Interpretation of saddle-splay and the Oseen-Frank free energy in liquid crystals

Abstract

This article re-examines a classic question in liquid-crystal physics: What are the elastic modes of a nematic liquid crystal? The analysis uses a recent mathematical construction, which breaks the director gradient tensor into four distinct types of mathematical objects, representing splay, twist, bend, and a fourth deformation mode. With this construction, the Oseen-Frank free energy can be written as the sum of squares of the four modes, and saddle-splay can be regarded as bulk rather than surface elasticity. This interpretation leads to an alternative way to think about several previous results in liquid-crystal physics, including: (1) free energy balance between cholesteric and blue phases, (2) director deformations in hybrid-aligned-nematic cells, (3) spontaneous twist of achiral liquid crystals confined in a torus or cylinder, and (4) curvature of smectic layers. arXiv:1901.06306v1 [cond-mat.soft]

Liquid crystal distortions revealed by an octupolar tensor

“…It is yet another manifestation of the monkey saddle defined in [21, p. 191]. 10 The corresponding points in the space of distortion characteristics (S, b, β) 10 In its original realization, the monkey saddle is a surface in 3D with three depressions, instead of the two needed for a human rider (the third accommodating the monkey's tail).…”

“…Describing the distortion in space of a nematic director field n is easier said than done. Motivated by a fresh look [10] into the classical topic of nematic elasticity, which originated with the works of Oseen [44] and Frank [45], we devised a mathematical construct that can easily identify various independent elastic modes, especially the biaxial splay, which is contending the role traditionally played in liquid crystal science by saddle-splay elasticity [10]. Our construct is an octupolar tensor A, that is, a third-rank, fully symmetric and traceless tensor built from n and ∇n.…”

“…where K 11 , K 22 , K 33 , and K 24 are the splay, twist, bend, and saddle-splay constants, respectively, each associated with a corresponding elastic mode. 1 The decomposition of F in independent elastic modes proposed in [10] is achieved through a new decomposition of ∇n. If we denote by P(n) and W(n) the projection onto the plane orthogonal to n and the skew-symmetric tensor with axial vector n, respectively, then…”

“…Greek counterpart ∆ used in both [11] and [10]. The only other point where our notation differs from that of [10] is in calling b the bend vector, which there was B.…”

Constant spacing in filament bundles

“…), in its nonzero 2D block, following [48,49]. The modes of zero H, then, have both locally isotropic gradients of t (vanishing biaxial splay), and constant cross-sectional area per filament (vanishing splay).…”