Kinetics of photoinduced ordering in azo-dye films:two-state and diffusion models

Kinetics of photoinduced ordering in azo-dye films:
two-state and diffusion models

Alexei D. Kiselev kiselev@iop.kiev.ua Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Institute of Physics of National Academy of Sciences of Ukraine, prospekt Nauki 46, 03028 Kyïv, Ukraine    Vladimir G. Chigrinov eechigr@ust.hk    Hoi-Sing Kwok eekwok@ust.hk Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
July 16, 2019
Abstract

We theoretically study the kinetics of photoinduced ordering in azo-dye photoaligning layers and present the results of modeling performed using two different phenomenological approaches. A phenomenological two state model is deduced from the master equation for the one-particle distribution functions of an ensemble of two-level molecular systems by specifying the angular redistribution probabilities and by expressing the order parameter correlation functions in terms of the order parameter tensor. Using an alternative approach that describes light induced reorientation of azo-dye molecules in terms of a rotational Brownian motion, we formulate the two-dimensional (2D) diffusion model as the free energy Fokker-Planck equation simplified for the limiting regime of purely in-plane reorientation. The models are employed to interpret the irradiation time dependence of the absorption order parameters defined in terms of the the principal extinction (absorption) coefficients. Using the exact solution to the light transmission problem for a biaxially anisotropic absorbing layer, these coefficients are extracted from the absorbance-vs-incidence angle curves measured at different irradiation doses for the probe light linearly polarized parallel and perpendicular to the plane of incidence. It is found that, in the azo-dye films, the transient photoinduced structures are biaxially anisotropic whereas the photosteady and the initial states are uniaxial.

photoinduced anisotropy; kinetic equations; azo-dye films
pacs:
61.30.Gd, 42.70.Gi, 82.50.Hp

I Introduction

It has long been known that some photosensitive materials such as compounds containing azobenzene and its derivatives may become dichroic and birefringent under the action of light. This phenomenon — the so-called effect of photoinduced optical anisotropy (POA) — has a long history dating back almost nine decades to the paper by Weigert Weigert (1919).

The Weigert effect (POA) has been attracted much attention over the past few decades because of its technological importance in providing tools to produce the light-controlled anisotropy. For example, the materials that exhibit POA are very promising for use in many photonic applications Eich et al. (1987); Natansohn et al. (1992); Prasad et al. (1995); Blinov et al. (1999).

It is also well known that producing substrates with anisotropic anchoring properties is one of the key procedures in the fabrication of liquid crystal electrooptic devices. The traditional method widely used to align liquid crystal display cells involves mechanical rubbing of aligning layers and has a number of the well known difficulties Chigrinov (1999). The photoalignment technique suggested in Refs. Gibbons et al. (1991); Schadt et al. (1992); Dyadyusha et al. (1992) is an alternative method that avoids the drawbacks of the mechanical surface treatment by using linearly polarized ultraviolet (UV) light to induce anisotropy of the angular distribution of molecules in a photosensitive film O’Neill and Kelly (2000); Chigrinov et al. (2008). Thus the phenomenon of POA (the Weigert effect) is at the heart of the photoalignment method.

Light induced ordering in photosensitive materials, though not being understood very well, can generally occur by a variety of photochemically induced processes. These typically may involve such transformations as photoisomerization, crosslinking, photodimerization and photodecomposition (a recent review can be found in Ref. Chigrinov et al. (2003a); Chigrinov et al. (2008)).

So, the mechanism underlying POA and its properties cannot be universal. Rather they crucially depend on the material in question and on a number of additional factors such as irradiation conditions, surface interactions etc. In particular, these factors combined with the action of light may result in different regimes of the photoinduced ordering kinetics leading to the formation of various photoinduced orientational structures (uniaxial, biaxial, splayed).

