**2. Methods of the local concentration and the distance measurement from EPR spectra**

It is well known from the basic theory of EPR spectroscopy that the value of magnetic dipole-dipole interaction between electron spins of paramagnetic centers depends strongly on the distance between them (Abragam, 1961, Blumenfeld et al., 1962, Altshuler & Kozirev, 1964), and this information is very important for understanding the structure and properties of such systems. There are several independent methods for measuring distances, which have been analyzed in detail in many publications, for example, in the following books: Likhtenshtein, 2008, Moebius & Savitsky, 2009, Berliner et al., 2001, Schweiger & Jeschke, 2001, Eaton & Eaton, 2004, Weil & Bolton, 2007, Tsvetkov et al., 2008, Parmon et al., 1980, Lebedev & Muromtsev, 1971.

The most complete analysis of different approaches and techniques allow measuring distances with high accuracy was presented by Berliner et al., 2001 and Eaton & Eaton, 2004. Below, for better understanding, we shortly report about some of them widely used. Complete information about these approaches one can read in works cited above. It should be mentioned that methods based on double electron-electron resonance, pulsed EPR and spin echo measurements, high frequency and high field EPR are usually used in the case of pairwise interaction between two spins distributed in an immobilized sample (Berliner et


0,0

0,4

al., 2001). As a rule, a distance between two interacting spins is much less than between different pairs, for example, in the case of nitroxide biradicals in diluted solid solutions or spin labeled proteins.

### **2.1. Dipolar interaction measured by EPR**

Fig. 3 shows typical changes in EPR spectra of R6OH dissolved in 1-butanol at three different concentrations. Broadening of CW X-band EPR spectra of nitroxide radicals in frozen solutions, due to dipolar interaction, results in changes of the whole spectrum shape which can be characterized with changes in widths of the EPR spectrum lines and in their relative intensities.

**Figure 3.** Experimental EPR spectra of R6OH dissolved in 1-butanol at 0.001 (1), 0.1 (2), and 0.4 mol/l (3) at 77 K. Asterisks \* show the 3rd and 4th lines of Mn2+ ions in MgO matrix

The line width of the separate line and the shape of whole EPR spectra depend on several various factors such as the distance between paramagnetics, type of their spatial distribution, temperature, polarity, viscosity and organization of the solvent, longitudinal relaxation time T1, etc. On practice, there are two most common types of distribution: random and pairwise. A quantitative structural characteristic for the first one is the local concertration, Cloc. The mean distance among interacting spins, <r>, can be calculated from Cloc in an assumption of the distribution type, for the simplest example: <r> = (Cloc) 1/3 for cubic regular lattice. For distribution of spins in pairs, for instance, in stable nitroxide biradicals, if they are dissolved at low concentration in solvents glazed under freezing in liquid nitrogen at 77 K, the biradical structure can be completely determined by the certain distance r between two interacting unpaired electrons localized at >NO bonds, and angles of their mutual spatial orientation (Parmon et al., 1977a, 1980). At high temperatures, if radicals are dissolved in low-viscous liquids with fast rotational and translational mobility, the dipolar coupling is averaged decreasing up to zero.

Dipole-dipole interaction between paramagnetic centers is manifested in dipolar splitting of EPR lines in the case of two coupled spins (biradicals) or in dipolar broadening of EPR lines in the case of interacting of several spins at random distribution. It was shown for the dipolar broadening H that (Abragam, 1961, Lebedev & Muromtsev, 1971):

116 Nitroxides – Theory, Experiment and Applications

**2.1. Dipolar interaction measured by EPR** 

spin labeled proteins.


0,0

0,4

intensities.

al., 2001). As a rule, a distance between two interacting spins is much less than between different pairs, for example, in the case of nitroxide biradicals in diluted solid solutions or

Fig. 3 shows typical changes in EPR spectra of R6OH dissolved in 1-butanol at three different concentrations. Broadening of CW X-band EPR spectra of nitroxide radicals in frozen solutions, due to dipolar interaction, results in changes of the whole spectrum shape which can be characterized with changes in widths of the EPR spectrum lines and in their relative

\*

**Figure 3.** Experimental EPR spectra of R6OH dissolved in 1-butanol at 0.001 (1), 0.1 (2), and 0.4 mol/l (3)

325 330 335 340

*B* , m T

The line width of the separate line and the shape of whole EPR spectra depend on several various factors such as the distance between paramagnetics, type of their spatial distribution, temperature, polarity, viscosity and organization of the solvent, longitudinal relaxation time T1, etc. On practice, there are two most common types of distribution: random and pairwise. A quantitative structural characteristic for the first one is the local concertration, Cloc. The mean distance among interacting spins, <r>, can be calculated from Cloc in an assumption of the distribution type, for the simplest example: <r> = (Cloc)

cubic regular lattice. For distribution of spins in pairs, for instance, in stable nitroxide biradicals, if they are dissolved at low concentration in solvents glazed under freezing in liquid nitrogen at 77 K, the biradical structure can be completely determined by the certain distance r between two interacting unpaired electrons localized at >NO bonds, and angles of their mutual spatial orientation (Parmon et al., 1977a, 1980). At high temperatures, if radicals are dissolved in low-viscous liquids with fast rotational and translational mobility,

1/3 for

3

2

1

at 77 K. Asterisks \* show the 3rd and 4th lines of Mn2+ ions in MgO matrix

\*

the dipolar coupling is averaged decreasing up to zero.

$$
\delta \mathsf{SH} = \Delta \mathsf{H} - \Delta \mathsf{H}\_0 = \mathsf{A} \bullet \mathsf{C} \tag{1}
$$

Here ΔH is the width of the homogeneous individual EPR spectrum line, ΔH0 is a linewidth at the absence of the dipolar coupling, C is concentration, and A is a coefficient, which depends on the character of the spatial distribution of the paramagnetic centres in the sample, the shape of the individual line, and the longitudinal relaxation time T1. Values of A for different cases of radical distributions (regular, random, in pairs, etc.) are published in (Lebedev & Muromtsev, 1971). For example, the theoretical value Atheor = 5.8•1020 G/cm3 = 34.8 G•l/mol for the width at a half-height ΔH1/2 for the Gaussian line shape (Grinberg et al., 1969). Eq.(1) is valid at not too high concentrations, such as (4/3)r03C << 1, where r0 is the characteristic size of the paramagnetic particle (a distance of the closest approach). Values of r0 calculated from the experiment were equal to 6.2 0.5 for R6OCOC6H5 in toluene, 5.8 0.6, and 6.0 0.4 for R6OH in ethanol and 50% H2O-glycerol mixture (Kokorin, 1974).

If the experimental Aexp value measured in a wide concentration interval, coincide or close to the Atheor value, it is the objective confirmation that paramagnetic species (radicals) are distributed in the whole volume of the sample. In the case of rather often real situation when paramagnetic species are localized in a certain part of the sample only (spin probes in emulsions, in lipid vesicles, spin labels attached to polymer macromolecules, etc.), the experimentally measured value of the parameter Aexp = H/Cloc characterize the magnitude of the dipolar broadening and can be significantly greater the value of Atheor. At the same time, if it is known or can be assumed that for paramagnetic centres used, the type of spatial distribution in the areas of their localization remains the same (random, regular, etc.), the real value of Aexp in these areas does not change and has to be equal to Atheor. Therefore, one can calculate Cloc values by the equation analogues to Eq. (1) (Kokorin, 1992):

Cloc = H/A (2)

Eq. (2) is valid for nitroxide radicals and paramagnetic metal ions with long T1 > 1010 s (Kokorin, 1992).

It has been experimentally shown that in accordance with Eq. 1, the dipolar broadening of the low-field "parallel" line H1/2 and of the central complex line Hpp (see Fig. 2) depends on the radical concentration linearly but with different values of ΔH0 and A for ΔH1/2 and ΔHpp (Fig. 4).

From these experimental dependences and Eq. 1, one can calculate: App = 24.9 0.5 G•l/mol, ΔH(0)pp = 10.2 0.07 G for all solvents studied; A1/2 = 40.6 0.7 G•l/mol, ΔH(0)1/2 is equal to 6.0 0.07, 7.6 0.08, 8.6 0.1 G for R6OH in frozen at 77 K toluene, ethanol, and 50% glycerol water mixture correspondingly. Very important contribution concerning correct

**Figure 4.** H1/2 (1) and Hpp (2) as a function of concentration at 77 K: R6OH in glycerol:H2O = 1:1 mixture (○) and in ethanol (), R6OCOC6H5 in toluene ()

determination of different impacts (dipole-dipole interaction, spin exchange coupling, electron spin relaxation, etc.) to the line broadening of EPR spectra was recently done by Salikhov, 2010.

#### **2.2. Second central moment measured by EPR spectra**

Any EPR spectrum can be characterized by its second central moment, which can be determined as:

$$\mathbf{M}2 = \mathbf{j}(\mathbf{H} - \mathbf{H}b)^2 \mathbf{F}(\mathbf{H})\mathbf{d}\mathbf{H}/\mathbf{F}(\mathbf{H})\mathbf{d}\mathbf{H},\tag{3}$$

where H0 is the value of the spectrum centre, F(H) gives the shape of the spectrum absorption line as a function of the magnetic field H, and ∫F(H)dH is the normalization condition of the EPR spectrum. The value of H0 one can find from the equation:

$$\int (\mathbf{H} - \mathbf{H}\mathbf{\hat{o}})\mathbf{F}(\mathbf{H})\mathbf{d}\mathbf{H} = 0\tag{4}$$

The classical theory of spin-spin interaction connected the value of M2 with the distance between interacting spins S (Van Vleck, 1948, Pryce & Stevens, 1950):

$$\mathbf{M}\mathbf{z} = (\mathbf{\hat{3}/4})\mathbf{\hat{g}}^2 \mathbf{\hat{3}} \mathbf{\hat{S}} (\mathbf{\hat{S}} + 1) \mathbf{\hat{1}} \mathbf{\hat{1}} [(1 - \mathbf{\hat{3}}\cos \theta \mathbf{\hat{}}\mu)^2 / \mathbf{\hat{n}}\mu^6] \tag{5}$$

Here rj,k is the distance between spins j and k, j,k is the angle between the line connecting these spins and the external magnetic field H, g is a g-factor, and is Bohr magneton. Later, it was shown by Lebedev, 1969, that the second central moment of the absorption EPR spectrum in the case of the random distribution of paramagnetic centres in a solid matrix depends linearly on their concentration C:

$$\mathbf{Mz} = (2\pi/15)\xi^2 \mathbf{g}^2 \mathcal{O}^2 \mathbf{C} \mathbf{C} \tag{6}$$

