Microscopic Theory of Light-Induced Deformation in Amorphous Side

5032

J. Phys. Chem. B 2009, 113, 5032–5045

Microscopic Theory of Light-Induced Deformation in Amorphous Side-Chain Azobenzene Polymers V. Toshchevikov,*,†,‡ M. Saphiannikova,† and G. Heinrich† Leibniz Institute of Polymer Research Dresden, 01069 Dresden, Germany, and Institute of Macromolecular Compounds, 199004 Saint-Petersburg, Russia ReceiVed: September 3, 2008; ReVised Manuscript ReceiVed: January 14, 2009

We propose a microscopic theory of light-induced deformation of side-chain azobenzene polymers taking into account the internal structure of polymer chains. Our theory is based on the fact that interaction of chromophores with the polarized light leads to the orientation anisotropy of azobenzene macromolecules which is accompanied by the appearance of mechanical stress. It is the first microscopic theory which provides the value of the light-induced stress larger than the yield stress. This result explains a possibility for the inscription of surface relief gratings in glassy side-chain azobenzene polymers. For some chemical architectures, elongation of a sample demonstrates a nonmonotonic behavior with the light intensity and can change its sign (a stretched sample starts to be uniaxially compressed), in agreement with experiments. Using a viscoplastic approach, we show that the irreversible strain of a sample, which remains after the light is switched off, decreases with increasing temperature and can disappear at certain temperature below the glass transition temperature. This theoretical prediction is also confirmed by recent experiments. 1. Introduction The role played by azobenzene polymers in modern photonic, electronic, and optomechanical applications cannot be underestimated. These polymers are successfully used to produce alignment layers for liquid crystalline fluorescent polymers in the display and semiconductor technology,1,2 to build waveguides and waveguide couplers,3,4 and as data storage media.5,6 The effect of photomechanical bending of azobenzene polymers opens up possibilities to use these polymers in fluidic actuation.7,8 A very hot topic in modern research is design of light-driven artificial muscles based on azobenzene elastomers.9,10 Incorporation of azobenzene chromophores into polymer systems via covalent bonding gives rise to a number of unusual effects under visible light irradiation. The most amazing effect is the inscription of surface relief gratings (SRGs) onto thin azobenzene polymer films.11,12 Another interesting effect is photoinduced deformation of azobenzene polymer films floating on a water surface. It has been shown by Bublitz et al.13 that different types of azobenzene side-chain polyesters undergo an opposite change of shape under illumination with linear polarized light. This result is in agreement with the observation that the phase of the SRGs, relative to that of the optical grating, strongly depends on the chemical architecture of the particular azobenzene polymer.3,14 Several models15-22 have been proposed to explain the origin of the inscribing force. Most models15-17,19,21,22 provide values of the light-induced stress (σ < 1 kPa) much smaller than values of the yield stress typical for glassy polymers, σ ∼ 10 MPa.23,24 Therefore, to explain the origin of the inscription of SRGs, many authors use a concept of photoinduced softening originally proposed by Tripathy and co-workers.16 According to this concept, illumination of an azobenzene polymer layer with * To whom correspondence should be addressed. E-mail: toshchevikov@ imc.macro.ru. † Leibniz Institute of Polymer Research Dresden. ‡ Institute of Macromolecular Compounds.

visible light induces a transition into a macroscopic low-viscosity melt and such a “molten” polymer can be then irreversibly stretched upon application of a very weak light-induced stress, σ < 1 kPa. However, it was shown recently with the help of three different experimental techniques that illumination with visible light does not affect material properties of an azobenzene polymer such as bulk compliance, Young modulus, and viscosity.24-27 Therefore, one can certainly claim that the polymer remains in a glassy state and, hence, the theories which need a concept of photoinduced softening should be discarded. On the other side, Saphiannikova and Neher28 have recently proposed an alternative thermodynamic approach which explains the formation of surface relief gratings in the absence of photoinduced softening. This approach is based on the striction effect, namely, that orientation anisotropy of molecules in solids is accompanied by the appearance of mechanical stress. A wellknown example is the electrostriction effect for solid dielectrics,29 where deformation is caused by external electric field. Another important example is spontaneous stretching of liquid crystalline elastomers where the elongation is due to intermolecular orientation interactions.30,31 In the case of azobenzene polymers, the orientation anisotropy of polymer chains is caused by the photoinduced reorientation of chromophores. In the thermodynamic approach of Saphiannikova and Neher,28 starting from an anisotropic orientation distribution of chromophores, the light-induced stress comparable with the yield stress typical for glassy polymers has been obtained at some strengths of the light-induced alignment. However, it has not been shown how the strength of light-induced alignment depends on intensity of the laser light used for the inscription of SRGs. Moreover, neither the recent thermodynamic theory28 nor the previous theories15-22 are able to explain the well-known fact that photoelastic properties of azobenzene polymers are very sensitive to the chemical architecture of macromolecules. In order to overcome these drawbacks, we propose here a microscopic approach for description of light-induced deformations in side-chain azobenzene polymers. It is the first time when

10.1021/jp8078265 CCC: $40.75  2009 American Chemical Society Published on Web 03/23/2009

Amorphous Side-Chain Azobenzene Polymers

J. Phys. Chem. B, Vol. 113, No. 15, 2009 5033

the internal architecture of azobenzene macromolecules, i.e., orientation distribution of chromophores with respect to the main chain, is explicitly taken into account. We start from effective potential which was introduced in ref 32 for chromophores in the field of the laser light and obtain values of the light-induced stress higher than the yield stress. The sign of the light-induced stress is found to be very sensitive to the internal architecture of side-chain azobenzene macromolecules due to the elastic coupling of chromophores to the polymer backbones. The paper is organized as follows. In section 2 we develop a general approach for calculating the Helmholtz free energy and equilibrium elongation of a homogeneous sample of an azobenzene (glassy) polymer affected by a homogeneous and linearly polarized laser light. The free energy in our approach contains three parts: orientation entropy of macromolecules, elasticity of a solid sample, and effective potential of chromophores in the field of the light. We find that a sample can be either stretched or uniaxially compressed along the polarization direction of the light depending on the molecular architecture of azobenzene chromophores inside the macromolecules. In section 3 we obtain a condition for the sign of elongation; this condition is shown to be qualitatively in agreement with both computer simulations and experimental results. We estimate the value of the light-induced stress which turns out to be higher than the value of the yield stress expected for conventional glassy polymers. In order to describe the irreversible elongation of a sample under the stress higher than the yield stress we use in section 4 a viscoplastic approach. We demonstrate that the magnitude of the irreversible elongation decreases with increasing temperature and can disappear at certain temperature which can be lower than the glass-transition temperature. These findings are also confirmed by experiments.33 In section 5 we describe briefly how it is possible to apply our theory (developed here for homogeneous light illumination) to description of the SRGs where the light intensity (or the direction of the light polarization) varies along the surface of a polymer.