POA was initially studied in viscous solutions of azodyes Neporent and Stolbova (1963) and in azodye-polymer blends Todorov et al. (1984), where the anisotropy was found to be rather unstable. This is the case where the photoinduced anisotropy disappears after switching off the irradiation Neporent and Stolbova (1963); Todorov et al. (1984); Dumont and Sekkat (1993); Dumont (1996); Sekkat et al. (2002); Raschellà et al. (2007). By contrast to this case, POA can be long term stable.

The stable POA was observed in polymers containing chemically linked azochromophores (azopolymers) Eich et al. (1987). It turned out that stable anisotropy can be induced in both amorphous and liquid crystalline (LC) azopolymers Eich et al. (1987); Natansohn et al. (1992); Holme et al. (1996); Petry et al. (1993); Wiesner et al. (1992); Blinov et al. (1998); Yaroshchuk et al. (2001).

The photoalignment has also been studied in a number of similar polymer systems including dye doped polymer layers Gibbons et al. (1991); Furumi et al. (1999), cinnamate polymer derivatives Schadt et al. (1992); Dyadyusha et al. (1992); Galabova et al. (1996); Perny et al. (2000) and side chain azopolymers Petry et al. (1993); Holme et al. (1996); Blinov et al. (1998); Wu et al. (2000). In addition, the films containing photochemically stable azo dye structures (azobenzene sulfuric dyes) were recently investigated as new photoaligning materials for nematic liquid crystal (NLC) cells Chigrinov et al. (2002); Chigrinov et al. (2003b); Kiselev et al. (2005).

In Ref. Chigrinov et al. (2003b), it was found that, owing to high degree of the photoinduced ordering, these films used as aligning substrates are characterized by the anchoring energy strengths comparable to the rubbed polyimide films. For these materials, the voltage holding ratio and thermal stability of the alignment turned out to be high. The azo-dye films are thus promising materials for applications in liquid crystal devices.

According to Ref. Kiselev et al. (2005), the anchoring characteristics of the azo-dye films such as the polar and azimuthal anchoring energies are strongly influenced by the photoinduced ordering. In this paper the kinetics of such ordering will be of our primary interest. More specifically, we deal with theoretical approaches and related phenomenological models describing how amount of the photoinduced anisotropy characterized by absorption dichroism evolves in time upon illumination and after switching it off.

There are a number of models Pedersen and Michael (1997); Pedersen et al. (1998); Puchkovs’ka et al. (1998); Sajti et al. (2001); Yaroshchuk et al. (2001); Kiselev (2002a); Sekkat et al. (2002) formulated for azocompounds exhibiting POA driven by the trans-cis photoisomerization. Generally, in these models, a sample is treated as an ensemble of two level molecular systems: the stable trans isomers characterized by elongated rod-like molecular conformation can be regarded as the ground state molecules whereas the bent banana-like shaped cis isomers are represented by the excited molecules.

The photoisomerization mechanism assumes that the key processes behind the orientational ordering of azo-dye molecules are photochemically induced trans-cis isomerization and subsequent thermal and/or photochemical cis-trans back isomerization of azobenzene chromophores.

Owing to pronounced absorption dichroism of photoactive groups, the rate of the photoinduced isomerization strongly depends on orientation of the azo-dye molecules relative to the polarization vector of the actinic light, . Since the optical transition dipole moment is approximately directed along the long molecular axis, the molecules oriented perpendicular to are almost inactive.

When the cis isomers are short-living, the cis state becomes temporary populated during photoisomerization but reacts immediately back to the stable trans isomeric form. The trans-cis-trans isomerization cycles are accompanied by rotations of the azo-dye molecules that tend to minimize the absorption and become oriented along directions normal to the polarization vector of the exciting light . Non-photoactive groups may then undergo reorientation due to cooperative motion Holme et al. (1996); Natansohn et al. (1998); Puchkovs’ka et al. (1998); Kiselev (2002a); Sekkat et al. (2002).

