**2. Determination of the translation diffusion coefficients of the paramagnetic molecules from the analysis of temperature dependence of EPR spectra concentration broadening**

It is well known that the spin probe technique can be used to measure molecular rotational mobility, as the rotational movements are reflected in the width and shape of EPR spectrum lines [1-5]. On the contrary, translational molecular mobility of paramagnetic molecules is studied by EPR technique very rarely. Meanwhile, it is known that translation of radicals influences EPR spectrum. There are two underlying mechanisms of this influence: dipoledipole interaction and spin (Heisenberg) exchange. Both of these mechanisms lead to broadening of the spectral lines. At low temperatures, when translational mobility is substantially hindered, the main cause of spectral line broadening is the dipole-dipole interaction of paramagnetic molecules. At high temperatures, the intensive translational movements average the dipole-dipole interaction of radicals while increasing spin exchange. It is obvious that there exists a temperature region where the contributions of dipole-dipole and exchange interactions to line broadening are comparable. A theoretic research [13] showed earnestly the difficulty of a direct analysis of the EPR spectra with the purpose of determining the translational diffusion coefficients. Nevertheless, the translational diffusion coefficients may be estimated by an analysis of the broadening of EPR spectra. For this purpose, the method described in [12, 14-16] can be used. This method allows estimating contributions of dipole-dipole and exchange interactions to the line width by means of analyzing the temperature dependence of the concentration broadening. The concentration broadening can be represented as follows:

$$
\delta H = [A \exp(-E\_{tr}^{a} / kT) + B \exp(E\_{tr}^{a} / kT)] \delta c \,, \tag{1}
$$

where *δH* is the line broadening, *Ea* tr is the effective activation energy of translational movement, and *δc* is the difference in concentrations of two radical solutions. The first summand in Eq. (1) describes the effect of the exchange broadening, and the second term describes the effect of the dipole-dipole interaction.

58 Nitroxides – Theory, Experiment and Applications

advantages and drawbacks of both approaches are discussed.

**EPR spectra concentration broadening** 

broadening can be represented as follows:

**2. Determination of the translation diffusion coefficients of the** 

**paramagnetic molecules from the analysis of temperature dependence of** 

It is well known that the spin probe technique can be used to measure molecular rotational mobility, as the rotational movements are reflected in the width and shape of EPR spectrum lines [1-5]. On the contrary, translational molecular mobility of paramagnetic molecules is studied by EPR technique very rarely. Meanwhile, it is known that translation of radicals influences EPR spectrum. There are two underlying mechanisms of this influence: dipoledipole interaction and spin (Heisenberg) exchange. Both of these mechanisms lead to broadening of the spectral lines. At low temperatures, when translational mobility is substantially hindered, the main cause of spectral line broadening is the dipole-dipole interaction of paramagnetic molecules. At high temperatures, the intensive translational movements average the dipole-dipole interaction of radicals while increasing spin exchange. It is obvious that there exists a temperature region where the contributions of dipole-dipole and exchange interactions to line broadening are comparable. A theoretic research [13] showed earnestly the difficulty of a direct analysis of the EPR spectra with the purpose of determining the translational diffusion coefficients. Nevertheless, the translational diffusion coefficients may be estimated by an analysis of the broadening of EPR spectra. For this purpose, the method described in [12, 14-16] can be used. This method allows estimating contributions of dipole-dipole and exchange interactions to the line width by means of analyzing the temperature dependence of the concentration broadening. The concentration

[ exp( / ) exp( / )]