V0 is negative, the effective potential (1) for chromophores in the electric field E of the light is characterized by positive value of the field strength, V0 > 0. This reflects the fact that long axes of chromophores affected by the light are preferably oriented perpendicular to the vector E.34,35 The value of V0 is determined by the intensity of the laser light Ip and can be estimated as32,33

2. Free Energy and Equilibrium Elongation of a Side-Chain Azobenzene Polymer under Illumination with Linear Polarized Light

1 V0 ) RυτIp ≡ C · Ip 2

We start from referring to the experimental fact that interaction of chromophores with the electric field E of linearly polarized light leads to the perpendicular alignment of long axes of the chromophores with respect to the vector E.34 Only in the case of such orientation further absorption of photons is prevented and, hence, a steady state is reached.35 Physically, such anisotropy is caused by a complicated quantum-mechanical process of multiple trans-cis-trans photoisomerization of chromophores in the electric field E. The probability P(Θ) that a chromophore is oriented at an angle Θ with respect to the vector E is shown to be described by the exponential function P(Θ) = C exp[-V(Θ)/kT], where T is absolute temperature, k is the Boltzmann constant, and effective potential V(Θ) has the following form32,36

V(Θ) ) V0 cos2 Θ

(1)

where V0 is the strength of potential. Potentials of such a kind when the energy of an orienting unit is proportional to cos2 Θ are called potentials of “quadrupole” symmetry.37 Examples are nematic liquid crystals (LC) where the quadrupole potential is caused by intermolecular interactions between mesogens.30,31 However, in contrast to the nematic LC for which the value of

Figure 1. Molecular architecture of a side-chain azobenzene polymer with (a) short and (b) long spacers used in recent computer simulations.36 Below the molecular architecture in Figure 1b we show possible orientations of chromophores with respect to the main chain which correspond to the local minima of the symmetrical potential of internal rotations for a polyethylene’s spacer (R ) 30°).

(2)

where R is the absorption coefficient, υ is the volume of azobenzene, and τ is the effective transition time between two isomer states. For characteristic laser intensities Ip ∼ 1 W/cm2 and for the room temperature one can find in the literature, V0 ∼ 10-19 J,32,33,36 so that the coefficient of proportionality C ∼ 10-19 J · cm2/W. Interaction of chromophores with the polarized light results in the orientation anisotropy of polymer chains due to covalent bonding of chromophores to the chain backbones. In Figure 1, we illustrate typical architectures of azobenzene macromolecules bearing chromophores in side chains which are connected via either short (Figure 1a) or long (Figure 1b) spacers to the backbones. We model here each azobenzene macromolecule as a rigid rod which bears a number of rodlike chromophores in side chains (Figure 2a,b). For short macromolecules, whose length is comparable with the Kuhn segment (such macromolecules were used, for example, in experiments23,38 and computer simulations36), the rods in our model describe orientation of stiff oligomers. For long azobenzene macromolecules,25,26,39 the rods imitate the orientations of freely jointed Kuhn segments which form polymer chains. Thus, the formalism developed below can be applied for side-chain azobenzene polymers of different molecular weights.

5034 J. Phys. Chem. B, Vol. 113, No. 15, 2009

Toshchevikov et al. azobenzene polymers for concrete architectures of chromophores inside oligomers. Now, we stop our consideration to outline the basic idea for calculating the mechanical stress appearing in a polymer sample under illumination with linear polarized light. The idea is based on the well-known fact29-31 that orientation anisotropy of molecules in a system leads to the appearance of the mechanical stress. The value of the stress, σ, is determined by the free energy of the system, F ) F(ε), which is found as a function of the strain, ε:40

σ(ε) ) -

Figure 2. (a) Spatial orientation of a rigid rod which models a stiff azobenzene oligomer bearing chromophores in side chains. Axis Oξ is perpendicular to the main chain and defines the frame of reference related to the oligomer; axis Oξ can be fixed on a plane of symmetry of the oligomer. (b) Internal orientation of chromophores inside an oligomer; R is the angle between a chromophore and the main chain, β is the angle between a projection of a chromophore on a plane perpendicular to the main chain and the axis Oξ (see part a).

Effective potential U of an oligomer in the electric field E is a sum of energies of chromophores inside the oligomer in the field E. Orientation of an oligomer in space is defined by the angles θ, φ, and ψ (Figure 2a): θ is the angle between a long axis of an oligomer and the vector E, and φ determines rotation of the oligomer around the vector E. We note that due to axial symmetry of the system, the potential U is independent of the angle φ and this angle is omitted below. The angle ψ defines rotation of the oligomer around its long axis; ψ is the angle between a fixed axis Oξ perpendicular to the long axis of the oligomer and the plane formed by the vector E and the long axis of the oligomer (Figure 2a). Now, using eq 1, the potential U(θ,ψ) which is a sum of energies of chromophores inside the oligomer in the electric field E can be written as

U(θ, ψ) ) NchV0u(θ, ψ)

(3)

Here Nch is the number of chromophores in each oligomer and

u(θ, ψ) ≡ 〈cos2 Θ(θ, ψ)〉

(4)

where Θ is the angle of a given chromophore with respect to the vector E and averaging runs over all chromophores inside the oligomer, whose orientation is defined by the angles θ and ψ. The function u(θ,ψ) is determined by the orientation distribution of chromophores inside the oligomer (Figure 2b). In the present section we consider the phenomenon using general relationship (3) for the function U(θ,ψ) and in the next section we will discuss the photoelastic properties of

∂F(ε) ∂ε

(5)

It should be noted that due to homogeneity of the light irradiation, no internal stress gradient exists which could lead to the translation movement of molecules in a homogeneous sample. Thus, one can certainly claim that the stress appearing in a homogeneous sample affected by a homogeneous laser beam is of “orientation” origin only. In such approach, the free energy is determined only by the orientation distribution of the oligomers, f(θ,ψ), and does not include contributions from translational movement of molecules. Hence, we can write the free energy as

F ) nkT

2

∫ dΩ f(Ω) ln f(Ω) + n ∫ dΩ f(Ω)U(Ω) + Eε2

(6) where n is the number of oligomers in a unite volume. Here we set Ω ≡ (θ,ψ). Integration in eq 6 runs over the angles θ and ψ: dΩ ) sin θ dθ dψ. The first term in eq 6 is the orientation entropy of oligomers. The second and third terms are the energy contributions. The second term is the energy of all chromophores in the field E of the light. The third term is caused by the elasticity of a glassy polymer, with E being the elastic Young modulus. We note here that the relative elongation, ε, of an azobenzene polymer is also of the “orientation” origin. Elongation of the sample is accompanied by the reorientation of oligomers and it is this reorientation which demands energy costs Eε2/2 due to intermolecular interactions between oligomers, whose rotational mobility is prevented in a glassy state. Therefore, ε is uniquely determined by the orientation distribution function of the oligomers, f(θ,ψ), and we can write ε in general form as follows:

ε)

∫ dΩ f(Ω)w(θ)

(7)

The function w(θ) is a “microscopic” analogue of the strain: w(θ0) shows how much the sample would be stretched if all oligomers were oriented on the angle θ0 with respect to the vector E. Because of the axial symmetry of the system around the vector E, function w(θ) is independent of both angles ψ and φ (Figure 2). In this section we will develop our consideration for the function w(θ) in a general form. Let us consider some general features of the function w(θ). First, in the isotropic state, a sample should be unstretched, so that

〈w(θ)〉I ≡ (4π)-1

∫ dΩ w(θ) ) 0

(8)



It is also obvious that 〈w(θ)〉 should be positive (i.e., ε > 0 and the sample is stretched) if the distribution function f(θ,ψ) has a maximum at θ ) 0 and monotonically decreases with increasing angle θ from 0° up to 90°. Moreover, for small strains, one can estimate the function w(θ) as follows. At small deformations the stress is known to be proportional to the order parameter, S,

S ) 〈P2(θ)〉 ≡

∫ dΩ f(Ω) 3 cos 2θ - 1 2

(9)

This fact is a consequence of Brewster’s relation41 that the birefringence of a stretched glassy sample is proportional to the mechanical stress, ∆n ) Kσ, with K being Brewster’s constant. On the other side, the birefringence is proportional to the order parameter S and the stress is proportional to the strain. Therefore, we have at small strains ε = εmaxS, where εmax is the maximal strain corresponding to the fully oriented sample with S ) 1. From the ratio ε = εmaxS and from eqs 7 and 9, we obtain for the function w(θ)

w(θ) = εmaxP2(θ)

(10)

Below, in section 3, we will use the last relationship for numerical estimations. Now, we focus on obtaining the free energy as a function of the strain ε. For this we have to find the distribution function f(Ω) which satisfies the condition of minimum of the free energy as well as eq 7 and the normalization condition:

∫ dΩ f(Ω) ) 1

(11)

In terms of the calculus of variations, the conditions for the distribution function f(Ω) can be reformulated as follows. At any variations, δf, which satisfy eqs 7 and 11, i.e.

