Theory of Light-Induced Deformation of Azobenzene Elastomers

Article pubs.acs.org/JPCB

Theory of Light-Induced Deformation of Azobenzene Elastomers: Effects of the Liquid-Crystalline Interactions and Biaxiality Vladimir Toshchevikov*,†,‡ and Marina Saphiannikova† †

Leibniz-Institut für Polymerforschung, Hohe Str. 6, 01069 Dresden, Germany Institute of Macromolecular Compounds, Bolshoi pr. 31, Saint-Petersburg 199004, Russia

‡

ABSTRACT: We study light-induced deformation of azobenzene elastomers which can display liquid-crystalline (LC) order. It is shown that photomechanical behavior of azobenzene elastomers is determined by the strength of the LC interactions, which is proportional to the density of rodlike azobenzene chromophores. At weak LC interactions, a uniaxial order and uniaxial deformation of azobenzene elastomers along the polarization vector of the light E is observed. At strong LC interactions, the light is able to induce a phase transition from the uniaxial to the biaxial state, with two axes being related to the vector E and to a preferable alignment of the chromophores in the plane perpendicular to E. The phase transition can be of either the first or the second order. Azobenzene elastomers can demonstrate elongation or contraction along the polarization vector E, depending on the orientation distribution of chromophores around the main chains of network strands. The results of the theory are in a qualitative agreement with experiments and computer simulations, which demonstrate biaxial ordering in azo-containing polymers.

1. INTRODUCTION Azobenzene elastomers belong to a class of smart functional materials which are able to change their shape under light illumination.1−15 Typically, these compounds represent a twocomponent system consisting of a weakly cross-linked polymer network and the azobenzene chromophores attached covalently to the network. Although the external light field acts only on the chromophores, this light action is transferred to the polymer matrix because of mechanical coupling between the phases. As a result, the azobenzene elastomers exhibit large-magnitude photoinduced deformation. Obviously, the azobenzene elastomers have a great potential for practical applications serving as lightcontrollable sensors, actuators, microrobots, micropumps, artificial muscles, etc. Therefore, understanding of the complex structure−property relationships in these compounds is very important for targeted synthesis of new light-controllable materials with predefined photomechanical properties. Azobenzene chromophores change their shape from the rodlike trans-state to the bent cis-state under light illumination in a wide range of wavelengths (360−540 nm). This effect is widely used to produce photodeformation in liquid crystalline (LC) azobenzene networks. The bent cis-isomers destabilize the LC phase and induce a transition of the LC elastomer from the nematic to isotropic state; this transition is accompanied by a contraction of the sample with respect to the nematic director.3−15 On the other side, the photodeformation is also possible for azobenzene networks which are macroscopically isotropic1,8,10,11 and can be even in the glassy state.16 In the latter case, azobenzene networks are deformed with respect to the © 2014 American Chemical Society

polarization direction of the light, E. The deformation is caused by the anisotropic character of photoisomerization: maximal probability for a transition from the trans- to cis-state is achieved at an orientation of the rodlike chromophore parallel to the electric vector of the light E. Under visible light irradiation, the rodlike chromophores are reoriented preferably perpendicular to the vector E after multiple trans−cis−trans photoisomerization cycles.17−21 Reorientation of chromophores results in the lightinduced mechanical stress and deformation.22−28 Under ultraviolet (UV) irradiation, most of the trans-isomers are transformed to the cis-state and orientation anisotropy appears without the cyclic trans−cis−trans photoisomerisation process.29−31 The present study deals with photomechanical behavior of azobenzene polymers under visible light irradiation, when the cyclic trans−cis−trans photoisomerization takes place. To describe theoretically the reorientation of azobenzene chromophores under visible light illumination, an effective light-induced orientation potential was introduced independently by Chigrinov et al.20 and by one of the authors.21 The strength of the orientation potential is proportional to the intensity of the light. Recently, this approach was used to study light-induced deformation of both un-crosslinked glassy azobenzene polymers22−24 and cross-linked azobenzene networks.25−28 Contribution of the light-induced orientation potential into the orientation free energy has been taken into account in ref 22, and the mechanical stress has been Received: June 25, 2014 Revised: September 23, 2014 Published: September 25, 2014 12297

