Neighbor Effect in Complexation of a Conjugated Polymer - The

Article pubs.acs.org/JPCB

Neighbor Effect in Complexation of a Conjugated Polymer Andrey Sosorev and Sergey Zapunidi* Faculty of Physics and International Laser Center, M.V. Lomonosov Moscow State University, Moscow 119991, Russia S Supporting Information *

ABSTRACT: Charge-transfer complex (CTC) formation between a conjugated polymer and low-molecular-weight organic acceptor is proposed to be driven by the neighbor effect. Formation of a CTC on the polymer chain results in an increased probability of new CTC formation near the existing one. We present an analytical model for CTC distribution considering the neighbor effect, based on the principles of statistical mechanics. This model explains the experimentally observed threshold-like dependence of the CTC concentration on the acceptor content in a polymer:acceptor blend. It also allows us to evaluate binding energies of the complexes.

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INTRODUCTION Blends of conjugated polymers (CPs) with low-molecular-weight organic acceptors are widely investigated as active layers of polymer solar cells. In some blends, a charge-transfer complex (CTC) is formed in the ground electronic state.1,2 Polymer CTCs can significantly absorb in the red and near-IR regions1,3 and generate mobile charges1,4,5 highlighting the CTC potential as low-bandgap materials for organic solar cells. Furthermore, the ground-state interaction between the donor and acceptor in CTCs could influence the conformation of the polymer chains3,6,7 and morphology of the blend.8 The pathway from excitons to free charges generally includes the CTC states as key intermediates,4,9 even if the CTC ground-state absorption is low (for example, in P3HT:PCBM blend).10−15 As the film photophysics is to the large extent determined by the solution properties,16,17 it seems reasonable to investigate the peculiarities of CTC formation in solution. Unlike the low-molecular-weight compounds, the CTC formation in donor−acceptor blends of CPs can show a threshold-like character. Below some threshold acceptor concentration, the CTC concentration is low, while, above that, intensive CTC formation begins. Such threshold-like dependence was observed in the blend of CP poly(methoxy,5-(2′-ethyl-hexyloxy-1,4-phenylene-vinylene)) (MEH-PPV) with 2,4,7-trinitrofluorenone (TNF).3 It was also shown that the threshold concentration is independent of the donor concentration in the range 0.25−7.0 mM.3 Furthermore, photoluminescence (PL) quenching experiments in MEH-PPV:TNF solution indicate an inhomogeneous CTC distribution over polymer chains.18 An exciton can migrate efficiently along the polymer chain,19,20 reach a quencher (for example, CTC), and become quenched.21,22 Hence, the PL intensity of a CP is very sensitive to the number and location of quenching sites (CTCs) on the chain. However, a dramatic increase of the CTC concentration above the threshold results in only moderate enhancement of PL quenching.18 It was concluded © 2013 American Chemical Society

that the CTCs are inhomogeneously distributed and form intrachain aggregates, making PL less sensitive to their number.18 Both the threshold-like CTC formation and intrachain CTC aggregation could be explained by the hypothesis of positive feedback suggested in ref 3. It assumes that new CTCs are preferentially formed near the existing ones. According to ref 3, CTC includes one acceptor molecule and two segments of CP chains, so that acceptor molecule is sandwiched between the segments (Figure 1). Upon the CTC formation, these conjugated

Figure 1. The illustration of the positive feedback hypothesis from ref 3.

polymer segments become more planar.6 The planarization can facilitate π-orbital overlapping between the chains involved in the CTC and nearby acceptor molecules. Therefore, one can expect a higher probability of new CTC formation near the existing one, which would result in cluster-like CTC distribution. However, the interchain mechanism of the positive feedback proposed in ref 3 is not necessary. The threshold acceptor concentration (after which intensive CTC formation begins) does not depend on the donor concentration in the MEHPPV:TNF blend.3 We consider therefore that the positive feedback in CP complexation can have an intrachain origin. One of the possible intrachain mechanisms, also mentioned in ref 3, is the following. CTC formation results in chain planarization.6 Received: April 8, 2013 Revised: July 2, 2013 Published: August 9, 2013 10913

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solubility of a particular compound can show a significant temperature and solvent dependence, the overall single CTC binding energy E0 also depends on the temperature and solvent. The neighbor effect is considered as follows (Figure 2). If two CTCs are formed on the adjacent segments, the energy