The above scenario, initially suggested in Ref. Neporent and Stolbova (1963), is known as the regime of photoorientation (angular redistribution) where the lifetime of cis isomers is short and POA is mainly due to the angular redistribution of the long axes of the trans molecules during the trans–cis–trans photoisomerization cycles. Note that, in the opposite case of long-living cis isomers, the regime of angular hole burning (photoselection) occurs so that the anisotropy is caused by angular selective burning of mesogenic trans isomers due to stimulated transitions to non-mesogenic cis form Dumont et al. (1994); Blinov et al. (1998); Kiselev (2002a).

From the above it might be concluded that, whichever regime of the ordering takes place, the photoinduced orientational structure results from preferential alignment of azo-dye molecules along the directions perpendicular to the polarization vector of the actinic light, , determined by the dependence of the photoisomerization rate on the angle between and the long molecular axis. So, it can be expected that the structure will be uniaxially anisotropic with the optical axis directed along the polarization vector.

Experimentally, this is, however, not the case. For example, constraints imposed by a medium may suppress out-of-plane reorientation of the azobezene chromophores giving rise to the structures with strongly preferred in-plane alignment Yaroshchuk et al. (2001). Another symmetry breaking effect induced by polymeric environment is that the photoinduced orientational structures can be biaxial Wiesner et al. (1992); Buffeteau and Pézolet (1998); Kiselev et al. (2001); Yaroshchuk et al. (2001); Kiselev (2002a); Yaroshchuk et al. (2003) (a recent review concerning medium effects on photochemical processes can be found in Kaanumalle et al. (2005)).

It was recently found that, similar to the polymer systems, the long-term stable POA in the azo-dye SD1 films is characterized by the biaxial photoinduced structures with favored in-plane alignment Kiselev et al. (2008a). Unlike azopolymers, photochromism in these films is extremely weak so that it is very difficult to unambiguously detect the presence of a noticeable fraction of cis isomers.

As compared to the polymer systems, modeling of photoinduced ordering in the azo-dye films has received little attention. In this paper we intend to fill the gap and describe the symmetry breaking and biaxiality effects using phenomenological models formulated on the basis of a unified approach to the kinetics of POA Yaroshchuk et al. (2001); Kiselev (2002a). The layout of the paper is as follows.

In Sec. II.1, we introduce necessary notations and discuss the relationship between the order parameter and the absorption tensors. Then, in Sec. II.2, we recapitulate the theory Kiselev (2002a) by assuming that the azo-dye molecules can be represented by two level molecular systems. This theoretical approach is based on the master equation combined with the kinetic equation for the additional (matrix) system, which phenomenologically accounts for the presence of long-living anisotropic (angular) correlations.

In Sec. II.3, a phenomenological two state model is introduced by specifying the angular redistribution probabilities and by expressing the order parameter correlation functions in terms of the order parameter tensor. In this model, the regime of photoorientation with short living excited molecules is characterized by weak photochromism and negligibly small fraction of cis isomers that rapidly decays after switching off irradiation.

According to Ref. Chigrinov et al. (2004), when the photochemical processes underlying photoisomerization are hindered, the process of photoinduced reorientation can be alternatively described as rotational diffusion of azo-dye molecules under the action of the polarized light.

In Sec. III.1, we show that diffusion models of POA can be formulated as the free energy Fokker-Planck equation Frank (2005) describing light induced reorientation of azo-dye molecules as rotational Brownian motion governed by the effective mean field potential. Using this approach, the diffusion model suggested in Chigrinov et al. (2004) can be easily extended to the case of biaxial orientational structures. In Sec. III.2, we introduce and study the simplified two-dimensional (2D) diffusion model that can be regarded as the first approximation representing the regime of purely in-plane reorientation.

The two state and 2D diffusion models are employed to interpret the experimental data in Sec. IV. Finally, in Sec. V we present our results and make some concluding remarks. Technical details on solving the light transmission problem for a biaxially anisotropic absorbing layer and on using the analytical result to extract the extinction coefficients from the measured dependence of absorbance on the incidence angle are relegated to Appendix.