Here = 3/2 in the case of the equivalent spins, and C0 = r0 3 means the characteristic volume occupied by a paramagnetic molecule in the matrix. Hence, linear dependence between M2 and radical concentration C in solid solutions is evidence of a random distribution of spins in the matrix. Dipole-dipole interaction contributes to the broadening of the spectrum and its second moment M2:

118 Nitroxides – Theory, Experiment and Applications

**Figure 4.** H1/2 (1) and Hpp (2) as a function of concentration at 77 K: R6OH in glycerol:H2O = 1:1

determination of different impacts (dipole-dipole interaction, spin exchange coupling, electron spin relaxation, etc.) to the line broadening of EPR spectra was recently done by

0,00 0,05 0,10 0,15 0,20 0,25

C , m ol/l

2

1

Any EPR spectrum can be characterized by its second central moment, which can be

M2 = ∫(H H0)2F(H)dH/∫F(H)dH, (3)

where H0 is the value of the spectrum centre, F(H) gives the shape of the spectrum absorption line as a function of the magnetic field H, and ∫F(H)dH is the normalization

The classical theory of spin-spin interaction connected the value of M2 with the distance

M2 = (3/4)g22S(S + 1)•∑[(1 3cos2j,k)2/rj,k6] (5)

Here rj,k is the distance between spins j and k, j,k is the angle between the line connecting these spins and the external magnetic field H, g is a g-factor, and is Bohr magneton. Later, it was shown by Lebedev, 1969, that the second central moment of the absorption EPR spectrum in the case of the random distribution of paramagnetic centres in a solid matrix

∫(H H0)F(H)dH = 0 (4)

condition of the EPR spectrum. The value of H0 one can find from the equation:

between interacting spins S (Van Vleck, 1948, Pryce & Stevens, 1950):

depends linearly on their concentration C:

mixture (○) and in ethanol (), R6OCOC6H5 in toluene ()

**2.2. Second central moment measured by EPR spectra** 

Salikhov, 2010.

0

2

H

1/2,

H

pp, G

4

6

8

1 0

determined as:

$$\text{M2} = \text{M2(O)} + \text{B} \bullet \text{C} \tag{7}$$

where M2(0) is the M2 value at the absence of dipolar broadening, and a coefficient B is a characteristic of a certain solid matrix, for instance, in magnetically diluted frozen solutions. This dependence is illustrated well in Fig. 5:

**Figure 5.** M2 as a function of concentration at 77 K: R6OH in glycerol:H2O = 1:1 mixture (○) and in ethanol (), R6OCOC6H5 in toluene (), R5NCHCOCH2I in ethanol (), R5NCH2Br in ethanol (▼)

As in the case of dipolar broadening (section 2.1) the local spin concentrations can also be measured from the spectral second central moment M2 by Eq. (8) in the agreement with Fig. 5:

$$
\Delta \mathbf{M} \mathbf{2} = \mathbf{M} \mathbf{2} - \mathbf{M} \mathbf{2} (\mathbf{0}) = \mathbf{B} \bullet \mathbf{C}\_{\mathrm{loc}} \tag{8}
$$

The following values were calculated: B = 910 30 G2•l/mol, and M2(0) is equal to 340 ± 10, 390 ± 12 G2 for R6OH in frozen at 77 K ethanol and 50% glycerol-water mixture correspondingly; 305 ± 6 G2 for R6OCOC6H5 in toluene, 250 10 G2 for R5NCH2Br, and 270 12 G2 for R5NCHCOCH2I (in ethanol both). M2(0) is a characteristic of the solvent and the radical structure, while ΔM2 depends on the magnitude of the dipolar interaction.

Several other useful relations between various spectroscopic parameters are presented in section 3.

### **2.3. Measurements based on the relaxation time and saturation effects**

A method for estimation distances between nitroxide radicals or between spins of the spin label and paramagnetic metal center, based on the quantitative analysis of saturation curves of spin label EPR spectra recorded at 77 K, was suggested by Kulikov & Likhtenstein, 1974. This approach has been tested using haemoglobin molecule labeled by SH groups with various nitroxide radicals. Values of the distances between labels and iron ion in haem estimated from the saturation curve parameters (values of T1 1, T2 1, and T1 1•T2 1) were compared with distances measured by d1/d parameter and estimated from the X-ray data of haemoglobin (Kulikov, 1976). Here, T1 and T2 are the longitudinal, T1, and transverse, T2, relaxation times of the nitroxide electron spin. Results obtained were in reasonable agreement. In the case of rapid spin relaxation of the metal paramagnetic center, this method allows one determine rather long distances up to 2.0 nm. Serious limitation of this method is the following: for structural investigations of haem containing and other proteins, one have to know the exact value of spin relaxation time T1 of the metal paramagnetic center from independent measurements (Kulikov, 1976).

Theoretical aspects of the method based on spin relaxation of nitroxide radicals in solid matrix (frozen solutions) were considered in the first part of the review by Kulikov & Likhtenstein 1977. The distance between the spins of the radical and the other paramagnetic centre can be determined from the change of the transverse, T2, and especially from the longitudinal, T1, relaxation times of the radical due to the dipole-dipole interaction between the radical and the metal ion. The study of structure of several metal-containing proteins: haemoglobin, myosin and nitrogenase, has been carried out by the method of spin labels. The interesting approach, so called method "spin label – paramagnetic probe" has been carefully examined by spin relaxation in solutions of spin labeled proteins in the presence of inert paramagnetics, readily diffused in the solution by measuring T2 values of the label. The influence of the probes on labels T2 values depends on the frequency of collisions between a label and a probe, and therefore this approach can be used for quantitative study of the factors, which effect the frequency of collisions: microviscosity, steric hindrances, the presence of electrostatic charges (Kulikov & Likhtenstein 1977).

Later, many scientists began to work on developing various modifications of the relaxation times method for investigation structural peculiarities and conformational dynamics of biological macromolecules, proteins first of all, their aggregates and bio-membranes. One of the most serious and complete reviews in this field to my opinion was written by Eaton & Eaton, 2001a & 2001b. This review contains theoretical and experimental fundamentals of the method in solid and fluid solutions, technical details, a lot of experimental data, their deep analysis, interesting applications.

### **2.4. Double electron-electron resonance (ELDOR)**

A theoretical basis of the method is perfectly described in (Saxena & Freed, 1997). For calculating double quantum two dimensional electron spin resonance spectra in the rigid limit, that correspond to the experimental spectra obtained from a nitroxide biradical, a specific formalism has been developed. The theory includes the dipolar interaction between the nitroxide moieties as well as the fully asymmetric *g* and hyperfine tensors and the angular geometry of the biradical. The effects of arbitrary strong pulses are included by adapting the recently introduced spin-Hamiltonian theory for numerical simulations. Creation of "forbidden" coherence pathways by arbitrary pulses in magnetic resonance, and their role in ELDOR is discussed. The high sensitivity of these ELDOR signals to the strength of the dipolar interaction was demonstrated and rationalized in terms of the orientational selectivity of the "forbidden" pathways. This selectivity also provides constraints on the structural geometry (i.e., the orientations of the nitroxide moieties) of the biradicals. The theory was applied to the double quantum modulation ELDOR experiment on an end-labeled poly-proline peptide biradical. A distance of 1.85 nm between the ends is found for this biradical (Saxena & Freed, 1997).

120 Nitroxides – Theory, Experiment and Applications

**2.3. Measurements based on the relaxation time and saturation effects** 

estimated from the saturation curve parameters (values of T1

paramagnetic center from independent measurements (Kulikov, 1976).

presence of electrostatic charges (Kulikov & Likhtenstein 1977).

**2.4. Double electron-electron resonance (ELDOR)** 

deep analysis, interesting applications.

A method for estimation distances between nitroxide radicals or between spins of the spin label and paramagnetic metal center, based on the quantitative analysis of saturation curves of spin label EPR spectra recorded at 77 K, was suggested by Kulikov & Likhtenstein, 1974. This approach has been tested using haemoglobin molecule labeled by SH groups with various nitroxide radicals. Values of the distances between labels and iron ion in haem

were compared with distances measured by d1/d parameter and estimated from the X-ray data of haemoglobin (Kulikov, 1976). Here, T1 and T2 are the longitudinal, T1, and transverse, T2, relaxation times of the nitroxide electron spin. Results obtained were in reasonable agreement. In the case of rapid spin relaxation of the metal paramagnetic center, this method allows one determine rather long distances up to 2.0 nm. Serious limitation of this method is the following: for structural investigations of haem containing and other proteins, one have to know the exact value of spin relaxation time T1 of the metal

Theoretical aspects of the method based on spin relaxation of nitroxide radicals in solid matrix (frozen solutions) were considered in the first part of the review by Kulikov & Likhtenstein 1977. The distance between the spins of the radical and the other paramagnetic centre can be determined from the change of the transverse, T2, and especially from the longitudinal, T1, relaxation times of the radical due to the dipole-dipole interaction between the radical and the metal ion. The study of structure of several metal-containing proteins: haemoglobin, myosin and nitrogenase, has been carried out by the method of spin labels. The interesting approach, so called method "spin label – paramagnetic probe" has been carefully examined by spin relaxation in solutions of spin labeled proteins in the presence of inert paramagnetics, readily diffused in the solution by measuring T2 values of the label. The influence of the probes on labels T2 values depends on the frequency of collisions between a label and a probe, and therefore this approach can be used for quantitative study of the factors, which effect the frequency of collisions: microviscosity, steric hindrances, the

Later, many scientists began to work on developing various modifications of the relaxation times method for investigation structural peculiarities and conformational dynamics of biological macromolecules, proteins first of all, their aggregates and bio-membranes. One of the most serious and complete reviews in this field to my opinion was written by Eaton & Eaton, 2001a & 2001b. This review contains theoretical and experimental fundamentals of the method in solid and fluid solutions, technical details, a lot of experimental data, their

A theoretical basis of the method is perfectly described in (Saxena & Freed, 1997). For calculating double quantum two dimensional electron spin resonance spectra in the rigid

1, T2

1, and T1

1•T2 1)

> The results of development of pulse electron-electron double resonance (PELDOR) technique and its applications in structural studies were summarized and described systematically in a review by Tsvetkov et al., 2008. The foundations of the theory of the method were discribed, some experimental features and applications were considered, in particular, determination of the distances between spin labels in the nanometre range for nitroxide biradicals, spin-labeled biological macromolecules, radical-ion pairs, and peptidemembrane complexes. The authors attention was focused on radical systems arising upon self-assembly of nanosized complexes, in particular, from peptides, spatial effects, and radical pairs formation in photolysis and photosynthesis. The position of PELDOR among other structural EPR techniques was analyzed (Tsvetkov et al., 2008).