*a a H A E kT B E kT c tr tr* , (1)

does not require spectra simulation. It was suggested in [12] about 20 years ago but has not found wide application. We illustrate this technique on samples of ordinary solvents and ionic liquids. The methods of spectra simulation are considered in section III. First, in the rigid-limit spectra of the isotropic samples, we showed the possibility to achieve agreement between experimental and calculated spectra within errors of spectrum recording. The magnetic and line width characteristics of the nitroxide probe are determined in the course of this simulation. The obtained values are necessary for the analysis of spectra recorded in more complicated conditions. The spectra simulation and determination of the rotational characteristics of the probes in polymer matrices are presented as examples. The quantitative description of EPR spectra in a wide temperature range up to the glass transition point was found to require consideration of quasi-libration movements. Section IV deals with the study of orientation alignment of paramagnetic probes. The section describes the orientation distribution functions that can be determined by simulation of spectrum angular dependencies. Liquid crystalline, polymeric, and low-molecular glassy systems are considered. The model-free method of characterizing orientation distribution function is compared with the mean-force approach realized in the known software [4, 5]. The

The widths of the spectral lines are determined as the distances between the points of the maximal slope of the absorption lines (peak-to-peak distances of the first derivatives of the absorption lines). In the paper [12], the application of this method in determining the translational diffusion coefficients of the various spin probes in liquid crystal matrices was demonstrated. In the joint work with our colleagues from the University of Graz (Austria), we showed the possibility of using the method to investigate the translation of radicals in standard low-molecular-weight solvents and room-temperature ionic liquids [17].

In Figure 1, one can see the EPR spectra recorded for two different concentrations of radical TEMPOL-d17 in glycerol at various temperatures. It is obvious that the spectra of the solution with a larger concentration are broadened noticeably in comparison with the spectra of the less concentrated solution. Because the rotational mobility of the radicals does not depend on concentration, the difference observed was therefore caused by the dipole-

**Figure 1.** The EPR spectra of TEMPOL-d17 in glycerol: (a) 7.1· 10−2 mol/L and (b) 3.0· 10−3 mol/L, recorded at temperatures 333K (a1, b1), 318K (a2, b2), 303K (a3, b3), and 295K (a4, b4). The red lines are the results of the simulation of the spectra according to the method described in [4,5].

dipole and exchange interactions of the paramagnetic molecules. The result of the computer simulation of the spectra for the solution with low concentration of the radicals is also presented in Figure 1. The simulation was performed according to [5] and was used to determine the rotation diffusion coefficients of TEMPOL-d17 in glycerol at different temperatures and estimation of the effective activation energy of the rotational movements.

Figure 2(a) presents the temperature dependence of the line broadening *δH* normalized on the difference of concentrations *δc* for radical TEMPOL-d17 in glycerol and the result of fitting the experimental data according to Eq. (1). The modeling was performed by the least squares method, with simultaneous variation of three parameters: *A*, *B*, and *Ea* tr. In this figure, one can also see the calculated contributions of the exchange and dipole-dipole broadening. Obviously, at temperatures 320–330K, the line broadening is caused mainly by the dipole-dipole interactions of the radicals, whereas at 360–370K, the lines are broadened basically by spin exchange. At the intermediate temperature range 340–360K, the contributions of dipole-dipole and exchange broadening are comparable. The effective activation energies of the rotational and translational movements of the radicals TEMPOLd17 in glycerol are presented in Table 1.

**Figure 2.** *δH/δc* as a function of temperature for TEMPOL-d17 in glycerol (a), TEMPOL in omimBF4 (b), 1-propanol (c), and cumene (d); experimental broadening (solid circles); contribution of spin exchange (open squares); and contribution of dipole-dipole interaction (open triangles).


\* Radical TEMPOL-d17

60 Nitroxides – Theory, Experiment and Applications

d17 in glycerol are presented in Table 1.

310 320 330 340 350 360 370 380

220 240 260 280 300 320

(open squares); and contribution of dipole-dipole interaction (open triangles).

exchange dipol-dipol

movements.

