Static shear layers in nematic liquid crystals
This paper attempts an explanation of an apparent conflict between continuum theory and an observation by \textit{P. E. Cladis}, and \textit{S. Torza} [(*) Phys. Rev. Lett. 35, 1283 ff. (1975)] of nematic layers sustaining a shear stress without deformation. To describe a possible mechanism for this behaviour, the authors consider the problem of a layer confined between parallel plates situated at \(y=\pm b\). A relaxation in the 'strong anchoring' condition is required as is achieved by introducing line disclinations in the surface of the plates parallel to the z-axis. Seeking two-dimensional static solutions for the director n of the form \(n_ x=\cos \vartheta (x;y),\quad n_ y=\sin \vartheta (x,y),\quad n_ z=0\) and assuming the splay and bend constants are equal, the direction equation becomes \(\nabla^ 2\vartheta =0\). The authors first consider a combination of solutions associated with regularly spaced arrays of line disclinations in the surfaces \(y=\pm b\) given by \[ \vartheta =\vartheta_ 0+N\sum^{\infty}_{n=-\infty}(-1)^ n\{\tan^{- 1}(\frac{y-b}{x-a-nc}\quad)-\tan^{-1}(\frac{y+b}{x+a-nc})\}, \] where \(\vartheta_ 0\), a,b,c are constants and N is an integer representing the disclination index. The shear stress component \(t_{xy}\) is found to be periodic in x and a non-zero average shear stress is obtained. To satisfy the boundary value problem with solid surfaces at \(y=\pm b\) it is necessay to consider the image system which consists of an infinite set of line arrays. The 'strong anchoring' condition is assumed to hold everywhere except at points where there is a disclination line. A non- zero average shear stress is obtained which takes a constant value on all planes parallel to and lying between \(y=\pm b\). Despite several simplifying assumptions in the model and lack of justification for certain steps in the analysis, the authors demonstrate that the continuum equations allow the possibility of layers sustaining shear stress without deforming. Finally it is shown that the model yields a shear stress magnitude compatible with that employed by Cladis and Torza (*). This is an interesting paper that is well written and should be of interest to researchers in continuum mechanics.
regularly spaced arrays of line disclinations
scientific article; zbMATH DE number 3963378