> PELDOR measures via the dipolar electron–electron coupling distances in the nanometre range, currently 1.5–8 nm, with high precision and reliability (Reginsson & Schiemann, 2011). Depending on the quality of the data, the error can be as small as 0.1 nm. Beyond measuring mean distances, PELDOR yields distance distributions, which provide access to conformational distributions and dynamics. The method was also used to count the number of monomers in a complex and allowed determination of the orientations of spin centres with respect to each other. If, in addition to the dipolar through-space coupling, a throughbond exchange coupling mechanism contributes to the overall coupling both mechanisms can be separated and quantified (Reginsson & Schiemann, 2011). This is of principle interest for researchers in many real cases.

> Interesting implication of ELDOR to polymer science has been done by Bird et al., 2008.They demonstrated on a series of spin-labeled oligomers quantitative determination the end-toend lengths and distance distributions. In case of oligomers with well-defined threedimensional structures (seven different macromolecules, each containing eight monomers) which were labeled with nitroxide radicals, the quantitative information about the shapes and flexibility of the oligomers was obtained, and end-to-end distances were calculared. The shapes of the EPR-derived population distributions allowed authors to compare the flexibility of these spiro-ladder oligomers.

Additional information concerning this very powerful technique one can find in Berliner et al., 2001, Webb, 2006, Reginsson & Schiemann, 2011.

## **2.5. High frequency/high field EPR spectroscopy**

High frequency (high field) EPR spectroscopy opened new approaches for investigating the structure, properties, dynamics, conformational transitions and functioning of many chemical and biological systems. Authors of the recent book considering this method: Möbius & Savitsky, 2009, presented the state-of-the-art capabilities and future perspectives of electron-spin triangulation by high-field/high-frequency dipolar EPR techniques designed for determining the three-dimensional structure of large supra-molecular complexes dissolved in disordered solids. These techniques combine double site-directed spin labeling with orientation-resolving PELDOR spectroscopy. In one of the last publications of this topic, the prospects of angular triangulation, which extends the more familiar distance triangulation was appraise (Savitsky et al., 2011). The three-dimensional structures of two nitroxide biradicals with rather stiff bridging blocks and deuterated nitroxide headgroups have been chosen as a model for spin-labeled proteins. The 95 GHz high-field electron dipolar EPR spectroscopy with the microwave pulse-sequence configurations for PELDOR and relaxation-induced dipolar modulation enhancement (RIDME) has been used. The approach showed good agreement with other structure-determining magnetic-resonance methods, and seems to be one of the most precise orientation-resolving EPR spin triangulation methods for protein structure determination (Savitsky et al., 2011).

To those who want to know about the high field/high frequency approach in detail, I recommend several additional books: Grinberg & Berliner, 2011, Misra, S. K., 2011, Eaton et al., 2010, Hanson & Berliner, 2010.

At the end of this section it is necessary to attract attention to two recent works. The first article considers joint analysis of EPR line shapes and 1H nuclear magnetic relaxation dispersion (NMRD) profiles of DOTA-Gd derivatives by means of the slow motion theory (Kruk et al., 2011). NMRD profiles have been extended to ESR spectral analysis, including in addition g-tensor anisotropy effects. The extended theory has been applied to interpret in a consistent way NMRD profiles and ESR spectra at 95 and 237 GHz for two Gd(III) complexes. The goal was to verify the applicability of the commonly used pseudorotational model of the transient zero field splitting, which was described by a tensor of a constant amplitude, defined in its own principal axes system. The unified interpretation of the EPR and NMRD leads to reasonable agreement with the experimental data. Seems, this approach to the electron spin dynamics can be also effectively used for quantitative description in the case of nitroxide spin probes and labels (Kruk et al., 2011).

One of very important questions in the spin label/probe method is connected with the distance distributions <r> between site-directedly attached spin labels, which obtained by measuring their dipole–dipole interaction in systems under investigation by EPR. As it was shown in (Köhler et al., 2011), the analysis of these distance distributions can be misleading particularly for broad distributions of <r>, because the most probable distance deviates from the distance between the most probable label positions. The authors studied this effect using numerically generated spin label positions, molecular dynamics simulations, and experimental data of a model systems. An approach involving Rice distributions is proposed to overcome this problem (Köhler et al., 2011).

### **3. The empirical d1/d parameter**

122 Nitroxides – Theory, Experiment and Applications

al., 2010, Hanson & Berliner, 2010.

case of nitroxide spin probes and labels (Kruk et al., 2011).

al., 2001, Webb, 2006, Reginsson & Schiemann, 2011.

**2.5. High frequency/high field EPR spectroscopy** 

Additional information concerning this very powerful technique one can find in Berliner et

High frequency (high field) EPR spectroscopy opened new approaches for investigating the structure, properties, dynamics, conformational transitions and functioning of many chemical and biological systems. Authors of the recent book considering this method: Möbius & Savitsky, 2009, presented the state-of-the-art capabilities and future perspectives of electron-spin triangulation by high-field/high-frequency dipolar EPR techniques designed for determining the three-dimensional structure of large supra-molecular complexes dissolved in disordered solids. These techniques combine double site-directed spin labeling with orientation-resolving PELDOR spectroscopy. In one of the last publications of this topic, the prospects of angular triangulation, which extends the more familiar distance triangulation was appraise (Savitsky et al., 2011). The three-dimensional structures of two nitroxide biradicals with rather stiff bridging blocks and deuterated nitroxide headgroups have been chosen as a model for spin-labeled proteins. The 95 GHz high-field electron dipolar EPR spectroscopy with the microwave pulse-sequence configurations for PELDOR and relaxation-induced dipolar modulation enhancement (RIDME) has been used. The approach showed good agreement with other structure-determining magnetic-resonance methods, and seems to be one of the most precise orientation-resolving EPR spin

triangulation methods for protein structure determination (Savitsky et al., 2011).

To those who want to know about the high field/high frequency approach in detail, I recommend several additional books: Grinberg & Berliner, 2011, Misra, S. K., 2011, Eaton et

At the end of this section it is necessary to attract attention to two recent works. The first article considers joint analysis of EPR line shapes and 1H nuclear magnetic relaxation dispersion (NMRD) profiles of DOTA-Gd derivatives by means of the slow motion theory (Kruk et al., 2011). NMRD profiles have been extended to ESR spectral analysis, including in addition g-tensor anisotropy effects. The extended theory has been applied to interpret in a consistent way NMRD profiles and ESR spectra at 95 and 237 GHz for two Gd(III) complexes. The goal was to verify the applicability of the commonly used pseudorotational model of the transient zero field splitting, which was described by a tensor of a constant amplitude, defined in its own principal axes system. The unified interpretation of the EPR and NMRD leads to reasonable agreement with the experimental data. Seems, this approach to the electron spin dynamics can be also effectively used for quantitative description in the

One of very important questions in the spin label/probe method is connected with the distance distributions <r> between site-directedly attached spin labels, which obtained by measuring their dipole–dipole interaction in systems under investigation by EPR. As it was shown in (Köhler et al., 2011), the analysis of these distance distributions can be misleading One of the most informative methods for investigating the structure, spatial organization and physical-chemical properties of complex and supramolecular systems on the microscopic, molecular level is EPR spectroscopy in its spin label/probe technique variant (Berliner, 1976; Buchachenko & Wasserman, 1976; Likhtenstein, 1976). Usually, nitroxide radicals of different structure were used for studying of structural peculiarities of spin labeled proteins and the spatial distribution of probes in them. We will discuss shortly the most important results obtained by different authors below.

With the increase of concentration, X-band EPR spectra of nitroxide radicals in frozen solutions reflect not only dipolar broadening of spectral lines and increasing of the M2 value, but also noticeable changes of the whole spectrum shape which can be characterized with the empirical "shape parameter" d1/d (Figs. 2, 3). Anisotropic "frozen" EPR spectra of nitroxide radicals at different widths of individual lines have been simulated (Parmon & Kokorin, 1976). They are shown in Fig. 6.

**Figure 6.** Simulated anisotropic EPR spectra of nitroxide radical at different widths of individual lines: 3 (1), 5 (2), 7 (3), 10 (4), and 13 G (5). The following values were used for simulations: gx = 2.0089, gy = 2.0061, gz = 2.0027; Ax = 7, Ay = 5, Az = 33 G

It is evidently seen from Figs. 3 and 6 that the line widths and the relative intensities of the center and outer lines in the spectra are changed with the increase of radical concentration. The experimental dependence of d1/d parameter, characterizing changes of the whole EPR spectrum shape of nitroxides, is not linear (Fig. 7):

**Figure 7.** Parameter as a function of nitroxide radical concentration at 77 K: R6OH in glycerol:H2O = 1:1 mixture (○) and in ethanol (), R6OCOC6H5 in toluene (), R5NCHCOCH2I in ethanol () and in toluene (▼)

Similar nonlinear dependence have been obtained from simulations of the anisotropic Xband EPR spectra of nitroxides (Parmon & Kokorin, 1976, Kolbanovsky et al., 1992a). For all nitroxide radicals frozen at 77 K in various glazed solvents studied, one can observe, as it was published by Kokorin et al., 1972, 1975, that:

$$\mathbf{q} \wedge \mathbf{q} = \mathbf{q} (\mathbf{b} \wedge \mathbf{b}) \mathbf{b} + \nabla \mathbf{q} \wedge \mathbf{b} \tag{9}$$

Here Δ is a contribution of the dipole-dipole interaction between radicals, and (d1/d)0 is a characteristic of the solvent and radical itself (Kokorin, 1974, Kokorin et al., 1975). This equation was verified by precise computer simulations of experimental EPR spectra (Kolbanovsky et al., 1992a). Results obtained for three radicals: R6OH, R6CH2CH2Br (both in ethanol), and R6H (in toluene) are given in Table 1. These data confirmed the correctness of usage the d1/d parameter for quantitative characterization the spatial distribution of spin labels, in the case of their random distribution, in magnetically diluted solid solutions. These results confirmed the possibility of applications of d1/d parameter for quantitative studies (Kolbanovsky et al., 1992a).

Several experimental correlations between spectral parameters useful for practical applications have been found for nitroxide radicals in solid solutions. They are shown in Figs. 8-10.


**Table 1.** Experimental and calculated from simulated spectra values of parameter

124 Nitroxides – Theory, Experiment and Applications

toluene (▼)

0,1

0,2

0,3

0,4

spectrum shape of nitroxides, is not linear (Fig. 7):

was published by Kokorin et al., 1972, 1975, that:

(Kolbanovsky et al., 1992a).

Figs. 8-10.

