Librations of Probe Molecules in Polymeric Matrixes - The Journal of

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Librations of Probe Molecules in Polymeric Matrixes S. Yu. Grebenkin* and V. M. Syutkin Institute of Chemical Kinetics and Combustion, Novosibirsk, 630090, Russian Federation ABSTRACT: Formation of optical anisotropy under linearly polarized light has been studied in films of poly(n-butyl methacrylate) and poly(n-hexyl methacrylate) doped with azo compound (1-naphthyl-p-azo-methoxybenzene). Time profiles of the absorption anisotropy and absorbance can not be fitted with a commonly used model of anisotropy formation, which includes the transition dipole turns upon cis−trans isomerization and the rotational diffusion: the experimental values of anisotropy happen to be much lower than the calculated ones. To explain the low anisotropy level, fast molecular librations, i.e., stochastic pendulum-like oscillations of azo molecules about their average orientations were introduced. Numerical simulation of anisotropy and absorbance time profiles with the modified model gives estimates for average libration amplitude of 45−60° at temperatures 25−65 K below Tg.

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INTRODUCTION The light-induced optical anisotropy caused by the cis−trans isomerization of azo compound in polymer matrix (polyvinyl alcohol) was reported for the first time by Neporent and Stolbova1 in the early 1960s. In the 1990s, the investigation of light-induced behavior of azo compounds in polymeric matrixes got a new impetus because of the development of devices for information storage and other optical instruments. A great number of original papers devoted to the problem of lightinduced changes in azo-containing polymeric and liquid-crystal systems and a number of reviews2−9 were published. Extensive investigations were carried out to design a new azo-containing polymers in which a large and long-life optical anisotropy can be generated.10−12 From the scientific aspect, works were focused on the anisotropy formation mechanism13−21 and on condensed medium properties.17,22−30 In the latter case, the anisotropy formation was used as a tool to probe the environment of azo molecules. General models proposed to describe the anisotropy generation in amorphous matrix13,31,32 take into account the following processes: (i) the trans → cis isomerization (this process stands for hole-burning) and cis → trans isomerization, (ii) the reorientation of molecules in the trans−cis−trans photoisomerization cycle (named angular redistribution), and (iii) the rotational diffusion. The corresponding system of two integro-differential equations for angular distribution functions of trans- and cis-isomers can be solved using Legendre polynomials or direct computer simulation. In this paper, we report the results of a study on the formation of optical anisotropy induced by linearly polarized light in amorphous matrixes of poly(n-butyl methacrylate) (PnBMA) and poly(n-hexyl methacrylate) (PnHexMA) doped with azo compound 1-naphthyl-p-azo-methoxybenzene (NAMB). The time profiles of anisotropy and absorbance were simulated with the commonly used model described, e.g., in refs 13 and 31. To decrease a number of fitting parameters of © 2014 American Chemical Society

the model, the rotational diffusion constants were determined in special experiments. We have found that the experimental anisotropy value is much lower than the model predicts. To explain the low anisotropy level, molecular librations were introduced. The equations for the angular distribution functions were modified in order to take into account the librations. The average amplitude of librations was estimated.

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EXPERIMENTAL SECTION Sample Preparation. NAMB was synthesized in our laboratory in accordance with ref 33. Its chemical structure is shown in Figure 1. PnBMA (Mw = 337 000 by GPC, Tg = 288 K) and PnHexMA (approx. Mw = 400 000, Tg = 268 K) were purchased from Aldrich and Scientific Polymer Products, Inc., respectively. The films of PnBMA were prepared as follows. The solution of NAMB and PnBMA in chloroform was dried for 20 min on a quartz plate at room temperature to form a film (25 ± 5 μm thick). Then the film was dried under vacuum for 5 h at 313 K. The sample was assembled from seven films. Prior to measurements, the sample was kept in the dark in air at room temperature for 2 years. The azo concentration in the polymer films was about 5 × 10−3 mol/L. The sample of PnHexMA was prepared from a solution of azo and polymer in toluene. The solution was coated onto a quartz plate and dried completely in vacuum at 333 K for 8 h. The resulting concentration of azo and the thickness of polymer layer were close to those of the PnBMA sample. Before each measurement, the samples were kept for at least 16 h at 295 K and then warmed up for 1 h at Tg + 25 K. As a result of warming, the azo compound was completely converted into the trans-form, and the memory of matrix of its history was erased. Then the samples were aged for 1 h at Tg − 6 K and Received: December 20, 2013 Revised: February 7, 2014 Published: February 12, 2014 2568

