Local stability of biaxial nematic phases between two cylinders
This paper examines the stability of equilibrium configurations in a sample of nematic liquid crystal confined between two circular cylinders which induce a uniaxial anchoring of the optic axis aligned everywhere radially on these bounding surfaces. Unlike earlier studies, the authors allow the nematic to become biaxial in the bulk of the sample and seek configurations which minimize the free energy functional \(F[{\mathbf Q}]:= \int_V\{\kappa|\nabla{\mathbf Q}|^2+ \sigma({\mathbf Q})\}dV\), where \(V\) is a circular annulus of height \(h\), \(\kappa\) is a positive elastic constant, \(\sigma\) is an internal potential which allows for biaxiality, and \({\mathbf Q}\) is the order tensor. After defining a particular admissible class of order tensors which contains both uniaxial and biaxial special states, the authors examine the case when the potential well has the special form \(\sigma:= k\{\eta(s- s_0)^2+ \alpha^2\}\), where \(k\) is a positive constant, \(s\) is the degree of orientation, \(s_0\) is the boundary value of \(s\), \(\alpha\) represents the degree of biaxiality and \(\eta\) is a positive parameter which represents the anisotropy of the well. The resulting variational problem is to find minimizers of \(F(s,\alpha,\phi)\), where \(\phi\) is the polar angle in the \((e_r,e_z)\) plane, which satisfy the conditions \(\alpha= \phi= 0\), \(s= s_0\) on the lateral surfaces of the annulus. Of the various solutions of the resulting equilibrium equations, the authors concentrate on those with \(\phi\equiv 0\), and they present an explicit solution in terms of Bessel functions in the special case when \(\eta= {1\over 3}\). They then use a comparison theorem due to Fubini to derive an asymptotic criterion for this so-called radial solution. Although explicit solutions are not possible when \(\eta\neq{1\over 3}\), approximate solutions are given for the two extreme cases of weak and strong potential wells, and numerical solutions are presented, in graphical form, for a variety of values of \(\eta\) and \(\lambda\) \((\equiv(\kappa/k)^{{1\over 2}})\). It is found that the qualitative features of the solution do not depend on the specific value of \(\eta\).
asymptotic criterion for radial solution
scientific article; zbMATH DE number 1080340