In its turn, chain planarization can increase the energy of the highest occupied molecular orbital (HOMO) of the donor segments adjacent to the complexed one. Hence, the difference between the polymer HOMO and acceptor lowest unoccupied molecular orbital (LUMO) will be lower. This can result in an increase of CTC binding energy according to the Mulliken model.23 Both of the explanations, interchain and intrachain positive feedback, are only qualitative. The reasonable step now is to describe quantitatively the CTC distribution over the conjugated polymer chains. In this study, we address the issue of such a description in the frames of the positive feedback hypothesis regardless of its particular microscopic mechanism. The positive feedback can be regarded as a particular case of the neighbor effect that is widely known in polymer chemistry.24−27 The notion “neighbor effect” means that the probability of a reaction (CTC formation in our case) with a polymer unit depends on the state of neighbor units, i.e., whether they have reacted or not. The theory of neighbor effect in polymer-analogous reactions was thoroughly developed in the middle of the 20th century and is described in detail in ref 27. Several methods were suggested for the description of kinetics, the fraction of the reacted sites, and their distribution.28−34 These methods can be divided into two groups. In one of them, balance equations are solved for groups of repeat units with 0, 1, and 2 reacted neighbors.30−32 The other group deals with the relations between the probabilities to find reacted units’ sequences of different length.27−29,34 However, equations in both approaches are very difficult to be solved analytically even with some approximations being made.27 In the current study, we suggest an original approach to describe the dynamic equilibrium between the CTCs and free donors and acceptors considering the neighbor effect. Our model allows us to obtain the equilibrium CTC concentration and distribution over polymer chains. It is based on the basic laws of statistical mechanics, namely, on the Gibbs canonical distribution. This approach has a clear physical background, as it operates in terms of energy and number of available microstates and mathematically is rather simple. Hence, in the future, it can be generalized to take into account complex polymer effects including interchain interaction and conformational changes. We show that our model predicts a threshold character of the CCTC(Ca) dependence and an inhomogeneous CTC distribution. The experimental CCTC(Ca) dependence for MEH-PPV:TNF blend solution can be precisely described by the model. We also utilize numerical simulation to investigate the validity of the approximations and provide a visualization of the complexation process.

Figure 2. An illustration of the neighbor effect model of CTC formation. Violet circles depict CTCs on the CP chain. For each of the CTCs, binding energy is shown.

gain is Ene as compared to the formation of the two isolated CTCs. The binding energy of each CTC also increases by Ene, as dissociation of either CTC results in energy change Ene. In other words, it is energetically more favorable for the two CTCs to be formed nearby. The effect is considered to be additive: a cluster of K CTCs provides energy gain of (K − 1)Ene, which corresponds to the number of “bonds” between complexed chain segments. If N CTCs on the chain are distributed into m groups, the total energy gain due to the neighbor effect will be Ene(N − m). We also assume that the probabilities to find a CTC cluster on the chain end and at any position inside the chain are equal. Although the origin of the neighbor effect is unknown, we expect that Ene does not depend on the temperature or solvent. The system is dynamic; i.e., the process of CTC formation competes with the process of CTC dissociation. We will investigate the equilibrium state, where these two processes compensate each other. Let the microstate define the particular distribution of donors, acceptors, and CTCs. We will attempt to find the probability of existence of N CTCs on the chain that are distributed into m clusters (it will be a macrostate). According to the statistical mechanics, the probability p of finding a macrostate “i” can be obtained using the Gibbs canonical distribution: p(i ) =

Γ(i)e−E(i)/ kT ∑j Γ(j)e−E(j)/ kT

(1)

where Γ(i) is the statistical weight of the macrostate, E(i) is the energy of the macrostate, T is the temperature, k is the Boltzmann constant, and summation in the denominator runs over all the available macrostates. Thus, the probability to find N CTCs on the chain distributed into m clusters is