dx.doi.org/10.1021/jp412476s | J. Phys. Chem. B 2014, 118, 2568−2575

The Journal of Physical Chemistry B

Article

nm) using probe beams polarized along Y and Z axes, respectively (see Figure 2). The values AbsY and AbsZ were combined to give the optical anisotropy Anisotropy(t ) = Abs Y (t ) − AbsZ(t )

and the isotropic absorbance Abs(t ) =

Abs Z(t ) + 2Abs Y (t ) 3

which is independent of angular molecule distribution and is proportional to the isomeric composition. Also, the Abs(t) can be measured using single probe beam polarized at the magic angle (≈ 54.7°) relative to the Z axis; however, to get the anisotropy, we have to measure the absorbance using two different polarizations of the probe beam. Here and below, for simplicity of the presentation, we use the definition of anisotropy which differs from the conventional one by the sign. Figure 1 shows the optical absorption spectra of NAMB in PnBMA before irradiation and after irradiation for 10 min with linearly polarized light of 546 nm. The difference in the spectra before and after irradiation is caused mainly by the difference in isomer ratios. The difference in the spectra recorded using probe light of different polarizations results from an anisotropic distribution of azo molecules. The steady-state fraction of transisomer was estimated to be 75% under irradiation with wavelength of 546 nm at 253 K. Here, we will use the absorbance and anisotropy normalized to the absorbance before irradiation, Abs0 ≡ Abs(0). The normalized values depend on the ratio between the absorption 382 cross sections of isomers, σ382 trans/σcis , at a probe wavelength (at maximum absorption of trans-isomer, 382 nm). This ratio for NAMB was taken to be equal to that for 1-phenylazonaphthalene at the wavelength of maximum absorption of transisomer that amounts to 5.35 The rate constant of dark cis → trans isomerization of NAMB in PnBMA was measured in the range 283−313 K and extrapolated to lower temperatures. Its value was estimated to be 1 × 10−6 and 3 × 10−8 s−1 at 263 and 243 K, respectively. Thus, the dark isomerization is quite negligible in the temperature range studied and does not affect the anisotropy.

Figure 1. UV−vis absorption spectra of NAMB in PnBMA before irradiation (top curve) and after irradiation for 10 min with Zpolarized light of 546 nm: probe light is polarized along the Y axis (middle curve) and the Z axis (bottom curve), T = 233 K. The inset shows the chemical structure of NAMB.

cooled to the temperature of measurement at a rate of about 2 K/min. In accordance with the results of dielectric measurements34 the time of α-relaxation in PnBMA at Tg − 6 K is about 600 s. Structural relaxation slows down strongly as the temperature decreases, therefore we believe that the structure of the matrixes changes only slightly during cooling below Tg − 6 K. Irradiation Setup. To generate optical anisotropy, the samples were irradiated with linearly polarized light with a wavelength of 546 nm (polarizer from LOMO PLC with the extinction ratio of 800 at 546 nm). The line of mercury spectrum was isolated from the radiation of a 500 W highpressure mercury arc lamp. The photon flux (measured with the power meter Thorlabs PM120) at 546 nm was (6.2 ± 0.7) × 1016 photons s−1 cm−2. The light absorption by the sample did not exceed 1%, therefore, the light intensity was considered constant throughout the sample. The polymer films were oriented at an angle of 45° to both the irradiation and probe beams, which were transverse to each other (see Figure 2). The sample temperature was kept constant to within 0.1 K with the measuring accuracy of ±0.5 K. Measurement Routine. To monitor anisotropy formation, the sample absorbances AbsY and AbsZ were measured at the wavelength of the absorption maximum of trans-isomer (382