dx.doi.org/10.1021/jp5063226 | J. Phys. Chem. B 2014, 118, 12297−12309

The Journal of Physical Chemistry B

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calculated. It was shown22 that the mechanical stress appearing at typical light intensities used in experiments (Ip ∼ 0.1 W/cm2) is comparable with the yield stress (σY ∼ 50 MPa) typical for azobenzene polymers and can be even higher than the yield stress. Note that recent experimental investigation32 confirms this theoretical finding and indicates that the light-induced stress can be even of the order of 100 MPa at the light intensity Ip = 0.1 W/cm2. Thus, the proposed orientation approach allows the explanation of the possibility for irreversible light-induced deformation of glassy azobenzene polymers, especially the possibility for inscription of surface relief gratings33,34 onto azo-containing glassy polymer films. Furthermore, it was shown, in agreement with experiments35−38 and computer simulations,39−41 that depending on the chemical structure the azo-containing polymers can either expand or contract along the polarization direction E.22−28 The orientation approach was used recently28 to study the light-induced bending deformation. Bending, as a three-dimensional movement, represents a special interest for design of artificial muscles and was observed in many experimental works.8−16 As was shown in ref 28, theoretically calculated bending angle as a function of the number density of azo-moieties is in a good agreement with the experimental data presented in ref 16. Thus, agreement of the orientation approach with both experiments and computer simulations demonstrates its great potential strength in studying the photomechanical properties of azobenzene polymers of different structures. Note that in the previous theoretical studies22−28 only effects of reorientation of chromophores with respect to the polarization direction E were taken into account, whereas possible interactions between the chromophores were neglected. As a result, only a uniaxial order with respect to the vector E was found for the azobenzene chromophores under homogeneous light illumination. At the same time, many experiments demonstrate the light-induced biaxial orientation order in azo-containing polymers, as was detected with the aim of UV, Fourier transform infrared spectroscopy, and polarizing optical microscopy techniques.42−46 To our knowledge, there are no theoretical works that investigate the light-induced biaxial order and biaxial deformation of azopolymers under irradiation with linearly polarized visible light. To overcome these drawbacks, we develop in this study the previous approaches22−28 by taking into account both the reorientation of chromophores with respect to the polarization direction of the visible light and the orientation LC interactions between rodlike azobenzene chromophores. We demonstrate that the orientation interactions are able to induce an additional order of chromophores in the plane perpendicular to the polarization vector of the light and that the polymer is transformed into the biaxial state. It will be shown that the light-induced biaxiality can be observed not only for liquid-crystalline polymers but also for isotropic polymers, in which the orientation interaction between chromophores is not enough to produce the LC phase in the absence of the light.

Figure 1. Network model built from freely jointed chains consisting of N rodlike Kuhn segments of length l. Each Kuhn segment includes a single azobenzene chromophore.

azobenzene chromophores in side chains.41 Orientation of the chromophores around the backbones is defined by the distribution function, W(α), where α is the angle between the long axis of a chromophore and a Kuhn segment (Figure 1). Varying the form of the function W(α), one can describe both the main-chain azobenzene elastomers (when α = 0 for all chromophores) and the side-chain azobenzene elastomers. In the latter case, the function W(α) is defined by the potentials of internal rotations and by the length of spacers connecting the chromophores with the main chains. Under illumination with the linearly polarized visible light of a proper wavelength (400−540 nm), the chromophores demonstrate cyclic trans−cis−trans photoisomerization that leads to their preferable orientation perpendicular to the polarization vector E of the light.17−19 As in refs 20−28, the light-induced orientation anisotropy of chromophores is described by introducing an effective orientation potential acting on each chromophore V (θ ) = V0 cos2 θ

(1)

where θ is the angle between the long axis of the rodlike trans isomer and the polarization vector E. The strength of the potential, V0, is related to the light intensity Ip20,21 V0 = CIp

(2)

The coefficient C is estimated at room temperature as C ≅ 10−19 J·cm2/W and is related to the optical parameters of chromophores: C = γυτ/2, where γ is the absorption coefficient, υ the volume of azobenzene, and τ the effective transition time between two isomer states.20,21,28 Interaction of the chromophores with the light changes the conformations of network strands because of covalent coupling of the chromophores to the network strands and leads to deformation of azobenzene elastomers. To study the light-induced deformation, we use the Gaussian approach for the statistics of network strands. In this approach, the end-to-end distance of network strands is approximated by the Gaussian distribution

2. MODEL OF AN AZOBENZENE ELASTOMER. ORIENTATION FREE ENERGY OF THE CHROMOPHORES To study photomechanical properties of azobenzene elastomers, a network model built from freely jointed chains consisting of N rodlike Kuhn segments of length l is used (Figure 1). It is assumed that each Kuhn segment bears only a single azobenzene chromophore. Such polymer chain statistics was found in computer simulations for polyethylene macromolecules containing 12298



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μ defines an additional ordering in the plane perpendicular to the vector E. The value of μ can be rewritten as μ = ⟨cos 2θy⟩ − ⟨cos 2θz⟩, where cos θβ denotes the cosine of the angle between the long axis of the chromophore and the β-axis (β = x, y, z): cos θx = cos θ, cos θy = sin θ cos φ, and cos θz = sin θ sin φ. Thus, the order parameter μ determines the orientation biaxiality around the vector E: μ = 0 for the uniaxial state, ⟨cos2 θy⟩ = ⟨cos2 θz⟩, whereas μ ≠ 0 for the biaxial state, ⟨cos2 θy⟩ ≠ ⟨cos2 θz⟩. The order parameter μ takes its maximal value μ = 1 for the anisotropic state with fully oriented chromophores along the y-axis, when ⟨cos2 θy⟩ = 1 and ⟨cos2 θx⟩ = ⟨cos2 θz⟩ = 0. The term ULC can be related with the ordering matrix Ŝ:47

Figure 2. Euler angles θ and φ defining the orientation of an azobenzene chromophore.

function, which neglects the finite extensibility of network strands. It was shown that the Gaussian approach describes very well the light-induced deformation of azobenzene elastomers built from long enough network strands (N ≫ 1) in a broad region of deformation which can reach even 100% of relative deformation. 25−28 The light-induced deformation of an azobenzene elastomer in the framework of the Gaussian approach is determined by the orientation order of azobenzene chromophores with respect to the polarization vector E.25−28 We consider below the orientation ordering of the chromophores under light illumination from analysis of the free energy taking into account the orientation liquid crystalline interactions between the rodlike chromophores. The orientation potential (eq 1) reorients the chromophores preferably perpendicular to the vector E. At the same time, orientation interactions between the chromophores can result in additional preferable orientation of the chromophores along some axis inside the plane perpendicular to the vector E. We introduce a rectangular frame of references, in which the x-axis is parallel to the vector E and the y-axis reflects the preferable orientation of chromophores in the plane perpendicular to the vector E (Figure 2). The orientation free energy per chromophore now reads as follows: F = kT