Figure 1: Frame of reference: the axis is normal to the substrate and the polarization vector of the activating light is directed along the axis.

Ii Master equation and two-state models

ii.1 Order parameters, absorption tensor and biaxiality

We assume that azo-dye molecules are cylindrically symmetric and orientation of a molecule in the azo-dye film can be specified by the unit vector, , directed along the long molecular axis. Quadrupolar orientational ordering of the molecules is then characterized using the traceless symmetric second-rank tensor de Gennes and Prost (1993)

(1)

where is the identity matrix.

The dyadic (1) averaged over orientation of molecules with the one-particle distribution function , describing the orientation-density profile of azo-dye molecules, is proportional to the order parameter tensor

(2)

where , , is the density profile and is the normalized angular distribution.

Throughout the paper we restrict ourselves to the case of spatially homogeneous systems with . For such systems, the order parameter tensor is given by

(3)
(4)

where , and are the eigenvalues of the order parameter tensor ; the eigenvector corresponding to the largest in magnitude eigenvalue, , , is the unit vector known as the director; is the biaxiality parameter and the eigenvectors form a right-handed orthonormal tripod.

In our case, the axis is directed along the polarization vector of the activating UV light, , the axis is normal to the substrates and the unit vector is parallel to the axis (see Fig. 1). On symmetry grounds, it can be expected that the basis vectors define the principal axes of the order parameter tensor. So, the tensor is given by

(5)

Then the dielectric tensor, , can also be written in the diagonal form

(6)

In the presence of absorption, the tensor (6) is complex-valued and its principal values, , are expressed in terms of the refractive indices, and the extinction coefficients, , as follows Born and Wolf (1999):

(7)

We can now define the absorption order parameters through the relation

(8)

where the optical densities are proportional to the extinction coefficients: Note that the optical density [] can be determined experimentally by measuring the absorption coefficient for a testing beam which is propagating along the normal to the film substrate (the axis) and is linearly polarized parallel [perpendicular] to the polarization vector of the activating UV light (the axis).

Now, following Ref. Yaroshchuk et al. (2003); Kiselev et al. (2005), we dwell briefly on the relation between the orientational and the absorption order parameters defined in Eq. (5) and Eq. (8), respectively. To this end, we begin with the absorption tensor of an azo-dye molecule

(9)

which is assumed to be uniaxially anisotropic. Its orientational average takes the following matrix form

(10)
(11)

where the angular brackets denote orientational averaging (see Eq. (3)).

In the low concentration approximation, the optical densities are proportional to the corresponding components of the tensor (10)

(12a)
(12b)
(12c)

and on substituting the expressions for the optical densities (12) into Eq. (8) we obtain

(13)

where . So, the absorption order parameters (8) are equal to the corresponding elements of the order parameter tensor (5) only in the limiting case where absorption of waves propagating along the long molecular axis is negligibly small: and . Note that the average optical density is proportional to and thus typically does not depend on the irradiation dose.

ii.2 Master equation

We shall assume that the azo-dye molecules can be represented by the two-level molecular systems with the two states: the ground state and the excited state. Angular distribution of the molecules in the ground state at time is characterized by the number distribution function , where is the volume and is the corresponding one-particle distribution function.

Similarly, the azo-dye molecules in the excited state are characterized by the function: . Then the number of molecules in the ground and excited states is given by

(14)
(15)

where is the total number of molecules; and are the concentrations of non-excited (ground state) and excited molecules, respectively; .

The normalized angular distribution functions, , of the ground state () and the excited () molecules can be conveniently defined through the relation

(16)

linking the one-particle distribution function, , and the corresponding concentration, .

The presence of long-living angular correlations coming from anisotropic interactions between azo-dye molecules and collective modes of confining environment can be taken into account by using the phenomenological approach suggested in Refs. Kiselev (2002b); Yaroshchuk et al. (2003). In this approach, the effective anisotropic field, that results in the long-term stability effect and determines angular distribution of the molecules in the stationary regime, is introduced through the additional angular distribution function, . characterizing the additional subsystem that, for brevity, will be referred to as the matrix system.

