# Design of effective bulk potentials for nematic liquid crystals via colloidal homogenisation

## TL;DR

This paper develops a homogenised Landau-de Gennes model for nematic liquid crystals with colloidal inclusions, showing how surface anchoring can tune bulk potential coefficients and phase transition properties.

## Contribution

It introduces a rigorous homogenisation approach to derive effective free energy in nematic colloidal suspensions, enabling control over bulk potential parameters.

## Key findings

- Effective free energy derived for colloidal nematic suspensions
- Surface anchoring influences phase transition temperature and potential coefficients
- Homogenisation results in tunable bulk properties through surface conditions

## Abstract

We consider a Landau-de Gennes model for a suspension of small colloidal inclusions in a nematic host. We impose suitable anchoring conditions at the boundary of the inclusions, and we work in the dilute regime - i.e., the size of the inclusions is much smaller than the typical separation distance between them, so that the total volume occupied by the inclusions is small. By studying the homogenised limit, and proving rigorous convergence results for local minimisers, we compute the effective free energy for the doped material. In particular, we show that not only the phase transition temperature, but any coefficient of the quartic Landau-de Gennes bulk potential can be tuned, by suitably choosing the surface anchoring energy density.