Multiple minimum-energy paths and scenarios of unwinding transitions in chiral nematic liquid crystals

We apply the minimum-energy paths (MEPs) approach to study the helix unwinding transition in chiral nematic liquid crystals. A mechanism of the transition is determined by a MEP passing through a first order saddle point on the free energy surface. The energy difference between the saddle point and the initial state gives the energy barrier of the transition. Two starting approximations for the paths are used to find the MEPs representing different transition scenarios: (a) the director slippage approximation with in-plane helical structures and (b) the anchoring breaking approximation that involves the structures with profound out-of-plane director deviations. It is shown that, at sufficiently low voltages, the unwinding transition is solely governed by the director slippage mechanism with the planar saddle-point structures. When the applied voltage exceeds its critical value below the threshold of the Fréedericksz transition, the additional scenario through the anchoring breaking transitions is found to come into play. For these transitions, the saddle-point structure is characterized by out-of-plane deformations localized near the bounding surface. The energy barriers for different paths of transitions are computed as a function of the voltage and the anchoring energy strengths.

Figure 1

Field free states: (a) the metastable left-handed helical (twisted) structure; (b) the metastable right-handed helical (twisted) structure; (c) the stable untwisted (nematic) structure.

Figure 2

Energy per unit area along the MEP for the director slippage transition computed at $U=0.53\mathrm{V}$ and $r_W=1.0$. The filled circles correspond to the images of the system used in the GNEB calculation. The reaction coordinate is defined as the displacement along the path normalized by its total length.

Figure 3

Profiles of the azimuthal angle for each image in the MEP for the director slippage transition.

Figure 4

(a) Free energy surface for the helical structures (1). (b) Energy along the path passing through the saddle point. The energy barrier is $\mathrm{\Delta }E\approx 22.6\mu \mathrm{J}/{\mathrm{m}}^2$.

Figure 5

Energy barrier map in the $r_W-U$ plane for the director slippage transitions. The green line with squares indicates the threshold voltage of the Fréedericksz transition $U_{\text{th}}$ and the energy barrier scale is given in $\mu \mathrm{J}/{\mathrm{m}}^2$.

Figure 6

Energy per unit area along the MEP for the anchoring breaking transition computed at $U=0.53\mathrm{V}$ and $r_W=1.25$.

Figure 7

Profiles of (a) the azimuthal and (b) the polar angles for each image in the MEP for the anchoring breaking transition.

Figure 8

Energy barrier map in the $r_W-U$ plane for the anchoring breaking transitions. The green line with squares indicates the threshold voltage of the Fréedericksz transition $U_{\mathrm{th}}$ and the energy barrier scale is given in $\mu \mathrm{J}/{\mathrm{m}}^2$.

Figure 9

Map of the minimum polar angle ${\theta }_{\min }$ [$\equiv {min}_z\theta \left(z\right)$] at the saddle point (transition state) for (a) the director slippage transition and (b) the anchoring breaking transition in the $r_W-U$ plane. For each contour line, digits in squares indicate the value of ${\theta }_{\min }$ measured in degrees.

Figure 10

Map of the ratio of the energy barriers for the anchoring breaking and the director slippage transitions in the $r_W-U$ plane.

Figure 11

Energy per unit area along the MEP for the transition above the Fréedericksz threshold computed at $r_W=2$ and $U=0.64\mathrm{V}$.

Figure 12

Profiles of (a) the azimuthal and (b) the polar angles for the images along the MEP calculated above the Fréedericksz transition at $r_W=2$ and $U=0.64\mathrm{V}$.