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RESULTS AND DISCUSSION Anisotropy Decay. Rotational diffusion strongly affects the anisotropy, therefore, to model the anisotropy formation, rotational diffusion constants should be known. They were found from the curves of anisotropy decay after sample irradiation with linearly polarized light of 405 nm for 30 s. Light of 405 nm was chosen because it provides a higher anisotropy level in a shorter period of time. We have compared two anisotropy decay kinetics: after irradiation of a sample of PnHexMA at 223 K with wavelength of 405 nm (for 30 s) and with wavelength of 546 nm (for 10 min). No difference was found, therefore we have concluded that the decay kinetics does not depend on wavelength of preirradiation. Typical curves of anisotropy decay are shown in Figure 3. The curves are strongly nonexponential, which indicates a wide molecule distribution over rotational times. The distribution of probe molecules over rotational times reflects heterogeneity of a polymeric matrix and was observed both below and above Tg.36−40 The duration of irradiation influences the amplitude of anisotropy and has no effect on the shape of decay curve: the time profiles of decay following different irradiation periods can

Figure 2. Schematic representation of the experimental setup, view from above. The irradiation and probe beams are transmitted along the axes Y and X, respectively. Sample S is oriented at an angle of 45° to both axes. Polarizer P1 passes light of 546 nm polarized along Z axis. Polarizer P2, depending on its orientation, passes probe light polarized either along Y axis or along Z axis. 2569



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Figure 3. Time profiles of anisotropy decay recorded at probe wavelength of 382 nm following irradiation with linearly polarized light of 405 nm, NAMB in PnBMA. ‘Anis’ stands for Anisotropy. The lines are fitting curves. Figure 4. Arrhenius plot for the rotational times of NAMB in PnBMA (top) and PnHexMA (bottom). The straight lines represent the best linear fit. The respective values of activation energy are listed in the figure in kJ mol−1.

be superposed by scaling vertical axis. This fact indicates that (i) the molecules with quite different rotational times have close quantum yields of trans → cis isomerization and (ii) the rotational diffusion constants of cis and trans-molecules are close to each other (since the isomer ratio varies with the irradiation duration). The decay curves at different temperatures (except the highest temperature) were fitted simultaneously by a sum of four exponential functions:

usually are out of equilibrium. As a result, below Tg, a more weak Arrhenius-like temperature dependence is often observed.41−44 The Arrhenius-like temperature dependence of τi shown in Figure 4 is a sign of a nonequilibrium structure of the matrixes; due to extremely slow structural relaxation, the structure of the matrixes changes negligibly in the range studied. The average activation energy for the rotation of NAMB in PnBMA (92.0 kJ mol−1) is greater than that for PnHexMA (76.0 kJ mol−1). The values corresponding to the individual ensembles are listed in Figure 4. The weights, Ai, reflect a distribution of azo molecules over polymeric environments with different dynamics. The values of 0.34, 0.31, 0.21, and 0.14 were obtained for A1, ..., A4, respectively, in the range from 233 to 253 K for PnBMA. For PnHexMA, the respective values of Ai were found to be 0.24, 0.27, 0.24, and 0.25 in the range from 193 to 223 K . We consider these values of Ai as estimates for the fractions of dynamically different environments. The rotational diffusion constants were estimated using the equation 1 Drot,i = , i = 1, ..., 4. 6τi (2)

Anisotropy(t , T )/Anisotropy(0, T ) 4

=

∑ Ai exp(−t /τi(T )) i=1

(1)

where Ai are the temperature independent weights and τi(T) are the rotational times. A smaller number of exponential functions does not provide a satisfactory description of the data set. The fitting parameters were the weights Ai and the temperature-dependent times, τi. At the lowest temperatures (233 K for PnBMA, 193 and 203 K for PnHexMA), the slowest rotational time is too long to be determined for experimental period. Therefore, we arbitrarily set the slowest time equal to 107 s, which is much greater than the experimental time window. Besides, at the highest temperature (263 K for PnBMA and 233 K for PnHexMA), the initial anisotropy value is noticeably smaller than that at lower temperatures. We assigned this decrease to the rotation of molecules from the two ‘fastest’ ensembles during preirradiation. By this reason, at the highest temperature, the weights Ai for the two ‘fastest’ ensembles were set as fitting parameters, and at the same time the anisotropy was normalized to the initial anisotropy at adjacent temperature point (253 K for PnBMA and 223 K for PnHexMA). The fitting curves are shown in Figure 3 by lines. Figure 4 shows the Arrhenius plot for the rotational times, τi. The lines represent the best linear fits to the data. It is known, that in equilibrium polymeric matrixes, the temperature dependence of rotational times can be described by the Williams−Landel− Ferry equation.41−43 Below Tg, the characteristic time of structural relaxation becomes extremely long in comparison with an experimental period; therefore, below Tg, samples

