the preceding section is used to obtain the solution for the problem of concern. In obtaining the numerical results, graphs and solution of the nonlinear finite element model, MATLAB program is used. The nanobeam is taken to be made of epoxy (E ¼ 1,44 GPa, <sup>ν</sup> <sup>¼</sup> <sup>0</sup>:38, l <sup>¼</sup> <sup>17</sup>:<sup>6</sup> <sup>μ</sup>m, <sup>ρ</sup> <sup>¼</sup> <sup>1600</sup> kg m3). In the numerical calculations, the number of finite elements is taken as 100. In the numerical integrations, five-point Gauss integration rule is used.

In order to establish the accuracy of the present formulation and the computer program developed by the author, the results obtained from the present study are compared with the available results in the literature. For this purpose, the static deflections shapes of a simple supported beam with rectangular cross section, which is subjected to a point load, are calculated for MCST and compared with those of Alashti and Abolghasemi [41] by inserting the material and load properties used in this reference. It is clearly seen that the curves of Figure 3 of the present study are very close to those of Alashti and Abolghasemi [41].

Figure 3. Comparison study: static deflections shape of the simple supported beam based on the MCST.

Figure 4a–c shows the effects of the thickness (h) on the static, vibration, and buckling responses of the nanobeam, respectively, in both the CBT and MCST. In these figures, static deflections, fundamental frequencies, and critical buckling loads are calculated and plotted for different values of the h for b ¼ l and L ¼ 30 l. In the calculation of Figure 3a, the nanobeam is subjected to a transversal point load (P ¼ 100 μN) at the midpoint of the beam in the transverse direction.

Figure 4. Effect of the thickness on the static, vibration, and buckling responses of the nanobeam for CBT and MCST; (a) static analysis, (b) free vibration analysis, and (c) buckling analysis.

As seen from Figure 4, the difference between the results of the MCST and CBT decreases significantly with the increase in the thickness of the nanobeam. Increase in the thickness of the nanobeam leads to a decline on effects of size effect and difference between the results of MCST and CBT.

In order to see the effect of material length scale parameter (l) on the static, vibration and buckling of the nanobeam, static deflections, fundamental frequencies, and critical buckling loads are displayed with different value of the dimensionless material length scale parameters (l/h) in both the CBT and MCST in Figure 5 for b ¼ 1 μm and L ¼ 30 μm. In this figure, for different values of the dimensionless material length scale parameters (l/h), the material length scale parameter (l) is varied when the thickness of the nanobeam (h) is kept constant as 1 μm.

It is seen from Figure 5 that with an increase in the dimensionless material length scale parameter l/h leads to a rise on the difference between the results of the MCST and CBT. Also, the dimensionless material length scale parameter has no effect on the mechanical responses for the classical theory, which is unable to capture the size effect. It is found that the deflections of the nanobeam by the CBT are always larger than those by the MCST. However, fundamental frequencies and critical buckling loads by the MCST are always larger than those by the CBT. It can be seen from figures that the difference between the CBT and MCST is very large when the h and l/h increase. The material parameter and dimension of the nanobeam have a very important role on the mechanical behavior of nanobeams.

Figure 5. Effect of the dimensionless material length scale parameter (l/h) on the static, vibration, and buckling responses of the nanobeam for CBT and MCST; (a) static analysis, (b) free vibration analysis, and (c) buckling analysis.

#### 4. Conclusions

Figure 4a–c shows the effects of the thickness (h) on the static, vibration, and buckling responses of the nanobeam, respectively, in both the CBT and MCST. In these figures, static deflections, fundamental frequencies, and critical buckling loads are calculated and plotted for different values of the h for b ¼ l and L ¼ 30 l. In the calculation of Figure 3a, the nanobeam is subjected to a transversal point load (P ¼ 100 μN) at the midpoint of the beam in the transverse

As seen from Figure 4, the difference between the results of the MCST and CBT decreases significantly with the increase in the thickness of the nanobeam. Increase in the thickness of the nanobeam leads to a decline on effects of size effect and difference between the results of

Figure 4. Effect of the thickness on the static, vibration, and buckling responses of the nanobeam for CBT and MCST; (a)

static analysis, (b) free vibration analysis, and (c) buckling analysis.

In order to see the effect of material length scale parameter (l) on the static, vibration and buckling of the nanobeam, static deflections, fundamental frequencies, and critical buckling loads are displayed with different value of the dimensionless material length scale parameters (l/h) in both the CBT and MCST in Figure 5 for b ¼ 1 μm and L ¼ 30 μm. In this figure, for different values of the dimensionless material length scale parameters (l/h), the material length scale parameter (l) is varied when the thickness of the nanobeam (h) is kept constant as 1 μm. It is seen from Figure 5 that with an increase in the dimensionless material length scale parameter l/h leads to a rise on the difference between the results of the MCST and CBT. Also, the dimensionless material length scale parameter has no effect on the mechanical responses for the classical theory, which is unable to capture the size effect. It is found that the deflections

direction.

132 Nanomechanics

MCST and CBT.

Finite element solution of nanobeams is investigated based on modified couple stress theory within the Euler-Bernoulli beam theory. The finite element formulations are derived for static, free vibration, and buckling problems of nanobeams. The effect of the material length scale parameter and geometry parameters on the static, vibration, and buckling responses of the nanobeam is presented and discussed in the numerical study. Also, the difference between the classical beam theory and modified couple stress theory is investigated.

It is observed from the investigations that the material length scale parameter and dimension of the nanobeam have a big influence on the static, free vibration, and buckling behaviors of nanobeams. With the increase in the thickness of the nanobeam (h) and decrease in the dimensionless material length scale parameter (l/h), the difference between the classical beam theory and modified couple stress theory decrease considerably. It is observed from the results that modified couple stress theory must be used instead of the classical beam theory for small values of nanobeam height and high values of the material length scale parameter.

## Author details

Şeref Doğuşcan Akbaş

Address all correspondence to: serefda@yahoo.com

Department of Civil Engineering, Bursa Technical University, Bursa, Turkey

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Author details

134 Nanomechanics

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Address all correspondence to: serefda@yahoo.com

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## **Green Synthesis of Metallic and Carbon Nanostructures**

R. Britto Hurtado, G. Calderon-Ayala, M. Cortez-Valadez, L.P. Ramírez-Rodríguez and M. Flores-Acosta

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68483

#### **Abstract**

The technological and biomedical applications of low toxicity and eco-friendly organic compounds are nowadays increasingly attracting the attention of researchers in nanoscience, who are aiming for more biocompatible and nanostructured systems for their application in antineoplastic therapies. This study presents the significance of "green components" in the production of graphene, metallic, and semiconductor nanoparticles, due to their antioxidant and antitumor properties. The formation of nanostructures is caused during green synthesis methods by organic molecules or carboxylic acid groups present in some plant extracts; for this reason, we include here a recapitulation and analysis of the role of carboxylic acids in those systems (organic). Furthermore, we propose the use of the extract from *Opuntia ficus-indica* cladodes to obtain metallic and carbon nanostructures, as an alternative biosynthesis method for the development of future nanobiotechnological applications.

**Keywords:** green synthesis, nanoparticles synthesis, carbon nanostructures, carboxylic acid

#### **1. Introduction**

Several nanoparticle synthesis methods are applied nowadays in different scientific fields; furthermore, based on the type of process (physical, chemical, or biological) and the conditions the synthesis is undertaken, they allow a control of the shape of the material, thus managing their application more accurately [1, 2]. With the objective of adopting eco-friendly methods that help to reduce the pollution caused by some toxic compounds and to exploit the local natural resources, the biosynthesis of nanostructured materials (green methods)

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

has been undertaken through antioxidant microorganisms and agents obtained from local plant extracts [3–5]. Thus, the use of green synthesis in nanotechnology is fundamental in scientific research in nanoscience, mainly, in order to find medical applications of the nanostructures that reduce toxicity risks while being biocompatible [6–8]. There is evidence of nanoparticles obtained through size-tunable biosynthesis [9, 10] and different technological and biocompatible applications. For instance, Ag nanoparticles show antibacterial and antitumor properties and have potential use in antineoplastic treatments [11–14]; furthermore, they are applicable in SERS (surface-enhanced Raman spectroscopy) [15]. Gold nanoparticles between 20 and 25 nm present catalytic activity [16], and ZnO nanoparticles with a range between 9.6 and 25.5 nm present antibacterial and photocatalytic applications [17]. Pooja et al. synthesized biocompatible gold nanoparticles using karaya gum, which can be used in the elaboration of antineoplastic medications [18]. Biocompatible silver nanoparticles were obtained from the extract of *Rosa damascen* petals, which have anticarcinogenic properties against pulmonary adenocarcinoma [19]. Patra et al. synthesized gold and silver nanoparticles with *Butea monosperma* leaves; this nanoparticle system inhibits the growth of cancer cells, and these authors consider that the synthesis of these nanoparticles is important in biomedicine for the development of cancer treatments [20]. The biosynthesis of nanostructured systems thus appears to be a valuable tool for nano-biotechnological applications in nanoscience around the world.

Similarly, the green synthesis is used in the graphene and bimetallic nanoalloys obtained. Coconut water and pomegranate juice were reported as reducing and capping agents of graphite oxide (OG) to obtained graphene [21, 22]. Other authors have obtained Au-Ag bimetallic nanoparticles with pomegranate juice [23] and extract from leaf of mahogany [24], as well as the bioreduction synthesis in bimetallic nanostructure-type core/shell of Ti/Ni between 1 and 4 nm employing *Medicago sativa* [25]. Sheny and collaborators used plant extract of *Anacardium occidentale* for the formation of bimetallic nanoparticles of Au-Ag, considering that polyols play an important role in the reduction of metal ions [26].


