## 2 Model for Pressure and Strain Induced by Photoisomerization
### Dynamics of Volume Change and Pressure Build-up
Note, from the expression of the reduced volume, \(v_{R}\), that no volume change in the irradiated sample is possible if the volume of the azobenzene molecule remains unchanged upon trans and cis isomerization, i.e., \(V=v_{t}\), and the difference in the occupied volume and the volume of the trans isomer is proportional to the difference in volumes of the two isomers, that is \((V-v_{t})=\alpha\)\((v_{c}-v_{t})\). Even though this model ignores free volume and expansion, both of which could be introduced into the equations of a more elaborate model, it does, however, describe reversible volume change in the material, and, more importantly, it shows that the differing volumes of the isomers build a mechanical pressure inside the material (_vide infra_), and it preserves the physics and simplifies the discussion. Many of the azo-polymers studied for photomechanical properties contain the azobenzene molecule of which the trans and cis isomers are spectrally distinguishable. Transazobenzene (t-AB) presents an absorption band in the UV region of the spectrum around 360 nm, and cis-azobenzene (c-AB) in the visible region around 450 nm. In addition, the lifetime of the c-AB is generally on the order of hours, depending on the polarity of the host material. Therefore, c-AB can be considered stable on a time scale of minutes, i.e., the time scale of the experiments. In such systems, the dynamics of \(trans\to cis\) photoisomerization and photo-orientation under polarized UV light irradiation are described by double exponentials for both the population, i.e., \(\alpha\), and the order parameter of \(A_{2}^{t}\) and \(A_{2}^{\zeta}\) of t-AB and c-AB, respectively [36]. To describe the intermolecular interaction of the azobenzene molecules in LC materials, one may assume a Maire-Saupe type potential and introduce it into the rate equations of photo-orientation of azobenzene under polarized UV light. Assuming linearly polarized irradiation and t-AB as a rodlike molecule, and the rates of the cis\(\rightarrow\)trans thermal reaction and the diffusion, i.e., stochastic thermally activated molecular motion, in the cis and trans forms small, a feature which is true in solid polymers, and if the irradiating light is turned on at the time \(t=0\), \(\alpha\) and \(v_{R}\) of irradiated samples are given by [36]  
\[v_{R}=\alpha\ \cong\ (1-\frac{1}{2}\left\{0.66 exp(-k_{2}t)+1.22 exp(-k_{0}t) \right\})\, \tag{1}\]  
\[k_{2}=2.23\ I_{A}\phi_{tc}\sigma_{t}, \tag{2}\]  
and  
\[k_{0}=0.35\ I_{A}\phi_{tc}\sigma_{t} \tag{3}\]  
\(k_{0,2}\) and \(t\) are expressed in units of \(s^{-1}\) and \(s\), respectively. This equation is obtained by substituting the expression of \(\alpha\) in \(V\), where \(I_{A}\) is the absorbed irradiation intensity, and \(\phi_{tc}\) is the isomerization quantum yield of the trans\(\rightarrow\)cis photoisomerization, and \(\sigma_{t}\) is the absorption cross section of t-AB. Both \(\sigma_{t}\) and \(\phi_{tc}\) are at the irradiation wavelength. Equation 1 shows analytically that the volume of the sample evolves with two time constants that mainly depend on the absorbed irradiation light intensity and the parameters of \(trans\to cis\) photoisomerization. Assuming an isotropic volume change, we have \(P=K\)\((\Delta V/V)\)[38], with \(\Delta V\) as the volume change of the material upon irradiation, e.g., the volume at irradiation minus the initial volume, \(P\) is the optical pressure, and \(K\) the bulk modulus of the polymer. So, using the expression of \(V\), the pressure reads  
\[P=K(V-v_{t})/v_{t}=K(v_{c}/v_{t}-1)\alpha \tag{4}\]The pressure builds in the azo-polymer as the content of the cis isomer increases, and the larger the volume change in the chromophore, upon trans\(\rightarrow\)cis isomerization, the larger the pressure (Figure 1). It is noteworthy that volume expansion of film samples was experimentally observed in glassy polymers by different authors using ellipsometry [39] as well as actuation [40].