∫ dΩ δf (Ω)w(θ) ) 0

and

∫ dΩ δf (Ω) ) 0 (12)

the equilibrium distribution function f(Ω) should provide variations of the free energy equal to zero, δF ) 0. Using eqs 6 and 7, the condition δF ) 0 can be rewritten as

δF )

∫ dΩ δf(Ω)[nkT(1 + ln f(Ω)) + nNchV0u(Ω) + Eεw(θ)] ) 0

(13)

Here we have used eq 3 for the function U(Ω). According to the Lagrange variation principle, conditions (12) and (13) are satisfied simultaneously when

nkT[1 + ln f(Ω)] + nNchV0u(Ω) + Eεw(θ) + A + Bw(θ) ) 0

(14)

where A and B are Lagrange multipliers which fulfill the conditions (7) and (11) for the function f(Ω). From eq 14 we obtain the following expression for the function f(Ω):

[

f(Ω) ) Z-1 exp -

]

NchV0u(Ω) B + Eε w(θ) kT nkT

(15)

Figure 3. Schematic illustration of the dependence of the stress on the strain σ(ε). Two cases are possible, when the striction stress σstr is either (1) positive or (2) negative.

where Z ) Z(ε,V0) is the partition function which represents the reformulated Lagrange multiplier A and satisfies the normalization condition (11), that gives for Z ) Z(ε,V0):

Z(ε, V0) )

[

∫ dΩ exp -

]

NchV0u(Ω) B(ε, V0) + Eε w(θ) kT nkT (16)

The Lagrange multiplier B ) B(ε,V0) is determined by eq 7. Now, substituting eq 15 into eq 6 we obtain the free energy as a function of ε and V0:

F(ε, V0) ) -nkT ln Z(ε, V0) - B(ε, V0)ε -

Eε2 (17) 2

It is now a simple matter to determine the total stress, σ ) -∂F/∂ε, eq 5. Using eq 16, which gives nkT(∂ ln Z/∂ε) ) (-∂B/ ∂ε - E)ε, we obtain from eq 17

σ ) B(ε, V0)

(18)

We note that B(ε,V0) is a complicated function of the variables ε and V0 and is determined by the solution of eq 7 for the parameter B, with f(Ω) being given by eqs 15 and 16. Calculating the derivatives of the left- and right-hand sides of eq 7 with respect to ε and using eqs 15, 16, and 18, we obtain:

[

∂σ nkT )- E+ 2 ∂ε 〈w 〉 - 〈w〉2

]

(19)

where the averaging is carried out with the distribution function j )2〉 in eq 19 is positive f(Ω). The quantity 〈w2〉 - 〈w〉2 ≡ 〈(w - w and, therefore, ∂σ/∂ε < 0 for any system. Here lies a general picture for the stress-strain behavior. As long as σ(ε,V0) decreases monotonically with ε due to ∂σ/∂ε < 0, the curve σ(ε) crosses always an abscissa’s axis at a certain point ε* (Figure 3):

σ(ε*, V0) ) 0

(20)


Toshchevikov et al.

u(θ, ψ) ) a1 cos2 θ + sin 2θ(a2 sin ψ - a3 cos ψ) + sin2 θ(a4 cos2 ψ - a5 sin 2ψ + a6 sin2 ψ)

(23)

where the constants aj are related with the averaged orientation characteristics of chromophores inside an oligomer:

1 a2 ) 〈sin 2R cos β〉W, 2 1 a3 ) 〈sin 2R sin β〉W 2 1 a4 ) 〈sin2 R cos2 β〉W, a5 ) 〈sin2 R sin 2β〉W, 2 a6 ) 〈sin2 R sin2 β〉W (24)

a1 ) 〈cos2 R〉W,

Figure 4. Schematic illustration of the dependence of the free energy on the strain F(ε) for the curves 1 and 2 presented in Figure 3.

Due to the ratio ∂F/∂ε ) -σ, we have a general conclusion that F decreases at ε < ε* (since here σ > 0 and, consequently, ∂F/∂ε < 0) and increases at ε > ε*; see Figure 4. Thus, the value ε* is an equilibrium elongation of a sample under illumination of the laser light. The equilibrium elongation is a function of the field strength, ε* ) ε*(V0). This complicated function is given by a solution of eq 20. Note that the sign of the equilibrium elongation coincides with the sign of the so-called striction stress σstr(V0) ≡ σ(0,V0) at ε ) 0 (Figure 3). Estimation of the value of the striction stress is one of the important tasks of our theory. 3. Estimation of the Striction Stress The value of the striction stress, σstr(V0) ≡ σ(0,V0), is determined as a solution of eq 7 for the parameter B, where ε ) 0 and the function f(Ω) is given by eqs 15 and 16. This condition for σstr(V0) ≡ B(0,V0) can be rewritten as

0)

[

∫ dΩ w(θ) exp -

]

NchV0 σstr(V0) u(Ω) w(θ) kT nkT

(21)

The solution of the last equation for σstr(V0) depends on the concrete form of structural functions u(Ω) and w(Ω). The function u(Ω) is determined by the orientation distribution of chromophores inside the oligomer, W(R,β). Here the angles R and β define the orientation of a chromophore in the frame of reference related to the oligomer (Figure 2b); R is the angle between the long axes of a given chromophore and of the oligomer; the angle β is measured from a fixed axis Oξ perpendicular to the main chain of the oligomer (Figure 2, a and b). Now, we can write the cosine of the angle Θ between a given chromophore and the electric field E as a function of the angles R, β, θ, and ψ (see Figure 2, a and b):

cos Θ ) cos θ cos R - sin θ sin R cos(ψ + β) (22) From eq 4 we have for u(θ,ψ)

Here averaging runs over the angles R and β with the distribution function W(R,β). Thus, the value of σstr(V0) appears to be a complicated function of the parameters aj of the potential u(θ,ψ). However, it turns out that for small values of V0 it is possible to get general conclusions about the dependence σstr(V0) for different architectures of oligomers. 3.1. Striction Stress at Small Laser Intensities, NchV0/kT < 1. The function σstr(V0) at small strengths of light-induced potential V0 is determined by the derivative, (∂σ/∂V0)V0)0:

σstr(V0) =

∂σstr(0) V ∂V0 0

(25)

Taking the derivative of the left- and right-hand sides of eq 21 with respect to V0 and substituting V0 ) 0 we obtain

0 ) -nN0〈u(Ω)w(θ)〉I -

∂σstr(0) 2 〈w (θ)〉I ∂V0

(26)

where the index I denotes the averaging with the isotropic distribution function. Let us consider the quantity 〈u(Ω)w(θ)〉I. The function u(Ω) is given by eq 23 where after averaging with isotropic distribution function with respect to the angle ψ we have

〈sin ψ〉I ) 〈cos ψ〉I ) 〈sin 2ψ〉I ) 0, 〈sin2 ψ〉I ) 〈cos2 ψ〉I ) 1/2

(27)

Using eqs 23 and 27 and the relationship (a4 + a6) ) 〈sin2 R〉W which follows from eq 24, we obtain for 〈u(Ω)w(θ)〉I:

〈u(Ω)w(θ)〉I ) 〈cos2 R〉W〈w(θ)cos2 θ〉I + 1 〈sin2 R〉W〈w(θ) sin2 θ〉I 2

(28)

Using the equality 〈w(θ)〉I ) 0, eq 8, which leads to the relationship 〈w(θ) sin2 θ〉I ) -〈w(θ) cos2 θ〉I, we can rewrite eq 28 in the following form:

3 〈u(Ω)w(θ)〉I ) -〈w(θ)cos2 θ〉I 〈sin2 R〉W - 1 2

[

]

(29)



Substituting eq 29 into 26 and using eq 25, we obtain finally for small V0

σstr(V0) = n0V0

3〈w(θ)cos2 θ〉I 2〈w2(θ)〉I

[〈sin R〉 2

W

-

2 3

]

(30)

where n0 ) nNch is the number of chromophores in a unit volume. Note that the quantity 〈w(θ) cos2 θ〉I in eq 30 is always positive because of general features of the function w(θ) mentioned in the text after eq 8. Moreover, it is obvious that 〈w2(θ)〉I > 0. For instance, 〈w(θ) cos2 θ〉I ) 2εmax/15 and 〈w2(θ)〉I ) εmax2/5 for the function w(θ) ) εmaxP2(θ) given by eq 10. Thus, the sign of the strain ε which determines the direction of a sample elongation at small values of V0 (stretching, ε > 0, or uniaxial compression, ε < 0) coincides with the sign of a simple factor, [〈sin2R〉W - 2/3], according to eq 30. This means that the chemical architecture of oligomers (i.e., orientation distribution of chromophores with respect to the main chain) affects strongly the photoelastic behavior of an azobenzene polymer. For instance, if chromophores are attached to the backbone via short and rigid spacers so that the angle R is fixed, R ) 90°, Figure 1a, we have 〈sin2 R〉W ) 1, i.e., [〈sin2 R〉W 2/3] > 0 and a sample is stretched along the vector E. This kind of behavior was found recently in computer simulations of azobenzene molecules with rigid spacers.42 If the spacers are long enough, Figure 1b, chromophores can be positioned at some equiprobable angles R* and 180° - R* with respect to the main direction of the backbone. In this case 〈R〉 ) 90°, but 〈sin2 R〉W ) sin2 R* < 1, so that for long enough spacers when R* < 54,7° we have [〈sin2 R〉W - 2/3] < 0 and a sample is uniaxially compressed along the vector E in accordance with computer simulations for molecules with long enough spacers.36 Thus, the striction stress obeys a linear dependence on the field strength σstr = const · n0V0 at NchV0/kT < 1, eq 30. The condition NchV0/kT < 1 corresponds to the light intensities Ip < 5 mW/cm2 at the room temperature and for Nch ∼ 20, as it follows from eq 2. These findings are in agreement with an experiment of Fukuda et al. 43 which demonstrates a linear dependence of sample elongation on the light intensity at Ip < 5 mW/cm2. However, at higher light intensities, a nonmonotonic dependence of sample elongation on the light intensity has been found.43,44 Below, we explain such nontrivial photoelastic behavior of azobenzene polymers under illumination with the laser light of high intensity. 3.2. Striction Stress at High Laser Intensities, NchV0/kT > 1. To begin with, we remark that photoelastic behavior of azobenzene polymers at small laser intensity is determined by the average angle R between chromophores and the main chain and does not depend on the distribution of chromophores with respect to the azimuthal angle β (eq 30). It turns out, however, that photoelastic behavior of azobenzene polymers at high laser intensities is strongly sensitive to the azimuthal distribution of chromophores inside the oligomers. To show this, we consider in this section two particular examples of the oligomer architectures, namely, the structures having either (i) isotropic azimuthal distribution of chromophores around the backbone or (ii) anisotropic but symmetric azimuthal distribution, which is typical for these compounds. For the function w(θ) we will use here the approximation w(θ) ) εmaxP2(θ), given by eq 10. 3.2.1. Molecules with Isotropic Azimuthal Distribution of Chromophores around the Main Chain. Let us consider the architecture of oligomers with isotropic distribution for the angle β but with anisotropic distribution W(R) for the angle R between

chromophores and the main chain. In this case, expression for the effective potential u(θ,ψ) is considerably simplified. After averaging over the angle β in eqs 24 with isotropic distribution function we have a2 ) a3 ) a5 ) 0 and a4 ) a6 ) 〈sin2 R〉W/2. Then, the expression for u(θ,ψ) can be rewritten as

u(θ, ψ) )

[ 23 〈sin R〉 2

W

]

- 1 sin2 θ

(31)

Here we have discarded a term that is independent of the angles θ and ψ. This term appears in the exponent in the right-hand side of eq 21 and can be canceled out. Substituting eq 31 into eq 21 and using the approximation given by eq 10 for w(θ), we obtain the following conditions for σstr ) σstr(V0):

0)

∫ dΩ P2(θ) exp[x sin2 θ]

(32)

where

3σstrεmax NchV0[3〈sin2 R〉W - 2] x) 2nkT 2kT

(33)

It is evident that eq 32 can be only satisfied if x ) 0. Hence

σstr ) n0V0

〈sin2 R〉W - 2/3 εmax

(34)

where n0 ) nNch is the number of chromophores in a unit volume. Note that the striction stress for molecules with isotropic azimuthal distribution of chromophores admits an exact analytical expression given by eq 34. Moreover, using 〈w(θ) cos2 θ〉isot ) 2εmax/15 and 〈w2(θ)〉isot ) εmax2/5 for the function w(θ) ) εmaxP2(θ), one can see that eq 34 is an extension of the linear dependence σstr(V0) given by eq 30 from small values of V0 to large ones. In other words, for the molecules with isotropic azimuthal distribution of chromophores around the main chain, the dependence σstr(V0) is exactly linear and is given by eq 34 for any strengths of the lightinduced potential V0. Now, one can estimate the value of the striction stress with the help of eqs 2 and 34. For the laser intensity Ip ∼ 1 W/cm2 and for the number density of azobenzenes n0 = 1.5 × 1021 cm-3 which is a typical value for these materials,28,45 we obtain σstr ∼ n0V0 ∼ 100 MPa. It is important to point out that our microscopic theory predicts the values for the striction stress comparable with the typical values of the yield stress, σY ∼ 10-100 MPa, for glassy polymers (e.g., σY ∼ 50 MPa for PMMA). At stresses σstr ∼ 100 MPa greater than σY, an azobenzene polymer is deformed irreversibly and the deformation is fixed after the light is switched off. Thus, our approach explains for the first time on the molecular level the possibility of irreversible deformation of an azobenzene polymer under homogeneous illumination with the polarized light and, consequently, explains the possibility for irreversible inscription of surface relief gratings (SRGs) onto thin azobenzene polymer films. Note that the present section covers also description of molecular glasses made of isolated chromophores.46,47 Each chromophore can be treated as a “degenerated oligomer” with a single chromophore inside (Nch ) 1) which lies along the long axis of the “oligomer” (R ) 0). Inserting the values Nch ) 1


Figure 5. Three areas of the values of the structural angles R* and β* (Figure 2b) for three typical behaviors of the dependence σstr(V0).

and R ) 0 into eqs 3 and 31 one can see that the expression 31 for the potential of such a “degenerated oligomer” is transformed into the expression 1 for the potential of an isolated chromophore (the constants which are independent of the angles θ and ψ can be discarded). Since the results discussed above hold for any values of Nch and 〈sin2 R〉W, they are applicable also to a system of isolated chromophores (molecular glasses46,47). Thus, one can conclude that the light-induced stress can be comparable with the yield stress in a system of isolated chromophores; therefore, irreversible inscription of SRGs is also possible in such a system. Moreover, as it follows from eq 34, the value of the striction stress should be negative in a system of isolated