**Table 1** lists a variety of plants that have been recently used for the synthesis of metallic, bimetallic, and semiconductor nanoparticles.


has been undertaken through antioxidant microorganisms and agents obtained from local plant extracts [3–5]. Thus, the use of green synthesis in nanotechnology is fundamental in scientific research in nanoscience, mainly, in order to find medical applications of the nanostructures that reduce toxicity risks while being biocompatible [6–8]. There is evidence of nanoparticles obtained through size-tunable biosynthesis [9, 10] and different technological and biocompatible applications. For instance, Ag nanoparticles show antibacterial and antitumor properties and have potential use in antineoplastic treatments [11–14]; furthermore, they are applicable in SERS (surface-enhanced Raman spectroscopy) [15]. Gold nanoparticles between 20 and 25 nm present catalytic activity [16], and ZnO nanoparticles with a range between 9.6 and 25.5 nm present antibacterial and photocatalytic applications [17]. Pooja et al. synthesized biocompatible gold nanoparticles using karaya gum, which can be used in the elaboration of antineoplastic medications [18]. Biocompatible silver nanoparticles were obtained from the extract of *Rosa damascen* petals, which have anticarcinogenic properties against pulmonary adenocarcinoma [19]. Patra et al. synthesized gold and silver nanoparticles with *Butea monosperma* leaves; this nanoparticle system inhibits the growth of cancer cells, and these authors consider that the synthesis of these nanoparticles is important in biomedicine for the development of cancer treatments [20]. The biosynthesis of nanostructured systems thus appears to be a valuable tool for nano-biotechnological applications in

Similarly, the green synthesis is used in the graphene and bimetallic nanoalloys obtained. Coconut water and pomegranate juice were reported as reducing and capping agents of graphite oxide (OG) to obtained graphene [21, 22]. Other authors have obtained Au-Ag bimetallic nanoparticles with pomegranate juice [23] and extract from leaf of mahogany [24], as well as the bioreduction synthesis in bimetallic nanostructure-type core/shell of Ti/Ni between 1 and 4 nm employing *Medicago sativa* [25]. Sheny and collaborators used plant extract of *Anacardium occidentale* for the formation of bimetallic nanoparticles of Au-Ag, considering

**Table 1** lists a variety of plants that have been recently used for the synthesis of metallic, bime-

*Artocarpus heterophyllus Lam*. 10.78 nm [28] *Vigna sp. L* 24.35 nm [29] *Hydrangea paniculata* 36–75 nm [30] *Andrographis paniculata* 13–27 nm [31] *Ficus religiosa* 5–35 nm [32] *Alternanthera sessilis Linn* 20–30 nm [33] *Lycium barbarum* 3–15 nm [34] *Osmanthus fragrans* 2–30 nm [35] *Sambucus nigra* 26 nm [36]

that polyols play an important role in the reduction of metal ions [26].

**Nanoparticle Plant extract Size Reference** Ag *Atrocarpus altilis* 34 nm; 38 nm [27]

nanoscience around the world.

140 Nanomechanics

tallic, and semiconductor nanoparticles.

**Table 1.** Plant extracts used in the synthesis of metallic, bimetallic, and semiconductor nanoparticles.

#### **2. The role of carboxylic acids in nanoparticles**

Nano-biosynthesis is classified as a chemical method that promotes the growth of a system by the aggregation of the metallic atoms reduced from the atoms in precursor solutions. Literature reports show that certain organic compounds are responsible for oxidation-reduction (redox) reactions, which trigger the formation and stabilization of nanoparticles. For instance, in the cases of the biometallic alloys Au, Ag, and Au/Ag, it is observed that polyphenols and polyoles that carry antioxidant properties present in plants have an important role in the formation of nanostructures [24, 72–74]. Quercetin (a flavonol with high antioxidant activity [75]) also appears in the reports as a main component in the formation of metallic nanostructures [76]. It is thus important to highlight the role that the antioxidant activity of the substances used in biosynthesis has on the fabrication of nanoparticles. Similarly, literature reports that carboxylic acids are commonly used in biological methods as reducing and sometimes stabilizing agents in the production and application of these materials. Yoosaf et al. show that it is possible to stabilize nanoparticles through electrostatic interactions with carboxylic groups (using gallic acid), which adhere to the surface of the nanoparticles [77]. This argument is supported by Amornkitbamrung et al. who attribute the reduction of Pd+2→Pd0 to the functionality of the carboxylate ion (R-COO-) [78]. On the other hand, Hosseini-M et al. address that carboxylic acids are crucial in the morphology, size, and distribution of Fe3 O4 nanoparticles; furthermore, they present a co-catalyst effect [79]. Au nanoparticles were synthesized with dicarboxylic acids (oxalic, malonic, succinic, glutaric, and adipic) as reducing agents of HAuCl4 , without the presence of any other surfactant agents, the synthesis resulted in different morphologies and SERS applications [80]. Similarly, other reports reiterate the importance of the carboxylic groups in the formation of nanoparticles [81, 82]. On nonmetals, Dwivedi et al. obtained selenium nanoparticles of 40–100 nm using carboxylic acids (acetic, oxalic, and gallic acids) for the synthesis method [83]. Propionic acid is used as a stabilizing agent in the fabrication of ZnO quantum dots (3.6–5.2 nm) [84]; similarly, carboxylic acids were used in manganese oxide nanoparticles, which work as catalysts in the conversion of CO to CO2 [85]; additionally, pimelic dicarboxylic acid is used as a nucleating agent for the synthesis of TiO2 nanoparticles [86]. Thus, the carboxylic acid groups stick to nanoparticles transferring stability (**Figure 1**), as reported by Zhi-Mei Qi et al. who synthesized gold nanoparticles through infrared (IR) spectroscopy [81].

Carboxylic acids are the most common type of organic acids in the carboxylic group (made by the fusion of one hydroxyl and carboxyl group) at the extreme end of the carbon chain. Under certain conditions, proton donors transfer H+ hydrons through heterolysis. The general formula of the carboxylic acid group is R-COOH, where R is a monovalent functional group (one hydrogen or carbon chain), when the carbon structure is replaced by t

wo functional carboxylic groups, the acid is dicarboxylic acid (HOOC-R-COOH). When the proton H+ is transferred to the remaining ion, the formula changes into R-COO- carboxylate [87]. These acids are used in the food and pharmaceutical industries and in the manufacture of detergent and surfactant agents, among other applications [88]. Recent reports have demonstrated that when COOH groups are applied to certain biological complexes, they present excellent antitumor and antioxidant activity [89]; furthermore, other reports indicate that the carboxylic acids in *Rhinacanthus nasutus* show antiviral activity [90].

**Figure 1.** Representation of the electrostatic interaction of the carboxylic acid group on the nanoparticles.

**2. The role of carboxylic acids in nanoparticles**

Amornkitbamrung et al. who attribute the reduction of Pd+2→Pd0

(oxalic, malonic, succinic, glutaric, and adipic) as reducing agents of HAuCl4

acids are crucial in the morphology, size, and distribution of Fe3

which work as catalysts in the conversion of CO to CO2

Under certain conditions, proton donors transfer H+

proton H+

142 Nanomechanics

acid is used as a nucleating agent for the synthesis of TiO2

Nano-biosynthesis is classified as a chemical method that promotes the growth of a system by the aggregation of the metallic atoms reduced from the atoms in precursor solutions. Literature reports show that certain organic compounds are responsible for oxidation-reduction (redox) reactions, which trigger the formation and stabilization of nanoparticles. For instance, in the cases of the biometallic alloys Au, Ag, and Au/Ag, it is observed that polyphenols and polyoles that carry antioxidant properties present in plants have an important role in the formation of nanostructures [24, 72–74]. Quercetin (a flavonol with high antioxidant activity [75]) also appears in the reports as a main component in the formation of metallic nanostructures [76]. It is thus important to highlight the role that the antioxidant activity of the substances used in biosynthesis has on the fabrication of nanoparticles. Similarly, literature reports that carboxylic acids are commonly used in biological methods as reducing and sometimes stabilizing agents in the production and application of these materials. Yoosaf et al. show that it is possible to stabilize nanoparticles through electrostatic interactions with carboxylic groups (using gallic acid), which adhere to the surface of the nanoparticles [77]. This argument is supported by

carboxylate ion (R-COO-) [78]. On the other hand, Hosseini-M et al. address that carboxylic

they present a co-catalyst effect [79]. Au nanoparticles were synthesized with dicarboxylic acids

ence of any other surfactant agents, the synthesis resulted in different morphologies and SERS applications [80]. Similarly, other reports reiterate the importance of the carboxylic groups in the formation of nanoparticles [81, 82]. On nonmetals, Dwivedi et al. obtained selenium nanoparticles of 40–100 nm using carboxylic acids (acetic, oxalic, and gallic acids) for the synthesis method [83]. Propionic acid is used as a stabilizing agent in the fabrication of ZnO quantum dots (3.6–5.2 nm) [84]; similarly, carboxylic acids were used in manganese oxide nanoparticles,

ylic acid groups stick to nanoparticles transferring stability (**Figure 1**), as reported by Zhi-Mei

Carboxylic acids are the most common type of organic acids in the carboxylic group (made by the fusion of one hydroxyl and carboxyl group) at the extreme end of the carbon chain.

formula of the carboxylic acid group is R-COOH, where R is a monovalent functional group

wo functional carboxylic groups, the acid is dicarboxylic acid (HOOC-R-COOH). When the

[87]. These acids are used in the food and pharmaceutical industries and in the manufacture of detergent and surfactant agents, among other applications [88]. Recent reports have demonstrated that when COOH groups are applied to certain biological complexes, they present excellent antitumor and antioxidant activity [89]; furthermore, other reports indicate that the

is transferred to the remaining ion, the formula changes into R-COO- carboxylate

Qi et al. who synthesized gold nanoparticles through infrared (IR) spectroscopy [81].

(one hydrogen or carbon chain), when the carbon structure is replaced by t

carboxylic acids in *Rhinacanthus nasutus* show antiviral activity [90].

to the functionality of the

nanoparticles; furthermore,

[85]; additionally, pimelic dicarboxylic

hydrons through heterolysis. The general

nanoparticles [86]. Thus, the carbox-

, without the pres-

O4

The use of these compounds in the synthesis processes of metallic and nonmetallic nanoparticles is increasing due to the biocompatible and antioxidant properties of carboxylic acids.

**Table 2** shows scientific reports on carboxylic acids used in biosynthesis for the production of nanomaterials; the table focuses on the different applications of the acids. In regard to the reduction-oxidation process of metallic ions, Zoya Zaheer and Rafiuddin [91] propose a reduction



**Table 2.** Influence of some carboxylic and dicarboxylic acids on the synthesis of nanoparticles.

mechanism of Ag+ by oxalic acid (HOOC-COO-) with CTAB as the stabilizing agent; the reduction mechanism takes place in an aqueous solution with a pH control of pK = 1.2 and pK2 = 4.2.

Since the organic compounds of carboxylic acids can be found in nature (for instance, in plants such as fungi) and they are not harmful for human consumption or the environment, they are used in the food and pharmaceutical industries. Studies on the synthesis of metallic and semiconductor nanoparticles are beginning to be used in these fields in order to improve processes that are beneficial to the environment and human beings. It is important to mention that some carboxylic acids functionalized with nanoparticles are also used in technological applications (**Scheme 1**).

A recent study shows a simpler unsaturated carboxylic acid (acrylic acid) that works with silver nanoparticles in the process of membrane filtration that avoids nanoparticles to adhere to the surface of the modified membranes, which at the same time show antibacterial/bacteriostatic properties [109]. The main reaction of acrylic acid is polymerization; thus, it is commonly used in the production of plastics, paints, and adhesives. Due to these characteristics, Ag nanoparticles modified with polyacrylic acid was produced by the redox method showing excellent water solubility, stability, and biocompatibility, as well as antibacterial properties against *Escherichia coli, Staphylococcus aureus*, and *Pseudomonas aeruginosa* [110]. Nanoparticles and formic acid have a similar role, since the electrochemical oxidation process of formic acid is an efficient energy supplier in direct formic acid fuel cells (DFAFCs) for mobile and portable applications [111–113]. The biodegradability and low cost of formic acid makes it a valuable resource in energy storage, in spite of its toxicity [114–117]. Thus, metallic and bimetallic nanoparticles Pt [118], Pt-Cu [119], Pt-Au [120, 121], Pd-Ag [122], and so on are used as catalysts in the electro-oxidation of formic acid.