It bears close resemblance to the equilibrium distribution of the mean field theories of photoinduced optical anisotropy Pedersen and Michael (1997); Pedersen et al. (1998); Sajti et al. (2001). In these theories, this distribution has been assumed to be proportional to , where is the mean-field potential that depends on the order parameter tensor.

We shall write the kinetic rate equations for in the general form of master equation Gardiner (1985); van Kampen (1984); Breuer and Petruccione (2002):

(17)

where .

The first term on the right hand side of Eq. (17) is due to rotational diffusion of azo-dye molecules. In frictionless models this term is absent. It will be considered later on in Sec. III.

Now we need to specify the rate of the transition stimulated by the incident UV light. For the electromagnetic wave linearly polarized along the –axis the transition rate can be written as follows Dumont and Sekkat (1992); Dumont (1996):

(18)
(19)

where is the tensor of absorption cross section for the molecule in the ground state oriented along : ; is the absorption anisotropy parameter; is the photon energy; is the quantum yield of the process and describes the angular redistribution of the molecules in the excited state; is the pumping intensity and .

Similar line of reasoning applies to the transition to yield the expression for the rate:

(20)

where and , is the lifetime of the excited state and the anisotropic part of the absorption cross section is disregarded, .

Equation (20)) implies that the process of angular redistribution for spontaneous and stimulated transitions can be different. All the angular redistribution probabilities are normalized so as to meet the standard normalization condition for probability densities:

(21)

Using the system (17) and the relations (18)-(20) it is not difficult to deduce the equation for :

(22)

where the angular brackets stand for averaging over the angles with the distribution function . Owing to the condition (21), this equation does not depend on the form of the angular redistribution probabilities.

The last square bracketed term on the right hand side of (17) describes the process that equilibrates the absorbing molecules and the matrix system in the absence of irradiation. The angular redistribution probabilities meet the normalization condition, so that thermal relaxation does not change the total fractions and . If there is no angular redistribution, then and both equilibrium angular distributions and are equal to .

The latter is the case for the mean field models considered in Pedersen and Michael (1997); Pedersen et al. (1998); Sajti et al. (2001). In these models the excited molecules (cis fragments) are assumed to be long-living with and . We can now recover the models by setting the angular redistribution probabilities and equal to the equilibrium distribution, , determined by the mean-field potential : . So, the mean field approach introduces the angular redistribution operators acting as projectors onto the angular distribution of the matrix system. This is the order parameter dependent distribution that characterizes orientation of the azo-molecules after excitation.

An alternative and a more general approach is to determine the distribution function from the kinetic equation that can be written in the following form Kiselev (2002a):

(23)

Equations for the angular distribution functions and can be derived from (17) by using the relations (18)–(22). The result is as follows

(24)
(25)

The system of equations (22) and (23)–(25) can be used as a starting point to formulate a number of phenomenological models of POA. We have already shown how the mean field theories of Pedersen and Michael (1997); Pedersen et al. (1998); Sajti et al. (2001) can be reformulated in terms of the angular redistribution probabilities.

ii.3 Two-state model

We can now describe our two state model. To this end, we follow the line of reasoning presented in Refs. Kiselev (2002a); Yaroshchuk et al. (2003).

In this model, the angular redistribution probabilities and are both assumed to be isotropic:

(26)

Since we have neglected anisotropy of the excited molecules, it is reasonable to suppose that the equilibrium distribution of such molecules is also isotropic, , so that

(27)

From the other hand, we assume that there is no angular redistribution

(28)

and the equilibrium angular distribution of molecules in the ground state is determined by the matrix system: .

Equilibrium properties of excited and ground state molecules are thus characterized by two different equilibrium angular distributions: and , respectively. It means that in our model the anisotropic field represented by does not influence the angular distribution of non-mesogenic excited molecules.