It is evidently seen from Figs. 3 and 6 that the line widths and the relative intensities of the center and outer lines in the spectra are changed with the increase of radical concentration. The experimental dependence of d1/d parameter, characterizing changes of the whole EPR

**Figure 7.** Parameter as a function of nitroxide radical concentration at 77 K: R6OH in glycerol:H2O = 1:1 mixture (○) and in ethanol (), R6OCOC6H5 in toluene (), R5NCHCOCH2I in ethanol () and in

0,00 0,05 0,10 0,15 0,20 0,25 0,0

C , m ol/l

Similar nonlinear dependence have been obtained from simulations of the anisotropic Xband EPR spectra of nitroxides (Parmon & Kokorin, 1976, Kolbanovsky et al., 1992a). For all nitroxide radicals frozen at 77 K in various glazed solvents studied, one can observe, as it

d1/d = (d1/d)0 + Δ (9)

Here Δ is a contribution of the dipole-dipole interaction between radicals, and (d1/d)0 is a characteristic of the solvent and radical itself (Kokorin, 1974, Kokorin et al., 1975). This equation was verified by precise computer simulations of experimental EPR spectra (Kolbanovsky et al., 1992a). Results obtained for three radicals: R6OH, R6CH2CH2Br (both in ethanol), and R6H (in toluene) are given in Table 1. These data confirmed the correctness of usage the d1/d parameter for quantitative characterization the spatial distribution of spin labels, in the case of their random distribution, in magnetically diluted solid solutions. These results confirmed the possibility of applications of d1/d parameter for quantitative studies

Several experimental correlations between spectral parameters useful for practical applications have been found for nitroxide radicals in solid solutions. They are shown in

**Figure 8.** (d1/d)0 as a function of hyperfine splitting constant A|| at 77 K for: R6=O (▲), R6H (), R6OH (○), R6NH2 (), and R6OCOC6H5 ()

**Figure 9.** M2(0) as a function of hyperfine splitting constant A|| (○), and of parameter (d1/d)0 () at 77 K for R6OH in glycerol:H2O = 1:1 mixture, ethanol, methanol, and R6OCOC6H5 in toluene

**Figure 10.** M2 as a function of parameter (), and parameter as a function of C1/2 at 77 K for R6OH in glycerol:H2O = 1:1 mixture (), in ethanol (), and R6OCOC6H5 in toluene (○)

The value of (d1/d)0 parameter depends on the hyperfine splitting constant A|| and on the nitroxide rihg structure (Parmon et al., 1977b). It is seen from Fig. 8 that

$$(\mathbf{d}\iota/\mathbf{d})\iota = 1.73 - 0.035 \bullet \mathbf{A} \iota \iota \tag{10}$$

These values, 1.73 0.06 and 0.035 0.002, obtained for R6H, R6OH, R6NH2, and R6OCOC6H5 dissolved in various solvents, practically coincide with values of 1.76 and 0.036 correspondingly, published earlier in Parmon et al., 1977b, 1980 and measured for R6OH radical only. For R6=O radical, these parameters are equal to 1.70 0.04 and 0.036 0.001, correspondingly.

Fig. 9 presents the value of M2(0) as a function of A|| and of (d1/d)0 parameter. This can be formalized by corresponding equations:

$$\mathbf{Mz}(0) = (25.3 \pm 0.6) \bullet \mathbf{A} \cup \dots \cup (560 \pm 25) \tag{11}$$

and

$$\text{Mn(0)} \bullet \text{(680} \pm 10) - \text{(705} \pm 17) \bullet \text{(dol/d)} \tag{12}$$

Previously (Parmon et al., 1977b, 1980), values of M2(0) = 25.5•A|| 570 has been reported.

Parameter ΔM2 also correlates with the values of parameter Δ, as it follows from Fig. 10.

$$
\Delta \mathbf{M} \mathbf{2} = (410 \pm 20) \mathbf{\hat{}} \,\mathrm{\Delta} \tag{13}
$$

It was reported (Kokorin, 1986) that parameter Δ, which is not linear on concentration, can be presented as a rather linear plot in other coordinates:

$$
\Delta = (0.89 \pm 0.05) \bullet \text{C}^{1/2} - (0.025 \pm 0.013) \tag{14}
$$

It should be stressed that this linear plot is valid in the concentration range between 0.02 and 0.4 mol/l, i.e., for 0.04 Δ 0.45.

Relations presented above were tested with different objects, and can be recommended for application in the case of random or regular distribution of spin probes in chemical and biological systems.

### **4. Pairwise interaction between two nitroxide spins**

126 Nitroxides – Theory, Experiment and Applications

correspondingly.


0,0

0,1

0,2

0,3

and

formalized by corresponding equations:

**Figure 10.** M2 as a function of parameter (), and parameter as a function of C1/2 at 77 K for R6OH

0,0 0,1 0,2 0,3 0,4 0,5

0,0 0,1 0,2 0,3 0,4 0,5

C 1/2 , (m ol/l) 1/2

The value of (d1/d)0 parameter depends on the hyperfine splitting constant A|| and on the

(d1/d)0 = 1.73 0.035•A|| (10)

These values, 1.73 0.06 and 0.035 0.002, obtained for R6H, R6OH, R6NH2, and R6OCOC6H5 dissolved in various solvents, practically coincide with values of 1.76 and 0.036 correspondingly, published earlier in Parmon et al., 1977b, 1980 and measured for R6OH radical only. For R6=O radical, these parameters are equal to 1.70 0.04 and 0.036 0.001,

Fig. 9 presents the value of M2(0) as a function of A|| and of (d1/d)0 parameter. This can be

M2(0) = (25.3 0.6)•A|| (560 25) (11)

M2(0) = (680 10) (705 17)•(d1/d)0 (12)

ΔM2 = (410 20)•Δ (13)


0

0,0

0,1

200

M2, G2

100

0,2

0,3

Previously (Parmon et al., 1977b, 1980), values of M2(0) = 25.5•A|| 570 has been reported.

Parameter ΔM2 also correlates with the values of parameter Δ, as it follows from Fig. 10.

in glycerol:H2O = 1:1 mixture (), in ethanol (), and R6OCOC6H5 in toluene (○)

nitroxide rihg structure (Parmon et al., 1977b). It is seen from Fig. 8 that

Pairwise distribution of the dipolar interacted spins, when a distance between them, r, is noticeably less than a mean distance between pairs, has been investigated in numerous works. This type of spatial distribution contains first of all nitroxide biradicals and doublelabeled proteins, peptides and oligomers. In case of short biradicals with r < 1.1 nm, when the dipolar splitting is observed in the EPR spectrum lines, the rD value, expressed in Eq. (15), can be easily determined from the spectrum simulation (Parmon et al., 1977a, 1980, Kokorin et al., 1984) or just from the experimental value of the dipolar splitting D by the equation (Kokorin et al., 1972):

$$\mathbf{r} \mathbf{o} = \mathbf{3} 0.3 \bullet \mathbf{D}\_{\perp}{}^{-1} \beta \,, \tag{15}$$

or from the relative intensity of the half-field ("forbidden", MS = 2) and normal-field (MS = 1) EPR transitions, rS (Lebedev & Muromtsev, 1972, Dubinskii et al., 1974):

$$\alpha = \text{I}\_2/\text{I}\_1 = (8/15)(\text{\textdegree g\S}/2\text{H}\text{or}\text{\textdegree}^3)^2 \approx \text{\textdegree } \text{\textdegree } \text{\textdegree } \text{\textdegree } \text{\textdegree} \tag{16}$$

where I2 and I1 are integrated intensities of EPR spectra of corresponding transitions, H0 320 mT is a value of the constant magnetic field in the center of MS = 1 EPR spectrum, and rS is measured in Å. In practice, the value of rS one can determine experimentally up to 1.2 nm.

The dipolar interaction impact to second central moment M2 = M2 M2(0) of EPR spectrum allows measuring the distance rM (in Å) in biradicals till ~1.8 nm by the equation (Kokorin et al., 1972, Kulikov et al., 1972):

$$\mathbf{r}\_{\mathsf{M}} = \mathsf{Z}\mathbf{3}.1^{\mathsf{a}} (\mathsf{A}\mathsf{M}\mathbf{2})^{-1/6} \tag{17}$$

Kokorin et al., 1972, reported about 9 biradicals for which distances r were measured independently by methods mentioned above and compared with d1/d parameter of these biradicals. Later, the number of such "reference" biradicals increased to thirteen, and allowed to plot the experimental dependence of r on presented in Fig. 11 based on the results given in Table 2:

**Figure 11.** (○) and 1 () as a function of the distance **r** between unpaired electrons in nitroxide biradicals in frozen solutions at 77 K


\* measured from the X-ray data as the distance between centres of NO bonds (Capiomont, 1972); rM measured from the second central moment ΔM2 (Kokorin et al., 1972, 1974, 1976, Kulikov et al., 1972); rS measured from the forbidden transitions ΔMS = 2 (Dubinskii et al., 1974), rcalc calculated from EPR spectra simulation at 77 (K (Kokorin et al., 1984)

**Table 2.** Distances between NO groups in nitroxide biradicals measured by different methods

Linear dependence of r on 1 obtained in the range 1.0 **r** 1.85 nm gives a correct, simple and rather precise method for the estimation of **r** values: (Kokorin et al., 1976)

$$\mathbf{r} = (9.6 \pm 0.2) + (0.75 \pm 0.02)/\Delta \tag{18}$$

Kokorin, 1974, Parmon et al., 1977b, 1980 published similar corresponding parameters equal to 9.3 0.25 and 0.77 0.03. One can see that these values are very close to those in Eq. (18). Therefore, the interval in which d1/d parameter is recommended for correct distance measurements is 1.2 **r** 2.5-2.7 nm.

