Scratching a 50-year itch with elongated rods

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 fieldtheoretic 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 K2 < K1 < K3. We also consider the case of disk-like particles, and then find in line with expectations that K3 < K1 < K2.

Hölder regularity and convergence for a non-local model of nematic liquid crystals in the large-domain limit

“…(1.9) An energy purely quadratic in ∇Q cannot give rise to three independent elastic constants in the Oseen-Frank model, with the so-called "cubic term" Q ij Q kl,i Q kl,j often being used to fill the degeneracy. Such a term does not arise from the model we will consider, although a more complex variant taking into account molecular length scales has been proposed to avoid this issue [11].…”