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Singular Damped Twist Waves in Chromonic Liquid Crystals

Silvia Paparini, Epifanio G. Virga

TL;DR

This work analyzes twist-wave dynamics in chromonic liquid crystals with a quartic elastic energy, focusing on how rotational viscosity affects finite-time shock formation. By reformulating the problem as a damped hyperbolic system and leveraging Riemann invariants, the authors derive a sufficient condition and an explicit critical-time estimate for shock formation in the presence of dissipation. The main result reduces to solving a time-dependent inequality involving a damping-weighted Riccati equation, yielding a bound $t_\lambda^* \le t_c(\lambda)$ and a maximal damping threshold $\lambda_{\max}$ under certain initial data. Applying the theory to a kink-like initial twist and validating with numerical simulations, they show that dissipation delays but does not generically prevent shocks, revealing a nuanced interplay between elasticity, damping, and nonlinear wave steepening with implications for chromonic LC dynamics.

Abstract

Chromonics are special classes of nematic liquid crystals, for which a quartic elastic theory seems to be more appropriate than the classical quadratic Oseen-Frank theory. The relaxation dynamics of twist director profiles are known to develop a shock wave in finite time in the inviscid case, where dissipation is neglected. This paper studies the dissipative case. We give a sufficient criterion for the formation of shocks in the presence of dissipation and we estimate the critical time at which these singularities develop. Both criterion and estimate depend on the initial director profile. We put our theory to the test on a class of initial kink profiles and we show how accurate our estimates are by comparing them to the outcomes of numerical solutions.

Singular Damped Twist Waves in Chromonic Liquid Crystals

TL;DR

This work analyzes twist-wave dynamics in chromonic liquid crystals with a quartic elastic energy, focusing on how rotational viscosity affects finite-time shock formation. By reformulating the problem as a damped hyperbolic system and leveraging Riemann invariants, the authors derive a sufficient condition and an explicit critical-time estimate for shock formation in the presence of dissipation. The main result reduces to solving a time-dependent inequality involving a damping-weighted Riccati equation, yielding a bound and a maximal damping threshold under certain initial data. Applying the theory to a kink-like initial twist and validating with numerical simulations, they show that dissipation delays but does not generically prevent shocks, revealing a nuanced interplay between elasticity, damping, and nonlinear wave steepening with implications for chromonic LC dynamics.

Abstract

Chromonics are special classes of nematic liquid crystals, for which a quartic elastic theory seems to be more appropriate than the classical quadratic Oseen-Frank theory. The relaxation dynamics of twist director profiles are known to develop a shock wave in finite time in the inviscid case, where dissipation is neglected. This paper studies the dissipative case. We give a sufficient criterion for the formation of shocks in the presence of dissipation and we estimate the critical time at which these singularities develop. Both criterion and estimate depend on the initial director profile. We put our theory to the test on a class of initial kink profiles and we show how accurate our estimates are by comparing them to the outcomes of numerical solutions.
Paper Structure (15 sections, 13 theorems, 161 equations, 8 figures)

This paper contains 15 sections, 13 theorems, 161 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Theoretical Recap
  3. Mathematical Methodology
  4. Problem Setup
  5. Boundedness of Riemann Functions
  6. Energy Dissipation and Mass Conservation
  7. Pre-Breakdown Properties
  8. Evolution along characteristics
  9. Damped Shock Waves
  10. Critical time estimates
  11. Auxiliary Lemmas
  12. Main Result
  13. Application
  14. Conclusions
  15. Auxiliary Results

Key Result

Lemma 1

Let $\varphi(t,x,y):=r(t,x)+\ell(t,y)$ and $\eta(t,x,y):=r(t,x)-\ell(t,y)$. Then, for a solution of the system eq:diagonal_system of class $\mathcal{C}^1$ over $[0,t_\ast)\times\mathbb{R}$,

Figures (8)

  • Figure 1: Functions $\widehat{u}_1$ and $k$ defined in \ref{['eq:u_1_rl']} and \ref{['eq:k_l_r']}, respectively, expressed in terms of $\eta$ as defined by \ref{['eq:eta_def']}.
  • Figure 2: Graph of the odd function $f(\eta)$. It is bounded between $-f_0$ and $+f_0$, which are the values attained by $f$ at $\eta=\mp\eta_0$, respectively. The numerical values are $\eta_0\doteq1.80$ and $f_0\doteq0.22$, which correspond to $u_0=-\sqrt{2/3}$ in \ref{['eq:f_parametric']}.
  • Figure 3: Initial profile of the twist angle $w_0$ represented by \ref{['eq:w_0_arctan']} for several values of the parameters $\kappa$ and $\zeta$, which describe the amplitude and width of the kink, respectively.
  • Figure 4: Initial profile of the Riemann function $r_0$ in \ref{['eq:r_0arctan']} corresponding to the initial twist $w_0$ illustrated in Fig. \ref{['fig:w_0arctan']}.
  • Figure 5: Graph of $\eta_\mathrm{L}^+(t,\alpha)$, as defined in \ref{['eq:lower_bound_eta_delta']}, plotted as a function of $t > 0$ for $\kappa = \frac{\pi}{4}$, $\zeta = \frac{1}{2}$, $\alpha=0.21$, and a sequence of values of $\lambda = 0, \, 0.05, \, 0.1, \, 0.13, \, 0.18$. The function is strictly decreasing and remains positive throughout the domain $t > 0$. The value $\alpha\doteq0.21$ minimizes $t_{\mathrm{c}}(\lambda,\alpha)$ for every selected $\lambda$.
  • ...and 3 more figures

Theorems & Definitions (50)

  • Remark 1
  • Remark 2
  • Definition 1
  • Remark 3
  • Definition 2
  • Remark 4
  • Remark 5
  • Lemma 1
  • Remark 6
  • Definition 3
  • ...and 40 more