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ANALYTICAL MODEL The Model Construction. We will examine an isolated polymer chain in solution and neglect interchain interactions assuming that they do not play the key role in the CP complexation process. Thus, imagine the following situation. There is a cubic lattice with one polymer chain and N0 cells where Na acceptor molecules can be distributed. As a CTC can include more than one repeat unit,6 we will describe the polymer chain length in “donor segments” each of which can form one CTC. The polymer length will be therefore L segments if maximum L CTCs can be formed on the chain. If an acceptor molecule is in one of the L cells in the immediate vicinity of the polymer chain, a CTC is formed. We consider that there is only one acceptor molecule per CTC. The single CTC binding energy is E0. It is worth noting that E0 is lower than the energy of donor−acceptor interaction in a vacuum Eda: E0 = Eda − dEs, where dEs is a decrease in the donor and acceptor solvation energy due to the CTC formation. As the

wNm =

Γch(L , N , m)·Γa(N0 , L , Na , N )·e[E0N + Ene(N − m)]/ kT ∑N ′ , m ′ Γch(L , N ′, m′)·Γa(N0 , L , Na , N ′)·e[E0N ′+ Ene(N ′− m ′)]/ kT (2)

where Γch(L, N, m) is a number of ways to distribute N CTCs into m groups on the chain and Γa(N0, L, Na, N) is a number of ways to distribute (Na − N) free acceptor molecules into (N0 − L) lattice cells. The latter is described by Γa(N0 , L , Na , N ) =

(N0 − L)! (N0 − L − Na + N ) ! (Na − N )! (3)

To obtain Γch, we should multiply the number of ways to place m CTC groups (clusters) on the chain by the number of ways to 10914

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The explicit form of the probability to find N CTCs

distribute N CTCs into these groups, considering that each group contains at least one CTC: (L − N + 1)! (N − 1)! Γch(L , N , m) = · m! (L − N − m + 1)! (m − 1)! (N − m)!

wNm =

distributed in m clusters, wNm, is

(4)

(N0 − L) ! (L − N + 1) ! (N − 1) ! · · ·e[E0N + Ene(N − m)]/ kT m ! (L − N − m + 1) ! (m − 1) ! (N − m) ! (N0 − L − Na + N ) ! (Na − N ) !

∑N ′ , m ′

(N0 − L) ! (L − N ′ + 1) ! (N ′ − 1) ! · · ·e[E0N ′+ Ene(N ′− m ′)]/ kT m ! (L − N ′ − m ′ + 1) ! (m ′ − 1) ! (N ′ − m ′) ! (N0 − L − Na + N ′) ! (Na − N ′) !

Expression 5 is cumbersome and can be simplified, as L (number of donor segments in the chain), Na (number of the acceptor molecules), and N (CTC number) are often much smaller than N0 (number of cells in the lattice); i.e., the solution is dilute. Further, if the donor concentration is low, the CTC concentration can be much lower than the acceptor one.

Therefore, we can neglect the decrease of free acceptor concentration due to its involvement into CTCs, i.e., replace Na − N ≈ Na. We will investigate this simplified model first, and then turn to the general case. Thus, in the case N ≪ Na and Na, L ≪ N0, eq 5 can be rewritten as

(L − N + 1) ! (N − 1) ! · ·e−(Ene·m)/ kT · m ! (L − N − m + 1) ! (m − 1) ! (N − m) !

wNm = ∑N ′ , m ′

(

Na (E0 + Ene)/ kT e N0

(L − N ′ + 1) ! (N ′ − 1) ! · ·e−(Ene·m ′)/ kT · m ′ ! (L − N ′ − m ′ + 1) ! (m ′ − 1) ! (N ′ − m ′) !

The fraction Na/N0 is the relative volume acceptor concentration, which can be expressed via the molar acceptor concentration Ca:

(

N

)

Na (E0 + Ene)/ kT e N0

N′

)

(6)

Using eq 7 and denoting Ca,half =

Na c·Ca = N0 Cs

(5)

Cs −(E0 + Ene)/ kT ·e c

(8)

(7)

where Cs is the molar solvent concentration and c is a coefficient describing the ratio of the acceptor molecular volume to that of the solvent molecule.

wNm(Ca ; Ca,half , L , Ene) =

eq 6 can be rewritten as

(L − N + 1) ! (N − 1) ! · ·e−(Ene·m)/ kT · m ! (L − N − m + 1) ! (m − 1) ! (N − m) !