Toshchevikov et al. chromophores, for which we have Nch ) 1 and R ) 0. This means that a sample should be uniaxially compressed along the polarization vector E, since the chromophores are oriented perpendicular to the vector E. We conclude the present paragraph by pointing out that for molecules with the isotropic azimuthal distribution of chromophores, W(R,β) ) W(R), the dependence σstr(V0) is monotonic (eq 34). In the next paragraph we will show that the dependence σstr(V0) is very sensitive to the azimuthal distribution W(β) of chromophores with respect to the angle β. For several types of the distribution W(β) the dependence σstr(V0) becomes nonmonotonic and can even change its sign on passing from small to high values of V0. 3.2.2. Molecules with Anisotropic Azimuthal Distribution of Chromophores around the Main Chain. To begin with, we note that effects of oligomer architecture on the photoelastic behavior of azobenzene polymers can be considered by varying the structural parameters, aj (j ) 1,..., 6) in eq 23. In the present paper for the sake of simplicity we consider the properties of oligomers with symmetric distribution of chromophores around the main chains, namely, W(R,β) ) W(R,-β) and W(R,β) ) W(180° - R,β). This case is of importance for practice, since azobenzene macromolecules possess, as a rule, a planar symmetry (see Figure 1), so that W(R,β) ) W(R,-β) where the angle β is measured from the plane of symmetry. Moreover, the condition W(R,β) ) W(180° - R,β) is satisfied for macromolecules having spacers with symmetrical potential of internal rotation, e.g., for polyethylene’s spacers (Figure 1b), which are widely used in azobenzene polymers.43,44,48 Important

Figure 6. Typical profiles of the potential u(θ,ψ) with corresponding architectures and orientations of oligomers with respect to the electric field E at high light intensities for different values of the structural angles R* and β*: (a) R* ) 50°, β* ) 30°; (b) R* ) 50°, β* ) 60°; (c) R* ) 70°, β* ) 30°; (d) R* ) 70°, β* ) 60°; (e) R* ) 70°, β* ) 5°; (f) R* ) 70°, β* ) 85°.



examples are molecules where chromophores are fixed at equiprobable angles β ) (β*, and R ) 90° (Figure 1a). Using the symmetry, we simplify the problem and demonstrate below how the azimuthal distribution of chromophores in oligomers affects the photoelastic behavior of azobenzene polymers. For the oligomers with such structural symmetry we have a2 ) a3 ) a5 ) 0 (eq 24), so that the orientation potential u(θ,ψ) can be rewritten as follows according to eq 23

u(θ, ψ) )

[ 23 〈sin R〉 2

W

-1+

1 〈sin2 R cos 2β〉W cos 2ψ sin2 θ 2

]

(35)

Note that at 〈sin2Rcos2β〉W ) 0 (e.g., when β is fixed at β ) (45°) the expression 35 is reduced to eq 31 for oligomers with isotropic azimuthal distribution of chromophores around the main chains. This means that the dependence σstr(V0) for oligomers with short spacers and β ) (45° is described by the linear relationship given by eq 34. Photoelastic behavior at high light intensities, NchV0/kT > 1, is determined mainly by minima of the potential u(θ,ψ). If the expression in the brackets in the right-hand side of eq 35 is positive at all values of ψ, then the minimum of the potential u(θ,ψ) is achieved at θ ) 0 and a sample is stretched at high light intensities, σstr > 0. It is obvious that this condition is satisfied when

3 1 〈sin2 R〉W - 1 > |〈sin2 R cos 2β〉W | 2 2

(36)

On the contrary, if the expression in the brackets in the righthand side of eq 34 is negative for some values of ψ, then the minimum of the potential u(θ,ψ) is achieved at θ ) 90° and a sample is uniaxially compressed along the electric vector E, σstr < 0. Such condition is satisfied when

Figure 7. Dependences of the striction stress on the field strength σstr(V0) at different values of the structural angles R* and β* (Figure 2b). Filled and open symbols correspond to R* ) 50° and R* ) 70°, respectively; different symbols present the results for different values of β*. Panels (a) and (b) present the same curves σstr(V0), but the scale is larger in panel (a).

3 1 〈sin2 R〉W - 1 < |〈sin2 R cos 2β〉W | 2 2

stretched with increasing light intensity both at small and at large V0. But if eq 37 is fulfilled at 〈sin2 R〉W > 2/3, sample elongation demonstrates nonmonotonic dependence on V0: at small V0 a sample is stretched, whereas it is uniaxially compressed at large V0. As examples, we illustrate such nontrivial photoelastic behavior for oligomers with fixed angles, β ) (β*, and R ) R*, 180° - R*. We have marked by indexes I, II, and III in Figure 5 three areas on the plane R* - β* where the abovementioned conditions for R and β take place. Figure 6a-e illustrate typical profiles of the potential u(θ,ψ) for these three areas I, II, and III. The dependences σstr(V0) for different values of R* and β* have been calculated by means of numerical solution of eq 21 where w(θ) ) εmaxP2(θ) and u(θ,ψ) is given by eq 35. The results of numerical solution are shown in Figure 7a,b for values of R* and β* belonging to the areas I, II, and III in Figure 5. Area I. When R* < 54,7°, i.e., 〈sin2R〉W < 2/3 (the area I in Figure 5), the potential u(θ,ψ) has a minimum at θ ) 90° which takes place either at ψ ) 90° (when β* < 45°, Figure 6a) or at ψ ) 0 (when 45° < β* < 90°, Figure 6b). One can see in Figure 7a that at large V0 the function σstr(V0) has negative values in this case and is approximated by a linear dependence in agreement with eq 38. The initial slope of the dependence σstr(V0) is also negative in this case according to eq 30. Thus, at R* < 54,7°, the striction stress is always negative, decreasing

(37)

Using the Laplace’s method for estimation of integrals with a large parameter,49 we have calculated the integral in the righthand side of eq 21) at NchV0/kT . 1 and obtained the following asymptotic relationship for the function σstr(V0) at NchV0/kT . 1 (see the Appendix):

3〈sin2 R〉W - 2 - |〈sin2 R cos 2β〉W | σstr(V0) = n0V0 3εmax

(38)

One can see from eq 38 that σstr > 0 if the condition (36) holds and σstr < 0 if the condition (37) takes place in agreement with the findings discussed above. Thus, we have three types of the photoelastic behavior of azobenzene polymers depending on their chemical architecture. If 〈sin2 R〉W < 2/3, then condition (37) is fulfilled, i.e., σstr < 0 at large V0. Moreover, initial slope of σstr(V0) is also negative in this case according to eq 30. Thus, if 〈sin2 R〉W < 2/3, a sample is monotonically compressed with increasing light intensity. If 〈sin2 R〉W > 2/3, the initial slope of σstr(V0) is positive, but the behavior of σstr(V0) at large V0 is determined by eqs 36 and 37. If eq 36 is satisfied at 〈sin2 R〉W > 2/3, a sample is monotonically


Toshchevikov et al.

monotonically with increase of V0; see curves with filled symbols in Figure 7a,b. These curves for different β* but at fixed R* ) 50° have the same initial slope, since the initial slope is determined only by the quantity 〈sin2 R〉W and is independent of the azimuthal distribution according to eq 30. At R* > 54,7°, the initial slope of the dependence σstr(V0) is positive at all values of β* according to eq 30. One can see in Figure 7b that at the fixed value R* ) 70° the curves with open symbols have the same positive initial slope at different β*. However, the limiting behavior and the sign of σstr(V0) at large values of V0 are determined by conditions (36) and (37), whose ranges of validity (the areas II or III, respectively) are separated by the straight curve in Figure 5. This curve is described by the following equation found from eqs 36 and 37

R* ) A(β*) ≡ arcsin

2 3 - |cos 2β* |

(39)