128 Nitroxides – Theory, Experiment and Applications

biradicals in frozen solutions at 77 K

1,0

1,2

1,4

r, nm

1,6

1,8

**Figure 11.** (○) and 1 () as a function of the distance **r** between unpaired electrons in nitroxide

0,1 0,2 0,3 0,4 0,5 0,6

1,0

1,2

1,4

1,6

1,8

0,1 0,2 0,3 0,4 0,5 0,6

2 468 10 12 <sup>1</sup>

Biradical Solvent rM, Ǻ rS, Ǻ rcalc, Ǻ OC(OR6)2 Toluene 11.2 11.7 11.3\* S(OR6)2 Toluene 11.1 10.8 11.6 O2S(OR6)2 Toluene 10.7 10.3 - CH2CH(O)P(OR6)2 Toluene 10.3 10.2 10.9 R6NH(CH2)2R6 Toluene 10.8 11.5 *m*-C6H4(COOR6)2 Toluene 15.5 - 15.7 *o*-C6H4(COOR6)2 Toluene 11.5 11.2 11.8 R6(CH2)4R6 Ethanol 12.9 - 13.1\* (CH2)4(HOR6)2 Ethanol 13.6 13.2\* 13.8 (CH2)6(COOR6)2 Ethanol - 18.5\* 17.8 *m*-N3C3Cl\*NHCH2R6)2 Methanol 13.8 - 13.0 *m*-N3C3Cl\*NHR6)2 Methanol 11.5 - 11.3

\* measured from the X-ray data as the distance between centres of NO bonds (Capiomont, 1972); rM measured from the second central moment ΔM2 (Kokorin et al., 1972, 1974, 1976, Kulikov et al., 1972); rS measured from the forbidden transitions ΔMS = 2 (Dubinskii et al., 1974), rcalc calculated from EPR spectra simulation at 77 (K (Kokorin et al., 1984) **Table 2.** Distances between NO groups in nitroxide biradicals measured by different methods

Linear dependence of r on 1 obtained in the range 1.0 **r** 1.85 nm gives a correct, simple

**r** = (9.6 0.2) + (0.75 0.02)/ (18)

and rather precise method for the estimation of **r** values: (Kokorin et al., 1976)

Values of <r> and **r** calculated from the EPR spectra simulation for nitroxide radicals and biradicals in the case of random and pairwise distributions according to recommendations of (Kokorin et al., 1984) were compared with experimental data and shown in Fig. 12. Good correlation between experimental and theoretical data is observed.

**Figure 12.** Parameter d1/d as a function of mean distances <r> for R6OH radical (, ▲), and distances r for biradicals (○, ) dissolved in frozen at 77 K solutions: experimental (▲, ) and calculated (, ○) from theoretical EPR spectra values

The EPR spectrum shape parameter d1/d has been used for studying the effect of the solvent on structural organization of nitroxide biradicals.

Results presented in Table 3 show that in glassy solid solutions frozen at 77 K in various solvents the conformational structure of biradicals can be changed by the influence of the solvent in case of non-rigid, rather long flexible molecules such as *m*-C6H4[COO(CH2)2R6]2, *o*-C6H4[COO(CH2)2R6]2, S[(CH2)2COOR6]2, while for rather short or more rigid molecules (R6NH(CH2)2R6, *m*-C6H4(COOR6)2) the solvent effect is not observed. A long flexible biradical (CH2)4[COO(CH2)2R6]2 was not also sensitive to changes in the solvent polarity (Table 3).

It should be stressed that modern EPR techniques allow researchers measuring distances longer 2.5-3.0 nm, which are out of the scale for d1/d. As an example, it can be illustrated by the work of Bird et al., 2008 in which authors demonstrated the synthesis of a series of spin-labeled curved oligomers and determined their end-to-end lengths and distance distributions using ELDOR technique of EPR spectroscopy. Spin labeled watersoluble, spiro-ladder oligomers with well-defined three-dimensional structures studied with


**Table 3.** Effect of solvent nature on the distances between unpaired electrons in nitroxide biradicals (Parmon et al., 1980)

ELDOR, provided to obtain quantitative information about the shapes and flexibility of the oligomers. The estimeted end-to-end distance of the oligomers ranges from 23 to 36 Å. The shapes of the EPR-derived population distributions allow the authors to compare the degree of shape persistence and flexibility of spiro-ladder oligomers (Bird et al., 2008).

The last works in this area based on high-frequency pulse EPR technique besides measuring distances allow to determine relative mutual orientation of paramagnetic >N–O groups at distances **r** > 3.0 nm (Savitsky et al., 2011). The 95 GHz high-field electron dipolar EPR spectroscopy with the microwave pulse-sequence configurations for PELDOR has been applied. It was concluded that due to the high detection sensitivity and spectral resolution the combination of site-directed spin labeling with high-field PELDOR stands out as an extremely powerful tool for 3D structure determination of large disordered systems. The authors approach compared with other structure-determining magnetic-resonance methods evidently showed its advantage. Angular constraints were provided in addition to distance constraints obtained for the same sample, and the number of necessary distance constraints was strongly reduced. The reduction of necessary distance constraints became another appealing aspect of orientation-resolving EPR spin triangulation which can be applied for protein structure determination (Savitsky et al., 2011).

Real advantage of d1/d method in comparison with modern ones is its simplicity, availability and quite good precision within the interval of its correctness (1.2-2.7 nm).

### **5. Interaction between radicals and paramagnetic metal ions**

130 Nitroxides – Theory, Experiment and Applications

R6NH(CH2)2R6

*m*-C6H4(COOR6)2

*o*-C6H4[COO(CH2)2R6]2

*m*-C6H4[COO(CH2)2R6]2

(CH2)4[COO(CH2)2R6]2

S[(CH2)2COOR6]2

(Parmon et al., 1980)

Biradical Solvent d1/d 0.01 r, Ǻ

**Table 3.** Effect of solvent nature on the distances between unpaired electrons in nitroxide biradicals

of shape persistence and flexibility of spiro-ladder oligomers (Bird et al., 2008).

protein structure determination (Savitsky et al., 2011).

ELDOR, provided to obtain quantitative information about the shapes and flexibility of the oligomers. The estimeted end-to-end distance of the oligomers ranges from 23 to 36 Å. The shapes of the EPR-derived population distributions allow the authors to compare the degree

The last works in this area based on high-frequency pulse EPR technique besides measuring distances allow to determine relative mutual orientation of paramagnetic >N–O groups at distances **r** > 3.0 nm (Savitsky et al., 2011). The 95 GHz high-field electron dipolar EPR spectroscopy with the microwave pulse-sequence configurations for PELDOR has been applied. It was concluded that due to the high detection sensitivity and spectral resolution the combination of site-directed spin labeling with high-field PELDOR stands out as an extremely powerful tool for 3D structure determination of large disordered systems. The authors approach compared with other structure-determining magnetic-resonance methods evidently showed its advantage. Angular constraints were provided in addition to distance constraints obtained for the same sample, and the number of necessary distance constraints was strongly reduced. The reduction of necessary distance constraints became another appealing aspect of orientation-resolving EPR spin triangulation which can be applied for

Toluene 1.03 11.1 0.2 Ethanol 0.94 11.0 H2O:glycerol = 1:1 0.865 11.0

Toluene 0.675 15.5 1-butanol 0.61 16.0 0.6 Ethanol 0.60 16.0 Methanol 0.595 15.7 0.5

Toluene 0.745 12.7 0.3 1-butanol 1.01 10.8 Ethanol 0.92 11.2 0.2 Methanol 0.82 11.6

Toluene 0.65 16.1 0.6 1-butanol 0.55 23.3 2.2 Ethanol 0.56 19.6 1.4 Methanol 0.55 19.0 1.2

Toluene 0.605 20.3 1.6 Ethanol 0.555 20.3

Toluene 0.66 15.7 0.5 Ethanol 0.57 18.4 1.0 It was experimentally revealed that d1/d parameter strongly depends on the longitudinal relaxation time T1 value, and a new method of measuring distances between spin labels and paramagnetic metal ions in macromolecules suggested (Kokorin & Formazyuk, 1981). The number of paramagnetic metal ions and complexes tested has been enlarged, and possible applications to biological systems discussed (Kokorin, 1986). Some later, this approach has been extended to spin-labeled metal-containing polymers (Kokorin et al., 1989).

Fig. 13 presents typical dependences of d1/d parameter of R6OH radical as a function of concentration of R6OH itself, and of some salts: Cu(NO3)2, Ni(en)2(NO3)2, CoSO4, MgSO4 dissolved in H2O;glycerol (1:1) mixture, and of Cr(acac)3 in methanol solution. One can see from Fig. 13 that the efficiency of d1/d increase is different for various metal ions. Indeed, it is known from theory that the dipolar broadening of EPR spectra depends besides concentration of paramagnetic centres on the value of its electron spin and the longitudinal relaxation time T1 (Abragam, 1961, Molin et al., 1980). Kokorin et al., 1981 suggested to characterize relative efficiency of dipole-dipole interaction between nitroxide radicals and paramagnetic metal ions with parameter \*:

$$
\alpha^\* = [4S(S+1)/\Im]^{-1/2} \bullet \prime \Delta \mu / \Delta \mathbb{R} \tag{19}
$$

Here M and R are the dipolar impacts to d1/d parameter measured at the same concentrations in cases of interaction between radicals, R, or between a radical and a metal ion, M, <M/R> is the averaging by all concentrations (Fig. 13), and coefficient = [4S(S+1)/3]–1/2 is used for metal ions with the electron spin S > ½, and takes into account that a spin probe interacts with several electron spins of the metal, S. In such case the dipolar broadening parameter, A\*, will be equal to:

$$\mathbf{A}^\* = \mathfrak{q} \bullet \delta \mathbf{H} / \mathbf{C} \mathbf{u}\_{\prime} \tag{20}$$

analogous to Eq. (1). This correction allows one to determine local concentrations of various paramagnetic metal complexes. A value of <M/R> parameter depends on the T1 value of the paramagnetic centres under investigation (Kokorin & Formazyuk, 1981). Other approaches for solving this problem as well as a perfect collection of experimentally measured values of T1 are collected in Eaton & Eaton, 2001a, 2001b.

In case of coupling between a nitroxide radical and a paramagnetic metal complex distributed in the matrix in pairs, interesting results were presented by (Fielding et al., 1986). Low-spin Fe(II1)-tetraphenylporphyrin complexes have been modified with seven nitroxide radicals of different length: CONHR5,6, CONHCH2R5,6, OCH2CONHR6, O(CH2)2CONHR6, O(CH2)4CONHR6. The spin labels were attached by amide or amide and ether linkages to the ortho position of one phenyl ring. The axial ligands were imidazole

**Figure 13.** d1/d as a function of concentration of R6OH radical (), Cu(NO3)2 (), Ni(en)2(NO3)2 (), CoSO4 (▲), MgSO4 () in H2O;glycerol = 1:1, and Cr(acac)3 (●) in methanol at 77 K. en is NH2CH2CH2NH2, and acac is acetylaceronate

or 1-methylimidazole. In frozen solution the complexes with amide linkages adopted two different conformations, and the populations of the conformations were solvent-dependent. Measured value of the exchange integral J was rather high, while the spin-spin interaction in the second conformation was much weaker than in the first conformation. Broadening of the nitroxyl signal in frozen solution was also observed for complexes with longer ether linkages between the phenyl ring and the nitroxyl. Distances between nitroxide NO groups and Fe(III) ions were estimated by d1/d parameter and by the Leigh method (Leigh, 1970). The EPR spectra, reported by other authors, of two spin labels coordinated to ferric cytochrome P450 were analyzed with the computer programs developed for the iron porphyrin model systems. The authors showed that electron-electron exchange interaction as well as dipolar interaction must be considered in analyzing the spectra of spin-labeled porphyrin-containing bio-macromolecules (Fielding et al., 1986).