(29a)
(29b)

where is the order parameter correlation function given by

(30)

The key point of the approach suggested in Ref. Kiselev (2002a) is the assumption that the correlators (30) which characterize response of azo-dye to the pumping light and enter the kinetic equations for the order parameter components (29), can be expressed in terms of the averaged order parameters .

(31a)
(31b)
(31c)
(31d)

where , , , and .

It was shown that the parabolic approximation used in Ref. Yaroshchuk et al. (2001) can be improved by rescaling the order parameter components: with computed from the condition that there are no fluctuations provided the molecules are perfectly aligned along the coordinate unit vector : at . In Ref. Kiselev (2002a) this heuristic procedure has also been found to provide a reasonably accurate approximation for the correlators calculated by assuming that the angular distribution of molecules can be taken in the form of distribution functions used in the variational mean field theories of liquid crystals de Gennes and Prost (1993); Chaikin and Lubensky (1995).

ii.3.1 Long term stability and photosteady state

Mathematically, our model is described by equations for the order parameters and the concentration given in Eq. (31) and Eq. (22), respectively. We may now pass on to discussing some of its general properties.

Our first remark concerns the effect of the long term stability of POA. It means that there is the amount of the photoinduced anisotropy preserved intact for long time after switching off the light. Clearly, this is a memory effect and the system does not relax back to the off state characterized by irradiation independent equilibrium values of the order parameters.

In order to see how this effect is described in our model, we assume that the activating light is switched off at time and consider subsequent evolution of the order parameters at . In the absence of irradiation, Eq. (31) decomposes into two decoupled identical systems of equations. for two pairs of the order parameters: and . So, without the loss of generality we may restrict ourselves to the evolution of the components of order parameters, governed by the equations

(32)
(33)

supplemented with the initial conditions

(34)

At , equation for the concentration (22) is easy to solve. So, for the initial value problem with , we have

(35)

From Eq. (35) it is clear that, in the limiting case of short living excited state with , the concentration of excited molecules, , rapidly decays to zero. In this regime, the first term on the right hand side of Eq. (32) is negligibly small and can be disregarded. Equations (36) and (37) can now be easily solved to yield the formulas

(36)
(37)

where and

(38)

Evidently, the order parameters defined in Eqs. (36) and (37) evolve in time approaching the stationary value (38). The memory effect manifests itself in the dependence of the stationary order parameter, , on the (initial) conditions (34) at the instant the activating light is switched off.

The photosteady states reached in the long irradiation time limit are represented by stationary solutions of the system (29) and the concentration equation (22).

The steady state concentration of excited molecules can be expressed in terms of the steady state order parameter, , through the relation

(39)

where . From Eq. (39) it can be seen that, in the case where the life time of the excited state is short and the ratio is large, , the fraction of the excited molecules is negligible, so that .

On substituting Eq. (19) into the steady state relation

(40)

derived from Eqs. (29) and (22) we obtain equation for the steady state order parameters

(41)

From Eq. (29b) the difference between the order parameters of the matrix system and the ground state azo-dye molecules dies out as the photosteady state is approached, at . Interestingly, Eq. (41) shows that the order parameters, , characterizing the regime of photosaturation are independent of the light intensity, .

For the specific form of the correlators used to obtain the system (31), the photosteady state is uniaxial with and the component of the order parameter tensor, , can be found by solving the equation

(42)

Iii Nonlinear Fokker-Planck equations and diffusion model

iii.1 Mean field Fokker-Planck equations

In this section we extend the diffusion model Chigrinov et al. (2004) by using the approach based on Fokker-Planck (F-P) equations of the following general form Risken (1989):

(43)

where and is the probability density (distribution function); is the drift vector and is the diffusion tensor.

