Density-functional theory of surfacelike elasticity of nematic liquid crystals

Abstract

Based on the density-functional theory, we show that the splay-bend surfacelike bulk elastic constant ${\mathrm{K}}_{13}$ is an artifact of the phenomenological as well as the gradient expansion construction of the curvature elastic free energy of nematic liquid crystals, while the saddle-splay constant ${\mathrm{K}}_{24}$ is real and approximately obeys an extended Nehring-Saupe relation ${\mathrm{K}}_{24}$ \ensuremath{\geqslant}(${\mathrm{K}}_{11}$ -${\mathrm{K}}_{22}$ )/2, with ${\mathrm{K}}_{11}$ and ${\mathrm{K}}_{22}$ being the splay and twist Frank constants, respectively. The result ${\mathrm{K}}_{13}$ =0 automatically resolves the Oldano-Barbero pathology, which inevitably accompanies a nonzero ${\mathrm{K}}_{13}$ , and gives a concrete rationale for the conventional approaches in the continuum theory disregarding the surfacelike elasticity. The source of an apparently nonzero ${\mathrm{K}}_{13}$ in previous microscopic theories is discussed in detail and is demonstrated to be a result of an inconsistent use of a nonlocal-to-local mapping of the elastic free-energy functional at the boundary. The absence of ${\mathrm{K}}_{13}$ can be regarded as a type of Cauchy relation in the nematic continuum theory in the sense that it is not directly rooted in any of the macroscopic symmetries existing in the nematic phase, but is a general consequence of the particular algebraic form of the nonlocal free-energy term from which ${\mathrm{K}}_{13}$ derives; its linearity in the distortion amplitude and the symmetry of the relevant direct correlation function with respect to the permutation of a molecular pair leads always to the vanishing ${\mathrm{K}}_{13}$ . In this respect, ${\mathrm{K}}_{13}$ =0 applies not only to nematic liquid crystals but also to a more general class of phases such as cholesteric liquid crystals, whose structure can be viewed as a weak modulation of a translationally invariant phase. We finally consider the elastic description of nematic liquid crystals in the presence of real interfaces. The present formulation allows a straightforward decomposition of the elastic excess free energy into the bulk contribution and the interfacial excess in the Gibbs sense. The bulk part yields the bulk Oseen-Frank elastic free-energy density along with the ${\mathrm{K}}_{24}$ term evaluated at the Gibbs dividing surface as an unambiguous local quantity. The interfacial excess, when gradient expanded, reduces to the surface free-energy density comprised of the anchoring energy, surface-excess Oseen-Frank elastic energy, ${\mathrm{K}}_{24}$ -like term, and elastic free-energy contributions reflecting the broken symmetry at the interface. The ${\mathrm{K}}_{24}$ -like term is formally similar to the bulk ${\mathrm{K}}_{24}$ term, but is no longer an intrinsic property of bulk nematic liquid crystal, as it depends also on the nature of the medium with which the nematic liquid crystal is in contact.