On a curvature surface energy for nematic liquid crystals (Q1382924)

When a nematic liquid crystal is in contact with a substrate or with an isotropic environment, there is necessary to account for the surface energy \(F_a\) which supplements the bulk elastic energy \(F_b\). Here, the authors propose a new form of the surface free energy, for which the boundary curvature \(\nu\), besides its normal \(\nabla_s\nu\), affects the energy density at the interface. First of all, the authors derive the most general form of the frame-indifferent density of surface energy which depends on \(\nu\) and \(n\), and is linear in both \(\nabla_s\nu\) and \(\nabla_sn\), where \(n\) is the orientation field. It is shown that each term in the corresponding representation is a surface null Lagrangian. Whereas the functional \(F_a\) is not bounded from below, the authors demonstrate that it is true for the full energy functional \(F= F_a+ F_b\), and derive equilibrium equations for \(F\). Finally, the developed theory is applied to special cases when the region \(B\) occupied by liquid crystal is a ball, when the shape of \(B\) is variable (drop), or \(\partial B\) contains edges.