As opposed to the linear case, in nonlinear F-P equations, either the drift vector, , or the diffusion tensor, , depend on the distribution function, : and . The theory and applications of such equations were recently reviewed in the monograph Frank (2005). Interestingly, according to Refs. Curado and Nobre (2003); Schwämmle et al. (2007), nonlinear F-P equations are derived by approximating the master equation with nonlinear effects introduced through the generalized transition rates.

More specifically, we concentrate on the special case of the so-called nonlinear mean-field Fokker-Planck (F-P) equations

(44)

that are characterized by the effective free energy functional, . In addition, the case of the rotational Brownian motion will be of our primary interest.

In order to derive the mean-field F-P equations describing the rotational diffusion, the nabla operator on the left-hand side of Eq. (44) , which is proportional to the linear momentum operator and represent the generators of spatial translations, should be replaced by the angular momentum operator representing the generators of rotations:  Favro (1960); Doi and Edwards (1988). This gives the rotational F-P equation in the following form

(45)

where is the rotational diffusion tensor and the components of the angular momentum operator, , expressed in terms of the Euler angles, , are given by Biedenharn and Louck (1981)

(46a)
(46b)
(46c)

When the effective free energy functional is a sum of two term that represent the contributions coming from the effective internal energy, , and the Boltzmann entropy term

(47)

the variational derivative of the free energy takes the form

(48)

where is the mean-field potential.

On substituting the relation (48) into Eq. (45) we obtain the mean-field F-P equation

(49)

describing the rotational diffusion governed by the mean-field potential (48). This equation can be conveniently cast into the form

(50)

where the right-hand side is rewritten using the operator

(51)

which is quadratic in the components of the angular momentum operator, .

In the linear case where the potential, , is independent of the angular distribution function, , the F-P equation (49) has been used to study dielectric and Kerr effect relaxation of polar liquids based on the rotational diffusion model the rotational motion of molecules in the presence of external fields Dejárdin et al. (1993); Dejárdin and Kalmykov (2000); Kalmykov and Quinn (1991); Felderhof (2002); Kalmykov (2001); Kalmykov and Titov (2007). Rotational diffusion of a probe molecule dissolved in a liquid crystal phase was investigated in Tarroni and Zannoni (1991); Berggren et al. (1993); Brognara et al. (2000).

When molecules and the orientational distribution function are cylindrically symmetric, the model can be described in terms the angle between the electric field and the molecular axis Dejárdin et al. (1993); Dejárdin and Kalmykov (2000), whereas angular distributions of a more general form require using both azimuthal and polar angles that characterize orientation of the molecules Kalmykov and Quinn (1991); Felderhof (2002). In this case, for uniaxial (rod-like, calamatic) molecules, the distribution function is independent of the Euler angle : , and the angular momentum operator can be expressed in terms of the azimuthal and zenithal (polar) angles, and , as follows

(52)

where

(53a)
(53b)
(53c)

When the rotational diffusion tensor is diagonal, , and and its elements are angular independent, the operator (51) can be written in the simplified form:

(54)

In the isotropic case with , we have

(55)

A more complicated biaxial case occurs for asymmetric top molecules Kalmykov (2001), macromolecules in liquid solutions Kalmykov and Titov (2007) and probes in the biaxial liquid crystal phase Berggren et al. (1993). For such low symmetry, analytical treatment cannot be simplified and involves the three Euler angles, .

Nonlinearity in the lowest order approximation can be introduced through the truncated expansion for the internal energy functional retaining one-particle (linear) and two-particle (quadratic) terms

(56)

where and is the symmetrized two-particle kernel. The the effective potential

(57)

is the sum of the external field potential, , and the contribution coming from the two-particle intermolecular interactions.

For rod-like azo-dye molecules, the one-particle part of the effective potential (57) can be written as a sum of the two terms:

(58)

where the light-induced contribution

(59)

comes from the interaction of azo-molecules with the activating UV light and the surface-induced potential

(60)

takes into account conditions at the bounding surfaces of the azo-dye layer.

Assuming that the two-particle interaction is of the Maier-Saupe form