# Oseen-Frank-type theories of ordered media as the $\Gamma$-limit of a non-local mean-field free energy

## TL;DR

This paper derives the classical Oseen-Frank theory for nematic liquid crystals as a limit of a non-local mean-field free energy model, accommodating general interactions and providing explicit formulas for elastic constants.

## Contribution

It establishes a rigorous Gamma-convergence framework connecting microscopic interactions to macroscopic elastic theories, including biaxial and polar molecules.

## Key findings

- Derivation of Oseen-Frank theory as a Gamma-limit of mean-field free energy.
- Explicit expressions for Frank constants in terms of interaction kernels.
- Extension of results to biaxial and polar molecular systems.

## Abstract

In this work we recover the Oseen-Frank theory of nematic liquid crystals as a $\Gamma$-limit of a particular mean-field free energy as the sample size becomes infinitely large. The Frank constants are not necessarily all equal. Our analysis takes place in a broader framework however, also providing results for more general systems such as biaxial or polar molecules. We start from a mesoscopic model describing a competition between entropy and a non-local pairwise interactions between molecules. We assume the interaction potential is separable so that the energy can be reduced to a model involving a macroscopic order parameter. We assume only integrability and decay properties of the macroscopic interaction, but no regularity assumptions are required. In particular, singular interactions are permitted. The analysis becomes significantly simpler on a translationally invariant domain, so we first consider periodic domains with increasing domain of periodicity. Then we tackle the problem on a Lipschitz domain with non-local boundary conditions by considering an asymptotically equivalent problem on periodic domains. We conclude by applying the results to a case which reduces to the Oseen-Frank model of elasticity, and give expressions for the Frank constants in terms of integrals of the interaction kernel.