Area II. When R* > A(β*), i.e., eq 36 is valid (the area II in Figure 5), the potential u(θ,ψ) has a minimum along the line θ ) 0° (Figure 6c,d), and σstr(V0) is positive at large V0. One can see from Figure 7a that at large V0 the striction stress obeys the linear dependence on V0 in agreement with eq 38. Thus, at R* > A(β*) the value of σstr is positive and increases monotonically with increasing V0; i.e., a sample is stretched monotonically with increasing V0. Area III. If 54,7° < R* < A(β*), i.e., the condition (37) is valid (the area III in Figure 5) and the values of σstr at large V0 are negative according to eq 38, whereas the initial slope of σstr(V0) is positive. In this case the dependence σstr(V0) is nonmonotonic; see Figure 7a,b. The potential u(θ,ψ) has in this case a minimum at θ ) 90° which takes place either at ψ ) 90° (when β* < 45°, Figure 6e) or at ψ ) 0 (when 45° < β* < 90°, Figure 6f). We note that such nonmonotonic photoelastic behavior of azobenzene polymers has been observed in experiments. For instance, one can see in Figures 5 and 6 of ref 43 a nonmonotonic behavior of the inscription rate with increasing light intensity: the inscription rate increases at Ip < 10 mW/cm2, whereas it starts to decrease at 20 mW/cm2 < Ip < 100 mW/ cm2. Tripathy and co-workers44 showed that the extension of a film can even change its sign at very high light intensities, Ip ∼ 100 W/cm2. One can see from Figures 4 and 5 of ref 44 that at Ip < 1 W/cm2 a film illuminated with a polarized Gaussian beam has a trench at the maximum of light intensity, i.e., a film is locally stretched along the light polarization E. However, at high light intensities Ip ∼ 100 W/cm2, a local peak is observed on a film surface; i.e., the film is locally compressed along the vector E. Note that Tripathy and co-workers44 did not propose any reason for explanation of such nontrivial results, whereas our theory shows unambiguously that the nonmonotonic photoelastic behavior is possible for azobenzene polymers with specific chemical architectures. We conclude this section by noting that for all values of R* and β* the absolute value of the striction stress σstr is described at large field strengths V0 by the linear relationship |σstr| ∼ n0V0 according to eq 38. This relationship provides the values of the striction stress σstr ∼ 100 MPa greater than the yield stress, σY ∼ 10 MPa, at Ip ) 1 W/cm2 and at the number density of azobenzenes n0 = 1.5 × 1021 cm-3 typical for these materials. Thus, our theory extended here for molecules with anisotropic azimuthal distribution of chromophores around the main chains explains the possibility for irreversible inscription of surface

relief onto azobenzene polymers of any chemical architecture. In order to describe dynamics of irreversible elongation of azobenzene polymer under illumination with the light (when σstr > σY) we propose in the next section a viscoplastic approach. This allows us to discuss temperature dependence of the degree of irreversible elongation in view of recent experimental data.33 4. Viscoplastic Approach for Describing the Irreversible Elongation of an Azobenzene Polymer under Visible Light Irradiation In the viscoplastic approach, response of a glassy material (i.e., its elongation, ε) on an external mechanical stress, σext, is determined from the following equation:33,50

σext )

{

,σext e σY Eε σY + σp ,σext > σY

(40)

For stresses below the yield stress, σY, the polymer response is pure elastic. Above the yield stress, the external stress can be considered as a sum of the yield stress σY and the plastic stress σp.33,50 The total stress on a boundary of an azobenzene polymer σ which we have analyzed above consists of two contributions and can be rewritten using eqs 5 and 6 as follows:

σ ) σext - Eε

(41)

The second term in the right-hand side of eq 41 appears from the third term in eq 6 and describes a pure elastic response of a sample on the stretching (external) force. The external force is determined by the orientation distribution of oligomers, f(Ω), which changes with stretching due to changes of energy of oligomers in the field of the light as well as due to changes of orientation entropy. Therefore, σext includes contributions of the first and second terms in eq 6. The equilibrium elongation, ε*, is determined from eq 20, σ(ε*) ) 0, which gives from eq 41 that σext ) Eε*. The last equation coincides with the first relationship in eq 40. Thus, the statistical theory developed in sections 2 and 3 provides only the value of reversible elongation, ε*, and does not take into account the effects of irreversible elongation of a sample at σext > σY. The aim of the present section is to consider these effects and to calculate the irreversible elongation at σext > σY. Let us consider the sample elongation as a function of time after applying at t ) 0 the light illumination of the constant intensity, i.e., V0 ) const. The striction stress, σstr, appearing at t ) 0 can be either positive or negative depending on the architecture of oligomers and on the value of V0. We will discuss below the case σstr > 0. The case σstr < 0 can be analyzed analogously. According to the viscoplastic approach,33,50 the total strain, ε, consists of two parts

ε ) ε* + εp

(42)

The contribution ε* is pure elastic; ε* is an initial jump of the strain, and it is independent of time and reversible: it disappears after the light is switched off. The contribution εp (or so-called plastic part) is irreversible. It describes the flow of a plastic material under the stresses larger than σY and is related to the plastic part σp in eq 40 as follows:


σp ) ηε˙ p


(43)

( dεdt )

)

t)0

where η is the viscosity of an amorphous material. Note that in our microscopic approach the irreversible elongation of a sample is caused by the irreversible reorientation of oligomers affected by the light of high intensity. At small light intensities, oligomers can rotate only in vicinities of their average positions due to volume interactions with their neighbors and the reorientation is reversible: after the light is switched off, the oligomers return to their initial orientations. But at high light intensities each oligomer can rotate on large angles, so that after the light is switched off, they become to be orientated in new directions and sample elongation is irreversible. Thus, in our approach the plastic flow of a sample has orientation origin. Substituting eqs 43 into 40 and using relationship ε˙ ) ε˙ p which follows from the fact that ε˙ * ) 0 we obtain a dynamic equation for the total elongation ε at t > 0:

σext(ε) ) σY + ηε˙

(44)

Thus, the dynamic equation for the elongation represents a differential equation of the first order. The initial condition at t ) 0 is

ε(0) ) ε*

(45)

Irreversible dynamics of elongation is determined by the function σext(ε) in the left-hand side of eq 44. Using eqs 19 and 42, we have for σext(ε)

∂σext nkT )- 2 ∂ε 〈w 〉 - 〈w〉2

(46)

j)2〉 in eq 46 is positive and, The quantity 〈w2〉 - 〈w〉2 ) 〈(w - w therefore, σext(ε) decreases with increasing ε for any system (Figure 8), since ∂σext/∂ε < 0. Now, we can obtain the condition when the irreversible elongation of a sample is possible. It can be seen from Figure 8 that the possibility of irreversible elongation is determined by the value of the external stress after the reversible jump of the strain up to ε*, i.e., by the value σext(ε*). If σext(ε*) < σY (point A in Figure 8), the first condition in eq 40 holds and the elongation is reversible: the strain remains constant ε ) ε* after the jump and it decreases up to zero after the light is switched off. But in the case σext(ε*) > σY (point B in Figure 8) the second condition in eq 40 is satisfied and a sample starts to flow in accordance with the dynamic eq 44. Thus, for irreversible elongation it is not enough that the striction stress is larger than the yield stress, σstr > σY. Even if σstr > σY, there can be a situation when σext(ε*) < σY (point A in Figure 8). In other words, after reversible jump of the strain up to the value ε*, the orientation distribution of oligomers changes and, as a result, the external stress, σext(ε), decreases from the value σstr up to σext(ε*). For irreversible elongation it is necessary that σext(ε*) > σY. Now, it is a simple matter to analyze the irreversible dynamics of elongation for any azobenzene polymer at σext(ε*) > σY. At small times, the strain linearly depends on time

ε(t) = ε* +

( dεdt )

t + ...

t)0

σext(ε*) - σY η

(48)

At σext(ε*) > σY, the initial slope is positive and the strain ε increases (a sample starts to be irreversibly stretched). When ε increases, the value σext(ε) monotonically decreases, since ∂σext/∂ε < 0, and the point B in Figure 8 shifts to the right-hand side. Therefore, the difference σext(ε) - σY decreases with time, which leads to the monotonic decrease of the slope, ε˙ (t), according to eq 44. It is obvious that the limiting value, ε∞, of the dependence ε(t) is achieved when ε˙ ) 0, i.e., according to eq 44 when