### **6. Applications to solid solutions and materials**

Nitroxide spin probes were successfully used for quantitative investigating the structure and micro-phase organization of frozen two-component solutions (Kokorin & Zamaraev, 1972). R6OH radical has been chosen as a spin probe to test the homogeneity of heptaneethanol, carbon tetrachloride-ethanol, and toluene-ethanol mixtures with different ethanol content, frozen at 77 K. EPR spectra of the probe showed non-linear changes of d1/d parameter, from which local concentrations Cloc were calculated. For explanation of the results observed the existence of two different phases in frozen mixed solutions was assumed. By the model suggested in (Khairutdinov & Zamaraev, 1970, Kokorin & Zamaraev, 1972), radicals were localized not in the whole volume of the sample (V0, C0) but only in one mixed phase of total volume V contained both components with local concentration Cloc. The second phase was crystallized and did not contain spin probes. Evidently, Cloc = C0•V0/V and it was obtained for the coefficient of non-uniformity of probes distribution = Cloc/C0:

132 Nitroxides – Theory, Experiment and Applications

NH2CH2CH2NH2, and acac is acetylaceronate

0,4

0,6

d1/d

0,8

1,0

**Figure 13.** d1/d as a function of concentration of R6OH radical (), Cu(NO3)2 (), Ni(en)2(NO3)2 (),

or 1-methylimidazole. In frozen solution the complexes with amide linkages adopted two different conformations, and the populations of the conformations were solvent-dependent. Measured value of the exchange integral J was rather high, while the spin-spin interaction in the second conformation was much weaker than in the first conformation. Broadening of the nitroxyl signal in frozen solution was also observed for complexes with longer ether linkages between the phenyl ring and the nitroxyl. Distances between nitroxide NO groups and Fe(III) ions were estimated by d1/d parameter and by the Leigh method (Leigh, 1970). The EPR spectra, reported by other authors, of two spin labels coordinated to ferric cytochrome P450 were analyzed with the computer programs developed for the iron porphyrin model systems. The authors showed that electron-electron exchange interaction as well as dipolar interaction must be considered in analyzing the spectra of spin-labeled

0,0 0,1 0,2 0,3 0,4 0,5

C, mol/l

Nitroxide spin probes were successfully used for quantitative investigating the structure and micro-phase organization of frozen two-component solutions (Kokorin & Zamaraev, 1972). R6OH radical has been chosen as a spin probe to test the homogeneity of heptaneethanol, carbon tetrachloride-ethanol, and toluene-ethanol mixtures with different ethanol content, frozen at 77 K. EPR spectra of the probe showed non-linear changes of d1/d parameter, from which local concentrations Cloc were calculated. For explanation of the results observed the existence of two different phases in frozen mixed solutions was assumed. By the model suggested in (Khairutdinov & Zamaraev, 1970, Kokorin &

CoSO4 (▲), MgSO4 () in H2O;glycerol = 1:1, and Cr(acac)3 (●) in methanol at 77 K. en is

porphyrin-containing bio-macromolecules (Fielding et al., 1986).

**6. Applications to solid solutions and materials** 

$$\rho = [(\mathbf{M}\_{\oplus}/\mathbf{d}\_{\oplus}) + \mathbf{n} \bullet (\mathbf{M}\_{\mathrm{cr}}/\mathbf{d}\_{\mathrm{cr}})]^{-1} \bullet \mathbf{C}\_{\mathrm{cr}}^{-1} \tag{21}$$

where Mgl, Mcr and dgl, dcr are molecular masses and densities of the glazed (ethanol) and crystallized (heptane, CCL4) components of the mixture; n is a number of molecules of the crystallized solvent per one molecule of the glazed one (ethanol) in the areas of the spin probe localization. CEt is the concentration of ethanol in the solution. From experimental dependences of on CEt–1 for mixtures heptane-ethanol and CCL4-ethanol a phase of nonpolar solvent and the binary mixture of constant composition were observed. The quantitative composition of binary mixtures was determined: 6.5 0.8 ethanol molecules per one heptane molecule, and 2.3 0.3 ethanol molecules per one CCl4 molecule. Binary toluene-ethanol mixtures were glassy at 77 K at all ratios of components, had complex nonlinear dependence of on CEt–1 but did not have phases of constant composition (Kokorin & Zamaraev, 1972).

Another interesting and important quantitative application of spin label technique and d1/d parameter can be illustrated by studies of gold nanoparticles with EPR spectroscopy.

Ionita et al., 2004, investigated the mechanism of a place-exchange reaction of ligandprotected gold nanoparticles using biradical disulfide spin labels which were chemically attached to the surface (Fig. 14). Analysis of reaction mixtures combined GPC and EPR technique allowed authors to determine concentration profile of spin probes and propose a kinetic model for the reaction. Local concentrations of spin labels and mean distances between them were measured using d1/d parameter. In the model suggested, only one branch of the disulfide ligand was adsorbed on the gold surface during exchange, and the other branch formed mixed disulfide with the outgoing ligand. The two branches of the disulfide ligand therefore did not adsorb in adjacent positions on the surface of gold nanoparticles. This was proven by the powder EPR spectra of frozen exchange reaction mixtures. The data presented also suggested the presence of different binding sites with different reactivity in the exchange reaction. It was assumed that the most-active sites are likely to be nanoparticle surface defects (Ionita et al., 2004).

**Figure 14.** Schematic localization of spin labels on the surface of gold nanoparticles (Ionita et al., 2005)

A series of gold nanoparticles modified with a nitroxide-functionalized ligands was synthesized with a range of spin-label coverage (Ionita et al., 2005). The X-band EPR spectra of frozen solutions of these nanoparticles showed coverage-dependent line-broadening due to dipole-dipole interactions between spin labels, and a noticeable increase of d1/d parameter. A methodology to analyze such spectra in terms of geometrical features of the nanoparticles (e.g., gold core size and the length of the spin-labeled ligand) was developed. The method was based on the assumption that the spectral line shape was determined by the average distance between nearest-neighboring spin labels adsorbed on the gold particle. Geometrical and statistical analysis related this distance to the line shape parameter d1/d, which was calibrated using a model system. A calibration curve was suggested as an empirical Eq. (22) (Ionita et al., 2005):

$$\mathbf{d}u/\mathbf{d} = a\mathbf{e} \exp\{-a\mathbf{z}(\mathbf{r}u - u\mathbf{z})\} + a\mathbf{u} + a\mathbf{v}/\mathbf{r}u \tag{22}$$

Here rn is the average distance between nearest-neighboring nitroxide labels. The values of empirical parameters *a*1-*a*5 were equal to 0.8050, 3.0150 nm1, 0.8736 nm, 0.5145, and 0.06824 nm, respectively, as obtained by nonlinear regression. Experimental and calculated values of d1/d parameter were compared and have been very close to each other. The interspin average distances rn between nearest-neighboring spins were determined in the range 1.46 rn 3.3 nm, and they decreased with increasing coverage of spin labels.

Application of this methodology to the experimental spectra provided information about the conformation of ligands on the gold surface. It was found that, if the spin-labeled ligand was substantially longer than the surrounding protecting layer, it did not adopt a fully stretched conformation but wrapped around the particle immediately above the layer of surrounding ligands. The results obtained also showed that the ligands were not adsorbed cooperatively on the gold surface (Ionita et al., 2005).

The lateral mobility of the thiolate ligands on the surface of gold nanoparticles was also probed by Ionita et al., 2008, using bisnitroxide ligands, which contained a disulfide group in the bridge (to ensure attachment to the gold surface) and a cleavable ester bridge connecting the two spin-labeled branches of the molecule. Upon adsorption of these ligands on the surface of gold particles, the two spin-labeled branches were held next to each other by the ester bridge as evidenced by the spin-spin interactions. Cleavage of the bridge removed the link that kept the branches together. CW and pulsed EPR (ELDOR) experiments showed that the average distance between the adjacent thiolate branches on the gold nanoparticle surface only slightly increased after cleaving the bridge and thermal treatment. This implied that the lateral diffusion of thiolate ligands on the nanoparticle surface was very slow at room temperature and took hours even at elevated temperatures (90°C). The changes in the distance distribution observed at high temperature were likely due to ligands hopping between the nanoparticles rather than diffusing on the particle surface (Ionita et al., 2008).

### **7. Applications to polymers**

Another quantitative application of d1/d parameter was suggested for determining local concentrations of chain units in macromolecular coils using the spin-label method (Kokorin et al., 1975). Labelling of poly-4-vinyl-pyridine (P4VP) with R6CH2CH2Br, R6OCOCH2Cl or R6NHCOCH2I radicals with the degree of alkylation of pyridine residues from 2 up to 35 mol.% allowed authors to determine such important structural characteristics of the polymer coil as local concentration of pyridine monomers CN, the effective volume <V> and effective radius Reff of the polymer coil, its local density loc:

134 Nitroxides – Theory, Experiment and Applications

empirical Eq. (22) (Ionita et al., 2005):

A series of gold nanoparticles modified with a nitroxide-functionalized ligands was synthesized with a range of spin-label coverage (Ionita et al., 2005). The X-band EPR spectra of frozen solutions of these nanoparticles showed coverage-dependent line-broadening due to dipole-dipole interactions between spin labels, and a noticeable increase of d1/d parameter. A methodology to analyze such spectra in terms of geometrical features of the nanoparticles (e.g., gold core size and the length of the spin-labeled ligand) was developed. The method was based on the assumption that the spectral line shape was determined by the average distance between nearest-neighboring spin labels adsorbed on the gold particle. Geometrical and statistical analysis related this distance to the line shape parameter d1/d, which was calibrated using a model system. A calibration curve was suggested as an

 d1/d = *a*1exp[*a*2(rn *a*3)] + *a*4 + *a*5/rn (22) Here rn is the average distance between nearest-neighboring nitroxide labels. The values of empirical parameters *a*1-*a*5 were equal to 0.8050, 3.0150 nm1, 0.8736 nm, 0.5145, and 0.06824 nm, respectively, as obtained by nonlinear regression. Experimental and calculated values of d1/d parameter were compared and have been very close to each other. The interspin average distances rn between nearest-neighboring spins were determined in the range 1.46

Application of this methodology to the experimental spectra provided information about the conformation of ligands on the gold surface. It was found that, if the spin-labeled ligand was substantially longer than the surrounding protecting layer, it did not adopt a fully stretched conformation but wrapped around the particle immediately above the layer of surrounding ligands. The results obtained also showed that the ligands were not adsorbed

The lateral mobility of the thiolate ligands on the surface of gold nanoparticles was also probed by Ionita et al., 2008, using bisnitroxide ligands, which contained a disulfide group in the bridge (to ensure attachment to the gold surface) and a cleavable ester bridge connecting the two spin-labeled branches of the molecule. Upon adsorption of these ligands on the surface of gold particles, the two spin-labeled branches were held next to each other by the ester bridge as evidenced by the spin-spin interactions. Cleavage of the bridge removed the link that kept the branches together. CW and pulsed EPR (ELDOR) experiments showed that the average distance between the adjacent thiolate branches on the gold nanoparticle surface only slightly increased after cleaving the bridge and thermal treatment. This implied that the lateral diffusion of thiolate ligands on the nanoparticle surface was very slow at room temperature and took hours even at elevated temperatures (90°C). The changes in the distance distribution observed at high temperature were likely due to ligands hopping between the nanoparticles

Another quantitative application of d1/d parameter was suggested for determining local concentrations of chain units in macromolecular coils using the spin-label method (Kokorin

rn 3.3 nm, and they decreased with increasing coverage of spin labels.

cooperatively on the gold surface (Ionita et al., 2005).

rather than diffusing on the particle surface (Ionita et al., 2008).

