# Regimes in the Response of Photomechanical Materials
## 2 Light Intensity and Photoisomerization Dynamics
### Asymptotic Dynamics  
In general, the intensity \(I\) varies with time because \(\rho_{t}\) changes with time. However, our numerical experiments suggest that keeping the intensity invariant with time would not result in a large discrepancy in \(\rho_{t}\). This implies that we can treat the intensity as a constant in time (This is reasonable for samples thin compared to the absorption length \(I_{d}\). ), which enables us to make qualitative estimates of the dye dynamics. In dimensionless quantities, Equation (7) can be integrated to give  
\[\rho_{t}=\frac{1}{I+1}(1+Ie^{-(I+1)t}), \tag{10}\]  
where the constant of integration has been chosen so that \(\rho_{t}(x,0)=1\); that is, all dyes are in the _trans_-state initially. It follows that  
\[\rho_{c}=\frac{I}{I+1}(1-e^{-(I+1)t}), \tag{11}\]  
and the system approaches the steady-state exponentially. Early; that is, when \(t\ll 1/(I+1)\), we have that  
\[\rho_{c}\simeq It, \tag{12}\]  
and the _cis_-number density \(\rho_{c}\) is proportional to the fluence of the incident light. The characteristic intensity plays a key role in the system response. Depending on the intensity, three different regimes can be distinguished. Regime 1. If \(I\ll 1\), then the _cis_-number density \(\rho_{c}\) is  
\[\rho_{c}\simeq I(1-e^{-t}). \tag{13}\]  
The time to reach the photostationary state is \(t\approx 1\), or, in dimensional units, \(\tau\) seconds. In the photostationary state,  
\[\rho_{c}\simeq I \tag{14}\]  
and the photostress is proportional to the intensity. If the illumination is removed, the _cis_-population will decay with the time constant \(\tau\) seconds; at the same rate as to reach the photostationary state. The total absorbed energy per volume by the sample to reach the photostationary state, from Equation (9), is \(\mathcal{E}_{p}\simeq\rho_{0}\frac{I}{I_{0}}h\
u\) in dimensional units, and in this regime, the absorbed energy is \(h\
u\) per molecule for the fraction \(I/I_{0}\) of dye molecules. we have that the absorption length is \(l_{d}=10^{-5}/1.3\) m = 7.7 \(\upmu\)m. From Equation (6) we obtain  
\[\sigma^{a}=\frac{1}{\rho_{0}l_{d}}=10^{-21}\;\text{m}^{2}. \tag{21}\]  
The relaxation time was obtained by measuring the transmittance of the photoexcited sample as a function of time. The transmittance, normalized to that of an identical but undyed sample, is shown in Figure 3a. The decay time \(\tau\) is 9030 s.  
The characteristic intensity can now be calculated as  
\[I_{0}=\frac{h\
u}{\sigma^{a}\tau}\simeq\frac{10^{-18}}{10^{-21}\times 10^{4}}=0.1 \;\text{W}/\text{m}^{2}. \tag{22}\]  
The estimate for \(I_{0}\) can be confirmed experimentally. For light traversing the sample in the photostationary state, from Equations (7) and (8), we write [8],  
\[I_{0}=\frac{I_{i}(1-T)}{\ln T+d\rho_{0}\sigma^{a}}. \tag{23}\]  
For incident light at 365 nm and using an attenuator to achieve low intensity, the incident intensity \(I_{i}\) was 0.05 mW/cm\({}^{2}\), (We use the units mW/cm\({}^{2}\) from here on to enable ready comparison with the values in the literature.) and the transmittance \(T\) of the photoisomerized sample was 0.8. This gives \(I_{0}=0.013\) mW/cm\({}^{2}\), in agreement with our earlier estimate. In Table 2, we summarize our results for the material parameters of the azo-acrylate system studied. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number Density** & **Decay Length** & **Absorption** & _cis_**-Lifetime** & **Characteristic** \\
**of Dye** & **Cross-Section** & **Intensity** & **Intensity** \\ \hline \(\rho_{0}\) (m\({}^{-3}\)) & \(l_{d}\) (m) & \(\sigma^{a}\) (m\({}^{2}\)) & \(\tau\) (s) & \(l_{0}\) (W/m\({}^{2}\)) \\ \hline \(10^{26}\) & \(10^{-5}\) & \(10^{-21}\) & \(10^{4}\) & \(10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material parameters of the azo-acrylate system. Figure 3: **(a)** Transmittance of the sample as a function of time. The normalization is against that of an undyed LCE sample with the same constituents except the azo acrylate.