Lattice elasticity of blue phases in cholesteric liquid crystals

TL;DR

This paper addresses the problem of quantifying the elastic properties of cubic blue phases in cholesteric liquid crystals by employing the Landau–de Gennes framework and focusing on the simplest cubic blue phase $O^5$. It develops three complementary modeling approaches—the rigid-tensor approximation, the free helicoid approximation, and an interpolation model—to compute the elastic moduli as functions of chirality $\kappa$ and reduced temperature $\tau$, using a Fourier representation of the order parameter. The authors show that the two limiting models yield the same bulk modulus $K=(128/3)\kappa^2 e^2$ but differ in shear behavior, with the rigid-tensor case predicting isotropy at $\eta=1$ and the free helicoid case satisfying a Cauchy-like relation $\lambda_{xxyy}=\lambda_{xyxy}$; the interpolation framework connects these limits and establishes applicability conditions (e.g., $\kappa^2\gg 1.0866$) for the purely gradient-driven regime. The work provides a quantitative route to estimating elastic constants for actual blue phases and related periodically ordered structures, highlighting how chirality controls elasticity and pointing to extensions to observed phases and to anharmonic and fluctuation effects relevant to weak crystallization.

Abstract

New theoretical approaches have been developed for studying and quantitatively describing the elastic properties of cubic blue phases in cholesteric liquid crystals. Within the framework of the Landau-de Gennes theory, using the simplest blue phase with spatial group $O^5$ ($I432$) as an example, calculations of the bulk modulus and two shear moduli were performed depending on the chirality strength and temperature below the crystallization point from isotropic liquid. It is shown that the used approximations of rigid tensors and free helicoids give qualitatively similar results but differ noticeably quantitatively, therefore further experimental studies and numerical modeling of blue phase elasticity are necessary.

Figures (3)

• Figure 1: Wave vectors and rotation planes of helicoids: 1 --- helicoid in undeformed crystal, 2 --- rigid tensor approximation $\hat{\chi}$, 3 --- free helicoid approximation
• Figure 2: Dependence of components $\lambda_{xxxx}$ and $\lambda_{xyxy}$ of the blue phase $O^5$ elastic tensor on chirality $\kappa$ in one-parameter approximation ($\eta = 1$) at $\tau = \tau_c$: 1 --- rigid tensor approximation, 2 --- free helicoid approximation, 3 --- interpolation model. For viewing convenience, all graphs are divided by $\kappa^2$. Parameter $\kappa = \tfrac{3}{2}$ corresponds to the boundary between high and low chirality.
• Figure 3: Temperature dependence of components $\lambda_{xxxx}$ and $\lambda_{xyxy}$ of the blue phase $O^5$ elasticity tensor in one-parameter approximation ($\eta = 1$) at $\kappa = 1$: 1 --- rigid tensor approximation, 2 --- free helicoid approximation, 3 --- interpolation model