0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

H/c) Gl/mol) a

c

H/c) Gl/mol)

dipole and exchange interactions of the paramagnetic molecules. The result of the computer simulation of the spectra for the solution with low concentration of the radicals is also presented in Figure 1. The simulation was performed according to [5] and was used to determine the rotation diffusion coefficients of TEMPOL-d17 in glycerol at different temperatures and estimation of the effective activation energy of the rotational

Figure 2(a) presents the temperature dependence of the line broadening *δH* normalized on the difference of concentrations *δc* for radical TEMPOL-d17 in glycerol and the result of fitting the experimental data according to Eq. (1). The modeling was performed by the least

figure, one can also see the calculated contributions of the exchange and dipole-dipole broadening. Obviously, at temperatures 320–330K, the line broadening is caused mainly by the dipole-dipole interactions of the radicals, whereas at 360–370K, the lines are broadened basically by spin exchange. At the intermediate temperature range 340–360K, the contributions of dipole-dipole and exchange broadening are comparable. The effective activation energies of the rotational and translational movements of the radicals TEMPOL-

tr. In this

T(K)

T(K)

280 300 320 340 360 380

200 220 240 260 280 300 320 340 360

squares method, with simultaneous variation of three parameters: *A*, *B*, and *Ea*

T(K)

T(K)

**Figure 2.** *δH/δc* as a function of temperature for TEMPOL-d17 in glycerol (a), TEMPOL in omimBF4 (b), 1-propanol (c), and cumene (d); experimental broadening (solid circles); contribution of spin exchange

H/c) Gl/mol) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

H/c)

Gl/mol)

0.05 0.10 0.15 0.20 0.25 0.30 0.35 b

d

**Table 1.** The effective activation energies of the rotational and translational moves of the radicals TEMPOL in different matrices

Consideration of spin-exchange contribution to spectral broadening makes it possible to calculate the spin-exchange constant as follows [18]:

$$k\_{exch} = \left[1.52 \cdot 10^7 \, (\text{g}/\text{}^\circ\text{2}) (2/\text{}\, \text{3}) (\delta H)\_{exch}\right] / \, \delta \text{c} \,\, \tag{2}$$

where *g* is the average value of the g-factor of the radical, *δH*exch is the contribution of spin exchange to the broadening of the spectra.

On the basis of the spin-exchange constant, it is easy to calculate the translation diffusion coefficient [18]:

$$D\_{tr} = k\_{exch} / 16f\pi r \,\text{ }.\tag{3}$$

where *f* is the steric factor reflecting different probability of spin exchange upon collisions with different mutual orientations of the radicals (for TEMPOL, *f* = 0.8 [19]) and *r* is the radius of the paramagnetic molecule.

The translational diffusion coefficients of the TEMPOL-d17 in glycerol, calculated at some temperatures, are presented in Table 2.

In Figure 2(b–d), the examples presented illustrate the application of the method [12] in determining the translational diffusion coefficients of the undeuterated probe TEMPOL in the ionic liquid omimBF4 and standard molecular solvents *n*-propanol and cumene. The

high temperature spectra of the diluted solutions of undeuterated TEMPOL demonstrate the additional hyperfine structure on protons. In these cases, we measured the width of the envelope of the spectral line. It can be seen that such method does not lead to distortion of the temperature dependence of the spectral line broadening. Indeed, intensive spin exchange at high temperatures and large concentration leads to substantial spectral line broadening. As a result, subtraction of a small value measured with some error from a big value measured exactly does not lead to a significant error in the result. The values of the effective activation energies for translation and rotation of the radical TEMPOL in different matrices are presented in Table 1.

In Figure 2(c, d), one can see the temperature dependence of the EPR spectra broadening for TEMPOL in the low-viscous solvents *n*-propanol and cumene. It is seen that in these cases, the temperature range in which the dipole-dipole broadening can be observed is confined significantly. At lower temperatures, the rotational mobility of the probe is so small that measurement of the spectral line widths becomes impossible. In the case of *n*-propanol, the parabolic dependence (1) has feebly marked the left dipole-dipole branch (viscosity at room temperature, *η*295 = 1.8 sP). In the case of cumene (*η*295 = 1.0 sP), we can observe practically only the exchange branch of the concentration broadening. However, even in this case, it is possible to distinguish the significantly different contributions of dipole-dipole interaction and spin exchange to broadening of the spectral lines by means of the method [12].