̂ = (3⟨cos θξ cos θβ⟩ − δξβ)/2 Sξβ

where ξ,β = x, y, z are indices referring to the laboratory frame and δξβ is the Kronecker symbol. Because of the symmetry with respect to the transformations ξ → (−ξ) for ξ = x, y, z, the matrix Ŝ takes the following diagonal form in the introduced frame of references: ⎛S ⎞ 0 0 ⎜ ⎟ ⎜ 0 − S + 3μ ⎟ 0 ⎟ Ŝ = ⎜ 2 4 ⎜ ⎟ 3μ ⎟ S ⎜⎜ 0 ⎟ 0 − − ⎝ 2 4 ⎠

(3)

where k is the Boltzmann constant and T is the absolute temperature. The first term determines the orientation entropy of the chromophores which is related to the orientation distribution function of chromophores, f(Ω) with respect to the Euler angles, Ω (θ, φ) (Figure 2). Here, φ is the angle between the y-axis and the projection of the long axis of the chromophore on the yz-plane. Integration in eq 3 is defined as follows: dΩ = sin θ dθ dφ, θ ∈ [0,π] and φ ∈ [0,2π]. The second term in eq 3 determines contribution of the light-induced potential V(Ω), given by eq 1, to the orientation free energy of chromophores. The third term in eq 3 is the energy of the orientation liquidcrystalline interactions between rodlike chromophores. The energy of orientation interactions for (possibly) biaxially ordered rodlike objects reads in the framework of the mean-field approach as follows:47,48 U 2 [S + 3μ2 /4] (4) 2 where U > 0 is the strength of the LC interactions; S and μ are two scalar order parameters ULC = −

2

∫ dΩ f (Ω) 3 cos 2θ − 1

μ=

∫ dΩ f (Ω) sin2 θ cos 2φ

(7)

2 2 One can see that the trace of the matrix Ŝ equals tr(Ŝ ) = 2 2 (3/2)[S + 3 μ /4]. Thus, the term ULC in eq 4 is directly 2 proportional to the first invariant of the matrix Ŝ . For the uniaxial state (μ = 0), the term ULC takes a standard form ∼US2/2 which appears in the classical mean-field Maier−Saupe theory.47,49 The parameter U in eq 4 characterizes the strength of the LC interactions. It is directly proportional to the density of rodlike azobenzene chromophores. The coefficient of proportionality is determined by the depth of the potential well for an effective pairwise interaction between rodlike azobenzene chromophores as well as by geometrical parameters of rodlike chromophores, e.g., by their aspect ratio. Note that under light illumination the density of rodlike chromophores can decrease because a part of rodlike trans-isomers is transformed into the bent cisisomers.4,17−19 Therefore, the parameter U can change its value under light illumination becoming a function of the light intensity. Below, we will vary the parameters V0 and U independently to obtain general results for the phase transitions in azobenzene elastomers. Using the general results, the phase transitions in azobenzene elastomers can be easily understood also if the parameters U and V0 change simultaneously with variation of the light intensity including the effects of strong dilution of the LC phase by the bent cis-isomers. The next step now is to find the free energy as a function of S and μ. When eq 1 is substituted into eq 3 and using eqs 4 and 5, the expression for the free energy is rewritten as

∫ dΩ f (Ω) ln f (Ω) + ∫ dΩ f (Ω) V (Ω) + ULC

S=

(6)

F = kT

∫ dΩ f (Ω) ln f (Ω) + V0 2S 3+ 1 − U2 [S2 + 3μ2 /4] (8)

(5)

At fixed values of S and μ, the distribution function f(Ω) should provide the minimum of the free energy. Moreover, f(Ω) should satisfy the normalization condition

The parameter S determines a usual uniaxial order of the chromophores with respect to the vector E, whereas the parameter 12299