σext(ε∞) ) σY

(49)

Figure 9 illustrates schematically the time dependence ε(t) which follows from the considerations given above. At t ) 0 the dependence ε(t) demonstrates a jump up to the value ε* and then the curve ε(t) increases monotonically with time. At that, the slope of the curve (ε˙ ) decreases monotonically and the dependence ε(t) tends to its limiting value ε∞; see Figure 9. After the light is switched off, the value of ε decreases from the value ε∞ by the reversible value, ε*, as marked by dashed line in Figure 9. Thus, the irreversible elongation, εir, which remains after the light illumination is

εir ) ε∞ - ε*

(50)

The higher the difference σext(ε*) - σY is, the larger is the value of the irreversible elongation εir. When σext(ε*) - σY e 0, one should use the first relationship in eq 40 which describes reversible elongation in this case. Therefore, εir ) 0 at σext(ε*) - σY e 0 and εir > 0 when σext(ε*) > σY. Note that the findings obtained above for the time behavior of the elongation (Figure 9) and for the value of εir are universal and valid for any azobenzene polymer. The time behavior illustrated in Figure 9 has been observed by the pulse-like inscription of surface relief gratings.33,39,48 It is a simple matter now to extend these findings to the case σstr < 0. In the latter

(47)

and the initial slope can be found from eqs 44 and 45 at t ) 0

Figure 8. Schematic illustration of the dependence of the external stress on the strain σext(ε).


Toshchevikov et al.

Figure 9. Schematic illustration of the time dependence of the strain ε(t) in the case σext(ε*) > σY in viscoplastic approach.

case, the initial jump, ε*, is negative, σext(ε*) < 0, and the irreversible elongation is possible when |σext(ε*)| > |σY| (point C in Figure 8). In this case, after the jump up to ε* < 0, the ε(t) starts to decrease (a sample is uniaxially compressed), since the initial slope is negative according to eq 48. The limiting value ε∞ < 0 is also determined by eq 49, where σY is negative: σY ) -|σY|, see Figure 8. The higher the difference |σext(ε*)| |σY| is, the larger is the absolute value of εir (εir < 0). The next important problem to study is the temperature dependence of the irreversible elongation, εir(T) ) ε∞(T) - ε*(T). The value ε* is determined from eq 20 which can be rewritten as follows using eqs 7, 15, 16, and 18:

NchV0u(Ω) Eε* dΩ w(θ) exp w(θ) kT nkT (51) NchV0u(Ω) Eε* dΩ exp w(θ) kT nkT

Figure 10. Temperature dependences of the irreversible strain, εir, at different values of the light intensity Ip calculated by using the approaches in which (a) V0 is independent of temperature and (b) V0 depends on temperature according to eqs 2 and 53. w(θ) ) εmaxP2(θ), εmax ) 0.5, R* ) 90°, β* ) 60°, E ) 4 GPa, σY ) 30 MPa, Nch ) 20, n0 ≡ Nchn ) 1.5 × 1021 cm-3, and Ea ) 32 kJ mol-1.

The value ε∞ is given by eq 49 and can be rewritten as follows using eqs 7, 15, 16, 18 and 41:

temperature C(T) can be related to the value C(T0) at the room temperature, T0, as follows:

ε* )

ε∞ )

∫

∫

[

[

[

]

NchV0u(Ω) σY w(θ) kT nkT (52) NchV0u(Ω) σY dΩ exp w(θ) kT nkT

∫ dΩ w(θ) exp ∫

]

]

[

]

Now, temperature dependence εir(T) ) ε∞(T) - ε*(T) is determined by the dependences σY(T) and V0(T). We consider here the phenomenon at temperatures below the glasstransition temperature of a material, T < Tg. Characteristic values of Tg for azobenzene polymers are Tg ≈ 90-120 °C.23,25 The value σY is independent of temperature at T < Tg. On the other hand, the estimation of the dependence V0(T) is an open question at the present. Therefore, we use here two approaches for calculation of V0(T). In the first approach, it is assumed that V0 is independent of temperature and is determined only by the light intensity Ip according to eq 2 with C ) 10-19 J · cm2/W ) const. In the second approach, we assume that the temperature dependence of the characteristic time τ in eq 2 can be described by the Arrhenius equation, τ(T) ) τ0 exp(Ea/RT), where Ea is the activation energy between trans- and cis- states of the azobenzene moieties and R ) 8.31 J K-1 mol-1 is the gas constant. In such an approach, the value of the constant C in eq 2 at given

C(T) ) C(T0) exp[(Ea /RT0){T0 /T - 1}]

(53)

with C(T0) ) 10-19 J · cm2/W. The results of numerical calculations of the values εir(T) ) ε∞(T) - ε*(T) by means of eqs 51 and 52 and using two approaches mentioned above are presented in Figure 10, a and b, for the oligomers with short ethylene’s spacers, for which R* ) 90° and β* ) 60°. For the calculations we have used the following values of the constants typical for azobenzene polymers, E ) 4 GPa, σY ) 50 MPa, Nch ) 20, n0 ≡ Nchn ) 1.5 × 1021 cm-3, and Ea ) 32 kJ mol-1.33 One can see from Figure 10a,b that both approaches predict the decrease of the value εir(T) with increasing temperature. However, the first approach with V0 independent of temperature provides a very slow change of the strain εir with temperature (Figure 10a), whereas the second approach with V0(T) dependent on temperature predicts much faster decrease of εir with temperature. In the latter case, at temperatures higher than a certain value, T*, the appearing stress becomes smaller than the yield stress due to decrease of the field strength V0(T) with temperature and, therefore, εirr disappears at T > T*. This means that the deformation becomes reversible and disappears after the light is switched off even at temperatures smaller than the glass transition temperature; see Figure 10b. These findings are



in agreement with the recent experiments of Veer et al.33 in which the grating amplitude was found to decrease rapidly with increasing temperature. Moreover, it has been shown that the grating entirely relaxes after the light is switched off at T ≈ 100 °C. We note, however, that the approximation given by eqs 2 and 53 provides very rapid change of εir with temperature as compared with the experimental data of Veer et al.33 Perhaps such disagreement is caused by the exponential factor in the right-hand side of eq 53 which is very sensitive to the value of the parameter Ea. For calculations, we have used the value Ea ) 32 kJ mol-1 estimated in ref 39 but maybe this value was overestimated. We note here that presently the estimation of the dependence V0(T) is a problem under discussion and it requires more detailed analysis of the temperature effects. We hope that our findings will stimulate further experimental efforts to solve the problem. For example, it would be important to study the effects of the light intensities and of the oligomer architectures on the temperature dependence of the surface relief grating. These efforts would allow us to precise the relationship 53. However, it is obvious that the main result of the present theory will hold: the critical temperature T* is independent in general of the glass-transition temperature Tg and is determined by the parameters, E, σY, n0, Ip, Ea, R*, and β*, the last two parameters being strongly dependent on the oligomer architecture. Thus, the inscription of surface relief gratings can disappear even at temperatures lower than the glass-transition temperature Tg, the fact which explains for the first time the recent experimental data.33

perpendicular to the gradient vector of the light intensity and the intensity along the vector E is constant. Thus, the stresses appearing in the neighboring parts along the vector E are the same and, therefore, the deformation in the s-s geometry is strongly suppressed due to the balance between stresses appearing in the neighboring parts. These reasons explain why the p-light is much more efficient than the s-light.3,51,52 Moreover, one can explain now the difference in the SRGs for azo-dye grafted and doped polymers.53,54 The deformation of a doped polymer system is caused by the orientation of chromophores, the orientation of the polymer chains being isotropic in this case. However, in grafted systems the orientation of chromophores leads also to the orientation of backbones of the chains due to rigid bonding between the chromophores and the backbones. Thus, in the grafted system the total orientation anisotropy both of chromophores and of polymer chains should be larger than the total orientation anisotropy in the doped system. Therefore, the deformation of a grafted system should be also larger than the deformation of a doped system; this conclusion is in agreement with experiment.53,54 We conclude the present section by noting that the consideration of the SRGs using the orientation approach developed here demands a more detailed effort which can be carried out in the future. In particular, it would be interesting to explain the spontaneous organization into SRGs upon uniform illumination,55 where, however, the effects of nonideality of the laser beam as well as the diffraction effects between the light beams reflected from the up- and down-surfaces of the polymer film can take place.