Reports also show oxalic acid functionalized with biocompatible magnetite nanobars (oxalic acid-Fe3 O4 ) prepared through the co-precipitation method for applications in electrochemical

**Scheme 1.** Some nanobiotechnological applications of carboxylic acids functionalized with nanoparticles.

mechanism of Ag+

Acetic acid CuS

144 Nanomechanics

Lauric acid Fe3

Glutaric acid Ag

CuInSe2

O4 Ag

NiO-Ni

5 400

9.4 ± 2.3 8

Trifluoroacetic acid TiO2 5 Trifluoroacetic acid—used as an electron

Benzoic acid ZnO 5–50 ZnO precursors—obtained by the intimate

**Table 2.** Influence of some carboxylic and dicarboxylic acids on the synthesis of nanoparticles.

30–50 22–41

by oxalic acid (HOOC-COO-) with CTAB as the stabilizing agent; the reduc-

tion mechanism takes place in an aqueous solution with a pH control of pK = 1.2 and pK2 = 4.2. Since the organic compounds of carboxylic acids can be found in nature (for instance, in plants such as fungi) and they are not harmful for human consumption or the environment,

**Carboxylic acid NPs Size (nm) Function report. used as: Reference**

silver nanoparticles.

protecting agent.

The CuInSe2

mineralizer.

lauric acid. Capping agents.

Decanoic acid ZnO 100 Zinc oxide (ZnO) nanoparticles were examined

extract is used as reducing agent and malic acid is used as capping agent of the bioreduced

CuS nanoparticles using carboxylic acid (acetic, propionic) as a solvent have been developed

cells were synthesized using acetic acid as a

The nanostructured material was coated with

scavenger and a morphological control agent.

mixing of zinc acetate dihydrate and carboxylic

The morphologies of silver nanoparticles are

Nanocomposite has been fabricated via the thermal decomposition of nickel salts by using

acids as capping agents.

impacted by glutaric acid.

glutaric acid as a spacer agent.

using surface modifiers (oleic acid and decanoic acid) in supercritical methanol.

nanoparticles for thin-film solar

[98]

[99]

[100] [101]

[102]

[103] [104]

[105]

[106]

[107] [108]

Malic acid Ag ~10 Citric acid present in *S. lycopersicums* fruit

Gallic acid Au@Pt ~50 nm Gallic acid (GA) as both a reducing and a

biosensors [123]. Additionally, oxalic acid and malic acid play an important role in the synthesis of tungstite nanoplates and nanoflowers (hydrated tungsten oxide: WO<sup>3</sup> ⋅H2 O) which show photocatalytic properties [124, 125]. Sedira et al. show that Ag nanoparticles oxidize easily in aqueous solutions, which in combination with acetic acid cause the liberation of Ag+ and improve the bactericidal effect in a higher range than with zinc oxide quantum dots [126]. Similarly, benzoic acid in the surface of TiO2 nanorods increases the power conversion efficiency of dye-sensitized solar cells (DSSCs), becoming at the same time an alternative and efficient method for the production of electrodes based on TiO<sup>2</sup> nanorods [127]. It has been shown recently that the synthesis of Se nanoparticles induced by carboxylic acids (acetic acid, pyruvic acid, and benzoic acid) has antitumor properties with good potential in the treatment of Dalton's lymphoma (DLA) cancer cells [128]. On the other hand, caffeic acid (containing the functional groups phenolic and acrylic) is used as a reducing and stabilizing agent in the preparation of silver nanoparticles, which show antitumor properties and works as an alternative agent in human hepatoma therapies [129]. Reports show that green synthesis of gold nanoparticles obtained with chlorogenic acid presents anti-inflammatory properties; additionally, it has promising applications in nanomedicine [130]. Maleic acid (dicarboxylic) functionalized with gold nanoparticles is used in the colorimetric detection of high efficiency of lead [131]; correspondingly, copper nanoparticles functionalized with carboxylic acid act as catalysts, when reducing 2-nitrophenol to 2-aminophenol in a few minutes [132]. Hence, it can be concluded that the different applications of metallic and nonmetallic nanoparticles in combination with organic agents such as carboxylic groups are of great importance in future medical applications for the development of antineoplastic therapies.

## **3. Biosynthesis of nanoparticles with** *Opuntia ficus-indica*

The cladodes from *O. ficus-indica* are characterized by their antioxidant properties, vitamin content, and by the presence of flavonoids and gallic acids [133–137], in addition to their content of uronic acid, a type of sugar acid with carbonyl and carboxylic functional groups [138]. In recent studies, gallic acid is used as a reducing and stabilizing agent in the mass production of silver nanoparticles with antioxidant properties and low cytotoxicity for normal cells [139, 140]. Similarly, biocompatible gold nanoparticles are synthesized at environment temperature through the reduction of HAuCl4 with gallic acid and poly-(N-vinyl-2-pyrrolidone) (PVP) [141]. Thus, carboxylic organic agents (gallic and uronic acids) in the cladode extract are responsible for the reduction and stabilization of the nanoparticles (**Figure 2**).

*O. ficus-indica* is well known for its anti-diarrheal, anti-inflammatory, antiviral, and anticarcinogen properties [142, 143], as well as for being used in treatments for diabetes and indigestion. The plant is commonly used as a nutritional complement, and its fruits and cladodes can be consumed in salads [144–146]. In general, this plant contains vitamins, minerals, and sugars, indispensable for human health. A recent study by E. Ramirez M et al. reports that *O. ficus-indica* cladodes improve the physicochemical properties of corn tortillas [147]; additionally, it increases the antioxidant activity in the blood and plasma of humans [148]. Furthermore, the extract has better anti-inflammatory potential than the drug indometacin [149]. Similarly, metallic and nonmetallic nanoparticles obtained with ecological methods are pioneering in the same fields as the extract *O. ficus-indica* (**Scheme 2**).

**Figure 2.** Carboxylic acids in *O. ficus-indica* play an important role in the formation of nanoparticles.

**Scheme 2.** Nanoparticles and cladodes from *O. ficus-indica* have similar range of technological applications.

For instance, silver nanoparticles obtained from the extracts of *R. indica* and *European black elderberry* show anti-inflammatory properties [150, 151]. Gold nanoparticles from *Inonotus obliquus* show antioxidant activity [152]. Synthesized ZnO nanoparticles from the root of *Polygala tenuifolia* show antioxidant and anti-inflammatory activity [153]. In the same way, Au and Ag nanoparticles obtained by biosynthesis are powerful nanomaterials in the control of diabetes [154–156]. Thus far, literature only reports in vitro studies with antibacterial activity using *O. ficus-indica* extract, applied in the biofabrication of silver nanoparticles, considering the synergic affects against *E. coli* and *S. aureus* [133]. As a consequence, the organic agents such as carboxylic acids, in the cladode extract of *O. ficus-indica*, and the nanoparticles obtained through eco-friendly methods, can improve the nanobiotechnological applications; furthermore, they secure innovative developments of the several application fields and the modern technology. It is still necessary to develop studies in order to validate the hypotheses presented in this study, concerning the nanoparticles obtained from the extracts from the cladodes.

### **4. Experimental section**

biosensors [123]. Additionally, oxalic acid and malic acid play an important role in the syn-

show photocatalytic properties [124, 125]. Sedira et al. show that Ag nanoparticles oxidize easily in aqueous solutions, which in combination with acetic acid cause the liberation of

efficiency of dye-sensitized solar cells (DSSCs), becoming at the same time an alternative and

shown recently that the synthesis of Se nanoparticles induced by carboxylic acids (acetic acid, pyruvic acid, and benzoic acid) has antitumor properties with good potential in the treatment of Dalton's lymphoma (DLA) cancer cells [128]. On the other hand, caffeic acid (containing the functional groups phenolic and acrylic) is used as a reducing and stabilizing agent in the preparation of silver nanoparticles, which show antitumor properties and works as an alternative agent in human hepatoma therapies [129]. Reports show that green synthesis of gold nanoparticles obtained with chlorogenic acid presents anti-inflammatory properties; additionally, it has promising applications in nanomedicine [130]. Maleic acid (dicarboxylic) functionalized with gold nanoparticles is used in the colorimetric detection of high efficiency of lead [131]; correspondingly, copper nanoparticles functionalized with carboxylic acid act as catalysts, when reducing 2-nitrophenol to 2-aminophenol in a few minutes [132]. Hence, it can be concluded that the different applications of metallic and nonmetallic nanoparticles in combination with organic agents such as carboxylic groups are of great importance in future

and improve the bactericidal effect in a higher range than with zinc oxide quantum dots

⋅H2

nanorods [127]. It has been

nanorods increases the power conversion

O) which

thesis of tungstite nanoplates and nanoflowers (hydrated tungsten oxide: WO<sup>3</sup>

[126]. Similarly, benzoic acid in the surface of TiO2

efficient method for the production of electrodes based on TiO<sup>2</sup>

medical applications for the development of antineoplastic therapies.

for the reduction and stabilization of the nanoparticles (**Figure 2**).

through the reduction of HAuCl4

**3. Biosynthesis of nanoparticles with** *Opuntia ficus-indica*

The cladodes from *O. ficus-indica* are characterized by their antioxidant properties, vitamin content, and by the presence of flavonoids and gallic acids [133–137], in addition to their content of uronic acid, a type of sugar acid with carbonyl and carboxylic functional groups [138]. In recent studies, gallic acid is used as a reducing and stabilizing agent in the mass production of silver nanoparticles with antioxidant properties and low cytotoxicity for normal cells [139, 140]. Similarly, biocompatible gold nanoparticles are synthesized at environment temperature

Thus, carboxylic organic agents (gallic and uronic acids) in the cladode extract are responsible

*O. ficus-indica* is well known for its anti-diarrheal, anti-inflammatory, antiviral, and anticarcinogen properties [142, 143], as well as for being used in treatments for diabetes and indigestion. The plant is commonly used as a nutritional complement, and its fruits and cladodes can be consumed in salads [144–146]. In general, this plant contains vitamins, minerals, and sugars, indispensable for human health. A recent study by E. Ramirez M et al. reports that *O. ficus-indica* cladodes improve the physicochemical properties of corn tortillas [147]; additionally, it increases the antioxidant activity in the blood and plasma of humans [148]. Furthermore, the extract has better anti-inflammatory potential than the drug indometacin [149]. Similarly, metallic and nonmetallic nanoparticles obtained with ecological methods are pioneering in the same fields as the extract *O. ficus-indica* (**Scheme 2**).

with gallic acid and poly-(N-vinyl-2-pyrrolidone) (PVP) [141].