∫ dΩ f (Ω) = 1

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3. BIAXIAL ORIENTATION ORDER OF AZOBENZENE CHROMOPHORES UNDER LIGHT ILLUMINATION Depending on the value of the parameter a, different types of the light-induced orientation order of azobenzene chromophores can be observed. One can distinguish three critical values of the parameter a. The critical values a2D = 8/3 ≈ 2.66 and a3D ≈ 4.542 correspond to the isotropic-to-anisotropic phase transitions of the rodlike objects in the 2D- and in the 3D-orientation space, respectively, due to the orientation interactions between the rods at the absence of any external fields.47−49 The phase transition in the 3D-space is of the first order (i.e., the order parameter changes stepwise at the phase transition), whereas the phase transition in the 2D space is of the second order (i.e., the order parameter changes continuously). As we will show below, the orientation potential (eq 1) can induce the phase transitions of the first and second orders in the vicinities of the values a3D and a2D, respectively. We have found that the critical value a∗ ≈ 3.8 determines the boundary between the areas of the parameter a, where the light-induced phase transitions are of the first order (a > a∗) or of the second order (a < a∗). Figure 3a−j illustrates the free energy F as a function of the order parameters S and μ at varying values of the parameters V0/kT and a. To distinguish different minima of the free energy, Figure 3a−j shows the difference between F and its minimal value Fmin in the logarithmic scale. Note that at a fixed value of S the biaxial order parameter μ changes in the range μ ∈ [−μmax, μmax], where μmax = 2(1 − S)/3. We use the reduced coordinate μ/μmax in Figure 3a−j to illustrate the dependence F = F(S,μ). Minimum of the free energy determines the equilibrium values of the order parameters S and μ. Figure 4a−d shows the equilibrium values of S and μ as functions of the parameters V0/kT and a. Depending on the values of the parameter a, four types of the light-induced ordering of chromophores can be distinguished. (i) If a < a2D, the orientation interactions are relatively weak and are not able to form an additional order in the plane perpendicular to the vector E at any values of V0/kT. In this case, the minimum of the free energy shifts to the range of negative values of S along the axis μ = 0 at increasing V0/kT (Figure 3a,b). Thus, the orientation distribution of chromophores in the yz-plane remains isotropic in this case. Figure 4a illustrates the dependences of the equilibrium values of the order parameters on the strength of the potential V0/kT at different values of a for a < a2D. One can see that μ = 0 in this regime at all values of V0/kT and a, i.e., the chromophores demonstrate a uniaxial orientation with respect to the vector E. The increase of both V0/kT and a leads to the increase of the magnitude of the negative order parameter S (Figure 4a). (ii) If a2D < a < a∗, the orientation interactions are strong enough to form an additional orientation order in the plane perpendicular to the vector E at a certain critical value Vc/kT, which depends on the value of the parameter a. In this regime, at small values V0 < Vc , the minimum of free energy shifts to negative values of S along the axis μ = 0 analogously to Figure 3a,b for the previous regime. However, in contrast to the previous regime, at the critical value V0 = Vc , the minimum of free energy splits into two minima with nonzero values of μ, as depicted in Figure 3c,d. The states with μ > 0 and μ < 0 are identical; their difference consists only in the difference between directions of spontaneous alignment of the chromophores inside the yz-plane: either along the y-axis (μ > 0) or along the z-axis (μ < 0). We have chosen the y-axis to be the direction of preferable orientation of

(9)

as well as the conditions given by eq 5. Thus, at any variation of the distribution function, δf(Ω), which satisfies eqs 5 and 9 at fixed values S and μ, i.e.

∫ dΩ δf (Ω)[cos2 θ − 1/3] = 0 ∫ dΩ δf (Ω) sin2 θ cos 2φ = 0 ∫ dΩ δf (Ω) = 0

(10)

the distribution function f(Ω) should provide the variation of the free energy equal to zero: δF = kT

∫ dΩ δf (Ω)[1 + ln f (Ω)] = 0.

(11)

According to the Lagrange variation principle, conditions 10 and 11 are satisfied simultaneously when 1 + ln f (Ω) = C1[cos2 θ − 1/3] + C 2 sin 2 θ cos 2φ + C3 (12)

where Ci (i = 1, 2, 3) are the Lagrange multipliers that satisfy eqs 5 and 9 for the function f(Ω). Using eq 12, we can rewrite the function f(Ω) in the following form: f (Ω) = Z −1 exp[C1 cos 2 θ + C2 sin 2 θ cos 2φ]

(13)

The normalization constant Z is related to the Lagrange multiplier C3 and is determined from condition 9 as follows: Z=

∫ dΩ exp[C1 cos2 θ + C2 sin2 θ cos 2φ]

(14)

The parameters C1 and C2 in eqs 13 and 14 have a simple physical meaning: they represent magnitudes of the selfconsistent orientation fields. The field C1 cos2 θ provides the orientation order with respect to the x-axis. The field C2 sin2 θ cos 2φ results in the additional orientation order in the xy-plane. The values C1 and C2 depend on the order parameters S and μ. The functions C1(S,μ) and C2(S,μ) are determined as a solution of the system of eq 5, where the function f(Ω) is given by eqs 13 and 14. The functions C1(S,μ) and C2(S,μ) were calculated numerically in this study. Substituting C1(S,μ) and C2(S,μ) into eq 14, we calculated numerically the dependence Z = Z(S,μ) as well. Now, using eqs 8 and 13, we obtain the free energy as a function of S and μ: F(S , μ) 2S + 1 = −ln Z(S , μ) + [C1(S , μ) + V0/kT ] kT 3 a + μC2(S , μ) − [S2 + 3μ2 /4] (15) 2

where a = U/kT is the reduced strength of the orientation interactions. Using eq 15, we have calculated the equilibrium values of the order parameters S and μ from the minima of the free energy. The equilibrium values of S and μ are functions of the strength of the light-induced potential V0/kT and of the reduced strength of the orientation interactions a = U/kT, which are included as parameters in eq 15. In the next section, we discuss the dependence F(S,μ) and present the equilibrium values of S and μ as functions of the parameters V0/kT and a. 12300



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Figure 3. Map of the free energy in the logarithmic scale as a function of the order parameters S and μ at different values of the parameters V0/kT and a.

