# Light-Induced Ring Pattern in a Dye-Doped Nematic Liquid Crystal
## 3 Mathematical Modeling for Photo-Isomerization in Dye-Doped Liquid Crystals  
Nematic liquid crystals are formed by rod-shaped molecules that--in a temperature range--can present an orientational rather than positional order. Then, the dynamics of the NLC can be described by a scalar order parameter \(S(\vec{r};t)\) that accounts for the alignment of the molecules along a given direction [7; 8; 9], defined by  
\[S(\vec{r};t)\equiv\frac{(3\langle\cos^{2}\theta\rangle-1)}{2}, \tag{1}\]  
where the brackets \(\langle\cdot\rangle\) mean the spatial average in a microscopic element volume at position \(\vec{r}\) and time \(t\), and \(\theta\) is the angle between the molecules and the local preferred direction [7; 8; 9]. Thus, \(S\) accounts for the dispersion of the molecules with respect to their average direction. The scalar order parameter for a perfectly aligned nematic phase is \(S=1\), and that for an isotropic phase is \(S=0\). Note that a negative and large S shows that the molecules are oriented, but the choice of the current orientation does not coincide with the molecular average orientation. Close to a phase transition, [PERSON] conjectured that the free energy can be written as a polynomial expansion of the order parameter [33]. Based on this type of approach, in the Landau-de Gennes theory, the transition between a nematic and an isotropic liquid state in a thin film is described by the dimensionless equation [7]  
\[\partial_{t}S(r_{\perp},t)=-AS+BS^{2}-S^{3}+\
abla^{2}S, \tag{2}\]  
where \(r_{\perp}\) accounts for the transversal coordinate of the liquid crystal layer and \(A\) is the bifurcation parameter, which is proportional to the difference between the current and critical temperature. Note that for large values of \(A\), the isotropic state is favored in comparison to the nematic state. \(B\) is a parameter that characterizes the size of the region of coexistence between the nematic and isotropic liquid state. The third and fourth terms on the right-hand side account for the nonlinear response of the medium and the spatial coupling of the order parameter, respectively. This coupling is diffusive in nature. Namely,  
Figure 4: Equilibria ring patterns were experimentally observed for different powers of the blue light beam. After a long period of evolution, snapshots were observed for different powers, as denoted in the lower part of each snapshot. the flow of the order parameter is proportional to its gradient. This model predicts that the nematic and the isotropic liquid transition is of a subcritical nature. On the other hand, the concentration of molecules in the _cis_-state \(C(\vec{r},t)\) at position \(\vec{r}\) in time \(t\) satisfies a relaxation and diffusion equation of the form [19]  
\[\partial_{t}C=-\lambda[C-C_{0}(I)]+\delta\
abla^{2}C, \tag{3}\]  
where \(\lambda\) is the decay rate related to the transition from a _cis_ to _trans_ state by thermal relaxation. \(C_{0}(I)\) is the equilibrium concentration of molecules in the _cis_ state that is proportional to the total intensity of the incident light \(I\). Indeed, \(C_{0}(I)\equiv\gamma I/(1+\eta I)\), where \(\gamma\) and \(\eta\) are dimensional parameters [19]. \(\delta\) is the diffusion coefficient of the concentration of the _cis_ state. As a result of the propagation of light, the intensity of the light has the following form:  
\[I=I_{0}e^{-r_{\perp}^{2}/w^{2}}, \tag{4}\]  
where \(w\) and \(I_{0}\) are the light beam waist and the light intensity at the beam's center, respectively. As we have mentioned, the incorporation of dye-dopants increases the nonlinear response of liquid crystals under the excitation of external fields [11; 18; 19; 20; 34]. Since the dye-dopant is not a liquid crystal and may even be immiscible, its excessive inclusion can prevent the mixture from being a liquid crystal; thus, a dye-dopant should be used in small amounts. We note that one of the reasons for considering methyl red methyl ester is that it is more miscible in E7 than other dopants; for example, methyl red.