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

TL;DR

The paper develops a modified Landau--de Gennes framework coupling a $\mathbf{Q}$-tensor with a scalar smectic density $\delta\rho$ to model temperature-driven transitions among cholesteric, helical smectic, and smectic C$^*$ phases. It establishes the existence of global minimizers in 3D with Dirichlet data, derives the Oseen--Frank limit via $\Gamma$-convergence, and proves convergence to a classical helical director in the large-elastic-constants regime; it then uses stability analysis and Crandall–Rabinowitz bifurcation theory to predict a sequence of symmetry-breaking transitions as temperature decreases, from cholesteric to helical smectic to Smectic-C$^*$, with numerical simulations validating the theoretical predictions. The work provides a rigorous foundation for two-parameter tensorial LC models, clarifies the interplay between nematic and smectic ordering, and offers a framework for exploring parameter regimes and boundary effects relevant to chiral smectic systems. It lays groundwork for future landscape analyses, parameter fitting, and potential integration with data-driven approaches to interpret complex chiral LC textures.

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.

Figures (4)

• Figure 1: Schematic illustration of the the Cholesteric-Helical Smectic phase transition with $d=T+10$, $e=0$, $f=10$, $k_1=k_2=k_3=k=0.025$, $h=2\pi$, $\sigma=q=4$, $\lambda_1=\lambda_2=0.001$, $\theta_0=\pi/9$, and the pitchfork bifurcation for $d+2\lambda_2 q^4 \cos^4\theta_0<0$. The solid black line denotes a stable phase, while the dashed black line denotes an unstable phase in all figures. We numerically calculate the minimizer ($\delta \rho_{min}$,$\theta_{min}$) of \ref{['energy theta final']} with various $d$. For better visualization, we plot the 3D $xy$-invariants: $\Tilde{\theta}(x,y,z)\equiv \theta(z)$ and $\Tilde{\delta \rho}(x,y,z) \equiv \delta \rho(z)$.
• Figure 2: $\theta^*$ versus $t$ in \ref{['optimal twist plane']} which represents the Helical smectic-Smectic C* phase transition with increasing $t$ (decreasing temperature). The director (represented by the white line segments) twists within the conical plane determined by $\theta^*$ , while the layer structure is visualized by $tsin(qz)$. Therefore, the larger the value of $t$, the more pronounced the layer structure becomes.
• Figure 3: Cholesteric--Helical Smectic--Smectic C* phase transitions of the energy functional \ref{['energy theta final']} for $T_2^*=-10$, $\alpha_2=1$ (i.e. $d=T+10$), $f=10$, $h=2\pi$, $q=\sigma=4$, $\lambda_1=\lambda_2=0.001$, $\theta_0=\pi/9$. We use $\delta \rho_{max}(T)$ and $\theta_{max}(T)$, where $\delta \rho_{max}(T)$ = $\max_{0\leqslant x \leqslant h} \delta \rho^*_T(x)$ and $\theta_{max}(T)$ = $\max_{0\leqslant x \leqslant h} \theta^*_T(x)$, to represent the global minimizer $(\delta \rho^*_T(x),\theta^*_T(x))$ of $F(\theta,\delta \rho)$ in \ref{['energy theta final']} at $T$. For better visualisation, we plot the 3D $xy$ invariants: $\Tilde{\theta}(x,y,z)\equiv \theta(z)$ and $\Tilde{\delta \rho}(x,y,z) \equiv \delta \rho(z)$.
• Figure 4: The effect of the nematic elastic constant and smectic elastic constant on the stable twist plane. $\bar{\theta}$ is the average out-of-plane angle, which is defined as $\bar{\theta}=\int_0^h \theta^*(z)/h \ \textrm{d} z$, where $(\delta \rho^*,\theta^*)$ is the global minimizer of $F(\theta,\delta \rho)$ in \ref{['energy theta final']}. In the left plot, we take parameters $k_1=k_2$, $d=-5, f=10, \lambda_1=\lambda_2=0.001, h=2\pi, q=\sigma=4$, and $\theta_0=\pi/9$. In the right plot, we take $\lambda_1=\lambda_2$, $d=-5, f=10, k_1=k_2=k_3=0.025, h=2\pi, q=\sigma=4, \theta_0=\pi/9$.

Theorems & Definitions (39)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Proposition 1
• proof
• Proposition 2
• proof