Ag+

146 Nanomechanics

#### **4.1. Metallic nanoparticles**

In this study, we synthesized metallic nanoparticles Ag, Au, Cd, Cu, Pb, and Ti with cladodes from *O. ficus-indica* in a colloidal medium. The synthesis presented excellent stability during long periods of time. The following were used as precursors during the synthesis processes: Nitrates AgNO3 ; Cu(NO3 )2 ; Pb(NO3 )2 ; Cd(NO3 ) 2 ; Chlorides: HAuCl<sup>4</sup> . Small fragments of metal underwent a thermal treatment in nitric acid for the synthesis of Ti nanoparticles (**Figure 2**). The method used in this study was made in collaboration with other authors [157]: 25 g of the cladode was mixed in 50 ml of deionized water; subsequently, the solution underwent thermal treatment at a constant temperature of 60°C, and magnetic agitation for 1 h. The resulting solution is then filtered obtaining the *O. ficus-indica* extract. Three milliliters of the extract is mixed with 25 ml of the precursor solutions (nitrate and chlorides) for the reduction of the metallic ions. The solution undergoes the same thermal treatment and magnetic agitation described above. The nanoparticles are formed and stabilized during these processes.

#### **4.2. Carbon nanostructures**

The organic molecules containing the *O. ficus-indica* extract may have hydrophilic properties. This represents of the extract of the plant a strong candidate for the obtaining of laminar materials of carbon and small quantum dots both in colloidal means.

To obtain a few graphene sheets from commercial graphite, 5 ml of the extract of *O. ficus-indica* was used as mentioned in the previous section. Subsequently, 2 g of commercial graphite was added to 50 ml of deionized water and mixed with 5-ml extract of *O. ficus-indica*. The mixture was kept in an ultrasonic bath for 30 min. Finally, floating material was collected on the liquid surface with a slightly bright hue, to be analyzed by transmission electron microscopy (TEM), Raman, and X-ray photoelectron spectroscopy (XPS). To obtain carbon quantum dots (CQDs), a mixture with the same components was maintained under magnetic stirring and at 50°C for 30 min and then subjected to an ultrasonic bath for 30 min. In the first minutes of subjecting the sample in the ultrasonic bath, a tone change in the surface of the liquid is observed. In the same way, floating material was collected on the surface of the liquid, finding small CQD.

### **5. Results and discussions**

Nanoparticles of the precursors mentioned in the experimental section were obtained and characterized by transmission electron microscopy. In the case of the silver nanoparticles, these mainly presented particle sizes that oscillate between 2 and 4 nm, and few cases are observed with sizes between 10 and 15 nm (**Figure 3a**). In both cases, morphologies of spherical type are observed. For the case of the precursor HAuCl4 after performing the synthesis process mentioned in the experimental section, gold nanoparticles with different morphologies such as triangular, pentagonal, hexagonal, and quasi-spherical were obtained. The mentioned macroscopic parameters allowed a diversity of morphologies for this precursor as seen in **Figure 3b**. In the case of cadmium nanoparticles (**Figure 3c**), an irregular shape with sizes located between 2 and 8 nm approximately was obtained.

In some cases, the organic molecules contained in the plant extract are manifested by interacting with the surface of the nanoparticle, possibly as a consequence of molecular affinity. As is the case with copper nanoparticles (**Figure 3d**), in these, a region between 2 and 3 nm in thickness is observed that surrounds a nanoparticle with a diameter of approximately 10 nm. Another similar case was presented when synthesizing titanium nanoparticles; in the TEM image (**Figure 3f**), we observed clusters of nanoparticles were stabilized by an organic

The method used in this study was made in collaboration with other authors [157]: 25 g of the cladode was mixed in 50 ml of deionized water; subsequently, the solution underwent thermal treatment at a constant temperature of 60°C, and magnetic agitation for 1 h. The resulting solution is then filtered obtaining the *O. ficus-indica* extract. Three milliliters of the extract is mixed with 25 ml of the precursor solutions (nitrate and chlorides) for the reduction of the metallic ions. The solution undergoes the same thermal treatment and magnetic agitation

described above. The nanoparticles are formed and stabilized during these processes.

materials of carbon and small quantum dots both in colloidal means.

spherical type are observed. For the case of the precursor HAuCl4

sizes located between 2 and 8 nm approximately was obtained.

The organic molecules containing the *O. ficus-indica* extract may have hydrophilic properties. This represents of the extract of the plant a strong candidate for the obtaining of laminar

To obtain a few graphene sheets from commercial graphite, 5 ml of the extract of *O. ficus-indica* was used as mentioned in the previous section. Subsequently, 2 g of commercial graphite was added to 50 ml of deionized water and mixed with 5-ml extract of *O. ficus-indica*. The mixture was kept in an ultrasonic bath for 30 min. Finally, floating material was collected on the liquid surface with a slightly bright hue, to be analyzed by transmission electron microscopy (TEM), Raman, and X-ray photoelectron spectroscopy (XPS). To obtain carbon quantum dots (CQDs), a mixture with the same components was maintained under magnetic stirring and at 50°C for 30 min and then subjected to an ultrasonic bath for 30 min. In the first minutes of subjecting the sample in the ultrasonic bath, a tone change in the surface of the liquid is observed. In the same way, floating material was collected on the surface of the liquid, finding small CQD.

Nanoparticles of the precursors mentioned in the experimental section were obtained and characterized by transmission electron microscopy. In the case of the silver nanoparticles, these mainly presented particle sizes that oscillate between 2 and 4 nm, and few cases are observed with sizes between 10 and 15 nm (**Figure 3a**). In both cases, morphologies of

thesis process mentioned in the experimental section, gold nanoparticles with different morphologies such as triangular, pentagonal, hexagonal, and quasi-spherical were obtained. The mentioned macroscopic parameters allowed a diversity of morphologies for this precursor as seen in **Figure 3b**. In the case of cadmium nanoparticles (**Figure 3c**), an irregular shape with

In some cases, the organic molecules contained in the plant extract are manifested by interacting with the surface of the nanoparticle, possibly as a consequence of molecular affinity. As is the case with copper nanoparticles (**Figure 3d**), in these, a region between 2 and 3 nm in thickness is observed that surrounds a nanoparticle with a diameter of approximately 10 nm. Another similar case was presented when synthesizing titanium nanoparticles; in the TEM image (**Figure 3f**), we observed clusters of nanoparticles were stabilized by an organic

after performing the syn-

**4.2. Carbon nanostructures**

148 Nanomechanics

**5. Results and discussions**

**Figure 3.** TEM images of metallic nanoparticles and carbon nanostructures synthesized with cladode extract from *O. ficus-indica*. a) Ag Nps, b) Au Nps, c) Cd Nps, d) Cu Nps, e) Pn Nps, f) Ti Nps, g) Graphene, h) Graphene, i) Carbon Quantum dots.

medium. The nanoparticles of titanium have sizes located at approximately 5 nm; we assume that this stabilizing medium may contain ascorbic acid, starches, proteins, and various vitamins naturally contained in the extract of the plant *O. ficus-indica*.

On the other hand, when using PbNO3 as a precursor of lead nanoparticles by the synthesis method presented, nanoparticles below 10 nm were obtained with well-defined crystalline phase as seen in **Figure 3e**. For the case of the synthesized metallic nanoparticles, we observed that the extract of *O. ficus-indica* facilitates the obtaining for a size smaller than 10 nm. This has several advantages for analyzing biomedical applications such as drug delivery, therapeutic applications, bioimaging, and magnetic energy storage [158–160].

For the laminar carbon nanostructures obtained by green synthesis methods, there are currently published results that start from graphite oxide as a precursor [161, 162]. In the present investigation, we use commercial graphite as a precursor, further reducing the costs of synthesis for the nanostructured laminar final product. **Figure 3g** and h show graphene layers made up of less than 10 layers. These were obtained by the simple ultrasonic sonication method shown in the experimental section. We assume that the method presented can be made repeatedly until a smaller number of graphene sheets are obtained because the hydrophilic components in the extract of *O. ficus-indica* favor the exfoliation.

The use of green synthesis to obtain CQD is rarely documented. Few numbers of articles show evidence of the synthesis of CQD using plant extract [163, 164]. By combining the *O. ficusindica* plant extract with a small amount of commercial graphite and maintaining the mixture at 60° C for 1 h, it was possible to collect the surface liquid from the solution for further analysis by TEM and to find CQD (**Figure 3i**). In the same way as the metallic nanoparticles, the extract allowed to obtain CQD with a size smaller than 5 nm. This favors the applications of chemical sensors, photodetectors, and so on [165, 166].

## **6. Optical properties in metallic nanoparticles**

As is well known, the metallic and semiconductor nanoparticles have new optical properties in relation to the bulk material. These properties can be detected by ultraviolet/visible spectroscopy (UV/Vis) and are associated with the existence of surface plasmon. Surface plasmon resonance (SPR) physically represents the oscillation of free electrons on the surface of the nanoparticles, constituting a characteristic fingerprint of each nanostructured material. The nanoparticles obtained using the extract of *O. ficus-indica* were analyzed experimentally by UV/Vis spectroscopy. All spectra were considered from 200 to 800 nm.

In the case of silver nanoparticles, these showed an absorption band centered on 390-nm characteristic of the SPR due to quantum confinement in silver, as seen in **Figure 4a**. The dependence of the location of the SPR is associated with the morphology of the nanostructures as well as the size. Silver nanostructures can be found in the literature with an SPR located at 408, 430, 440, and so on [167–169], associated with different sizes of silver nanoparticles.

The gold nanoparticles probably represent the most studied metal nanostructured in terms of behavior and shifts of SPR. The gold nanoparticles have absorption bands located at 500, 550, 800, and so on, for nanoparticles with different morphologies. This indicates the dependence and sensitivity of the location of the SPR with the morphology of the gold nanostructures [170–172]. The gold nanoparticles obtained in this work show a large amplitude absorption band associated with the presence of the SPR centered at 55 nm. We assume that this band implicitly considers the contribution of several SPRs associated with the different morphologies obtained, as shown in **Figure 4b**. This can be seen in the large amplitude of the absorption band, with a range from 500 to 670 nm. For cadmium nanoparticles, an absorption band centered at 232 nm is shown in **Figure 4c**. For lead, there are reports of nanocubes with absorption bands located at approximately 320 and 400 nm [173]. A band detected at 300 nm was associated with SPR due to the confinement in these lead nanoparticles (**Figure 4e**).

The obtained copper nanoparticles had a well-defined absorption band centered at 580 nm approximately as shown in **Figure 4d**. The synthesis of copper nanoparticles it faces to copper oxidation easily in a colloidal medium, the extract of *O. ficus-indica* facilitates the stabilization of these nanoparticles. Prabsash et al. obtained nanoparticles of copper with a size of 10 nm, using chemical reduction by the reducing agent sodium borohydride [174].

