## 2 Model for Pressure and Strain Induced by Photoisomerization
### Strain Induced by Photo-Orientation  
Next, we assume that the isomers are uniformly distributed in the irradiated volume, and that linearly polarized light, say along a direction denoted by \(\parallel\), uniformly irradiates the sample. \(\perp\) refers to a direction perpendicular to \(\parallel\). Photo-orientation occurs and it influences the deformation process. Indeed, \(N_{t,c,\parallel}\) and \(N_{t,c,\perp}\) denote the concentration distribution, and \(A_{2}^{tc}\) the geometrical order parameters of the isomers in the \(\parallel\) and \(\perp\) directions, respectively. The lengths of the sample in \(\parallel\) and \(\perp\) directions are given by  
\[L_{\parallel,\perp}=N_{t,\parallel,\perp}l_{t}+N_{c,\parallel,\perp}l_{c} \tag{5}\]  
\[N_{t,c,\parallel}=\ (1+2\mathbf{A}_{2}^{\mathbf{tc}})N_{t,c} \tag{6}\]  
\[N_{t,c,\perp}=\ (1-\mathbf{A}_{2}^{\mathbf{tc}})N_{t,c} \tag{7}\]  
Figure 1: Theoretical simulation of the time evolution of the pressure P, normalized by K, using Equations (1) and (2), for different normalized pump intensities, i.e., in units of \(I=I_{A}\phi_{tc}\sigma_{t}\): 0.2, 0.5, and 1.0, and for different ratios of the occupied volumes of the isomers: \(v_{c}/v_{t}=0.5,\ 1.0,\ 2.0\). Note that when \(v_{c}/v_{t}=1\), no pressure occurs, i.e., no isomeric volume change. Negative (positive) pressure occurs for \(v_{c}/v_{t}<1\) (\(v_{c}/v_{t}>1\)) and the larger the pump intensity, the faster the pressure build-up. \[A_{2}^{t}=\frac{1}{5}\{exp(-k_{2}t)-exp(-k_{0}t)\};A_{2}^{c}=-P_{2}^{tc}A_{2}^{t} \tag{8}\]  
The expressions of A\({}_{2}^{tc}\) are developed in [36]. \(l_{t,c}\) is the average length per molecule for an all-trans (all-cis) sample. The subscripts c and t refer to cis and trans, respectively. After rearrangement, the strains \(\epsilon_{\parallel,\perp}=\Big{(}L_{\parallel,\perp}-L\Big{)}/L\), with \(L\) the length of the sample prior to irradiation, are  
\[\epsilon_{\parallel}=\big{(}l_{c}/l_{t}-1\big{)}\alpha+2\big{(}1-P_{2}^{tc}l_ {c}/l_{t}\big{)}A_{2}^{t} \tag{9}\]  
\[\epsilon_{\perp}=\big{(}l_{c}/l_{t}-1\big{)}\alpha-\big{(}1-P_{2}^{tc}l_{c}/l_ {t}\big{)}A_{2}^{t} \tag{10}\]  
\[\epsilon_{\parallel}-\epsilon_{\perp}=3\big{(}1-P_{2}^{tc}l_{c}/l_{t}\big{)}A _{2}^{t} \tag{11}\]  
\(P_{2}^{tc}\) is a parameter that characterizes the change in the orientation of the azobenzene molecule after trans\(\rightarrow\)cis isomerization, and its absolute value is between zero and one. \(A_{2}^{t}\) is always negative and so is \(\epsilon_{\parallel}\); if \(l_{t,c}\) are assigned to the lengths of the long axis of the trans and cis isomers, i.e., 9 and 5.5 A, respectively, the length of the sample in the direction parallel to the irradiation light polarization \(L_{\parallel}\) is smaller than \(L\), a feature which is indicative of the shrinkage of the sample in the direction of the irradiating light polarization. \(\epsilon_{\perp}\) is also negative and the absolute value of \(\epsilon_{\parallel}\) is larger than that of \(\epsilon_{\perp}\) at any time, indicating that the sample also shrinks in the perpendicular direction but the amount of shrinkage in the parallel direction is larger than that in the perpendicular one (Figure 2). At saturation, e.g., at pss (\(\alpha_{PSS}\sim 1\)), photo-orientation disappears, e.g., \(A_{2}^{t}\sim 0\), and \(\epsilon_{\parallel}=\epsilon_{\perp}=\epsilon_{pss}\) is independent of light intensity, and it only depends on the relative difference between the lengths of the long axis of the trans and cis isomers. The system saturates at large irradiation doses, because even molecules that are oriented perpendicular to the irradiation light polarization present an appreciable probability of being photoisomerized, and the model of isotropic volume change holds. The orientation of mesogens in liquid crystalline films influences the bending direction of macroscopic samples.