Anisotropy Formation and Modeling Procedure. Figure 5 represents the time profiles of anisotropy generated by linearly polarized light with a wavelength of 546 nm in PnBMA and PnHexMA matrixes. The time profiles have a typical form (cf. refs 15, 16, 32). The fast initial growth is caused, to a large extent, by the angular hole burning2,13 which is due to the isomerization of molecules ‘aligned’ along the light polarization. At long times, the anisotropy increase is due to the angular redistribution only, since the absorbance remains constant. At the highest temperatures (233 K for PnHexMA and 263 K for PnBMA), the anisotropy reaches a steady state within the experimental period. Experimental curves were fitted with a conical model of angular redistribution,31 which is a special case derived from 2570



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Simulation Results. Numerical simulations have shown that the anisotropy and absorbance curves cannot be fitted simultaneously with the model used: if the absorbance is fitted well, then the simulated anisotropy greatly exceeds the measured one (see Figure 6).

Figure 5. Formation of optical anisotropy at a wavelength of 382 nm in PnBMA (top) and PnHexMA (bottom) under linearly polarized irradiation at 546 nm. The lines are fitting curves obtained with the libration model developed below. Figure 6. The example of simultaneous fitting of the absorbance (top) and anisotropy (bottom) time profiles with the standard conical model, NAMB in PnHexMA irradiated at a wavelength of 546 nm, 213 K; the absorbance was measured at 382 nm. The inset demonstrates the initial part of anisotropy curve. The lines are fitting curves.

general equations for angular distribution function.13,31 The conical model was chosen because it looks to be the most reasonable simple model. Additionally, it has been shown that different models of angular redistribution give similar anisotropy and absorbance time profiles if the average angle of transition dipole turns upon isomerization is the same.45 A brief description of the model is as follows. We assume that azo molecules absorb light by a purely polarized transition mechanism. We postulate that the transition dipole changes direction upon isomerization by an angle α with equal probability relative to the initial direction. Under these conditions, the kinetic equations for the angular distributions of isomers were derived in explicit form previously.46 On the basis of the fitting results of anisotropy decay, we must use a distribution of molecules over Drot, α, and rate parameters kt and kc. The parameters kt and kc are defined as kt = Iσtϕtc and kc = Iσcϕct, where I is the light intensity, σt and σc are the cross sections of trans- and cis-molecules, and ϕtc and ϕct are the quantum yields of trans → cis and cis → trans isomerization, respectively. We used a discrete distribution consisting of four groups of probe molecules. The weights of distribution and the values of Drot,i (i = 1, ..., 4) were set equal to those obtained from fitting the anisotropy decay. Thus, the fitting parameters are the effective rate constants, kt,i and kc,i, and the turn angles of transition dipole, αi, where i = 1, ..., 4. We considered that the transition moment directions for the π−π* (probe light, 382 nm) and n−π* (irradiation, 546 nm) transitions coincide (this is justified for trans-47 but debatable for cis-molecules). The absorbance and anisotropy time profiles were fitted simultaneously. The absorbance curve strongly depends on the rate parameters, kt,i and kc,i, and only slightly on αi. Therefore, to a first approximation, the values of kt,i and kc,i were determined from the absorbance curve. We imposed an additional restriction on the rate parameters: the equality of steady-state isomer ratios for all i. The anisotropy time profile strongly depends on all the parameters.

