Chirality Amplification and Chiral Segregation in Liquid Crystals

Abstract

Liquid crystal mesophases of achiral molecules are normally achiral, yet in a few materials they spontaneously segregate and form right- and left-handed chiral domains. One mechanism that drives chiral segregation is molecular shape fluctuations between axial chiral conformations, where molecular interactions favor matching chirality and promote helical twist. Cooperative chiral ordering may also play a role in chirality amplification, as when a tiny fraction of chiral dopant drives a nematic phase to become cholesteric. We present a model of cooperative chiral ordering in liquid crystals using Maier-Saupe theory, and predict a phase diagram with a segregated cholesteric phase with alternating domains of left- and right-handed chiral twist, with opposite enantiomeric excess, in addition to racemic nematic and isotropic phases. Our model also demonstrates how chiral molecular fluctuations influence the helical twisting power of dopants in the nematic phase, which may be observed even in materials where the segregated cholesteric phase is preempted by a transition to another phase. We compare these results with Monte Carlo simulation studies of the switchable chiral Lebwohl-Lasher model, where each spin switches between right- and left-handed chiral states. Simulation results validate the predicted phase diagram, demonstrate chiral amplification in the racemic nematic phase, and reveal complex coarsening dynamics in the segregated cholesteric phase. These results suggest that molecular fluctuations between degenerate chiral configurations may be a common mechanism to produce cooperative chiral order in achiral liquid crystals.

Figures (13)

• Figure 1: Left: Enantiomeric excess $\eta$ (blue -- inner curve) and ratio $x$ of the cholesteric wave number $q$ and the maximal value $q_0$ (red -- outer curve) in the co-existing segregated cholesteric phases as a function of the dimensionless magnitude of the chiral interaction between the nematogens $\chi$. Right: phase diagram as a function of the temperature $T/T_\mathrm{IN}$ scaled to the nematic transition temperature $T_\mathrm{IN}$, and the ratio of the chiral and nematic interaction strengths $|K|/J$. The vertical line in red indicates the isotropic-nematic transition. The curved line in blue demarcates the chiral transition between the segregated cholesteric phase and the racemic nematic phase for $T<T_\mathrm{IN}$ and the racemic isotropic phase for $T>T_\mathrm{IN}$, which cross at what we interpret to be a critical end point (CEP) indicated by the circle. See also the main text.
• Figure 2: Enantiomeric excess $\eta$ for a fixed strength of the chiral interaction $K/J = 0.333$, as a function of the dimensionless chiral biasing potential $g/k_\mathrm{B}T$ for different temperatures $T/T_\mathrm{IN}$. (a) Predictions from Maier-Saupe theory. The corresponding values of the chiral interaction strength $\chi$ are $0.825, 0.634, 0.412$ and $0.239$ from top to bottom. (b) Results of Monte Carlo simulations. Indicated is also the ideal hyperbolic tangent behavior expected in the absence of a chiral interaction and $K=0$.
• Figure 3: Phase diagrams as a function of the chiral interaction strength $K/J$ and scaled temperature $T/T_\mathrm{IN}$ obtained by means of Monte Carlo simulations. (a) Heat map of the logarithm of the specific heat, $\mathrm{log}_\mathrm{10} C_V$, and (b) that of the chiral susceptibility $\chi_\eta$. The latter cannot distinguish between the racemic isotropic and nematic phases.
• Figure 4: (a-b): Simulation results showing local orientation and chirality configurations at fixed temperature $T/T_{IN} =0.067$ and varied values of chiral interaction, $K/J$. A horizontal $XZ$ cross-section is shown with the local unit orientation field $\hat{u}_i$ colored blue and red corresponding to $\eta_i=\pm1$. Simulation box size $N = 128$. (c): Simulation of size $N = 480$ after $2\times 10^6$ steps showing alternating chiral domains. (d): Double-logarithmic plot of the average layer thickness, $\langle L(t) \rangle$, as a function of time, $t$, in number of steps for the box size $N = 480$. Stripes of alternating chirality where counted across the $101$ plane every $1\times 10^5$ steps. The dashed line shows a curve fit of $\langle L \rangle \propto t^\alpha$ producing a scaling exponent $\alpha = 0.379 \pm 0.016$.
• Figure S1: Monte Carlo simulation results showing the enantiomeric excess, $\eta$, as a function of the dimensionless chiral biasing potential $g/k_\mathrm{B}T$. Configurations of $128^3$ sites were annealed from $T/T_{IN} = 0.668$ to $T/T_{IN} = 0.267$ for $2 \times 10^6$ Monte Carlo steps at each temperature. Shown in solid black is the analytical function, $\eta = \tanh \frac{g}{k_\mathrm{B} T}$, representing the limit of zero chiral coupling, $K/J=0$.