\*The data were obtained by means of cyclic voltammetry

\*\*The data were obtained by means of chronoammetry

\*\*\*Radical TEMPOL-d17

**Table 2.** The translation diffusion coefficients (Dtr·107) of the radical TEMPOL at various temperatures

Recently it was supposed [20-22] that paramagnetic molecules in the solvent cage repeatedly collide and exchange their spin states. In such case the spin exchange does not reflect adequately the rate of translational diffusion of molecules in the medium. To check the correctness of the obtained data we compared the value of translation diffusion coefficient of TEMPOL in ionic liquid bmimBF4 with the values measured for the same system by two independent electrochemical methods - cyclic voltammetry and chrono-ammetry [23]. The results of all types of measurements are in agreement within experimental errors (Table 2).

62 Nitroxides – Theory, Experiment and Applications

matrices are presented in Table 1.

bmimBF4 − − − −

\*The data were obtained by means of cyclic voltammetry \*\*The data were obtained by means of chronoammetry

\*\*\*Radical TEMPOL-d17

high temperature spectra of the diluted solutions of undeuterated TEMPOL demonstrate the additional hyperfine structure on protons. In these cases, we measured the width of the envelope of the spectral line. It can be seen that such method does not lead to distortion of the temperature dependence of the spectral line broadening. Indeed, intensive spin exchange at high temperatures and large concentration leads to substantial spectral line broadening. As a result, subtraction of a small value measured with some error from a big value measured exactly does not lead to a significant error in the result. The values of the effective activation energies for translation and rotation of the radical TEMPOL in different

In Figure 2(c, d), one can see the temperature dependence of the EPR spectra broadening for TEMPOL in the low-viscous solvents *n*-propanol and cumene. It is seen that in these cases, the temperature range in which the dipole-dipole broadening can be observed is confined significantly. At lower temperatures, the rotational mobility of the probe is so small that measurement of the spectral line widths becomes impossible. In the case of *n*-propanol, the parabolic dependence (1) has feebly marked the left dipole-dipole branch (viscosity at room temperature, *η*295 = 1.8 sP). In the case of cumene (*η*295 = 1.0 sP), we can observe practically only the exchange branch of the concentration broadening. However, even in this case, it is possible to distinguish the significantly different contributions of dipole-dipole interaction

and spin exchange to broadening of the spectral lines by means of the method [12].

 220K 240K 260K 280K 295K 300K 320K 340K 360K 380K emimBF4 − − − − 1.9 2.0 2.5 3.0 3.6 4.2

omimBF4 − − − − 0.6 0.7 1.0 1.6 2.4 3.3 omimPF6 − − − − 0.9 1.0 1.5 2.1 2.8 3.7 omimCl − − − − 0.3 0.4 0.6 0.9 1.4 1.9 glycerol\*\*\* − − − − − − 1.0 1.7 2.7 4.1 cumene 3.0 5.5 9.2 14.3 19.2 21.0 29.4 39.6 51.5 − *n*-propanol 0.5 1.2 2.7 5.4 8.4 9.6 16.1 − − −

**Table 2.** The translation diffusion coefficients (Dtr·107) of the radical TEMPOL at various temperatures

Recently it was supposed [20-22] that paramagnetic molecules in the solvent cage repeatedly collide and exchange their spin states. In such case the spin exchange does not reflect adequately the rate of translational diffusion of molecules in the medium. To check the correctness of the obtained data we compared the value of translation diffusion coefficient

1.1 ± 0.4 0.8 ± 0.3\* 0.9 ± 0.3\*\*

1.3 1.9 2.7 3.7 4.8

From Table 2, it can be seen that in cases of viscous solvents, the effective values of activation energy for rotational mobility exceed noticeably the effective values of activation energy for translational movements. The reason of this phenomenon is not clear at the present time. Perhaps it is a result of microstructure of viscous solvents, such as glycerol, and all ionic liquids.

We hope that the method of determining the translation diffusion coefficients of paramagnetic molecules [12], which possibly does not possess high accuracy but is very simple to use and does not demand computer simulation of the spectra, will be widely applicable.
