A variational problem for nematic liquid crystals with variable degree of orientation
type: Item
instance of: scholarly article
title: A variational problem for nematic liquid crystals with variable degree of orientation
zbMATH Open document ID: 0754.76007
DOI: 10.1007/BF00375588
author: Victor J. Mizel
author: Diego Roccato
author: Epifanio G. Virga
published in: Archive for Rational Mechanics and Analysis
publication date: 1992-09-27T00:00:00Z
review text: This paper examines the problem in which a sample of nematic liquid crystal is contained in a circular cylinder of radius \(R\) and height \(H\). When the unit vector \({\mathbf n}\) that represents the optical axis is aligned everywhere radially on the lateral surface of the cylinder, the classical disclination solution, \({\mathbf n}={\mathbf e}_ r\), of Frank's model [\textit{F. C. Frank}, Discuss. Faraday Soc., 25, 19-28 (1958)], has the unfortunate feature of being associated with an infinite energy. Seeking minimizers of the one constant approximation of Frank's energy integral having the form \({\mathbf n}=(\cos \phi){\mathbf e}_ r+(\sin \phi){\mathbf e}_ z\), where \(\phi\) depends only on the radial distance from the cylinder's axis, Calids and Kléman and Meyer obtained the continuous solution \(\phi=\pi/2-2\arctan(r/R)\) which is associated with a finite energy. Here the authors adopt a simplified form of Ericksen's model [\textit{J. L. Ericksen}, Arch. Rational Mech. Anal., 113, No. 2, 97-120 (1991; Zbl 0729.76008)] by assuming that the energy functional has the form \[ F[s,{\mathbf n}]= \alpha\int_ V\{k|\nabla s|^ 2+s^ 2| \nabla{\mathbf n}|^ 2\}dV, \] where \(\alpha\) and \(k\) are positive constants, \(V\) denotes the volume occupied by the liquid crystal and \(s\) is an additional scalar variable, the degree of orientation, which describes the local microstructure of the nematic material. Assuming that \({\mathbf n}=(\cos \phi){\mathbf e}_ r+(\sin\phi){\mathbf e}_ z\), where the axis of the cylinder is the \(z\)-axis of a set of cylindrical polar coordinates, and that \(s\) and \(\phi\) depend only upon the radial coordinate \(r\), the energy functional takes the relatively simple form \[ F[s,\phi]=\int^ R_ 0\left\{k\left ({ds\over dr}\right )^ 2+s^ 2\left ({d\phi\over dr}\right )^ 2+{\cos^ 2\phi\over r^ 2}\right\}rdr. \] Appropriate boundary conditions on the lateral surface of the cylinder are \[ {\mathbf n}={\mathbf e}_ r\text{ and } s(r)=s_ 0\;(\te…
reviewed by: P. J. Barratt
Mathematics Subject Classification ID: 76A15
Mathematics Subject Classification ID: 76M30
zbMATH DE Number: 58003
zbMATH Keywords: Frank's model
zbMATH Keywords: Ericksen's model
zbMATH Keywords: local microstructure
zbMATH Keywords: Euler equations
zbMATH Keywords: du Bois-Raymond equation
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: phase plane analysis
MaRDI profile type: Publication
cites work: Liquid crystals with variable degree of orientation
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cites work: Weakly Lipschitzian mappings and restricted uniqueness of solutions of ordinary differential equations
full work available at URL: https://doi.org/10.1007/bf00375588
OpenAlex ID: W2025263506
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description: scientific article; zbMATH DE number 58003