L

⟨N (Ca ; Ca,half , L , Ene)⟩ =

N′ ⎤

⎥ ⎦

(9)

N m=1

(11)

where wNm is determined by eq 9. The average CTC number per chain can be easily converted into the equilibrium CTC concentration in solution, which will be shown in the data fitting section. The model (eqs 9 and 11) has three parameters: Ca,half, Ene, and L. It is worth noting that, instead of this set of parameters, another one can be utilized: E0, Ene, and L. This follows from eq 8 which provides an unequivocal relationship between the two parameter sets. The latter set seems to be more convenient for data fitting, as it provides a direct estimation of the isolated CTC binding energy. General Case. If a significant part of the acceptors are involved in CTCs (CCTC ≪ Ca is not fulfilled), we should write

∑ wNm(Ca ; Ca,half , L , Ene) m=1

(10)

The average number of CTCs on the chain is

wNm =

Ca Ca,half

∑ (N · ∑ wNm(Ca ; Ca,half , L , Ene)) N =0

N

wN (Ca ; Ca,half , L , Ene) =

Ca,half

⎡ (L − N ′ + 1) ! (N ′ − 1) ! ∑N ′ , m ′ ⎢ m ′ ! (L − N ′ − m ′ + 1) ! · (m ′ − 1) ! (N ′ − m ′) ! ·e−(Ene·m ′)/ kT ⎣

It turns out that Ca,half has a clear physical meaning: it is the acceptor concentration where half of the chain is complexed (see the Supporting Information, section 1 and Figure 4S). The probability wNm to find N CTCs on the chain distributed between m clusters (eq 9) can be used for calculating different properties of the blend, for example, PL quenching. In the current study, we will use the obtained probability for calculating the average CTC number on the chain. One can obtain the total probability to find N CTCs on the chain by summing eq 9 over m:

N

( ) ·( ) Ca

(L − N + 1) ! (N − 1) ! · ·e[(Ene + E0)·N − Ene·m]/ kT · m ! (L − N − m + 1) ! (m − 1) ! (N − m) !

(

⎡ (L − N ′ + 1) ! (N ′ − 1) ! ∑N ′ , m ′ ⎢ m ′ ! (L − N ′ − m ′ + 1) ! · (m ′ − 1) ! (N ′ − m ′) ! ·e[(Ene + E0)·N ′− Ene·m ′]/ kT · ⎣

10915

N − Na

) ( )

Na − N N0

Na − N ′ N0

N ′− Na ⎤

⎥ ⎦

(12)

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(3) The energy gain for the two neighboring CTCs Ene is the only parameter that determines the shape of the model curve. Figure 4 presents ⟨N⟩(Ca) dependencies for different Ene values

instead of eq 9. In contrast to the simple model, N0 should be explicitly calculated using the molar concentration of donor repeat units Cd and the molar solvent concentration Cs: N0 =

Cs·Nu c·Cd

(13)

In eq 13, c is a coefficient describing the ratio of the acceptor molecular volume to that of the solvent molecule and Nu is the number of repeat units in a polymer chain. The general model (eqs 11 and 12) has three fitting parameters: E0, Ene, and L. As compared with the simple model (eqs 9 and 11), it predicts lower CTC concentration values for each particular acceptor concentration. It is reasonable since acceptors’ involvement in CTCs depletes free acceptor concentration. The larger the donor concentration, the larger the difference between the ⟨N⟩(Ca) curves described by eqs 9 and 11 (see the Supporting Information, section 5). Analysis of the Model Curves. In this section, we will address the influence of the model parameters on the ⟨N⟩(Ca) curve. We will conduct this analysis for the simple model described by eqs 9 and 11, where N ≪ Na and the roles of parameters turn out to be particularly clear. Thus, the parameters L, Ene, and Ca,half have the following influence on the model curve ⟨N⟩(Ca) described by eqs 9 and 11: (1) The half-complexation acceptor concentration Ca,half is responsible for “x” axis scaling, and its variation does not affect the curve shape. It follows from eq 9 as wNm (and consequently ⟨N⟩) depends on Ca via the (Ca/Ca,half) ratio. (2) It turns out that ⟨N⟩ is approximately proportional to the number of available sites for CTC formation on the chain L for L ≫ 1 (see the Supporting Information, section 2). Figure 3

Figure 4. Influence of Ene varying on the ⟨N⟩(Ca) curve shape for the model (eqs 9 and 11).