5. Discussion

6. Conclusions

In the present section we describe briefly how it is possible to extend our theory (developed here for homogeneous light irradiation) to the SRGs where the light intensity (or the direction of light polarization) varies along the surface of polymer. The idea is based on dividing the surface into infinitesimal parts where the light intensity (or the direction of light polarization) is constant. Then, the local deformation of each part is a result of two factors: (i) interaction of the molecules with the homogeneous light in each infinitesimal part and (ii) a balance between stresses appearing in the neighboring parts. The first effects (i) are fully described by the present theory. Furthermore, orientation of molecules in an each part leads to the deformation of the part; as a result, the molecules are moved to another point with another light intensity (or light polarization). Therefore, the deformation of each local part results in the translational movement of molecules. Thus, the phenomenon can be treated as a combination of two approaches: “orientational approach” developed in the present theory and “a mass transport approach” proposed earlier, e.g., in refs 20 and 28. Note that the consideration of these two effects together demands a special effort which is out of the topics of the present work and may be carried out in the future. However, using the results of the present theory it is possible to explain qualitatively some experimental results for SRGs in azobenzene polymers. For instance, one can explain why inscription of SRGs is much more efficient in the p-p geometry compared to the s-s geometry.3,51,52 For p-p geometry the light polarization vector, E, is known to be oriented along the gradient vector of the light intensity. In this geometry, each infinitesimal part is able to deform along the vector E. Moreover, due to the condition of constant volume for a glassy polymer each infinitesimal part should deform in the direction perpendicular to the surface. On the other side, in the s-s geometry, polarization vector E lies

We have proposed a microscopic theory of light-induced deformation in amorphous side-chain azobenzene polymers taking the internal structure of azobenzene macromolecules explicitly into account. In our approach the mechanical stress appearing in the system under illumination with the polarized laser light is caused by reorientation of macromolecules due to interactions of chromophores in side chains with the light. Using this approach, we obtain the following results, all of them being in agreement with experiments and recent computer simulations. Depending on the architectures of oligomers, a sample can be either stretched or uniaxially compressed along the polarization direction of the laser light. For some architectures, elongation of a sample demonstrates nonmonotonic behavior with the light intensity and can even change its sign (a stretched sample starts to be compressed). The light-induced stress can be larger than the yield stress at characteristic light intensities used in experiments. This result explains the possibility of irreversible sample deformation under homogeneous illumination and, hence, it explains the possibility of the inscription of surface relief gratings. Using a viscoplastic approach, we have shown that the irreversible elongation of a sample decreases with the increasing temperature. The critical temperature, when the irreversible elongation disappears, is independent of the glasstransition temperature, Tg, and can be smaller than Tg. These results demonstrate a great potential strength of the proposed microscopic orientation approach for describing the photoelastic properties of different azobenzene polymers. This approach can be used in future for more detailed analysis of the temperature effects and effects of the molecular architecture on the grating formation in these materials. Acknowledgment. We cordially thank Prof. Valery Shibaev and Dr. Jaroslav Ilnytskyi for the helpful discussions. The help


Toshchevikov et al.

of Russian Foundation for Basic Research (grant 08-03-00150) is gratefully acknowledged. Appendix The value of the striction stress, σstr, at high light intensities, NchV0/kT . 1, can be found by means of the Laplace’s method for estimation of integrals with a large parameter.49 Introducing a large parameter λ ) NchV0/kT . 1, we can rewrite eq 21, which defines the value of σstr, in the following form using eqs 10 and 35

If a > |b|, the function Q(θ,ψ) has a minimum along the lines θ ) 0, π (Figure 6c,d). The vicinity θ < δ gives the following asymptotic relationship for the integral J(λ) in this case after a new variable is introduced, z ) λθ2:

∫02π dψ ∫0δ dθ θP2(0)e-λ(a+bcos2ψ)θ = λ-1 × ∫02π dψ ∫0∞ dze-(a+bcos2ψ)z ) λ-1 ∫02π a + bdψcos 2ψ )

J(λ) = 2

2

2π λ√a - b2

(A6)

2

J(λ) ≡

∫02π dψ ∫0π dθ sin θP2(θ) exp[-λQ(θ, ψ)] ) 0 (A1)

Here we set

Q(θ, ψ) ) [a + b cos 2ψ]sin2 θ

(A2)

and

Now, one can see that in this case (when a > |b|) the integral J(λ) has a positive value J(λ) ∼ λ-1 at λ > > 1, so that the equality (A1) cannot be satisfied at all values a > |b|. To sum up, eq A1 has no solutions for the parameter σstr at λ . 1 in both cases, when a < |b| and a > |b|. Therefore, eq A1 has a solution at λ . 1 in the only case when a ) |b|. Using the equality a ) |b| and eq A3, we obtain the asymptotic eq 38 for the function σstr(V0) at λ ) NchV0/kT . 1. References and Notes

3σstrεmax 3 a ) 〈sin2 R〉W - 1 , 2 2n0V0

1 b ) 〈sin2 R cos 2β〉W 2 (A3)

where n0 ) nNch is the number of chromophores in a unit volume. According to Laplace’s method,49 the asymptotic behavior of the integral J(λ) in eq A1 is determined by the vicinity of a point (θmin,ψmin), where the function Q(θ,ψ) has a minimum. Our idea is to calculate the integral J(λ) using Laplace’s method and, then, to find a condition for the parameters a and b when the equality J(λ) ) 0 is possible. Analysis of the function Q(θ,ψ) is similar to that for the function u(θ,ψ) given by eq 35. If a < |b|, then the minima of the function Q(θ,ψ) is achieved at θmin ) π/2 and either at ψmin ) π/2, 3π/2 (when b > 0, Figure 6a) or at ψmin ) 0, π (when b < 0, Figure 6b). Expanding the function Q(θ,ψ) into series in the vicinity of the point (θmin,ψmin), we have

Q(θmin + x, ψmin + y) = (a - |b|) + (|b|-a) · x2 + 2|b|y2 (A4) Now, the vicinity of the point (θmin,ψmin) (when |x| < δ and |y| < δ, where δ is a small parameter) gives the following asymptotic relationship for the integral J(λ) after introducing new variables ξ ) xλ, ζ ) yλ and using a standard procedure of Laplace’s method:49

∫-δδ dx sin θminP2(θmin)e-λ(|b|-a)x × λ(|b|-a) ∫-δδ dy e-λ2|b|y = - e λ ∫-∞∞ dξ e-(|b|-a)ξ × 2

J(λ) = 2eλ(|b|-a)

2

2

∫-∞∞ dζ e-2|b|ζ

2

)-

eλ(|b|-a)π λ√2|b|(|b|-a)

(A5)

One can see that at λ . 1 the integral J(λ) has a large negative value J(λ) ∼ - eλ(|b|-a)/λ in this case (when a < |b|), so that the equality (A1) cannot be satisfied at all values a < |b|. Thus, at a < |b|, eq A1 has no solutions for the parameter σstr at λ . 1.

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JP8078265

Microscopic Theory of Light-Induced Deformation in Amorphous Side

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