V0/kT at the value V0 = Vc which defines the phase transition from the uniaxial (μ = 0) to biaxial (μ ≠ 0) state. Thus, the phase transition is of the second order in this regime. It can be seen also

the chromophores in the yz-plane; thus, only equilibrium values μ > 0 are considered below. One can see from Figure 4b that equilibrium values of S and μ change continuously as functions of 12301



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Figure 4. Equilibrium values of the order parameters S and μ as well as the order parameter at the metastable state, Smet, as functions of the reduced strength of the light-induced potential V0/kT at different values of a: a < a2D (a), a2D < a < a∗ (b), a∗ < a < a3D (c), and a > a3D (d).

that the critical value Vc/kT decreases with increasing value of the parameter a (Figure 4b). The increase of both V0/kT and a leads to the increase of the magnitudes of both order parameters S and μ (Figure 4b). (iii) If a∗ < a < a 3D, the chromophores can also demonstrate the light-induced phase transition from the uniaxial to biaxial state, but the phase transition is of the first order in this case: both order parameters change stepwise as functions of V0/kT at the phase transition (Figure 4c). As in the previous two regimes, in this case the minimum of free energy shifts to negative values of S along the axis μ = 0 at small values V0 < Vc, analogously to Figure 3a,b. At the critical value Vc/kT, two additional minima of the free energy with nonzero values of μ and with strongly negative values of S appear, as depicted in Figure 3e,f. With further increase of the parameter V0/kT > Vc/kT, the biaxial state (with μ ≠ 0) becomes more favorable and both order parameters change stepwise at the point of phase transition (Figure 4c). Thus, the phase transition is of the first order in this regime. As in the previous regime, the critical value Vc/kT decreases with increasing values of the parameter a. The increase of both V0/kT and a leads to the increase of the magnitudes of both order parameters S and μ (Figure 4c). Interestingly, as was shown above, the light is able to induce the biaxial state even in isotropic elastomers, in which the orientation interactions are relatively weak and not enough to form the LC order at the absence of the light. Below, the photomechanical properties of LC networks with strong orientation interactions are considered. (iv) If a > a 3D, the orientation interactions are so strong that the chromophores demonstrate the nematic state in the absence

of the light. The nematic states are illustrated in Figure 3g,i by three minima of the free energy at S ≠ 0 which correspond to three identical LC states with spontaneous orientation of the nematic director n along the three principal directions (x-, y-, and z-axes). Moreover, in the range a3D < a < 5, the fourth shallow minimum of the free energy at S = μ = 0 is observed, as depicted in Figure 3g. Under illumination with the polarized light, the minima of the free energy with nonzero values of μ become deeper as compared with the minimum/minima along the axis μ = 0 (Figure 3h,j). Thus, the biaxial state becomes more favorable, whereupon the phase transition to the biaxial state depends on the initial orientation of the nematic director n with respect to the polarization direction of the light E. If n is perpendicular to E, then the initial state, which corresponds to the minimum of free energy at μ ≠ 0, becomes most favorable. Thus, the system remains in the initial state under light irradiation. Equilibrium values of the order parameters S and μ are shown by thick lines in Figure 4d in this case. One can see that the magnitudes of equilibrium values of S and μ increase with increasing V0/kT, the values of S and μ being negative and positive, respectively. We recall that the order parameter S defines the orientation order with respect to the vector E, whereas the orientation order with respect to the nematic director n is defined in this case by the order parameter Syy ≡ (3⟨cos2 θy⟩ − 1)/2 = 3μ/4 − S/2. One can deduce from previous findings that the order parameter Syy is positive and increases with increasing V 0/kT. Thus, the application of the electric vector of the light E perpendicular to the nematic director n stabilizes the initial state and raises the orientation order with respect to the nematic director n. 12302



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Dependence of the light-induced ordering on the density of chromophores was observed experimentally.42 In ref 42 it was shown that at low density of chromophores the azobenzene polymer demonstrates uniaxial orientation ordering inside experimental errors, whereas at high concentration of chromophores the biaxial ordering can be observed. This result is in agreement with our finding for types i, ii, and iii of the light-induced ordering. Furthermore, reorientation of the LC director perpendicular to the electric vector E in the LC polymers that leads to the biaxiality (type iv) was confirmed by recent computer simulations.41 Thus, the theoretical findings presented above have an explicit background both from experiments and from computer simulations. To conclude this section, we note that the LC interactions enrich the character of light-induced ordering of azobenzene chromophores. Depending on the degree of LC interactions, the chromophores can demonstrate either uniaxial order or phase transitions (of both first and second orders) from a uniaxial to a biaxial state. Below, we will show that orientation biaxiality of the chromophores results in the biaxial deformation of azobenzene elastomers.