and fixed values of L = 100 and E0 = 3 kT. It can be seen that the curves show threshold character: when Ca exceeds some threshold value, intensive complex formation starts. The larger the Ene, the steeper the threshold. The curves for different Ene values intersect at the point Ca = Ca,half, where half of the chain is complexed. Thus, eq 11 can be approximately rewritten as ⎛ C ⎞ ⟨N ⟩ ≈ L ·γE ⎜⎜ a ⎟⎟ ne C ⎝ a,half ⎠

(14)

where γ E ne (C a /C a,half ) is a function of the acceptor concentration normalized on Ca,half with only one parameter Ene (see the Supporting Information, section 2). Therefore, although the model has three parameters, only one of them determines the shape of the ⟨N⟩(Ca) curve, while the remaining two are responsible only for the axes scaling. If we use another set of parameters, E0, Ene, and L, the situation is similar: only Ene determines the curve shape, L determines the y-axis scale, and E0 (in combination with Ene) is responsible for x-axis scaling.

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NUMERICAL SIMULATION To verify our statistical approach and provide visualization of the clustering tendency, we have conducted Monte Carlo simulation of the polymer chain complexation process. The simulation details are given in the Supporting Information, section 4. The numerical solution perfectly corresponds to the analytical one (see the Supporting Information, section 4, Figures 2S and 3S). Figure 5 shows the dynamics of the chain complexation, corresponding to different Ene values and Ca = Ca,half. In this figure, the chain state at different time moments is presented. One can see that the average size of the clusters is determined by the Ene value. The larger the Ene, the longer is the average cluster and simultaneously the longer is the average free chain fragment between the CTCs. In other words, high Ene values result in significant inhomogeneity of the CTC distribution.

Figure 3. Influence of L varying on the curve shape for the model (eqs 9 and 11): (a) Ene = 3 kT; (b) Ene = 5 kT. Insets: the same curves magnified.

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DATA APPROXIMATION The model can be applied to the absorption data from ref 18 (the solution of the MEH-PPV:TNF blend in chlorobenzene at room temperature), where the threshold-like CCTC(Ca) dependence was observed. The average CTC number per chain can be obtained from the CTC concentration:

presents dependencies of the share of the complexed donor segments versus normalized acceptor concentration for Ene = 3 kT and Ene = 5 kT. It can be seen that the curves for different L values are rather close. Therefore, the curve shape practically does not depend on L within the parameters’ range under consideration, and L is responsible only for “y” axis scaling. Hence, the threshold acceptor concentration does not depend on the donor concentration in the solution. This is in correspondence with experimental observations in ref 3.

⟨N ⟩ = 10916

CCTC ·Nu Cd

(15)

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Figure 5. Visualization of the chain complexation at Ca = Ca,half for Ene = 1 kT (left), Ene = 3 kT (center), and Ene = 5 kT (right). The chain state at different time moments is presented. Green, complexed units; white, free (non-complexed) units.

where Cd is the donor repeat units’ concentration and Nu is the average number of repeat units in the chain. Nu was considered to be about 300 units, as the mean molecular weight of the MEHPPV chain used was about 300-fold weight of the repeat unit. The experimental ⟨N⟩(Ca) dependence is shown in Figure 6 with blue points.

