Nonlinear continuum theory of smectic-\(A\) liquid crystals
This paper presents a general nonlinear continuum theory for smectic-\(A\) liquid crystals. In particular, the author seeks to incorporate into the model a unique permeation process which is responsible for much of the unusual behaviour observed in these layered structures. Adopting a form of the free energy \(f\) proposed by \textit{M. Kleman} and \textit{O. Parodi} [J. Physique 36, 671-681 (1975)], namely \(2f= B(|\nabla\phi|- 1)^2+ k(\nabla\cdot {\mathbf n})^2\), where the isosurfaces of a layer variable \(\phi\) parametrize the smectic layers, \({\mathbf n}\) is the unit director and \(B\) and \(k\) are material constants, the author proceeds to present the hydrostatic part of the theory. A computation of the variation of \(f\) with respect to the layer position without deforming the material and the deformation of the material without altering the position of the layers leads to expressions for the `permeation force' and the static stress, respectively. To derive the hydrodynamic equations, the author starts with a set of basic variables consisting of density, velocity \({\mathbf u}\), total energy density, and layer variable \(\phi\). Then the familiar balance equations and entropy production inequality are written down together with an equation for \(\phi\). Restricting attention to a linear constitutive theory, the author proceeds to derive the final form of the equations for compressible materials by utilizing the parity of various quantities under time reversal. It is observed that in the isothermal case, the system of equations is among the simplest for nonlinear anisotropic fluids. An appropriate set of equations in the incompressible limit is then derived by two distinct methods. One employs the constraints on the fluid velocity \((\text{div }{\mathbf u}= 0)\) and the smectic layers \((|\nabla\phi|= 1)\) but otherwise follows the lines indicated for the compressible case; the other uses a systematic asymptotic analysis of the equations for a compressible material in the limit as the compression modulus \(B\to\infty\). Finally, the author presents a linear stability analysis of a simple planar structure and demonstrates that it is stable, in agreement with known results.
scientific article; zbMATH DE number 1045412