A Landau-de Gennes Type Theory for Cholesteric-Helical Smectic-Smectic C* Liquid Crystal Phase Transitions

TL;DR

This paper rigorously analyzes a modified Landau-de Gennes theory for phase transitions among cholesteric, helical smectic, and smectic C* liquid crystal phases, establishing existence, asymptotic behavior, and symmetry-breaking transitions.

Contribution

It introduces a coupled tensor and scalar order parameter model for these phases and provides rigorous mathematical analysis of energy minimizers and phase transitions.

Findings

Existence of energy minimizers in three dimensions.

Convergence of minimizers to Landau-de Gennes bulk energy in the Oseen--Frank limit.

Identification of symmetry-breaking transitions with decreasing temperature.

Abstract

We present a rigorous mathematical analysis of a modified Landau-de Gennes (LdG) theory modeling temperature-driven phase transitions between cholesteric, helical smectic, and smectic C* phases. This model couples a tensor-valued order parameter (nematic orientational order) with a real-valued order parameter (smectic layer modulation). We establish the existence of energy minimizers of the modified LdG energy in three dimensions, subject to Dirichlet conditions, and rigorously analyze the energy minimizers in two asymptotic limits. First, in the Oseen--Frank limit, we show that the global minimizer strongly converges to a minimizer of the Landau-de Gennes bulk energy. Second, in the limit of dominant elastic constants, we prove that the global minimizers converge to a classical helical director profile. Finally, through stability analysis and bifurcation theory, we derive the complete sequence of symmetry-breaking transitions with decreasing temperature-from the cholesteric phase (with in-plane twist and no layering) to an intermediate helical smectic phase (with in-plane twist and layering), and ultimately to the smectic C* phase (with out-of-plane twist and layering). These theoretical results are supported by numerical simulations.