A genetic algorithm for the response of twisted nematic liquid crystals to an applied field

Abstract

When an external field is applied across a liquid-crystal cell, the twist and tilt distributions cannot be calculated analytically and must be extracted numerically. In the standard approach, the Euler-Lagrange equations are derived from the minimization of the free energy of the system and then solved via finite-difference methods, often implemented in commercial software. These tools iterate from initial solutions that are compatible with the boundary conditions, providing limited to no flexibility for customization. Here, we present a genetic algorithm that outputs fast and accurate solutions to the integral form of the equations. In our approach, the evolutionary routine is sequentially applied at each position within the bulk of the cell, thus overcoming the necessity of assuming trial solutions. The predictions of our routine strongly support the experimental observations on different instances of spatially varying twisted nematic liquid-crystal cells, patterned with different topologies on the two alignment layers.

Figures (8)

• Figure 1: Integration parameter $\beta(\phi_m,V)$. The dots correspond to the numerically calculated values for maximum twist angles ${\phi_m=90^\circ,67.5^\circ,45^\circ,22.5^\circ,0^\circ}$. The vertical gradient line is the analytical $\beta_T(\phi_m)$ values from Eq. \ref{['eq:beta-threhold']}, whereas the cascading solid-coloured lines are the lines of best fit using Eq. \ref{['eq:beta-V']}.
• Figure 2: Maximum tilt angle $\theta_m(\phi_m,V)$. The dots correspond to the numerically calculated values for maximum twist angles ${\phi_m=90^\circ,67.5^\circ,45^\circ,22.5^\circ,0^\circ}$. In the bottom inset, the gradient line is the analytical $V_T(\phi_m)$ values from Eq. \ref{['eq:vt']}. The dashed lines in both insets connect the dots for visual ease.
• Figure 3: Fits for $\theta_m(\phi_m,V)$. With the ansatz of Eq. \ref{['eq:theM-ansatz-1']}, fits are produced using the ${\phi_m=0^\circ,45^\circ,90^\circ}$ data sets. Each fit is then used to produce the $\theta_m(\phi_m,V)$ curves for other $\phi_m$ values, as shown in each subplot.
• Figure 4: GA results for ${\phi_m=45^\circ}$.a. Twist $\phi(z)$, b. tilt $\theta(z)$, and c. phase retardation $\Gamma(z)$ distributions numerically calculated for a range of voltages between 1.0 V and 4.2 V. The $\Gamma(z)$ are computed using $L=35~\mu$m, $\Delta n=0.151$, and $\lambda = 632$ nm.
• Figure 5: GA results for different maximum twist angles. Twist, tilt, and phase retardation distributions at a. $V=1.061$ V, b. $V=2.0$ V, c. $V=4.0$ V.