**7. Applications to polymers** 

$$\text{CV} \succ \text{I} \succ \text{In}\_{\text{loc}} ; \text{R}\_{\text{eff}} \equiv \langle \text{Sv} / 4 \pi \text{C}\_{\text{loc}} \rangle^{1/3} ; \text{C} \succ \text{I} \newline \langle \text{P} - \text{n} \rangle \langle \text{n} \rangle \bullet \text{C}\_{\text{loc}} ; \text{p} \space \text{loc} = \text{P} \bullet \text{C}\_{\text{loc}} / \text{n} \tag{23}$$

Here P and n are a degree of polymerization (mean number of monomer units in the chain) and mean number of spin labels in the coil correspondingly. It should be stressed that this approach is correct if: a) distribution of spin labels in the coil is statistically random, b) the mean volume <V> occupied by spin labels is identical to the volume of the coil, and c) spin labeling does not change the conformation of macromolecules.

High values of CN equal to 0.3 0.1 mol/l, obtained for labeled P4VP molecules in dilute solutions, confirm the theoretical estimations made by Tanford, 1961. Table 4 contains some values obtained for spin labeled P4VP, polyethylenimine (PEI), polyglycidylmethacrylate (PGMA), poly(methacrylic acid) (PMAc), and its sodium salt (PMAcNa). <ĥ2> = 6•(3<V>/4)2/3 is the mean-square end-to-end distance for a Gaussian chain.

This approach was successfully used for analyzing the conformational state of PGMA macromolecules in diluted and concentrated polymer solutions (Shaulov et al., 1977). Determination of mean distances between monomer units in the chain or their local concentration, Cloc, in the effective macromolecular volume <V> as a function of the polymer concentration in solutions of different thermodynamic quality as well as in a solid amorphous powder was carried out. Both labelled and non-labelled polymers were used. It was revealed that within limits of the experimental conditions, the size of the coil considered to be Gaussian, exceeds theta-dimensions, while the coil size in solid polymer is close to -dimensions. Models of concentrated PGMA solutions were analyzed and the most probable one was chosen basing on the experiment (Shaulov et al., 1977).

The local density of monomer units of the macromolecule (local density of the host residues, host) in poly(4-vinyl pyridine) solutions in ethanol was determined by the spin label technique and d1/d parameter (Wasserman et al., 1979). In dilute solutions, host is considerably greater than the mean density of monomer units in the volume of the polymer coil. When the P4VP concentration increases from 0.5 to 65 wt%, the host increase does not exceed 30%. This fact indicates that the differences between polymer coil spatial organization (mutual positions of monomer units close to a labeled unit) in dilute and concentrated solutions are small. The local density of monomer units of neighbouring macromolecule coils (guest macromolecules, guest) is strongly dependent on the polymer concentration in solution. In dilute solutions, host >> guest; for polymer concentrations above 2–3 wt%, overlapping and interpenetration of macromolecular coils take place, local density of guest coils, guest; monotonously increases with polymer concentration growing up. The concentration dependence of the local rotational and translational mobility of chain units was also determined for spin-labeled P4VP (Wasserman et al., 1979).

Several works were published in which d1/d parameter was used for the study of intramolecular dynamics and of local density of P4VP units in diluted and concentrated liquid and frozen solutions with spin label method Wasserman et al., 1980a, 1980b). Temperature dependences of EPR spectra lines of spin labeled P4VP solutions in ethanol allowed estimate dipolar and spin exchange impacts, and to calculate mean local densities loc at 77 and 293 K. Measured parameters: 0.3 0.1 at 77 K and 0.25 0.05 mol/l at 293 K, were several times higher than the average concentrations of the units in solutions. Knowledge of these concentrations allowed to calculate correct diffusion coefficients of spin labels in the P4VP coil. These results are reasonably close to those listed in Table 4.


**Table 4.** Some parameters characterizing spin-labelled polymers (Kokorin, 1992, Wasserman et al., 1992)

The intramolecular mobility and local density of monomer units in spin labeled styrene copolymers with maleic anhydride has been studied (Aleksandrova et al, 1986). The estimated value of loc for these co-polymers dissolved in dimethylformamide equal 0.03 mol/l was tenfold less loc values measured for P4VP in ethanol or 50% H2O:ethanol mixtures (Table 4).

Quantitative measurements of non-crystallized (solvated) water molecules, based on the EPR study of the structure of frozen aqueous solutions of polyvinylpyrrolidone (PVP) and of polyvinylalcohol (PVA) have been reported (Mikhalev et al., 1985). Local concentrations of spin labels and spin probes were determined by d1/d parameter and Cloc values calculated using procedure suggested in Khairutdinov & Zamaraev, 1970, Kokorin & Zamaraev, 1972. In the presence of NaCl salt in frozen solutions of PVP, the formation of strong complexes between PVP links with water has been observed in the areas containing salt, polymer fragments and H2O molecules.

The application of EPR spin probe and spin label technique for solving two actual problems of polymer physical chemistry was considered in (Wasserman et al., *1996*). The first problem is the determination of conformational state and chain sizes in amorphous solid polymers. This determination is based on the analysis of the intramolecular dipolar broadening of EPR spectra of spin labelled macromolecules in glassy solvents or in the bulk of unlabeled polymers at 77 K with the use of d1/d parameter. The second problem is the determination of molecular dynamics and structure of polymer colloid systems: (a) the complex of colloidal silica and synthetic polycation macromolecule, and (b) polymer-surfactant micellar organized systems. Possible approaches were discussed.

136 Nitroxides – Theory, Experiment and Applications

fragments and H2O molecules.

Several works were published in which d1/d parameter was used for the study of intramolecular dynamics and of local density of P4VP units in diluted and concentrated liquid and frozen solutions with spin label method Wasserman et al., 1980a, 1980b). Temperature dependences of EPR spectra lines of spin labeled P4VP solutions in ethanol allowed estimate dipolar and spin exchange impacts, and to calculate mean local densities loc at 77 and 293 K. Measured parameters: 0.3 0.1 at 77 K and 0.25 0.05 mol/l at 293 K, were several times higher than the average concentrations of the units in solutions. Knowledge of these concentrations allowed to calculate correct diffusion coefficients of spin

labels in the P4VP coil. These results are reasonably close to those listed in Table 4.

Polymer P Cloc, mol/l , mol/l <V>, nm3 Reff, nm <ĥ2>, nm2 P4VP-25% 430 0.14 0.52 1370 6.8 280 P4VP-35% 270 0.17 0.49 930 6.1 220 P4VP-10% 1330 0.04 0.4 5540 11.0 726 P4VP-15% 1330 0.07 0.49 4570 10.3 636 P4VP-20% 1330 0.11 0.55 4030 9.9 590 P4VP-28% 1330 0.14 0.5 4430 10.2 624 P4VP-40% 1330 0.18 0.45 4930 10.6 670 PEI-5% 120 0.023 0.46 435 4.7 130 PEI-22% 120 0.14 0.62 320 4.25 110 PGMA-10% 690 0.033 0.33 3450 9.4 530 PGMA-16% 690 0.05 0.3 3830 9.7 560 PMAc-23% 1600 0.11 0.48 5580 9.3 520 PMAcNa-23% 1600 0.1 0.43 6130 9.6 550 **Table 4.** Some parameters characterizing spin-labelled polymers (Kokorin, 1992, Wasserman et al., 1992)

The intramolecular mobility and local density of monomer units in spin labeled styrene copolymers with maleic anhydride has been studied (Aleksandrova et al, 1986). The estimated value of loc for these co-polymers dissolved in dimethylformamide equal 0.03 mol/l was tenfold less loc values measured for P4VP in ethanol or 50% H2O:ethanol mixtures (Table 4).

Quantitative measurements of non-crystallized (solvated) water molecules, based on the EPR study of the structure of frozen aqueous solutions of polyvinylpyrrolidone (PVP) and of polyvinylalcohol (PVA) have been reported (Mikhalev et al., 1985). Local concentrations of spin labels and spin probes were determined by d1/d parameter and Cloc values calculated using procedure suggested in Khairutdinov & Zamaraev, 1970, Kokorin & Zamaraev, 1972. In the presence of NaCl salt in frozen solutions of PVP, the formation of strong complexes between PVP links with water has been observed in the areas containing salt, polymer

The application of EPR spin probe and spin label technique for solving two actual problems of polymer physical chemistry was considered in (Wasserman et al., *1996*). The first problem is the determination of conformational state and chain sizes in amorphous solid polymers. Spin labeling was used to investigate the topochemical characteristics of polymer carriers and immobilized metal complexes (Bravaya & Pomogailo, 2000). Functionalized polyethylene (PE) molecules such as PE-grafted-polyallylamine (PE-PAA), PE-graftedpolydiallylamine (PE-PDAA), and PE-grafted-poly-4-vinylpyridine (PE-P4VP), obtained by grafting polymerization of the corresponding monomers, were used as polymer carriers. Metal-containing polymers were synthesized by attaching to polymers either TiCl4 (PE-PDAA-Ti) or Al(C2H5)2Cl (PE-PDAA-Al). 2,2,6,6-tetramethyl-4-(2'-oxy-4',6'-dichlorotriazine) piperidine-1-oxyl nitroxyl radical (R1) was used for spin labeling PE-PDAA, while 2,2,5,5,tetramethyl-3-(N-acetoamidiiodine)pyrrollidine-1-oxyl (R2) was used for spin labeling PE-P4VP, and 2,2,6,6-tetramethyl-4-hydroxy-pyperidine-1-oxyl (R3) was bound to PE-PDAA-Ti and PE-PDAA-Al respectively. Estimation of the effective distances between the spin labels by d1/d parameter and the dynamic behavior of nitroxyl radicals in the functionalized polymer matrixes and metal-containing polymers revealed several important features of spin-labeled systems. Metal complex formation of functional polymers made them more accessible for spin labeling and had a considerable effect on the dynamic characteristics of the polymer matrix. Thermodynamic characteristics of the rotational diffusion of the labels were determined (Bravaya & Pomogailo, 2000).