On the other hand, titanium (metal) nanoparticles are difficult to find in the literature. There are reports from Mohammadi and Halali who used the electromagnetic levitation melting gas

Green Synthesis of Metallic and Carbon Nanostructures http://dx.doi.org/10.5772/intechopen.68483 151

**Figure 4.** Optical absorbance of metallic nanoparticles synthesized by *O. ficus-indica extract*.

The use of green synthesis to obtain CQD is rarely documented. Few numbers of articles show evidence of the synthesis of CQD using plant extract [163, 164]. By combining the *O. ficusindica* plant extract with a small amount of commercial graphite and maintaining the mixture at 60° C for 1 h, it was possible to collect the surface liquid from the solution for further analysis by TEM and to find CQD (**Figure 3i**). In the same way as the metallic nanoparticles, the extract allowed to obtain CQD with a size smaller than 5 nm. This favors the applications

As is well known, the metallic and semiconductor nanoparticles have new optical properties in relation to the bulk material. These properties can be detected by ultraviolet/visible spectroscopy (UV/Vis) and are associated with the existence of surface plasmon. Surface plasmon resonance (SPR) physically represents the oscillation of free electrons on the surface of the nanoparticles, constituting a characteristic fingerprint of each nanostructured material. The nanoparticles obtained using the extract of *O. ficus-indica* were analyzed experimentally by

In the case of silver nanoparticles, these showed an absorption band centered on 390-nm characteristic of the SPR due to quantum confinement in silver, as seen in **Figure 4a**. The dependence of the location of the SPR is associated with the morphology of the nanostructures as well as the size. Silver nanostructures can be found in the literature with an SPR located at 408, 430, 440, and so on [167–169], associated with different sizes of silver nanoparticles.

The gold nanoparticles probably represent the most studied metal nanostructured in terms of behavior and shifts of SPR. The gold nanoparticles have absorption bands located at 500, 550, 800, and so on, for nanoparticles with different morphologies. This indicates the dependence and sensitivity of the location of the SPR with the morphology of the gold nanostructures [170–172]. The gold nanoparticles obtained in this work show a large amplitude absorption band associated with the presence of the SPR centered at 55 nm. We assume that this band implicitly considers the contribution of several SPRs associated with the different morphologies obtained, as shown in **Figure 4b**. This can be seen in the large amplitude of the absorption band, with a range from 500 to 670 nm. For cadmium nanoparticles, an absorption band centered at 232 nm is shown in **Figure 4c**. For lead, there are reports of nanocubes with absorption bands located at approximately 320 and 400 nm [173]. A band detected at 300 nm was

associated with SPR due to the confinement in these lead nanoparticles (**Figure 4e**).

using chemical reduction by the reducing agent sodium borohydride [174].

The obtained copper nanoparticles had a well-defined absorption band centered at 580 nm approximately as shown in **Figure 4d**. The synthesis of copper nanoparticles it faces to copper oxidation easily in a colloidal medium, the extract of *O. ficus-indica* facilitates the stabilization of these nanoparticles. Prabsash et al. obtained nanoparticles of copper with a size of 10 nm,

On the other hand, titanium (metal) nanoparticles are difficult to find in the literature. There are reports from Mohammadi and Halali who used the electromagnetic levitation melting gas

of chemical sensors, photodetectors, and so on [165, 166].

150 Nanomechanics

**6. Optical properties in metallic nanoparticles**

UV/Vis spectroscopy. All spectra were considered from 200 to 800 nm.

method to evaporate titanium particles [175]; the nanoparticles obtained by them have sizes between 28 and 40 nm. Although the method presented is effective, it represents a requirement of special equipment to carry out the synthesis of titanium nanoparticles. The titanium nanoparticles obtained in the present work were based on the modified method presented by R. Britto et al. [176], varying slightly the amount of extract of the plant *O. ficus-indica*. The absorption band obtained for titanium nanoparticles (**Figure 4f**) was located at 350 nm, associated with the SPR of this metal.

## **7. Conclusions**

In conclusion, the current importance of organic agents functionalized with nanoparticles in the nanotechnological and biomedical fields has been exposed in this study. The properties of carboxylic acids make the fabrication of biocompatible nanostructured systems attractive for future attention. The biosynthesis process used in the fabrication of graphene layers and nanoparticles initiates an ecological and low-cost alternative in biocompatible applications for the treatment of diseases, mainly for antineoplastic therapies. The cladode extract from *O. ficus-indica* is highly efficient in the formation of nanoparticles, which can play an important role in the biocompatibility for the benefit of health and nutrition. Metallic and nonmetallic nanoparticles with nanobiotechnological applications synthesized with ecological methods (carboxylic groups) will be a fundamental tool for biocompatible applications in nanoscience and nanotechnology.

## **Author details**

Ricardo Britto Hurtado<sup>1</sup> , Gerardo Calderon-Ayala1 , Manuel Cortez-Valadez2 \*, Luis Patricio Ramírez-Rodríguez3 and Mario Flores-Acosta1

\*Address all correspondence to: manuelcortez@live.com

1 Departamento de Investigación en Física, Universidad de Sonora, Apdo, Hermosillo, Sonora, Mexico

2 CONACYT—Departamento de Investigación en Física, Universidad de Sonora, Apdo, Hermosillo, Sonora, Mexico

3 Departamento de Física, Universidad de Sonora, Apdo, Hermosillo, Sonora, Mexico

## **References**


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**7. Conclusions**

152 Nanomechanics

and nanotechnology.

**Author details**

Ricardo Britto Hurtado<sup>1</sup>

Hermosillo, Sonora, Mexico

Ramírez-Rodríguez3

Mexico

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In conclusion, the current importance of organic agents functionalized with nanoparticles in the nanotechnological and biomedical fields has been exposed in this study. The properties of carboxylic acids make the fabrication of biocompatible nanostructured systems attractive for future attention. The biosynthesis process used in the fabrication of graphene layers and nanoparticles initiates an ecological and low-cost alternative in biocompatible applications for the treatment of diseases, mainly for antineoplastic therapies. The cladode extract from *O. ficus-indica* is highly efficient in the formation of nanoparticles, which can play an important role in the biocompatibility for the benefit of health and nutrition. Metallic and nonmetallic nanoparticles with nanobiotechnological applications synthesized with ecological methods (carboxylic groups) will be a fundamental tool for biocompatible applications in nanoscience

, Gerardo Calderon-Ayala1

1 Departamento de Investigación en Física, Universidad de Sonora, Apdo, Hermosillo, Sonora,

2 CONACYT—Departamento de Investigación en Física, Universidad de Sonora, Apdo,

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164 Nanomechanics

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## **Perspective of the Structuration Process Use in the Optoelectronics, Solar Energy, and Biomedicine**

Natalia Vladimirovna Kamanina

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/68123

#### **Abstract**

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166 Nanomechanics

In the current chapter, the results of the optimized organic and inorganic materials features have been presented and discussed under the conditions when the material volumetric body and their surfaces have been structured. The dramatic change of the main characteristics of the inorganic matrix which surface has been modified with oriented carbon nanotubes and additionally treated by surface electromagnetic waves has been established. The transmittance and reflection spectral change, of the micro-hardness and of the wetting angle increase have been discussed due to the covalent bonding of the carbon nanotubes with the near-surface materials layers. Analytic and molecular dynamics simulations have supported the data. The essential change of the basic macro-parameters of the organic matrix, including the liquid crystal one, via their structuration with the nano- and/or bio-objects such as the fullerenes, carbon nanotubes, shungites, quantum dots, graphene oxides, DNA has been found. The spectral, photorefractive, and photoconductive characteristics modification has been discussed due to the drastic increase of the dipole moment. The laser-induced change of the refractive index has been considered as the indicator of the basic materials macro-parameters changing. It has been predicted that a scientific knowledge and the technology advances can be useful for the solar energy, display technique, for the system to absorb the gas and impurities, for the schemes with the compacted information recording as well as for the biomedicine.

**Keywords:** structuration process, carbon nanotubes, surfaces, coatings, conducting quasi-graphene layers, inorganic and organic materials, spectra, mechanical properties, refractive features, laser-mater interaction

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1. Introduction**

Careful investigations of the structural, refractive, conductive, and wetting properties of the innovative hybrid materials modified with effective nano- and bio-objects are timely and quite popular [1–6] because of the relevance study of new physical and chemical effects in them. Structured inorganic and organic materials, including liquid crystal ones, are able to optimize the technological processes for nano- and bio-technologies, as well as to extend the possible their application for the purposes of telecommunications, display, laser, biomedical engineering, industrial production, etc.

It should be said that the study of the structural and optical effects in the materials has a special place in the development of modern innovative technologies. This is due to the substantial increase in applications last, as the photon energy lies in the range of electronic and vibrational transitions in matter, and therefore, this circumstance allows the use of light and more particularly laser radiation, to obtain unique information about structural and dynamic refractive properties of the materials. Moreover, it permits to manifest new and classic confirmation of the effects in them and to use laser sources for modeling properties of the investigated matrix systems.

This chapter partially summarizes the prospects of the process of surface structuring (e.g., some inorganic materials) and the volume (e.g., some organic systems) in order to create and advocate new composites with unique improved their physical and chemical properties.

#### **1.1. Features of structuration of the surface of inorganic model materials**

Covalent bonding of the CNTs, as an indicator of enlightenment, the strengthening of materials, changing their resistance and laser strength.

Among other perspective nanoparticles and nanostructures, the carbon nanotubes (CNTs) occupy the special place. The basic features of carbon nanotubes are regarded to their branched surface, high conductivity, strong hardness of their C–C bonds, little value of the refractive index as well as to their complicated, and unique mechanisms of charge carrier moving [7–10]. These characteristics permit to apply the CNTs materials with good advantage in order to improve the physical and chemical properties of the inorganic systems especially when the CNTs have been deposited on the material surfaces.

It should be mentioned that to modify the properties of the inorganic materials via their surface treatment, the SWCNTs type #704121 with the diameter placed in the range of 0.7–1.1 nm purchased from Aldrich Co. has been used. These nanotubes have been deposited on the material surface using IR CO2 -laser with *p*-polarized irradiation at wavelength of 10.6 μm and power of 30 W. Moreover, CNTs have been placed at the materials interface under the conditions when an additional electric field of 100–600 V cm−1 has been applied in order to orient the nanotubes in the vertical position during the deposition process. This procedure has been partially shown in the RU patents [11, 12]. The spectra of the nanotubes-treated materials have been obtained using Perkin-Elmer Lambda 9 and Furrier FSM-1202 instruments as well as using VIS SF-26 device operated in the range of 250–1200 nm. Surface mechanical hardness has been revealed using the CM-55 instrument as well as via using of the micro-hardness device PMT-3M (produced by "LOMO", Saint-Petersburg, Russia) with the ability to vary an indenter forces. The laser strength has been checked with pulsed nanosecond Nd-laser. The special accent has been given to observe the relief at the material surface via checking the wetting angle. In this case, the camera with parameters as Compact F1.6 1/3 CS Mount 6.0–60 mm Manual Focal Iris Zoom Lens for CCTV Camera (Black) has been applied. The modified surface analysis has been made using Solver Next AFM (purchased from NT MDT Co., Zelenograd, Moscow region, Russia).

#### **1.2. ITO conducting layer: spectral parameters, wetting phenomenon, laser and mechanical characteristics**

**1. Introduction**

168 Nanomechanics

tigated matrix systems.

material surface using IR CO2

engineering, industrial production, etc.

als, changing their resistance and laser strength.

when the CNTs have been deposited on the material surfaces.