If n is parallel to E, then the initial state, which is defined by the values μ = 0 and S > 0, becomes unfavorable (metastable) (Figures 3g−j). Values of the order parameter S at the metastable states with μ = 0 are presented in Figure 4d by thin lines as functions of the parameter V0/kT. One can see that the order parameter S at the metastable state decreases with increasing V0/kT. Note that the order parameter S defines in this case the orientation order with respect to the nematic director n. Because the initial state is metastable, the system should be stepwise transformed as a function of V0/kT from the initial uniaxial state to the biaxial state with preferable alignment perpendicular to the vector E. Thus, application of the electric vector of the light E parallel to the nematic director n destabilizes the initial state, reduces the orientation order with respect to the initial direction of n, and transforms the chromophores into the LC state with preferable alignment perpendicular to the vector E at high values of V0/kT. It is interesting to point out that the chromophores can display at a3D < a < 5 two metastable uniaxial states with positive and slightly negative values of S; under light illumination, the state with a slightly negative value of S can become more favorable as compared with the state at S > 0 (Figure 3h). Thus, at small values of V0/kT, the chromophores can be transformed from the initial nematic state with positive value of S to a uniaxial metastable state with slightly negative S, as is depicted by arrow A in Figure 4d. At higher values of V0/kT, the minimum of free energy with slightly negative S disappears and the system of chromophores is transformed into a biaxial state, as is shown by arrow B in Figure 4d. In the range a > 5, the uniaxial state with slightly negative value of S is absent (Figure 3j) and the chromophores are directly transformed from the initial uniaxial state to the biaxial state, as is depicted by arrow C in Figure 4d. Here we note that the transformation of chromophores from the LC state to a uniaxial state with slightly negative S and then to the biaxial state with strongly negative S has been found recently in computer simulations (see Figure 4 of ref 41). Thereby, our theory explains for the first time the possibility of such nontrivial behavior. The agreement of our theory with computer simulations validates the results of our study. It is important to point out that the light-induced orientation order is determined by dimensionless parameters V0/kT and a = U/kT. These parameters can change their values additionally because of the heating of a sample under light illumination. Heating under light illumination or so-called photothermal effect was observed experimentally.9,50,51 Thus, the photothermal effects can also influence the light-induced ordering. Moreover, the number density of chromophores, n, can influence the type of the light-induced ordering (i−iv). As stated in the classical theory of the LC phase transitions,47−49 the parameter U is directly proportional to the number density of the rodlike LC groups; thus, a = U/kT ∝ n/T. Using the last relation and the critical value a3D ≈ 4.542, we can estimate the value a as follows: a = a3D·(n/T)·(Tc/nc), where the values Tc and nc correspond to the isotropic-to-nematic phase transition for a given azobenzene polymer. The values Tc and nc depend on the chemical structure of the polymer backbones as well as on the chemical structure of the spacers and can be determined experimentally (see, e.g. Table 1 of ref 44). Thus, using the typical values Tc and nc, one can estimate the characteristic values of T, n, and a, which correspond to different types of light-induced ordering (i−iv).

4. LIGHT-INDUCED DEFORMATION OF AZOBENZENE ELASTOMERS: PHASE TRANSITIONS AND BIAXIALITY Orientation phase transitions induced by the light in the subsystem of chromophores are accompanied by uniaxial and biaxial deformation of azobenzene elastomers due to rigid coupling between the chromophores and network strands. To calculate the elongation ratios λx, λy and λz along the x-, y-, and z-axes, we assume as in a classical theory of rubber elasticity52,53 that network strands deform affinely with the deformation of azobenzene elastomer: bx = bx(0)λx ,

by = b(0) y λy ,

bz = bz(0)λz

(16)

where b and b(0) are the end-to-end vectors of a given network strand in a deformed network under light illumination and in a undeformed network at the absence of the light, respectively (Figure 1). In the framework of the Gaussian approach, the distribution function of the end-to-end vectors b reads as follows:52−55 ⎡ ⎛ 2 ⎤ by2 bz2 ⎞⎥ b ⎟ + P(b) = P0 exp⎢ −⎜⎜ x 2 + ⎢⎣ ⎝ 2⟨bx ⟩ 2⟨by2⟩ 2⟨bz2⟩ ⎟⎠⎥⎦

(17)

where P0 is a normalization constant and ⟨b2β⟩ (β = x,y,z) is the mean-square projection of the end-to-end vectors of network strands on the β-axis in a deformed elastomer. The value ⟨b2β⟩ is related to the averaged projection of the Kuhn segments on the β-axes, ⟨l2β⟩: ⟨bβ2⟩ = N ⟨lβ2⟩

for β = x , y , z

(18)

The conformational free energy of the network strands, Fconf, reads Fconf (λ) = −kT ⟨ln P(b)⟩

(19)

where the averaging is performed over all strands. Substituting eq 17 into eq 19 and using the relationship between the vectors b and b(0) given by eq 16, we obtain the free energy 12303



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Figure 5. Elongation ratios λx, λy, and λz as functions of V0/kT for azobenzene elastomers with preferable orientation of the chromophores parallel to the main chains (q = 1) at different values of the parameter a.

Fconf as a function of elongation ratios λx and λy in the following form: Fconf (λx , λy) =

two equations can be simply solved with respect to the parameters λx and λy when using eq 20. The value λz is determined from the condition of incompressibility, λz = 1/(λxλy). As a result, we obtain the following equations for the elongation ratios:

⎡ 2 2 ⟨ly2,0⟩λy2 ⟨lz2,0⟩ ⎤ kT ⎢ ⟨lx ,0⟩λx ⎥ + + 2 ⎢⎣ ⟨lx2⟩ ⟨ly2⟩ ⟨lz2⟩λx2λy2 ⎥⎦

⎛ ⟨l 2⟩2 ⟨l 2 ⟩⟨l 2 ⟩ ⎞1/6 x y ,0 z ,0 λx = ⎜⎜ 2 2 2 2 ⎟⎟ ⎝ ⟨lx ,0⟩ ⟨ly ⟩⟨lz ⟩ ⎠

(20)

