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Interaction energies in nematic liquid crystal suspensions

Lia Bronsard, Xavier Lamy, Dominik Stantejsky, Raghavendra Venkatraman

TL;DR

This work provides a rigorous asymptotic expansion for the minimal Dirichlet energy of $\mathbb{S}^2$-valued maps in the exterior of finitely many small three-dimensional particles in a nematic liquid crystal, under general boundary anchoring. By coupling a precise upper-bound construction with a matching lower-bound analysis, the authors derive an expansion $E_\rho = \sum_j \mu_j - 4\pi\rho \sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} + o(\rho)$, where $\mu_j$ are single-particle energies and $v_j$ are torques determined by the far-field of each particle, yielding a Coulomb-like interaction between particle centers. The paper then provides a rigorous justification of the electrostatics analogy used in colloid-nematic physics and quantifies the error introduced by linearizing the nonlinear director equation away from the particles. The analysis hinges on a detailed study of harmonic extensions in exterior domains, sharp inner/outer energy decompositions, and a far-field expansion for rescaled minimizers, culminating in a robust two-sided estimate that captures the leading interaction term and its precise coefficient. This framework lays groundwork for continuum limits and many-particle extensions in nematic suspensions.

Abstract

We establish, as $ρ\to 0$, an asymptotic expansion for the minimal Dirichlet energy of $\mathbb S^2$-valued maps outside a finite number of three-dimensional particles of size $ρ$ with fixed centers $x_j\in\mathbb{R}^3$, under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form \begin{align*} E_ρ= \sum_j μ_j -4πρ\sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(ρ)\,, \end{align*} where $μ_j$ is the minimal energy after zooming in at scale $ρ$ around each particle, and $v_j\in\mathbb{R}^3$ is a torque determined by the far-field behavior of the corresponding single-particle minimizer. The above expansion highlights Coulomb-like interactions between the particle centers. This agrees with the \textit{electrostatics analogy} commonly used in the physics literature for colloid interactions in nematic liquid crystal. That analogy was pioneered by Brochard and de Gennes in 1970, based on a formal linearization argument. We obtain here for the first time a precise estimate of the energy error introduced by this linearization procedure.

Interaction energies in nematic liquid crystal suspensions

TL;DR

This work provides a rigorous asymptotic expansion for the minimal Dirichlet energy of -valued maps in the exterior of finitely many small three-dimensional particles in a nematic liquid crystal, under general boundary anchoring. By coupling a precise upper-bound construction with a matching lower-bound analysis, the authors derive an expansion , where are single-particle energies and are torques determined by the far-field of each particle, yielding a Coulomb-like interaction between particle centers. The paper then provides a rigorous justification of the electrostatics analogy used in colloid-nematic physics and quantifies the error introduced by linearizing the nonlinear director equation away from the particles. The analysis hinges on a detailed study of harmonic extensions in exterior domains, sharp inner/outer energy decompositions, and a far-field expansion for rescaled minimizers, culminating in a robust two-sided estimate that captures the leading interaction term and its precise coefficient. This framework lays groundwork for continuum limits and many-particle extensions in nematic suspensions.

Abstract

We establish, as , an asymptotic expansion for the minimal Dirichlet energy of -valued maps outside a finite number of three-dimensional particles of size with fixed centers , under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form \begin{align*} E_ρ= \sum_j μ_j -4πρ\sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(ρ)\,, \end{align*} where is the minimal energy after zooming in at scale around each particle, and is a torque determined by the far-field behavior of the corresponding single-particle minimizer. The above expansion highlights Coulomb-like interactions between the particle centers. This agrees with the \textit{electrostatics analogy} commonly used in the physics literature for colloid interactions in nematic liquid crystal. That analogy was pioneered by Brochard and de Gennes in 1970, based on a formal linearization argument. We obtain here for the first time a precise estimate of the energy error introduced by this linearization procedure.
Paper Structure (10 sections, 15 theorems, 232 equations, 2 figures)

This paper contains 10 sections, 15 theorems, 232 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Prelude: harmonic extensions outside a union of small spheres
  3. Upper bound
  4. A matching lower bound
  5. Pointwise estimates
  6. Compactness of rescaled minimizers
  7. Lower bound in terms of $\hat{n}_\rho^j-\hat{m}_j$
  8. Far field asymptotics of $\hat{n}_j^\rho$
  9. Decaying solutions of Poisson's equation
  10. Decay of harmonic functions in annuli

Key Result

Theorem 1.1

There exist minimizers $\hat{m}_j$ of the single-particle problems eq:muj such that the minimum of $E_\rho$ over maps $n\colon\Omega_\rho\to\mathbb{S}^{ 2}$ with far-field alignment eq:farfieldOmegarho satisfies where $\mu_j=\widehat{E}_j(\hat{m}_j)$ is the minimal single-particle energy eq:muj, and $v_j\in n_\infty^\perp$ is defined by the asymptotic expansion eq:vj of $\hat{m}_j$.

Figures (2)

  • Figure 1: General setup for Theorem \ref{['t:asympt']}
  • Figure 2: Structure of the competitor $n$ constructed in Proposition \ref{['p:up']}.

Theorems & Definitions (31)

  • Theorem 1.1
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['p:harmextspheres']}
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['p:up']}.
  • Proposition 4.1
  • proof : Proof of Theorem \ref{['t:asympt']}
  • Lemma 4.2
  • ...and 21 more