Careful investigations of the structural, refractive, conductive, and wetting properties of the innovative hybrid materials modified with effective nano- and bio-objects are timely and quite popular [1–6] because of the relevance study of new physical and chemical effects in them. Structured inorganic and organic materials, including liquid crystal ones, are able to optimize the technological processes for nano- and bio-technologies, as well as to extend the possible their application for the purposes of telecommunications, display, laser, biomedical

It should be said that the study of the structural and optical effects in the materials has a special place in the development of modern innovative technologies. This is due to the substantial increase in applications last, as the photon energy lies in the range of electronic and vibrational transitions in matter, and therefore, this circumstance allows the use of light and more particularly laser radiation, to obtain unique information about structural and dynamic refractive properties of the materials. Moreover, it permits to manifest new and classic confirmation of the effects in them and to use laser sources for modeling properties of the inves-

This chapter partially summarizes the prospects of the process of surface structuring (e.g., some inorganic materials) and the volume (e.g., some organic systems) in order to create and advocate new composites with unique improved their physical and chemical properties.

Covalent bonding of the CNTs, as an indicator of enlightenment, the strengthening of materi-

Among other perspective nanoparticles and nanostructures, the carbon nanotubes (CNTs) occupy the special place. The basic features of carbon nanotubes are regarded to their branched surface, high conductivity, strong hardness of their C–C bonds, little value of the refractive index as well as to their complicated, and unique mechanisms of charge carrier moving [7–10]. These characteristics permit to apply the CNTs materials with good advantage in order to improve the physical and chemical properties of the inorganic systems especially

It should be mentioned that to modify the properties of the inorganic materials via their surface treatment, the SWCNTs type #704121 with the diameter placed in the range of 0.7–1.1 nm purchased from Aldrich Co. has been used. These nanotubes have been deposited on the

power of 30 W. Moreover, CNTs have been placed at the materials interface under the conditions when an additional electric field of 100–600 V cm−1 has been applied in order to orient the nanotubes in the vertical position during the deposition process. This procedure has been partially shown in the RU patents [11, 12]. The spectra of the nanotubes-treated materials have been obtained using Perkin-Elmer Lambda 9 and Furrier FSM-1202 instruments as well as using VIS SF-26 device operated in the range of 250–1200 nm. Surface mechanical hardness


**1.1. Features of structuration of the surface of inorganic model materials**

It should be mentioned that an emphasis has been given on the ITO-conducting layers modification due to the reason that these structures have so broad area of the application in the general telecommunication systems, solar energy area, and biomedicine as well as in display, gas storage and laser technique [13–20]. For example, if one of the optoelectronic key element, such as electrically or light-addressed spatial light modulators (SLMs) can be considered, it should be said that besides of the solution to obtain the high resistivity and good speed, some other important problem should be resolved. It connected with the possible increase of the transparency, mechanical and laser strength of the SLM elements as well. Thus, nanostructuration process permits to find new ways to resolve the problem mentioned above with good advantage. In this case, the influence of the surface treatment procedure on the materials basic characteristics can be taken into account.

The data showed in **Figures 1** and **2** as well as in **Tables 1** and **2** support the spectral evidences, the mechanical properties, the laser features, the resistance values, and the wetting angles. It should be mentioned that an additional surface electromagnetic wave (SEW) treatment has

**Figure 1.** VIS (a) and near IR-range (b) transmittance spectra of the ITO-layer covered with the vertically aligned CNTs under the different electric field applied.

**Figure 2.** Possible CNTs placement on the ITO surfaces: general view (a) and the dimension one (b).

been made to modify the ITO relief in order to decrease the roughness and resistivity as well as to apply one as an orienting coating for the classical nematic liquid crystal cells [21, 22].

One can see from **Figure 1** data that the structuration of the ITO surfaces with oriented CNTs provokes the shift of the spectral parameters (a) and increase the transparency (b). The molecular dynamics method based on the LAMMPS program [23, 24] has been made to support these results. Data of this simulation have supported the bonding existing effect and testified that the one has been coincided with the diameter and speed of the CNTs under the laser-oriented deposition (LOD) conditions. The data presented in **Figure 2** show the possible bonding between carbon atoms and atomic surface layer.

It should be mentioned that this ITO surface modification reveled by LOD has provoked the dramatically decrease of the resistivity. It can be explained by the formation of the possible quasi-graphene layer due to the fact that CNTs reveal the large charge carrier moving from the core of the nanotubes and show the donor acceptor properties too. **Figure 3a** presented


**Table 1.** Comparative values of the resistivity of the pure ITO and of the transparent conducting layers under the nanostructure process treatment.


**Table 2.** Comparative values of the mechanical hardness and laser strength of the pure ITO and of the transparent conducting layers under the nanostructure process treatment.

the tentative model picture to form the quasi-graphene layers on the ITO surfaces structured by LOD. As an additional, the AFM-image is shown in **Figure 3b** in order to present the real structured optimized relief.

been made to modify the ITO relief in order to decrease the roughness and resistivity as well as to apply one as an orienting coating for the classical nematic liquid crystal cells [21, 22].

**Figure 2.** Possible CNTs placement on the ITO surfaces: general view (a) and the dimension one (b).

**Resistivity before treatment, Ohm Resistivity after treatment, Ohm**

One can see from **Figure 1** data that the structuration of the ITO surfaces with oriented CNTs provokes the shift of the spectral parameters (a) and increase the transparency (b). The molecular dynamics method based on the LAMMPS program [23, 24] has been made to support these results. Data of this simulation have supported the bonding existing effect and testified that the one has been coincided with the diameter and speed of the CNTs under the laser-oriented deposition (LOD) conditions. The data presented in **Figure 2** show the possible bonding

It should be mentioned that this ITO surface modification reveled by LOD has provoked the dramatically decrease of the resistivity. It can be explained by the formation of the possible quasi-graphene layer due to the fact that CNTs reveal the large charge carrier moving from the core of the nanotubes and show the donor acceptor properties too. **Figure 3a** presented

**Table 1.** Comparative values of the resistivity of the pure ITO and of the transparent conducting layers under the

between carbon atoms and atomic surface layer.

170 Nanomechanics

1570 700 1400 500 600 200 390 120 195 70 155 52 150 43 100 46 100 40

nanostructure process treatment.

To observe the change of the resistivity, one can see **Table 1** data. Thus, the resistivity parameters can be change up to 2–3 times. It can be important for the future design of the telecommunication systems, such as light-addressed and electrically addressed SLMs, electrically controlled display and holographic elements, etc., because it should provoke the essential decrease of the applied voltage. Indeed, the data presented in **Table 1** have been obtained for the initial different pure ITO coatings with different their thickness and treated at different annealing temperatures.

Among these advantages, the mechanical hardness and laser strength improvement have been shown as well. The data of **Table 2** support this fact. It should be testified that the mechanical hardness and laser strength of the ITO-conducting layer can be successfully increased via CNTs treatment. It should be mentioned that traditionally the HfO2 coatings have been

**Figure 3.** Possible model presentation to form the quasi-graphene layers on the ITO surfaces coated with oriented CNTs (a) and AFM-image of the structured ITO relief (b).

deposited on the ITO surface to eliminate their roughness and increase the strength, but the CNTs modification can provoke the better results via applying the LOD technique.

Some evidences to increase the wetting angle have been obtained too for the ITO relief studied. Thus, the pure ITO coatings additionally treated with the surface electromagnetic waves have revealed the wetting angle close to 70–75°. After nanostructuration process using LOD technique, the wetting angle has been increased and placed in the range of 85–89°.

Analyzing these evidences, it should be noticed that the ITO-relief obtained can be sufficiently used especially in the display and SLM technique due to the fact that the modified relief at the ITO surfaces can be considered as the conducting layer with the decreased resistivity as well as the orienting one to align the LC dipoles with good advantage. Moreover, these structured ITO can be prospectively using in the solar energy area and in the biomedicine as well as in the general optoelectronics too especially due to their increased transparency.

#### **1.3. MgF2 and relative structures: spectral parameters, wetting phenomenon, and mechanical hardness**

Using the mentioned above designed laser technologies based on the oriented deposition of the carbon nanotubes [11, 12] on the surface of optical materials including as well MgF<sup>2</sup> , CaF<sup>2</sup> , BaF<sup>2</sup> , LiF, NaCl, KBr, KCl, etc. materials operated in the UV and in the IR spectral range, the study of the spectrum of transmittance, micro-hardness, changing the ability to water absorption, etc. is important. Results for the obtained transmittance and the reflection spectra of the group of randomly selected samples are shown in **Figure 4**. One can observe the essential change of the spectral characteristics for the 200–500 nm and for the near-IR spectral range. Moreover, for the BaF<sup>2</sup> structures, the reflection spectrum change has been obtained up to 16 micrometers.

Analyzing these data, it is evident that the process of laser deposition of CNTs on the surface of the studied materials changes the spectral parameters, leading to a substantial increase of transmittance and decrease of Fresnel losses due to the covalent binding of the CNTs to the surface of the material [25, 26] with regard to the small value of the imaginary part of permittivity of CNTs and their small refractive index that can be <1.1.

Moreover, dramatically change of the micro-hardness and the wetting angle has been obtained for these systems. The comparative data before and after laser deposition of the CNTs are shown in **Table 3**. Thus, micro-hardness can be increased more than 6–10%, and the angle of wettability can be changed from 1.5 to 5 times. Of course, not only essential mechanical properties of carbon nanotubes should be considered when one analyzes these data, but also the formation of predicted covalent bonds between carbon nanotubes and the surface atoms of the investigated materials. It should be mentioned that the analytical calculations presented in the paper [26] for the model MgF<sup>2</sup> structure have shown that in order to destroy the structured material surface we should firstly to bend the vertically oriented nanotubes with energy of 1.8 × 10−20 J, which is in five times larger than the one, which can be applied to simply remove the nanotubes from the surface after

Perspective of the Structuration Process Use in the Optoelectronics, Solar Energy, and Biomedicine http://dx.doi.org/10.5772/68123 173

deposited on the ITO surface to eliminate their roughness and increase the strength, but the

Some evidences to increase the wetting angle have been obtained too for the ITO relief studied. Thus, the pure ITO coatings additionally treated with the surface electromagnetic waves have revealed the wetting angle close to 70–75°. After nanostructuration process using LOD

Analyzing these evidences, it should be noticed that the ITO-relief obtained can be sufficiently used especially in the display and SLM technique due to the fact that the modified relief at the ITO surfaces can be considered as the conducting layer with the decreased resistivity as well as the orienting one to align the LC dipoles with good advantage. Moreover, these structured ITO can be prospectively using in the solar energy area and in the biomedicine as well as in the general optoelectronics too especially due to their increased

 **and relative structures: spectral parameters, wetting phenomenon, and** 

Using the mentioned above designed laser technologies based on the oriented deposition of