To derive the last equation, the incompressibility condition for 2 elastomers was used, λxλyλz = 1, as well as the relationship ⟨(b(0) β ) ⟩= 2 2 N⟨lβ,0⟩, where ⟨lβ,0⟩ is the averaged projection of Kuhn segments on the β-axis (β = x, y, z) in the absence of the light. Note that ⟨l2x,0⟩ = ⟨l2y,0⟩ = ⟨l2z,0⟩ = l2/3 if the elastomer is in an isotropic state at the absence of light and ⟨l2x,0⟩ ≠ ⟨l2y,0⟩ = ⟨l2z,0⟩ if it is in the LC nematic state. Now it is a simple matter to calculate the equilibrium values of the elongation ratios λx and λy from the minimum of the free energy, ∂Fconf/∂λx = 0 and ∂Fconf/∂λy = 0. The system of the last

⎛ ⟨l 2⟩2 ⟨l 2 ⟩⟨l 2 ⟩ ⎞1/6 y x ,0 z ,0 λy = ⎜⎜ 2 2 2 2 ⎟⎟ ⟨ l ⟩ ⎝ y ,0 ⟨lx ⟩⟨lz ⟩ ⎠ ⎛ ⟨l 2⟩2 ⟨l 2 ⟩⟨l 2 ⟩ ⎞1/6 z x ,0 y ,0 λz = ⎜⎜ 2 2 2 2 ⎟⎟ ⎝ ⟨lz ,0⟩ ⟨lx ⟩⟨ly ⟩ ⎠ 12304

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Figure 6. Same as Figure 5 but for azobenzene elastomers with preferable orientation of chromophores perpendicular to the main chains (q = −1/2).

Averaging over the orientation of Kuhn segments around the long axes of chromophores according to the assumption given above, we obtain the following relationships:

We note that for a uniaxial elastomer produced from an isotropic polymer (i.e., when ⟨l2y ⟩ = ⟨l2z ⟩ and ⟨l2x,0⟩ = ⟨l2y,0⟩ = ⟨l2z,0⟩) the last equations are transformed to classical relationships:56 λx = (⟨l2x⟩/⟨l2y ⟩)1/3 and λy = λz = λ−1/2 . Importantly, a biaxial ordering x (i.e., ⟨l2x⟩ ≠ ⟨l2y ⟩ ≠ ⟨l2z ⟩) leads to biaxial deformation of an elastomer, λx ≠ λy ≠ λz, according to eq 21. Equation 21 contains the averaged quantities ⟨l2β⟩ and ⟨l2β,0⟩ which define the orientation order of the Kuhn segments. The latter can be related to the orientation order of the chromophores. To relate the quantities ⟨l2β⟩ with the order parameters for chromophores S and μ, we assume that the polar orientation of the long axes of Kuhn segments with respect to the long axes of chromophores attached to them is defined by the distribution function W(α), introduced in Section 2 (see Figure 1 as well). The azimuthal orientation of Kuhn segments around the long axes of chromophores is assumed to be random. The latter assumption is trivial for the main-chain liquid crystals; for side-chain liquid crystals it means that network strands are assumed to be not LC objects and are not influenced by the LC interactions.

⟨lx2⟩ =

l2 [1 + 2qS] 3

⟨ly2⟩ =

l2 [1 − q(S − 3μ/2)] 3

⟨lz2⟩ =

l2 [1 − q(S + 3μ/2)] 3

(22)

where the constant q is determined by the chemical structure of an azobenzene elastomer. It defines the orientation distribution of chromophores with respect to the network strands q=

3⟨cos2α⟩W − 1 2

(23)

Here, the averaging is performed with respect to the distribution function W(α). Parameter q can take its values in the range 12305



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Figure 7. Dependences of the elongation ratio λx on the reduced strength of the light-induced potential V0/kT at different values of the structural parameters a and q.

q ∈ [−1/2, 1]. The minimal value q = −1/2 corresponds to chemical structures with preferable orientation of the chromophores perpendicular to the main chains (α = 90°), whereas the maximal value q = 1 corresponds to the structures with preferable orientation of the chromophores parallel to the main chains (α = 0; e.g. for main-chain polymers, or for specifically prepared side-chain polymers). Because the equilibrium value of S is negative (see Section 3), we conclude from eqs 21 and 22 that azobenzene elastomers demonstrate expansion (λx > 1) or contraction (λx < 1) along the polarization direction E if q < 0 or q > 0, respectively. Thus, photomechanical behavior of azobenzene elastomers is very sensitive to the orientation distribution of the chromophores with respect to the main chains, which is determined by the chemical structure. Moreover, because the elongation ratios λx, λy, and λz are explicitly related to the order parameters S and μ according to eqs 21 and 22, the deformation of azobenzene elastomer must display the same phase behavior as the order parameters S and μ

investigated in the previous section: (i) uniaxial deformation, (ii) phase transition of the second order from the uniaxial to biaxial state, (iii) phase transition of the first order from the uniaxial to biaxial state, and (iv) biaxial deformation of LC elastomers. In the latter regime, the character of the light-induced deformation must depend on the orientation of the vector E with respect to the LC director n of an LC elastomer. The above-mentioned findings are illustrated in Figures 5 and 6, where the elongation ratios λx, λy, and λz are presented as functions of V0/kT for elastomers with preferable orientation of chromophores parallel (q = 1, Figure 5) and perpendicular (q = −1/2, Figure 6) to the network strands. Panels a, b, and c in Figures 5 and 6 illustrate the results for the regimes a < a2D, a2D < a < a∗, and a∗ < a < a3D, respectively; the panels d and e show the dependences λx,y,z(V0/kT) at a > a3D, when the electric vector of the light E is applied either perpendicular (panel d) or parallel (panel e) to the LC director n for azobenzene elastomers in the nematic state at a > a3D. In the regime a > a3D, only the results 12306