There is one obvious way to decrease the simulated anisotropy: to decrease the values of turn angles αi. However, if the asymptotic values of simulated anisotropy are decreased to the levels of experimental ones, then the simulated absorbances deviate strongly from the experimental values. Besides, at all values of turn angles, the initial growth of simulated anisotropy remains much greater than that of the experimental one (the inset in Figure 6 represents a typical example). This occurs because the initial growth is controlled mainly by the angular hole burning, which depends only slightly 382 on α but strongly depends on the ratio σ382 trans/σcis at a probe wavelength (382 nm). Then we conclude that there is an additional process that leads to a decrease in anisotropy. In our opinion, such a process is the fast librations of azo molecules. We considered that, due to the librations, the directions of transition dipole of a molecule are restricted by a cone with apex angle of 2Ω. All the directions inside this cone are equally probable. Thus, the librations result in averaging of the molecule absorbance over all the directions of transition dipole restricted by a cone with apex angle 2Ω. Modified equations for the absorbance and angular distribution functions are given in the Appendix. Figure 7 shows the anisotropy and absorbance time profiles simulated using the libration angles, Ωi, equal to 54, 54, 54 and 41° for i = 1, ...,4, respectively. The values of αi are equal to 68, 45, 8.4 and 4°, respectively. Except the libration angles, all the fitting parameters are the same as they were used for simulation the curves shown in Figure 6. The residual curves in Figure 7 demonstrate satisfactory fitting quality. Using the modified model, the anisotropy formation kinetics in PnHexMA and PnBMA matrixes were fitted satisfactorily (see Figure 5). In PnBMA, at all temperatures, the librations amplitudes were determined to be 58, 58, 58 and 46° for i = 1, ..., 4, respectively. 2571



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absorption of cis isomer, the anisotropy is greater than in the case of a rod-like one. Moreover, we have simulated the anisotropy for the cases of (i) the isotropic cis-isomer absorption and (ii) the disc-like cisisomer absorption with the disc oriented transversely to NN bond. In all the cases, the deviation of the cis absorption from the perfectly anisotropic rod-like form leads to an increase in anisotropy. Then we conclude that the presence of an additional absorption component of cis-isomer cannot explain the low level of anisotropy. A nonrod-like absorption of trans-isomer could be a reason for low anisotropy. Equations for absorbances (AbsZ and AbsY) in the case of nonlibrating ‘ellipsoid-like’ molecules are similar to the equations in case of librating ‘rod-like’ molecules (the latter equations are demonstrated in the Appendix). We have deduced that the absorption in the form of symmetrical ellipsoid with the cross sections ratio, σy/σz, of 0.29 for both isomers, at probe and irradiation wavelengths, results in the same anisotropy as the librations of rod-like molecules with an amplitude of 55° (here, σy and σz are the cross sections expressed in principal axes). However, according to the data on n → π*47,49,50 and π → π*50 transitions in trans-azobenzene, and the data on the same transitions in trans-cyano-methoxy-azobenzene48 the absorption of trans-isomer has a rod-like form. Then we conclude that a nonrod-like absorption of azo is not the reason of the low anisotropy observed. An additional argument in favor of librational interpretation of the low anisotropy is the following fact: in a glassy matrix of o-terphenyl doped with NAMB, anisotropy induced by 546 nm is not anomalously low and can be described without invoking librations.46 This fact also points to the dependence of libration amplitude on medium.

Figure 7. The example of simultaneous fitting of the absorbance (top) and anisotropy (bottom) time profiles with consideration for librations, NAMB in PnHexMA, 213 K (the same experimental data as in Figure 6). The lines are fitting curves. The differences between the experimental and fitting curves are shown in the bottom parts of each figure.

Average values of the rotational angle, ⟨α⟩, and libration amplitude, ⟨Ω⟩, are plotted in Figure 8 versus temperature. In

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CONCLUSIONS

It has been shown that in matrixes of PnBMA and PnHexMA doped with azo compound, the value of optical anisotropy induced by linearly polarized light is much less than it follows from modeling the anisotropy and absorbance time profiles. In our opinion, the most probable reason of the low anisotropy is the librations of azo molecules. The model, which takes into account molecular librations, satisfactorily describes the experimental time profiles of optical anisotropy and absorbance for both polymers. The amplitude of librations in PnHexMA increases with temperature. In the range from Tg − 65 K to Tg − 35 K, the average amplitude of librations in PnHexMA was estimated to lie within the range from 45 to 60°. The same value for PnBMA was estimated to be 56°, at Tg − 25 K. We conclude that molecular librations can strongly affect the value of optical anisotropy generated in glassy solids.