Analyzing these data, it is evident that the process of laser deposition of CNTs on the surface of the studied materials changes the spectral parameters, leading to a substantial increase of transmittance and decrease of Fresnel losses due to the covalent binding of the CNTs to the surface of the material [25, 26] with regard to the small value of the imaginary part of permit-

Moreover, dramatically change of the micro-hardness and the wetting angle has been obtained for these systems. The comparative data before and after laser deposition of the CNTs are shown in **Table 3**. Thus, micro-hardness can be increased more than 6–10%, and the angle of wettability can be changed from 1.5 to 5 times. Of course, not only essential mechanical properties of carbon nanotubes should be considered when one analyzes these data, but also the formation of predicted covalent bonds between carbon nanotubes and the surface atoms of the investigated materials. It should be mentioned that the analytical

in order to destroy the structured material surface we should firstly to bend the vertically oriented nanotubes with energy of 1.8 × 10−20 J, which is in five times larger than the one, which can be applied to simply remove the nanotubes from the surface after

tivity of CNTs and their small refractive index that can be <1.1.

calculations presented in the paper [26] for the model MgF<sup>2</sup>

, LiF, NaCl, KBr, KCl, etc. materials operated in the UV and in the IR spectral range, the study of the spectrum of transmittance, micro-hardness, changing the ability to water absorption, etc. is important. Results for the obtained transmittance and the reflection spectra of the group of randomly selected samples are shown in **Figure 4**. One can observe the essential change of the spectral characteristics for the 200–500 nm and for the near-IR spectral range.

structures, the reflection spectrum change has been obtained up to

, CaF<sup>2</sup> ,

structure have shown that

the carbon nanotubes [11, 12] on the surface of optical materials including as well MgF<sup>2</sup>

CNTs modification can provoke the better results via applying the LOD technique.

technique, the wetting angle has been increased and placed in the range of 85–89°.

transparency.

**mechanical hardness**

Moreover, for the BaF<sup>2</sup>

16 micrometers.

**1.3. MgF2**

172 Nanomechanics

BaF<sup>2</sup>

**Figure 4.** Transmittance spectral change of the LiF and KBr (a); MgF<sup>2</sup> (b), BaF<sup>2</sup> materials (c), and reflection spectral parameters of BaF<sup>2</sup> (d). Curve 1 and curve 2 are regarded to the systems before and after nanotreatment. The sample thickness was ~2 mm.

destroying MgC bonding. This simple model can be used to explain the results of the dramatic increase of mechanical surface hardness of MgF<sup>2</sup> covered with nanotubes placed by LOD technique.


**Table 3.** Comparative values of the mechanical hardness of the UV-IR materials under the nanostructure process treatment.

Analyzing these evidences, one can propose that some hydrophilic properties of the functional UV-IR materials can be changed to reveal the hydrophobic features. That is very important in the Fourier spectrometry devices, for the microscopy area, as well as for the biomedicine. For example, the output windows of magnesium fluoride are used in the lamps for quartz treatment of medical institutions. It also can be required for the some substrate of the spatial light modulators which area of their operation should be extended to the UV and IR spectral ranges.

In addition, registered increase of the spectral characteristics, macro-hardness parameters and of the wetting phenomena too indicates the novel tendency to find unique way to improve the technical parameters of these materials under the condition of the oriented CNTs laser deposition technique for the purpose of the general optoelectronics, solar energy, and biomedicine systems.

#### **1.4. Metal and semiconductor materials: spectral parameters, wetting phenomenon, and mechanical hardness**

Some materials from the metal and semiconductor groups have been studied in order to accept the knowledge about the influence of the nanostructuration process on the basic physical parameters of this type of the materials. Partially, the data are presented in **Figure 5**. It should be noticed that in the case of the analysis of the spectra, one took into account the change of the spectrum with one-sided processing of the material and processing from two their sides.

**Figure 5.** Transmittance spectral change of the ZnS (a); ZnSe (b) materials, and reflection spectral parameters of Al (c), and Cu (d).

The micro-hardness results reveal the increase of these parameters too. Thus, for the Al, Cu, ZnS, ZnSe materials, the micro-hardness has been increased up to 7, 9, 5, 5%, respectively. It should be mentioned that the indenter with the forces of 30 and 20 g has been used to treat the metal and semiconductor materials, respectively.

The molecular dynamics method has supported the results of the spectral and the mechanical properties changing due to the incorporation of the CNTs into the materials surfaces and due to form the possible bonding between the carbon atom and the original material surface atoms of Al and ZnS materials. It should be mentioned that the penetration depth has been depended on the velocity of the CNTs and their diameters. The molecular dynamics results are shown in **Figure 6**. This analysis has been made using the procedure in the framework of the LAMMPS program [23, 24].

**Figure 6.** Dependence of the penetration on the CNTs velocity for the studied Al (a) and ZnS (b) materials under the laser-oriented deposition process using the different diameter of the CNTs.

**Figure 5.** Transmittance spectral change of the ZnS (a); ZnSe (b) materials, and reflection spectral parameters of Al (c),

Analyzing these evidences, one can propose that some hydrophilic properties of the functional UV-IR materials can be changed to reveal the hydrophobic features. That is very important in the Fourier spectrometry devices, for the microscopy area, as well as for the biomedicine. For example, the output windows of magnesium fluoride are used in the lamps for quartz treatment of medical institutions. It also can be required for the some substrate of the spatial light modulators which area of their operation should be extended to the UV and IR spectral ranges. In addition, registered increase of the spectral characteristics, macro-hardness parameters and of the wetting phenomena too indicates the novel tendency to find unique way to improve the technical parameters of these materials under the condition of the oriented CNTs laser deposition technique for the purpose of the general optoelectronics, solar energy, and biomedicine systems.

**1.4. Metal and semiconductor materials: spectral parameters, wetting phenomenon, and** 

Some materials from the metal and semiconductor groups have been studied in order to accept the knowledge about the influence of the nanostructuration process on the basic physical parameters of this type of the materials. Partially, the data are presented in **Figure 5**. It should be noticed that in the case of the analysis of the spectra, one took into account the change of the spectrum with one-sided processing of the material and processing from two

and Cu (d).

**mechanical hardness**

their sides.

174 Nanomechanics

It should be noticed that the interaction between Zn and S atoms in the penetration process of CNT into Zn-S surface was described using Tersoff many-body potential in which parameters are given in publication [27]. Interaction between Al atoms was described using embedded atom method (EAM) [28] parametrized according to Ref. [29].

To describe the interaction between CNT and Al substrate, the 12–6 potential *<sup>E</sup>* <sup>=</sup> <sup>4</sup>*ε*[(\_ *σ r*) 12 <sup>−</sup> (\_ *σ r*) 6 ] has been used. The interaction parameters were fit as: ε = 2.63 eV, σ = 1.91 Å based on interlayer distance between graphene and Al surface. To describe the interaction between CNT and Zn atoms of the substrate, the parameters for 12–6 potential were fitted as ε = 0.011 eV, σ = 1.80 Å. Interaction between C and S was taken two times smaller than for Zn-C. The interaction between carbon atoms in the CNT was described using Tersoff many-body potentials with parameters described in Ref. [23].

The molecular dynamics simulations have been carried out at constant temperature about 300 K. The total time of simulation has been of 60 ps with the time step of 1 fs. The velocities of the tubes have been varied from 100 to 600 m×s−1 according to the experiments.

Thus, using the consideration based on the ITO, MgF<sup>2</sup> and relative structures, Al, Cu, ZnS and ZnSe materials, the current study can propose the CNTs laser treatment as the innovative way to modify the important physical and chemical characteristics of these inorganic materials.

Analyzing these evidences, one can remember that Al is the material used to fabricate the aviation, aerospace, food processing and the production of tableware; copper is a material for the manufacture of wires for electrical industry; ZnS and ZnSe are the materials for the manufacture of space components and devices of technical sensors, etc. Thus, all of the mentioned above area of the application of the materials under the consideration can be essentially exceeded via using the LOD technique with good advantage.

## **2. Features of the structuring of the volume of model polymers and liquid crystals: change of the refractive parameters as indicator of the changes of the photoconductive, spectral and structural properties**

Using four-wave mixing of laser beams (see **Figure 7**) in the nanosecond range of durations, at the wavelength of the second harmonic of a neodymium laser λ = 532 nm, with variation, as energy density or spatial frequency λ, the investigation of the change of refractive properties of a large group of conjugated organic materials by recording the diffraction efficiency in the diffraction Raman-Nath has been made. The scheme and the conditions are coincided to that considered before in the papers [14, 30]. It should be mentioned that for all materials studied, the threshold value of the input energy density at which the reversible process has been transfer to the irreversible one has been detected. Some AFM-image is shown in **Figure 8**, which corresponded to thermal replica on the material surface to indicate this fact. One can see that the spatial frequency approximately equaled to 90–100 mm−1 has been formed.

Perspective of the Structuration Process Use in the Optoelectronics, Solar Energy, and Biomedicine http://dx.doi.org/10.5772/68123 177

It should be noticed that the interaction between Zn and S atoms in the penetration process of CNT into Zn-S surface was described using Tersoff many-body potential in which parameters are given in publication [27]. Interaction between Al atoms was described using embedded

has been used. The interaction parameters were fit as: ε = 2.63 eV, σ = 1.91 Å based on interlayer distance between graphene and Al surface. To describe the interaction between CNT and Zn atoms of the substrate, the parameters for 12–6 potential were fitted as ε = 0.011 eV, σ = 1.80 Å. Interaction between C and S was taken two times smaller than for Zn-C. The interaction between carbon atoms in the CNT was described using Tersoff many-body potentials

The molecular dynamics simulations have been carried out at constant temperature about 300 K. The total time of simulation has been of 60 ps with the time step of 1 fs. The velocities

ZnSe materials, the current study can propose the CNTs laser treatment as the innovative way to modify the important physical and chemical characteristics of these inorganic materials.

Analyzing these evidences, one can remember that Al is the material used to fabricate the aviation, aerospace, food processing and the production of tableware; copper is a material for the manufacture of wires for electrical industry; ZnS and ZnSe are the materials for the manufacture of space components and devices of technical sensors, etc. Thus, all of the mentioned above area of the application of the materials under the consideration can be essen-

**2. Features of the structuring of the volume of model polymers and liquid crystals: change of the refractive parameters as indicator of the changes of** 

Using four-wave mixing of laser beams (see **Figure 7**) in the nanosecond range of durations, at the wavelength of the second harmonic of a neodymium laser λ = 532 nm, with variation, as energy density or spatial frequency λ, the investigation of the change of refractive properties of a large group of conjugated organic materials by recording the diffraction efficiency in the diffraction Raman-Nath has been made. The scheme and the conditions are coincided to that considered before in the papers [14, 30]. It should be mentioned that for all materials studied, the threshold value of the input energy density at which the reversible process has been transfer to the irreversible one has been detected. Some AFM-image is shown in **Figure 8**, which corresponded to thermal replica on the material surface to indicate this fact. One can see that the spatial frequency approximately equaled to 90–100 mm−1

of the tubes have been varied from 100 to 600 m×s−1 according to the experiments.

