Scratching a 50-year itch with elongated rods

TL;DR

This paper revisits the elastic constants problem in liquid crystal theories, demonstrating that non-interacting rod-like particles exhibit three elastic constants with specific scaling and inequalities, aligning with classical and new theoretical insights.

Contribution

It provides a density functional theory approach to resolve the elastic constants problem, extending previous field-theoretic work to more accessible particle models.

Findings

Rod-like particles have three elastic constants scaling with particle length.

Ordered assemblies of rods satisfy inequalities K2 < K1 < K3.

Disc-like particles exhibit inequalities K3 < K1 < K2.

Abstract

The classical Oseen-Frank theory of liquid crystal elasticity is based on the experimentally verified fact that there are three independent modes of distortion, each with its associated elastic constant. On the other hand, the arguably more first-principles order parameter-based Landau-de Gennes theory only involves two independent elastic modes. The resulting 'elastic constants problem' has led to a considerable amount of vexation among theorists. In a series of papers at the turn of the century, Fukuda and Yokoyama suggested that the resolution of this problem could be found in the proper treatment of non-local effects in the ideal part of the free energy. They used an ingenious, but technically complex, technique based on a field-theoretic approach to semi-flexible polymers. Here we revisit their idea but now in the more accessible framework of density functional theory of rigid particles. Our work recovers their main results for rod-like particles, in that generically an ordered assembly of non-interacting rods has three independent elastic constants associated to it that all scale as the square of the length of the particles and obey the inequalities $K_2 < K_1 < K_3$. We also consider the case of disc-like particles, and then find in line with expectations that $K_3 < K_1 < K_2$.