# Regimes in the Response of Photomechanical Materials
## 2 Light Intensity and Photoisomerization Dynamics  
The system under consideration consists of a polymer network formed by cross-linked chains consisting of liquid crystal-forming mesogenic units and photoisomerizable azo dyes. A schematic of one system studied, and the molecular structure of its constituents are shown in Figure 1. The relative quantities of the constituents are given in Table 1.  
The sample shape considered is a rectangular film, consisting of the material in the nematic phase, where the director \(\mathbf{\hat{n}}\), the direction of average orientation of the mesogenic units, is in the plane of the film, parallel to the long edge. The azo dye units, which can be either in the elongated low-energy _trans_-isomer state or the more compact high-energy _cis_-isomer state, are randomly distributed throughout the sample. It is assumed that the _trans_-isomers are aligned along the nematic director \(\mathbf{\hat{n}}\), while the _cis_-isomers are assumed to be unaligned, effectively spherical in shape and randomly oriented. The key notion is that photons with energy \(\hbar v\), where \(h\) is Planck's constant and \(\
u\) is the frequency of peak UV absorption by the _trans_-isomers, are absorbed by the _trans_-isomers, which then undergo photoisomerization to the _cis_-form. The shape change of the azo dye constituents results in a shape change of the entire sample, and due to this shape change, the sample is capable of exerting pressure and performing mechanical work on a load. Due to thermal excitations, the photoexcited _cis_-isomers have a lifetime \(\tau\), after which they decay back to the low-energy _trans_-form. It is assumed that the rate of thermal excitation from _trans_- to _cis_-form is essentially negligible, but thermal excitation drives the _cis_- to _trans_-isomerization. A schematic of the absorbance, energy landscape, and the corresponding molecular shapes for the _trans_- and _cis_-isomers is shown in Figure 2.  
Denoting the number densities of the _trans_- and _cis_-isomers and a total number of dye molecules by \(\rho_{t}\), \(\rho_{c}\) and \(\rho_{0}\), we have  
\[\rho_{t}+\rho_{c}=\rho_{0}. \tag{1}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **RM82** & **2 Azo** & **Acrylates to Amine** & **Irgacure** \\  & **Molar Fraction** & **Molar Fraction** & **Molar Ratio** & **819** \\ \hline Sample & \(1-x\) & \(x\) & \(1.1\) & \(2.5\) wt\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The relative quantities of the constituents. Figure 1: Schematic and constituents of one studied system. If the sample is illuminated by normally incident UV, then the number densities depend on time \(t\) and position \(x\) in the sample, where \(x\) is measured along the normal film surface. The dynamics of the system are given by  
\[\frac{\partial\rho_{t}}{\partial t}=-I\frac{\rho_{t}\sigma^{a}}{h\
u}+\frac{ \rho_{c}}{\tau}, \tag{2}\]  
where \(I(x,t)\) is the UV light intensity, \(\
u\) is the peak absorption frequency, and \(\sigma^{a}\) is the absorption cross-section of the _trans_-isomers. It is useful to define the characteristic intensity (The characteristic intensity was identified by [PERSON] [7; 8], but its relation to material parameters has not been previously given explicitly.) \(I_{0}\), a material parameter, as  
\[I_{0}=\frac{h\
u}{\sigma^{a}\tau}, \tag{3}\]  
which sets the scale of the photoexcitation. This corresponds to a characteristic photon number density of  
\[\rho_{0}^{photon}=\frac{1}{c\sigma^{a}\tau}=\frac{I_{0}}{ch\
u}, \tag{4}\]  
where \(c\) is the speed of light. The dependence of intensity on distance is given by  
\[\frac{\partial I}{\partial x}=-I\rho_{t}\sigma^{a}. \tag{5}\]  
It is also useful to define a decay length \(l_{d}\) as  
\[l_{d}=\frac{1}{\rho_{0}\sigma^{a}}, \tag{6}\]  
in terms of the total dye number density. As can be seen from Equation (2), the system behavior is determined by the characteristic intensity \(I_{0}\) and the _cis_-decay time \(\tau\).