*σ r*) 12 <sup>−</sup> (\_ *σ r*) 6 ]

and relative structures, Al, Cu, ZnS and

To describe the interaction between CNT and Al substrate, the 12–6 potential *<sup>E</sup>* <sup>=</sup> <sup>4</sup>*ε*[(\_

atom method (EAM) [28] parametrized according to Ref. [29].

with parameters described in Ref. [23].

176 Nanomechanics

has been formed.

Thus, using the consideration based on the ITO, MgF<sup>2</sup>

tially exceeded via using the LOD technique with good advantage.

**the photoconductive, spectral and structural properties**

**Figure 7.** Experimental setup. 1—Nd-laser with the second harmonic convertor (532 nm), 2,7—filters, 3,4,8—mirrors, 5,6—photo detectors, 9—samples, 10—an additional Nd- or He-Ne-laser to read-out, 11—recorded diffractive grating.

The values of light-induced additives to the refractive index have been calculated, and the range of variation of cubic nonlinearity and nonlinear refraction has been defined. It is established that these values are comparable with those obtained for three-dimensional silicon structures. But, it should be noticed that the technology of obtaining thin films of conjugated organic materials significantly easier and their thickness is an order of magnitude less. The main results of the largest light-induced change of the refractive index Δ*n*<sup>i</sup> in a number of structured organic materials, for example, polyimide PI, 2-cyclooctylamino-5-nitropyridine (COANP) and nematic liquid crystals (NLC) with different concentrations of the sensitizers and treated at different energy densities are systematic and presented in the comparative **Table 4**.

As nano- and bio-sensitizers the fullerenes, shungites, carbon nanotubes (CNT), quantum dots (QD), oxides graphene, DNA has been used. The explanation of the change of the refractive properties of the organic conjugated systems with structuring volume of the medium is given in the framework of the significant growth of the polarizability, dipole moment, and the way of the transfer of charge carriers from intramolecular electron donor is not on the

**Figure 8.** AFM and optical images of the relief obtained on the material surfaces under the conditions when reversible process becomes the irreversible one.


**Table 4.** Refractive characteristics modification via checking of the laser-induced change of the refractive index.

intramolecular acceptor but to the intermolecular acceptor with high electron affinity and ability to delocalize of not one, but a larger number of carriers [31, 32]. Qualitative picture is shown in **Figure 9**.

Analyzing the basic experimental data presented in **Table 4**, it should be remembered that the different types of the sensitizers, their content, and energy density have been applied. Moreover, the experiments have been performed at the close spatial frequencies and at the same laser pulse width. One can see that the values of the laser-induced refractive index have been exceeded the ones obtained for the chosen pure matrix systems. It should be mentioned that the refractive features have been connected with the increase of the charge carrier mobility obtained via photoconductive experiments. The comparative analysis to check the increase of the charge carrier mobility has been made via procedure shown in the paper [33].

It should be said that the calculated values of nonlinear refraction *n*<sup>2</sup> and third-order nonlinear susceptibility χ(3) are included in a range of values: *n*<sup>2</sup>  = 10−10–10−9 cm<sup>2</sup> ×W−1 и χ(3) = 10−10– 10−9 cm<sup>3</sup> ×erg−1 [30, 31, 34–36]. The growth of refractive coefficients correlated with the change of the photoconductive (increase of mobility of charge carriers), and it is connected with the changes in the order parameter of the studied materials and with the infrared shift of their spectral characteristics. It has been registered to the photosensitivity and has been

**Figure 9.** Model to explain the formation of the intermolecular charge transfer complex between an organic matrix donor and nano- or bio-objects as an effective acceptor (*left*) and the picture of the observed diffractive grating (*right*).

calculated via checking the charge carrier mobility. Moreover, the tendency of the change of the nano-doping to bio-doping one has been discussed because the bio-doped organics have shown the same refractive parameters change [37, 38]. The correlations indicated above can be used, on the one hand, for the basis of the predicted paths to extend the use of organic conjugated structured materials for the industrial and the biomedical applications, on the other hand, for the assessing of the ability of the competing influence of bio-objects, in comparison with nano-objects, due to nontoxicity and easy renewability from the World Ocean these biological structures.

### **3. Conclusion**

intramolecular acceptor but to the intermolecular acceptor with high electron affinity and ability to delocalize of not one, but a larger number of carriers [31, 32]. Qualitative picture is

COANP **+** C60 5 0.9 6.89 × 10−3 LC **+** polyimide **+** C70 0.2 0.1 1.2 × 10−3 LC **+** COANP **+** C70 5 17.5 × 10−3 1.4 × 10−3 LC **+** COANP **+** C70 1 30 × 10−3 1.45 × 10−3 LC **+** QDs CdSe(ZnS) **+** DNA 0.1 0.1 1.35 × 10−3

**Table 4.** Refractive characteristics modification via checking of the laser-induced change of the refractive index.

5 0.9 6.21 × 10−3

**Materials Nano‐object content, wt.% Energy density, J cm−2 Δ***n***<sup>i</sup>** Pure polyimide 0 0.5–0.6 10−4–10−5 Polyimide + QDs CdSe(ZnS) 0.003 0.2–0.3 2.0 × 10−3 Polyimide + graphene oxide 0.05 0.2 2.7×10−3 Polyimide + graphene oxide 0.1 0.2 3.4 × 10−3 Polyimide + C60 0.2 0.5–0.6 4.2 × 10−3 Polyimide + C70 0.2 0.6 4.68 × 10−3 Polyimide **+** shungite 0.1 0.5 3.46 × 10−3 Polyimide **+** shungite 0.2 0.1 5.3 × 10−3 Polyimide **+** CNTs 0.1 0.5–0.6 5.7 × 10−3 Pure COANP 0 0.8–0.9 ~10−5

Analyzing the basic experimental data presented in **Table 4**, it should be remembered that the different types of the sensitizers, their content, and energy density have been applied. Moreover, the experiments have been performed at the close spatial frequencies and at the same laser pulse width. One can see that the values of the laser-induced refractive index have been exceeded the ones obtained for the chosen pure matrix systems. It should be mentioned that the refractive features have been connected with the increase of the charge carrier mobility obtained via photoconductive experiments. The comparative analysis to check the increase

×erg−1 [30, 31, 34–36]. The growth of refractive coefficients correlated with the change

of the photoconductive (increase of mobility of charge carriers), and it is connected with the changes in the order parameter of the studied materials and with the infrared shift of their spectral characteristics. It has been registered to the photosensitivity and has been

and third-order non-

×W−1 и χ(3) = 10−10–

 = 10−10–10−9 cm<sup>2</sup>

of the charge carrier mobility has been made via procedure shown in the paper [33].

It should be said that the calculated values of nonlinear refraction *n*<sup>2</sup>

linear susceptibility χ(3) are included in a range of values: *n*<sup>2</sup>

shown in **Figure 9**.

2-cyclooctylamino-5-nitropyridine

(COANP) **+** C60

178 Nanomechanics

10−9 cm<sup>3</sup>

Analyzing the obtained and discussed data, it should be concluded the followings:


#### **Acknowledgements**

The results have been obtained in the Department "Photophysics of media with nanoobjects" and have been partially supported by RFBR grant no.13-03-00044 (2013–2015) and project "Nanocoating-GOI" (2012–2015), as well as by international FP7 Program, Marie Curie Action, Project "BIOMOLEC" (2011-2015). Author wishes to thank their colleagues D.G. Kvashnin (Emanuel Institute of biochemical physics RAS, Institute of Science and Technology "MISiS") and P.B. Sorokin (Technological Institute for Superhard and Novel Carbon Materialstheir, Institute of Science and Technology "MISiS") for the help with the molecular dynamics calculations. The author appreciates their colleagues from the department for "Photophysics of media with nanoobjects," Vavilov State Optical Institute and abroad too for the helpful discussion at the seminars and conferences.

#### **Author details**

Natalia Vladimirovna Kamanina

Address all correspondence to: nvkamanina@mail.ru

1 Vavilov State Optical Institute, St. Petersburg, Russia

2 St. Petersburg Electrotechnical University ("LETI"), St. Petersburg, Russia

## **References**

**3.** Structuration of the inorganic material surfaces based on ITO via CNTs-oriented laser deposition technique can provoke to form the possible quasi-graphene layers, which can easy explain the change of the ITO resistivity due to large charge from the core of the CNTs and

**4.** Doping of the organic materials by studied nanoparticles has revealed the drastically change of the laser-induced refractive index that is larger than the ones obtained for the

**5.** Calculated refractive parameters of the nanoparticles doped organics can be compared with the analogous refractive parameters obtained for the classical inorganic bulk crystals.

**6.** The laser-induced change of the refractive index of the nano-doped organics can be compared with the analogous refractive parameters obtained for the bio-doped organics mate-

**7.** Applications of the materials, which body and interface can be modified with effective nanoparticles, can be extended essentially. The solar energy, gas storage systems, information technology, displays, and the biomedicine can accumulate the knowledge and the

The results have been obtained in the Department "Photophysics of media with nanoobjects" and have been partially supported by RFBR grant no.13-03-00044 (2013–2015) and project "Nanocoating-GOI" (2012–2015), as well as by international FP7 Program, Marie Curie Action, Project "BIOMOLEC" (2011-2015). Author wishes to thank their colleagues D.G. Kvashnin (Emanuel Institute of biochemical physics RAS, Institute of Science and Technology "MISiS") and P.B. Sorokin (Technological Institute for Superhard and Novel Carbon Materialstheir, Institute of Science and Technology "MISiS") for the help with the molecular dynamics calculations. The author appreciates their colleagues from the department for "Photophysics of media with nanoobjects," Vavilov State Optical Institute and abroad too for the helpful

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180 Nanomechanics

pure organic conjugated matrixes.

practical results.

**Acknowledgements**

**Author details**

Natalia Vladimirovna Kamanina

discussion at the seminars and conferences.

Address all correspondence to: nvkamanina@mail.ru 1 Vavilov State Optical Institute, St. Petersburg, Russia

2 St. Petersburg Electrotechnical University ("LETI"), St. Petersburg, Russia

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## *Edited by Alexander Vakhrushev*

In recent years, nanotechnology is the basis for the development of modern production. This determined the urgency of the intensive development of the new direction of mechanics and nanomechanics, for the scientific description of nanotechnological processes and the solution of several topical nanotechnology problems. Topics included in the book cover a wide range of research in the field of nanomechanics: thermomass theory of nanosystems; deformation of nanomaterials; interface mechanics of assembly carbon nanotube; nanomechanics on surface; molecular interactions and transformations; nanomechanical sensors, nanobeams, and micromembranes; nanostructural organic and inorganic materials; green synthesis of metallic nanoparticles. The main goal of these works is the establishment of the nanosystem macroparameter dependence on its nanoparameters using nanomechanics. This book will be useful for engineers, technologists, and researchers interested in methods of nanomechanics and in advanced nanomaterials with complex behavior and their applications.

Nanomechanics

Nanomechanics

*Edited by Alexander Vakhrushev*

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