A Unified Material Description for Light Induced Deformation in Azobenzene Polymers

Complex light-matter interactions in azobenzene polymers have limited our understanding of how photoisomerization induces deformation as a function of the underlying polymer network and form of the light excitation. A unified modeling framework is formulated to advance the understanding of surface deformation and bulk deformation of polymer films that are controlled by linear or circularly polarized light or vortex beams. It is shown that dipole forces strongly respond to polarized light in contrast to higher order quadrupole forces that are often used to describe surface relief grating deformation through a field gradient constitutive law. The modeling results and comparisons with a broad range of photomechanical data in the literature suggest that the molecular structure of the azobenzene monomers dramatically influences the photostrictive behavior. The results provide important insight for designing azobenzene monomers within a polymer network to achieve enhanced photo-responsive deformation.

tential that contains free space energy (L F ), kinetic and stored energy of the solid (L M ), electronic interactions (L I ), and dissipation due to light scattering and photochemical inefficiencies (D). These terms are given by The electromagnetic field Lagrangian density (L F ) includes E i as the electric field and B i as the magnetic flux density, both in the spatial frame. The free space permittivity and permeability are ǫ 0 and µ 0 , respectively. The interaction Lagrangian (L I ) includes the spatial frame current density J i , magnetic vector potential A i , bound charge density q, and electric potential φ. The matter Lagrangian density per undeformed volume (L M ) includes separate kinetic energies for the center-of-mass in the first term, a set of optical modes in the second term, and the last term is the stored energy. The kinetic energies are defined by the mass density per undeformed volume ρ 0 and velocityẋ i and the electronic kinetic energy written as a function of the material time derivatives of the electronic vector order parameters,ẏ α i , and the effective mass density with units of mass per undeformed volume, m α . Two order parameters (α = t, c) are considered to independently model the trans and cis states. The stored energy per undeformed volume (ρ 0 Σ) in L M is a function of the small strain tensor ε ij , the electronic coordinate (y α i ), and its gradient (y α i,j ) 1, 2 . The dissipative energy D describes losses associated with light scattering and photochemical reactions in terms of the velocity of the electronic coordinates and the damping parameter γ α .
The stored energy and the electronic coordinate vector order parameters are defined for characterizing material continuum length scales while retaining critical underlying light-matter characteristics.
The model assumes that the electronic coordinates are summed and averaged over a continuum representative volume element. This formulation is non-relativistic where material velocities must be much smaller than the speed of light. Within the visible and ultra-violet light spectra, the oscillation velocity of the charged particles is still approximately three orders of magnitude smaller than the speed of light. This assumes that the charged particles do not displace more than 10% of the size of an azobenzene molecule. An important component of this formulation is the choice of the stored energy function contained within L M , which is expressed in the following section.

Stored energy relations
The stored energy function Σ consists of four different contributions: Σ t and Σ c are non-convex energy functions for the trans and the cis states, respectively; Σ m is the elastic energy of the glassy polymer network, and Σ coupl defines the coupling between the trans vector coordinate and the polymer network deformation. Deformation attributed to the cis state is neglected since this phase normally leads to disorder. Furthermore, the trans coordinate is reduced in magnitude as the cis concentration increases. The energy function is written as in terms of the the small strain tensor ǫ ij , the electronic vector order parameters y α i and their gradients y α i,j . Each part of the stored energy density is defined by where the phenomenological parameters a α and b α (ŷ t 0 ) for α = t, c govern the evolution of the electronic coordinates, a α 0 is a penalty on gradients of y α i , b ijkl is a fourth order tensor that couples the trans coordinate vector order parameter to stress. It should be noted that we neglect any coupling between two vector order parameters and therefore conservation of concentration of the two material states is not necessarily guaranteed. In Eq. (3), the parameters a α < 0 and b α (ŷ t 0 ) > 0 which creates the non-convex function given in the main paper ( Figure 2). The higher order parameter, b α , is defined to be a function of a time averaged trans state,ŷ t 0 , to model the slower time dynamics of photoisomerization relative to dynamics that occur at visible and UV light frequencies. This equation is given in the following section. The higher order model parameter b α (ŷ t 0 ) changes during trans-cis photoisomerization such that the trans coordinate reduces in magnitude while the cis coordinate increases from near zero. This assumes a loss of nematic order as the concentration of the cis state increases. The functional forms of these phenomenological parameters are assumed to averaged electronic state is restricted to 0 <ŷ t 0 < 1 such that b t and b c are bounded andŷ t 0 = 1 and y t 0 = 0 denote the fully trans and the fully cis state, respectively. All the parameters, excluding the photostrictive coefficients, are summarized in Table 1. The photostrictive parameters are described in more detail in the subsequent section.

