Local stability of biaxial nematic phases between two cylinders

We consider a nematic liquid crystal confined between two cylinders of radii r(1) > 0 and r(2) > r(1). We suppose that the lateral surfaces induce a uniaxial anchoring with the optic axis aligned along the radial direction, and we study the equilibrium configurations which minimize the free energy functional. This problem has been already studied under the assumption that the nematic remains uniaxial in the whole tube, with fixed degree of orientation s. We allow the nematic to become biaxial between the surfaces, and include in the free-energy functional an internal potential which favours uniaxiality. We prove that, if the internal potential is neglected (i.e. if biaxiality can arise at no cost), the free energy minimizer is biaxial in the whole volume (except, of course, on the lateral surfaces, where it must be uniaxial). The minimizer is unique, and no bifurcation arises for any value of rho := r(1)/r(2). We arrive at new results also when the internal potential is at work: an exact solution, obtained in a special case, proves the existence of a bifurcation at a critical value of rho; approximate minimizers show how biaxiality fades away in the bulk as the potential is magnified, and numerical studies illustrate the features of the most general minimizers. (C) 1997 Elsevier Science Ltd.