Figure 8. (a): average rotational angle ⟨α⟩, (b): average angle of librations ⟨Ω⟩. Squares and circles relate to PnBMA and PnHexMA, respectively. At T − Tg of −45 and −35 K, in the first 100 s, the anisotropy formation in PnBMA is not described well; the respective points are denoted by crossed squares. The error bars reflect the fitting errors. The lines are a guide for the eye.

case of PnHexMA, both angles increase with temperature. No temperature dependence of the average angles is observed for PnBMA, within the accuracy of the analysis. Most probably, this is because the used model does not quite adequately describe anisotropy formation in PnBMA at T − Tg of −45 and −35 K at small times: in the first 100 s, the simulated anisotropy is less than the experimental one. An alternative explanation of the low level of anisotropy involves a disc- or ellipsoid-like absorption of azo molecules (in the former case the absorption is nonzero for two Cartesian components of light polarization, and in the latter case for three ones). Pedersen et al.48 have shown that in the range of n → π* transition the cis isomer of cyano-methoxy-azobenzene has approximately isotropic absorption in the plane containing C− NN−C, i.e., the absorption has a disc-like form. However, our simulations showed that in the case of the disc-like

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APPENDIX: INTRODUCING MOLECULAR LIBRATIONS IN GENERAL EQUATIONS We start from the assumption that the transition dipole of each molecule executes a libration-type chaotic motion about its principal (average) direction. The principal direction is defined by the polar, θ, and azimuthal, φ, angles in spherical coordinates. We postulate that the allowable transition dipole directions lie inside a cone with opening angle 2Ω. The librations are considered to be very fast compared with the experimental temporal resolution, therefore all the transition dipole directions inside the cone are considered to be equally 2572



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probable. The instantaneous direction of transition dipole inside the cone is defined by the polar, ϑ, and azimuthal, ψ, angles with respect to the cone axis. To calculate total absorbance, we should integrate absorbance of molecules over all principal directions (over θ and φ) and average over all directions which are accessible due to librations (over ϑ and ψ). We denote by σ the absorption cross-section of a molecule at probe wavelength for light polarized along the transition dipole and by n(θ,φ) the angular distribution function of molecules. The irradiation beam is considered to be polarized along Z, hence the angular distribution depends only on θ; then φ can be omitted from the denotation of distribution function, i.e., n(θ) ≡ n(θ,φ). 2πn(θ) sinθ dθ expresses the number of molecules in 1 cm3 whose transition dipole directions lie in the range from θ to θ + dθ. Starting equations for the absorbances of a 1 cm thick sample measured with Z and Y-polarized light are given by the following equations: Ω

2π

AbsZ = 0.434σ

2π

d n t (θ ) = −Iσtϕtc(C1 + C2 cos2 θ )nt(θ ) + Drot Δnt(θ ) dt Iσcϕct + π θ+α nc(θ′)(C1 + C2cos2 θ′) sin θ′dθ′

∫|θ−α|

(5)

Here, σt and σc are the absorption cross sections of trans- and cis-molecules, respectively, for light polarized along the transition dipole at the wavelength of irradiation, Drot is the rotational diffusion constant and Δ is the Laplacian. To obtain eq 5, we took into consideration that the transition moment directions at probe (382 nm, π−π* transition) and irradiation (546 nm, n−π*) wavelengths almost coincide.47 Then, taking into account eq 3, we replaced the squared cosines, cos2 θ and cos2 θ′, in eq A.6 of ref 46 with their average values C1 + C2 cos2 θ and C1 + C2 cos2 θ′. The kinetic equation for nc(θ) can be obtained by replacing the subscripts t and c in eq 5 with c and t, respectively.

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π

∫ψ =0 ∫ϑ=0 ∫φ=0 ∫θ=0 n(θ)

sin ϑd ϑdψ cos θin(θ , ϑ , ψ ) sin θ dθ dφ , 2π (1 − cos ϑ) 2

2π

Abs Y = 0.434σ

Ω

2π

*E-mail: [email protected]. Notes

π

The authors declare no competing financial interest.