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5. CONCLUSIONS Effects of the orientation LC interactions between azobenzene chromophores on the light-induced deformations of azocontaining elastomers have been studied in the framework of the mean-field approximation. Gaussian approach for the statistics of network strands has been used to calculate the lightinduced deformation of azobenzene elastomers. Reorientation of the chromophores after multiple trans−cis−trans photoizomerization cycles perpendicular to the polarization vector of the light E was modeled by an effective orientation potential introduced in previous studies.20−28 The strength of the potential is proportional to the intensity of the light. It is shown that orientation LC interactions between the chromophores can result in an additional alignment of the chromophores in the plane perpendicular to the polarization direction E and the biaxial order can be observed in agreement with experiments.42−46 The possibility for the biaxial state is determined by the strength of the LC interactions, which is proportional to the density of rodlike trans-isomers of azomoieties. With increasing strength of the LC interactions, the azobenzene elastomers can demonstrate the following photomechanical behavior: (i) uniaxial ordering and uniaxial deformation along the vector E, (ii) phase transition of the second order from the uniaxial to biaxial state, (iii) phase transition of the first order from the uniaxial to biaxial state, and (iv) biaxial ordering and deformation of LC elastomers. In the last regime, the light-induced deformation depends on the orientation of the vector E with respect to the nematic director n of an LC elastomer. If E is perpendicular to n, then the light irradiation stabilizes the initial LC state and the deformation changes continuously with increasing intensity of the light. If E is parallel to n, then the light illumination destabilizes the initial LC state and reorients the nematic director n perpendicular to the vector E in accordance with computer simulations.41 Depending on the orientation distribution of chromophores with respect to the main chains, which is determined by the chemical structure of network strands, azobenzene elastomers can demonstrate expansion or contraction along the polarization vector of the light in agreement with experiments.35−38 Agreement of our findings with experimental results and computer simulations validates the results of our theory.

for the equilibrium states, which correspond to the global minima of the free energy, are presented. One can see from Figures 5 and 6 that in accordance with above-mentioned findings, azobenzene elastomers demonstrate several types of light-induced deformation: (i) uniaxial deformation (λy = λz) with respect to the vector E at a < a2D (Figures 5a and 6a), (ii) phase transitions of the second order from the uniaxial state (λy = λz) to the biaxial state (λx ≠ λy ≠ λz) at a2D < a < a∗ (Figures 5b and 6b), (iii) phase transitions of the first order from the uniaxial state to the biaxial state at a∗ < a < a3D (Figures 5c and 6c), and (iv) biaxial light-induced deformation of LC azobenzene elastomers at a > a3D (Figures 5d,e and 6d,e). In the regime a > a3D, the light-induced deformation depends on the orientation of the polarization vector E with respect to the nematic director n. If E is applied perpendicular to n (Figures 5d and 6d), then the light irradiation stabilizes the initial LC state (cf. with Section 3) and the deformation changes continuously with increasing V0/kT. If E is applied parallel to n, then the light illumination destabilizes the initial LC state and reorients the nematic director n perpendicular to the vector E. Reorientation of the director is accompanied by stepwise deformation of an elastomer at the point V0 = 0, when the light is switched on, from the values λx,y = 1 up to some values λx,y ≠ 1 (Figures 5e and 6e). Note that in reality the light-induced deformation can change continuously in the vicinity of V0 = 0 because of the presence of metastable uniaxial phases mentioned in Section 3. Moreover, one can see from Figures 5a−e and 6a−e that azobenzene elastomers built from the polymer chains with a preferable orientation of the chromophores parallel or perpendicular to the main chains of network strands demonstrate contraction (λx < 1, Figure 5) or expansion (λx > 1, Figure 6), respectively, with respect to the polarization direction of the light E. To demonstrate in detail that the magnitude of light-induced deformation is very sensitive to the chemical structure of network strands, we present in Figures 7a−e the elongation ratio of an elastomer λx along the vector E as a function of V0/kT at varying values of the structural parameter q ∈ [−1/2, 1]. One can see that variation of the parameter q changes the magnitude of light-induced deformation λx at fixed values of V0/kT and a. Thus, our theory explains the experimental results35−38 that the sign and magnitude of light-induced deformation are very sensitive to the chemical structure of azobenzene polymers. The present theory generalizes the previous theoretical studies22−28 for azobenzene elastomers without LC interactions and demonstrates that the orientation LC interactions between chromophores enrich considerably the character of light-induced deformation of azobenzene polymers. They can display both uniaxial and biaxial order with respect to the electric vector of the light depending on the strength of the LC interactions. Note that light-induced biaxial order of azobenzene polymers is confirmed by experiments42−46 and by recent computer simulations.41 Agreement of our theoretical findings with experiments and computer simulations demonstrates a great strength of the proposed theoretical approach to study photomechanical properties of azobenzene polymers of different chemical structures. Because the proposed theory relates the photomechanical properties of azobenzene polymers with their chemical structure, it can be useful in further practical developments of these lightcontrollable smart materials.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support of the DFG Grant GR 3725/2-2 is gratefully acknowledged.

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REFERENCES

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