Elastic moduli of blue phases of cholesteric liquid crystals with low chirality

TL;DR

This paper develops a Landau–de Gennes-based theory for the elastic properties of cubic blue phases in cholesteric liquid crystals at low chirality, employing a rigid-tensor approximation to derive the lattice elastic moduli for BPI (O^8) and BPII (O^2). By working in the Fourier representation of the order-parameter χ and exploiting cubic symmetry, it yields explicit expressions for the bulk and shear moduli, showing isotropic elasticity and zero Lamé parameter in the one-constant limit (η = 1) with zero Poisson ratio, while allowing for anisotropy and auxetic behavior when η ≠ 1. The authors compute phase-specific gradient-bulk energy balances, determine the equilibrium reciprocal-lattice parameters, and provide closed-form formulas for K, E, and ν for the O^5, O^8, and O^2 blue phases, including the dependence on χ-harmonic ratios and the mean order-parameter magnitude X. The results reveal that, at low chirality, the gradient energy is subdominant and all moduli scale with X^2, offering insights into direction-dependent stiffness, potential auxetic behavior, and the prospects for linking elastic, optical, and electro-optical properties within a unified framework applicable to photonic-crystal-like blue phases.

Abstract

A new theoretical approach has been developed to describe the elastic properties of cubic blue phases of cholesteric liquid crystals (LCs). Blue phases are three-dimensional periodic chiral liquids with local anisotropy of the average orientation of molecules, and due to their periodicity, they have lattice elastic moduli characteristic of ordinary crystalline solids. The rigid tensor approximation, which works well at low chirality parameter ($κ\ll1$), was used to calculate the elastic moduli of the experimentally observed blue phases $O^8$ (BPI) and $O^2$ (BPII). It is shown that in the one-constant approximation for Frank moduli of LCs ($K_{11}=K_{22}=K_{33}$), the cubic lattice of blue phases has isotropic elasticity, and the Lamé's first parameter $λ_\mathrm{L}$ and Poisson's ratio $ν$ are equal to zero. It is found that the sign of the Poisson's ratio is determined by the ratio of elastic moduli $K_0/K_1$; in particular, when $K_0>K_1$, the Poisson's ratio is negative.

Figures (4)

• Figure 1: Distribution of the value of $\mathrm{Tr}(\hat{\chi}^3)$, which determines the free energy gain, in the unit cell of the ideal blue phase $O^5$. The condition $\mathrm{Tr}(\hat{\chi}^3)=0$ (yellow surface) defines the boundary of topological defects (disclination cores). Light blue surfaces correspond to $\mathrm{Tr}(\hat{\chi}^3)$ values equal to 20, 40, 60, and 80% of the maximum.
• Figure 2: Distribution of the value of $\mathrm{Tr}(\hat{\chi}^3)$, which determines the free energy gain, in the unit cell of the ideal blue phase $O^8$ (BPI). The condition $\mathrm{Tr}(\hat{\chi}^3)=0$ (yellow surface) defines the boundary of topological defects (disclination cores). Light blue surfaces correspond to $\mathrm{Tr}(\hat{\chi}^3)$ values equal to 20, 40, 60, and 80% of the maximum.
• Figure 3: Distribution of the value of $\mathrm{Tr}(\hat{\chi}^3)$, which determines the free energy gain, in the unit cell of the ideal blue phase $O^2$ (BPII). The condition $\mathrm{Tr}(\hat{\chi}^3)=0$ (yellow surface) defines the boundary of topological defects (disclination cores). Light blue surfaces correspond to $\mathrm{Tr}(\hat{\chi}^3)$ values equal to 20, 40, 60, and 80% of the maximum.
• Figure 4: Dependence of the ratios of Young's moduli $E_{\langle100\rangle}$ and $E_{\langle111\rangle}$ to the bulk modulus $K$ on the parameter $\eta$ for the blue phases $O^8$ (BPI) and $O^2$ (BPII) in the rigid tensor approximation.