Governing equations
Minimization of the Lagrangian energy densities and dissipative energy function from Eq.
where ν is t or c for the trans and cis vector order parameter, respectively. q α is the effective bound charge density per undeformed volume given by equation Eq. (1) in the main paper and q αν is the charge density associated with the higher order gradients on the electric field and magnetic flux density. The higher order terms were identified to be negligible and is therefore not used in the current study.
The time-averaged magnitude of the trans state,ŷ t 0 , which is approximately (ȳ t 1 ) 2 + (ȳ t 2 ) 2 + (ȳ t 3 ) 2 1/2 at a given time where the overbar represents time-average 3, 4 , is treated on the time scale of photoisomerization which is significantly slower than the optical wavelength time period. The time-averaged magnitude of the trans state is determined by where α = |y t · e| |y t ||e| and e is the unit vector representing the direction of light polarization. χ is a sensitivity parameter governing the amount of photoisomerization lying on the range of 0 ≤ χ < 1/(e · E) 2 .
For example, if χ = 0, the time averaged trans state is always one (zero photoisomerization). As χ increases, the time averaged trans state will be reduced if the electronic oscillation increases at a particular light wavelength. The time constant τ avg is defined to be on the order of the photoisomerization rate (∼10 ps). This equation must be solved simultaneously with all other governing equations to determine the evolution of the azobenzene state.

Maxwell's equations
Minimization of the field quantities leads to the Maxwell equations in the spatial configuration given by For the azo-LCN model, both the surface current and surface charge are set to zero and the charge density is where P i and Q ij are the polarization and the quadrupole density, respectively. The polarization and the quadrupole are determined from the internal electronic coordinates using for µ, ν = t, c which is valid in the small strain limit.
The current density, neglecting magnetization effects, is

Linear momentum equations
We assume linear elastic behavior of a glassy polymer coupled to quadratic dependence on the trans vector order parameter. Due to large disparity in time scales between optical waves and elastic waves and viscoelasticity within glassy polymer, we neglect the rate dependent deformation and solve the quasi-static form of linear momentum given by where the reference stress is σ R ij = −b ijkl y t0 k y t0 l and the superscript t0 denotes the initial microstructure configuration of the trans state. The sign of the photostrictive parameters determine elongation or contraction with respect to the trans vector order parameter. The two cases considered in the simulations uses the parameter values in Table 2. In this case, the coupling tensor b ijkl reduces to b ijkl = 0 Also note due to symmetry that b ijkl = b klij and b ijkl = b jikl = b ijlk . Similar symmetry exists for the elastic tensor c ijkl .
The total Cauchy stress denoted by σ ij contains components associated with deformation and the internal electronic order parameters. Assuming an isotropic elastic medium, the expanded form of the Cauchy stress is 5 Young's modulus, and ϑ = 0.45 is Poisson's ratio. Table 2 represents parameters used in the present study. An uniform bending of a monodomain thin film during the trans-cis photoisomerization is numerically performed using photostrictive parameters that assume long spacers otherwise surface relief deformation during the trans-cis-trans photoisomerization are performed using that in the case of short spacers.

Supplemental Computational Analysis
The computational domain for the two configurations simulated in the main text is shown in Figure 1. The domain in Figure 1(a) is used to simulate bending from uniform light exposure while   We further analyze the model by switching the sign of the topological charge to be ξ = −10 (see Figure 5). For this case, it is evident that the helical pattern on the surface is opposite to that in Figure 8(b) in the main paper.   Table 2: Parameters used in the model for shape deformation due to t-c-t photoisomerization process. (c)Ē z Figure 5: Deformation due to a linearly polarized vortex laser beam when the negative vortex topological charge ξ = −10 is used.