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2

sin θin(θ , ϑ , ψ ) cos φin(θ , φ , ϑ , ψ ) sin θ dθ sin ϑ dϑ dψ dφ 2π (1 − cos ϑ)

ACKNOWLEDGMENTS The authors are grateful to B. Bol’shakov for kindly providing NAMB. This work was supported by the Russian Foundation for Basic Research, project No 08-03-00632-a.

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where cos2 θin(θ,ϑ,ψ) ≡ (cos ϑ cos θ + sin ϑ sin θ cos ψ)2, ‘in’ means instantaneous. Integration of cos2 φin(θ,φ,ϑ,ψ) over φ gives π; it becomes clear if we express φin in the following form: φin(θ,φ,ϑ,ψ) = φ + Δφ(θ,ϑ,ψ). After integration over φ, ψ and ϑ, we obtain the final equations: Abs Z = 0.434 × 4πσ

∫0

π /2

∫0

n(θ )(C1 + C2 cos2 θ) sin θ dθ

π /2

n(θ )(C1 +

sin θ dθ

C2 sin 2 θ ) 2 (4)

with C1 =

2 − cos Ω(3 − cos2 Ω) , 6(1 − cos Ω)

C2 =

cos Ω sin 2 Ω . 2(1 − cos Ω)

REFERENCES

(1) Neporent, B.; Stolbova, O. The Orientational Photodichroism of Viscous Solutions. Opt. Spectrosc. 1961, 10, 146. (2) Dumont, M. Photoinduced Orientational Order in Dye-Doped Amorphous Polymeric Films. Mol. Cryst. Liq. Cryst. 1996, 282, 437− 450. (3) Atassi, Y.; Chauvin, J.; Delaire, J. A.; Delouis, J.-F.; FantonMaltey, I.; Nakatani, K. Photoinduced Manipulations of Photochromes in Polymers: Anisotropy, Modulation of the NLO Properties and Creation of Surface Gratings. Pure Appl. Chem. 1998, 70, 2157−2166. (4) Delaire, J. A.; Nakatani, K. Linear and Nonlinear Optical Properties of Photochromic Molecules and Materials. Chem. Rev. 2000, 100, 1817−1845. (5) Ichimura, K. Photoalignment of Liquid-Crystal Systems. Chem. Rev. 2000, 100, 1847−1873. (6) Yager, K. G.; Barrett, C. J. All-Optical Patterning of Azo Polymer Films. Curr. Opin. Sol. State Mater. Sci. 2001, 5, 487−494. (7) Yu, Y.; Ikeda, T. Alignment Modulation of AzobenzeneContaining Liquid Crystal Systems by Photochemical Reactions. J. Photochem. Photobiol. C 2004, 5, 247−265. (8) Sekkat, Z. In Photoreactive Organic Thin Films; Sekkat, Z., Knoll, W., Eds.; Academic Press: San Diego, CA, 2002; pp 63−104. (9) Sekkat, Z.; Knoll, W. In Photoreactive Organic Thin Films; Sekkat, Z., Knoll, W., Eds.; Academic Press: San Diego, CA, 2002; pp 107− 143. (10) Chen, J. P.; Labarthet, F. L.; Natansohn, A.; Rochon, P. Highly Stable Optically Induced Birefringence and Holographic Surface Gratings on a New Azocarbazole-Based Polyimide. Macromolecules 1999, 32, 8572−8579. (11) Wu, Y.; Natansohn, A.; Rochon, P. Photoinduced Birefringence and Surface Relief Gratings in Novel Polyurethanes with Azobenzene Groups in the Main Chain. Macromolecules 2001, 34, 7822−7828.

(3)

Abs Y = 0.434 × 4πσ

AUTHOR INFORMATION

Corresponding Author

∫ψ =0 ∫ϑ=0 ∫φ=0 ∫θ =0 n(θ)

2

sin 2 α sin 2 θ′ − (cos θ − cos α cos θ′)2

In the absence of librations (Ω = 0), C1 = 0 and C2 = 1. The kinetic equations for angular distribution functions of trans- and cis-isomers, nt(θ) and nc(θ), in the absence of librations, are derived elsewhere.46 Taking into account librations, we have the following equation for nt(θ): 2573



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Librations of Probe Molecules in Polymeric Matrixes - The Journal of

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