accordance with experimental results presented in ref 3, where it was estimated as 2 ≤ n ≤ 10. The value of E0 ≈ 3.3 kT ≈ 0.08 eV is the same order of magnitude as the typical binding energy of intermolecular complexes.35,36 The energy gain for the formation of the two nearby CTCs was evaluated to be Ene ≈ 4 kT ≈ 0.1 eV; i.e., the CTC binding energy doubles if one of the adjacent polymer segments is also complexed. The presence of CTCs on both of the neighboring segments triples the binding energy. As a result, the neighbor effect appears to play a crucial role in the CTC formation in conjugated polymer:acceptor solutions. It is worth noting that E0 and probably Ene depend on the temperature and solvent. Therefore, the absolute values and the relative CTC binding energy enhancement due to the neighbor effect can depend on these factors. The model implies assumptions Na, L ≪ N0, and was constructed for isolated polymer chains in solution. Therefore, it is not applicable at large donor and acceptor concentrations, especially in the films. Moreover, the model does not take into account an interplay between polymer−polymer, acceptor− acceptor, and polymer−acceptor interactions which actually determines the film morphology. However, it was frequently observed that the film morphology is to a large extent determined by the solution state in pristine conjugated polymers16,37 and polymer:acceptor blends.17 Namely, a polymer chain conformation in solution can determine its conformation in the film, which strongly affects its photophysics.16,38,39 Interchain aggregation which takes place in solution is also transferred to the film.37 Moreover, it was recently found that CTC formation results in polymer chain planarization in MEH-PPV:TNF blend solution, which is inherited in the film.6 This leads to the formation of ordered complexed polymer regions after the film drying.40 The same picture of CTC-induced polymer ordering was observed in refs 38 and 41 for a P3HT-based blend. Therefore, we expect that CTC intrachain aggregates (clusters) predicted by the model in solution will be preserved during film drying. Indeed, the acceptor concentration increases with the solvent evaporation, which is favorable for CTC formation according to the model. Interchain interactions can result in formation of 3D domains of the complexed units. As CTC formation results in polymer chain planarization,6,40 these hypothetic domains will be the areas of ordered (aligned) complexed polymer regions.40 The film can be therefore microscopically structured, and the photophysics will differ significantly for complexed and free areas. Understanding of the neighbor effect origin and evaluation of the domain size can be useful for the improvement of devices based on the CP blends where CTCs are formed.

Figure 6. Experimental data fitting with the model (eqs 11 and 12). Best-fit parameter values are L = 80 ± 12, Ene = 4.0 ± 0.6 kT, E0 = 3.3 ± 0.5 kT. Blue points, absorption data; magenta line, best-fit model curve.

We used the general model (eqs 11 and 12) for experimental data approximation, as according to the experimental conditions, free acceptor concentration decrease due to its involvement in CTCs cannot be neglected. Indeed, at high acceptor concentration, about 20% of the acceptor was involved in the CTCs.18 The model (eqs 11 and 12) has three fitting parameters, namely, L, E0, and Ene. To evaluate c in eq 8, we compared the specific volume of acceptor (TNF) molecule va with the specific volume vs of solvent molecule (chlorobenzene). These volumes were calculated from the density values: μ v= ρNA (16) where μ is the molar mass, ρ is the density, and NA is the Avogadro number. To calculate specific molecular volumes, the liquid state density value was taken for chlorobenzene (ρ = 1.11 g/cm3) and the solid state density was taken for TNF (ρ = 1.747 g/cm3). We have obtained c = 1.7 ± 0.2, assuming that the possible changes in solvent molecular packing due to acceptor addition can result in ca. 10% error. The best-fit values were found to be L = 80 ± 12, Ene = 4.0 ± 0.6 kT, and E0 = 3.3 ± 0.5 kT. The obtained L value corresponds to the involvement of about four polymer units in one CTC; i.e., the CTC stoichiometry is n ≈ 4. It is in 10917

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CONCLUSIONS In summary, the first quantitative model explaining the thresholdlike CTC concentration dependence on the acceptor content in MEH-PPV:TNF blend solution was suggested. We propose a three-parametrical analytical model based on the Gibbs distribution for polymer chain complexation considering the neighbor effect. The parameters are the number of available sites for the CTC formation on the chain, the binding energy of a single CTC, and the additional binding energy taking effect when CTCs are formed nearby. This model was successfully applied to the analysis of the absorption data in MEH-PPV:TNF chlorobenzene solution at room temperature, providing reasonable stoichiometry, a single CTC binding energy of 0.08 eV, and an additional binding energy of 0.1 eV. It means that the neighbor effect in this blend doubles the binding energy for CTCs with one neighbor and triples the binding energy for CTCs with both neighbors. It results in formation of the CTC clusters on the chain, which are expected to remain after the film drying and affect its morphology. It can be concluded therefore that the neighbor effect plays a crucial role in the process of CTC formation in conjugated polymer:acceptor blends.

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ASSOCIATED CONTENT

* Supporting Information S

Derivation of the Ca,half physical meaning and influence of the L variation on the model curve, spectral characteristics of CTCs with different numbers of neighbors, numerical simulation details, and analysis of the ⟨N⟩(Ca) curve for the general case model. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +7(495)9392228. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank Prof. Dr. D. Y. Paraschuk and Dr. O. D. Parashchuk for valuable discussions. This work is supported by Russian Ministry of Science and Education (Contract No. 16.740.11.0249).

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REFERENCES

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