The Oseen-Frank limit of Onsager's molecular theory for liquid crystals

TL;DR

This paper establishes a rigorous connection between Onsager's molecular theory and the Oseen-Frank continuum theory for nematic liquid crystals, showing how molecular interactions lead to classical elastic behavior in the limit.

Contribution

It proves the convergence of minimizers from Onsager's molecular free energy to the Oseen-Frank energy as the interaction distance shrinks to zero.

Findings

Existence of global minimizers under strong anchoring conditions.

Convergence of molecular minimizers to Oseen-Frank minimizers.

Validation of continuum theory as a limit of molecular interactions.

Abstract

We study the relationship between Onsager's molecular theory and the Oseen-Frank theory for nematic liquid crystals. Under the molecular setting, we consider the free energy that includes the effects of nonlocal molecular interactions. By imposing the strong anchoring boundary condition on the second moment of the number density function, we prove the existence of global minimizers for the free energy. Moreover, when the re-scaled interaction distance tends to zero, the corresponding global minimizers will converge to a uniaxial distribution whose orientation is described by a minimizer of Oseen-Frank energy.