Next serious approach to better understanding structural organization of spin labeled macromolecules in the amorphous solid state and their conformational transitions has been suggested by Khazanovich et al., 1992. The algorithm for EPR spectra computation was developed: it was assumed that molecular weights of labelled linear chains are high enough and their solid solution is diluted. It was shown that the scaling exponent which determines the dependence of mean-square end-to-end distance on molecular weight and stiffness parameter (mean-square length of monomer unit) may be extracted from the spectra. Simulated EPR spectra were compared with experimental ones, measured at 77 K, of diluted solutions of spin-labelled poly(4-vinyl pyridine), P4VP, of different molecular weights in methanol and non-labelled P4VP. The conformational state of the Gaussian coil, parameter of stiffness, and mean square radius of gyration <RG2>1/2 of spin-labeled P4VP macromolecule in frozen solutions were determined via measuring d1/d parameter, local density values loc of links, and parameters mentioned above were calculated from it. It was concluded that EPR spectroscopy may become a sensitive tool for studying chain conformation in solid polymers (Khazanovich et al., 1992, Kolbanovsky et al., 1992b).

This approach was successfully used to determine the conformational states of spin labeled P4VP, poly(methacrylic acid), PMAc, and its sodium salt in glassy methanol, ethanol, 1 propanol solutions and in the bulk matrix of unlabeled polymers at 77 K. All macromolecules had near-Gaussian coil conformations. The mean square lengths of the

repeating units, the characteristic ratios, and the mean-square end-to-end distances <R2>1/2 of the polymers were determined. Typical results are listed in Table 5 with corresponding values of (extracted from Wasserman et al., 1992).


\* P4VP-1,2,6 were labeled with R6CH2CH2Br, P4VP-4,5 – with R5N=CHCOCH2I, PMAc – with R6NH2. P4VP-1,2, P4VP-4,5 and P4VP-6 are of different molecular mass.

#### **Table 5.** Parameters and <R2>1/2 of spin labeled macromolecules

Analyzing developing of the area of spin probes and labels during a quarter of a century of its application to polymer studies, the author described in details history of the method, investigations of local and segmental mobility in polymers, and paid special attention to the approaches for determining local densities and translational dynamics of monomeric units in a coil (Kovarski, 1996).

It should be noted that EPR data on conformational state and dimensions of the polymer coil can substantially complement the data of other physical methods: neutron scattering, for example.

The spin label method was also used for studying the spatial organization of the labeled linear polyethyleneimine (PEI) macromolecules in glassy 50% water-ethanol solutions in the process of complex formation with transition metal ions (Kokorin et al., 1989). The mean local density of PEI chain units loc was measured by d1/d parameter, as well as PEI coil volume <V>, the mean coil radius Reff, and the average distance berween spin labels (Table 5). It is known, that if in the sample there are paramagnetic centres of different nature, total dipolar broadening of EPR spectrum lines is the sum of broadenings caused by paramagnetics of each type (Lebedev & Muromtsev, 1972). Analogously, in case of coordination of metal ions (paramagnetic Cu(II), Ni(II), Co(II) and diamagnetic Zn(II) and sodium ions) by the spin-labeled PEI, a procedure for separate determination the impacts of the dipole-dipole interaction between spins of radical labels LL and of radicals with metal ions LM was suggested. These impacts can be expressed as a sum to the experimentally measured value of parameter :

$$
\Delta = \Delta \Box + \Delta \Box \tag{24}
$$

The average distances RLM between a label and the nearest paramagnetic complex were determined by the method suggested by Leigh, 1970. The average local concentrations of complexes <CM> in the PEI coil volume were calculated using the data and coefficients obtained in Kokorin & Formazyuk, 1981. Changes in LL caused by the decrease of the coil volume as a result of polymer-metal complex formation was determined by measuring d1/d values in the case of PEI interaction with diamagnetic Zn(II) ions in the whole range of [M]/[L] ratios. As an example of this approach Fig. 15 shows the effective local concentration of metal complexes <CM> in the PEI coil vs. the metal-to-label ratio [M]/[L] for different metal ions at 77 K (Kokorin et al., 1989).

138 Nitroxides – Theory, Experiment and Applications

4,5 and P4VP-6 are of different molecular mass.

in a coil (Kovarski, 1996).

measured value of parameter :

example.

**Table 5.** Parameters and <R2>1/2 of spin labeled macromolecules

P4VP-1 P4VP-1 P4VP-2 P4VP-4 P4VP-5 P4VP-6 PMAc PMAcNa

values of (extracted from Wasserman et al., 1992).

repeating units, the characteristic ratios, and the mean-square end-to-end distances <R2>1/2 of the polymers were determined. Typical results are listed in Table 5 with corresponding

Polymer\* Solvent <R2>1/2, nm

\* P4VP-1,2,6 were labeled with R6CH2CH2Br, P4VP-4,5 – with R5N=CHCOCH2I, PMAc – with R6NH2. P4VP-1,2, P4VP-

Analyzing developing of the area of spin probes and labels during a quarter of a century of its application to polymer studies, the author described in details history of the method, investigations of local and segmental mobility in polymers, and paid special attention to the approaches for determining local densities and translational dynamics of monomeric units

It should be noted that EPR data on conformational state and dimensions of the polymer coil can substantially complement the data of other physical methods: neutron scattering, for

The spin label method was also used for studying the spatial organization of the labeled linear polyethyleneimine (PEI) macromolecules in glassy 50% water-ethanol solutions in the process of complex formation with transition metal ions (Kokorin et al., 1989). The mean local density of PEI chain units loc was measured by d1/d parameter, as well as PEI coil volume <V>, the mean coil radius Reff, and the average distance berween spin labels (Table 5). It is known, that if in the sample there are paramagnetic centres of different nature, total dipolar broadening of EPR spectrum lines is the sum of broadenings caused by paramagnetics of each type (Lebedev & Muromtsev, 1972). Analogously, in case of coordination of metal ions (paramagnetic Cu(II), Ni(II), Co(II) and diamagnetic Zn(II) and sodium ions) by the spin-labeled PEI, a procedure for separate determination the impacts of the dipole-dipole interaction between spins of radical labels LL and of radicals with metal ions LM was suggested. These impacts can be expressed as a sum to the experimentally

The average distances RLM between a label and the nearest paramagnetic complex were determined by the method suggested by Leigh, 1970. The average local concentrations of

0.24 0.29 0.14 0.28 0.17 0.13 0.25 0.23

= LL + LM (24)

Methanol 1-Propanol Methanol Ethanol Ethanol Methanol Methanol Methanol

**Figure 15.** Effective local concentration of metal complexes <CM> in PEI coil as a function of metal-tolabel ratio [M]/[L] at 77 K for: ●, - Cu(II), ▲, - Ni(II) and ,+ - Co(II). Concentration of spin labels [L] = 0.045 (●,▲,) and 0.012 mol/l (,,+)

It should be stressed that this procedure is correct only in the assumption that spatial distributions of spin labels and polymer-metal complexes in the coil are the same, random, and there are no areas of their specific localization. A fact that <CM> plots vs. [M]/[L] ratio are the same for all divalent ions studied allows conclude that the structure of metal-PEI coil for these ions is similar.

Determination of the nanostructure of polymer materials by EPR spectroscopy was considered as one of the few methods that can characterize structural features in the range between 1 and 5 nm in systems that lack long-range order (Jeschke, 2002). Approaches based on various techniques of EPR spectroscopy, such as CW X-band EPR, electron spin echo, ENDOR) provided good structural contrast even in complex materials, because the sites of interest could be selectively labeled or addressed by suitably functionalized spin probes using well established techniques. In the article, experiments on distance measurements on nanoscales in terms of the accessible distance range, precision, and sensitivity were discussed, and recommendations were derived for the proper choice of experiment. Both simple and sophisticated methods for data analysis are described and their limitations are evaluated. The approach of Khazanovich et al., 1992, based on d1/d and parameters was used for characterization of the chain conformation was described. The conformational organization of the polymer chain and the structure of ionomers based on diblock copolymers were analyzed (Jeschke, 2002).

Bird et al., 2008, demonstrated modern possibilities of ELDOR and computing methods on a series of spin-labeled oligomers to determine their end-to-end lengths, Ree, and distance distributions. Seven different shape-persistent macromolecules from conformationally restricted, asymmetric monomers that are coupled through pairs of amide bonds to create water-soluble, spiro-ladder oligomers with well-defined three-dimensional structures were synthesized and investigated. The ends of these oligomers were labeled with nitroxide radicals. ELDOR experiments were carried out to obtain quantitative information about the shapes and flexibility of the oligomers. The most probable Ree distance of the oligomers ranges from 2.3 to 3.6 nm. The relative distances measured for the oligomers confirm that, by varying the sequence of an oligomer, one can control its shape. The shapes of the EPRderived population distributions allowed the authors to compare the degree of shape persistence and flexibility of spiro-ladder oligomers to other well-studied nanoscale molecular structures such as *p*-phenylethynylenes (Bird et al., 2008).

Interesting application of spin label method and d1/d parameter was presented by Kozlov et al., 1981, for investigation the oligomers in solutions where long-chain flexible nitroxide biradicals were used as a model. Measuring distances **r** between N–O groups in oligomers, the dependence of **r** on the number of units in the chain, n, was experimentally obtained for 12 biradicals of different length, and the equation (Flory, 1969):

$$<\mathfrak{a}> = \alpha 2\beta 2\mathfrak{n},\tag{25}$$

where is a Flory-Fox constant, and is the characteristic length. It was shown that = 0.56 nm for hydrocarbon oligomers, = 0.534 nm for dimethylsiloxane ones, and also = 0.452 and 0.405 nm for poly(methylene) and poly(dimethylsiloxane) chains correspondingly (Kozlov et al., 1981). Fig. 16 Illustrates Eq. () well.

**Figure 16.** Distance <r2>1/2 as a function of n1/2 in toluene solutions at 77 K for (CH2)k(COOR6)2, k = 6-8, 10, 14 (●), and R6O–[Si(CH3)2O]m–R6, m = 2-6 ()

Additional information on applications of spin label technique for investigation structural properties of synthetic polymers is described in detail in monographs by Wasserman & Kovarsky, 1986, Kovarski, 1996, Schlick, 2006.
