Symmetry and Thermodynamic Bounds on Cross-Coupling Transport in Chiral Liquid Crystals

TL;DR

The paper addresses cross-coupled transport in chiral liquid crystals by embedding nematic order in the Q-tensor and applying linear response alongside the thermodynamic uncertainty relation to bound Leslie-type coefficients. It shows cross-coupling terms depend on the scalar order parameter $S$ and that new terms like $ ext{μ}'^{N}$ can generate torque from a heat current parallel to the director, even in isotropic-leaning configurations. The results unify symmetry, thermodynamics, and experimental observations, reproducing classical Leslie inequalities in appropriate limits while predicting sign relationships tied to molecular shape and chirality. This framework offers principled bounds and predictions for energy conversion and torque generation in chiral soft matter, with implications for soft actuators and thermal Edelstein-like phenomena in condensed matter systems.

Abstract

We reformulate the Leslie effects that describe the dynamic cross-couplings in chiral liquid crystals driven by the transport of heat, electric charge, and mass. The Ericksen--Leslie model is extended in the linear response framework by representing nematic order with the Q-tensor. Subsequently, the thermodynamic uncertainty relation is applied to identify the upper bounds of the Leslie cross-coupling coefficients. We reveal that the cross-coupling coefficients are dependent on the scalar order parameter and vanish in the isotropic phase. In addition, the chirality of the phase allows torque induced by a transport current parallel to the director. The mutual signs of the Leslie thermohydrodynamic and thermomechanical coefficients are likely to be opposite in calamitic liquid crystals, as suggested by recent experimental observations. Our model is applicable to the thermal, chemical, and electrical Leslie effects. The present arguments suggest that a common underlying principle may govern both the Leslie effects and the thermal Edelstein effect in chiral solid crystals attributed to chiral phonons.

Figures (5)

• Figure 1: Mechanisms of the Lehmann effect. (a) Classification of theoretical models. Synonyms of the Lehmann effect are presented in the gray callout. The thermal Leslie effect and related extensions by de Gennes de_Gennes are also included. (b) Stylistic format "Leslie A+B C" for denoting the Leslie effects in this article, where A, B, and C represent allowable terms for slots. (c) Symmetry of collinear temperature gradient and torque under parity $\mathcal{P}$ (spatial inversion) and time reversal $\mathcal{T}$. (d) Terms of the stress tensors in the thermal Leslie effect. No specific designation has been assigned to the $\mu'^{\text{N}}$-term derived in this article.
• Figure 2: Consistency of the thermodynamic bound with the experiments. The black plots show the experimentally obtained Leslie thermomechanical coefficient $\nu$: the closed circles in the N$^\star$ phase and the open squares in the N$^\star$ --I coexistence state. We calculated $|Y|$ from $\nu$ and supplemented $\gamma_1$ and $\sigma_\perp$ (and temperature) as shown in Table \ref{['tab:nu_exp']}. The white line represents the regression curve $\log_{10}{|Y|} = 1.0\, \log_{10}{|\nu|} + 2.6$ for all the experiments, and the black dotted lines represent the $95\%$ prediction interval. The colored plots show the estimation (Table \ref{['tab:nu_MDsim']}) by MD simulation in Ref. Sarman2016. The red hexagons, the blue diamonds, and the green circles are system I (disklike molecules), II (rodlike molecules), and III (mixture of chiral and achiral rodlike molecules), respectively. The gray dashed line is the upper limit of $|Y|$ imposed by the TUR, and the gray region is consistent with the thermodynamics as shown in Eq. \ref{['eq:bound_|Y|']}.
• Figure 3: Coefficients $X,Y,Z$ that are consistent with thermodynamics. (a)--(c) Objects locating in $XYZ$-space whose interior is consistent with thermodynamics. (a) Object $A$ showing Eqs. \ref{['eq:bound_|X|']}--\ref{['eq:bound_|Z|']}. (b) Object $B$ showing Eq. \ref{['eq:bound_X2+Y2+Z2-2XYZ']}. (c) The intersection $A \cap B$ showing Eqs. \ref{['eq:bound_|X|']}--\ref{['eq:bound_X2+Y2+Z2-2XYZ']}, all inequalities to be concerned. Objects $A,\ B$ and $A \cap B$ are centered at the origin ($X=Y=Z=0$). (d)--(h) The cross sections of $A \cap B$ at a fixed $Z$. Probable $X,Y$ is included in the ellipse. In the ellipse, the regions where $X/Y > 0$ and $X/Y < 0$ are highlighted in orange and blue, respectively. On the perimeter of the ellipse, the Carnot efficiency is attained. The black square circumscribing the ellipse represents the perimeter of $|X| \le 1 \land |Y| \le 1$. (i) Area of the ellipse and its parts as a function of $Z$. The black line is the area of entire ellipse, the orange and blue lines are areas of $X/Y > 0$ and $X/Y < 0$ parts, respectively.
• Figure 4: Thermodynamic bounds for the chemical Leslie effect in smectic-C$^\star$ phase investigated in Refs. Bunel2023nonsingularBunel2023singular. The oblique ellipse shows thermodynamically acceptable Leslie chemohydrodynamic and chemomechanical cross-coupling coefficients. The perimeter of the ellipse intersect the dashed line that represents $X/Y = -2.5/\sqrt{2}$ at two points.
• Figure 5: The irreversible stress for rodlike and disklike molecules. (a) The rotational stress induced by an irrotational flow, depending on the shapes and orientations of molecules. The arrows extending from the square represent the forces acting on the face at each arrow’s base. Arrows are black when the coefficient is positive and gray when negative. The streamlines and the director lie in the plane. Each of the three panels on the left and right displays rod‑shaped and disc‐shaped molecules, respectively. (b) The deviatoric stress induced by a rotational flow. (c) The deviatoric stress induced by the Leslie effects. The coefficient $\mu^{\text{A}}$ considers a case where it is negative $(X < 0)$ in the rodlike molecule and positive $(X > 0)$ in the disklike molecule as an example. The heat current $\bm{\mathrm{q}}$ is directed out of the page. Under these conditions, the thermodynamically favored rotational stress (for $Y > 0$) is shown in (d) and (e).