Defects in nematic membranes can buckle into pseudospheres.

Many factors influence the shapes of living and manufactured membranes. In addition to boundary conditions, surface tension, and curvature, the ordering of particles embedded in or attached to a membrane can strongly influence its equilibrium shape. As a simple model of such ordering, we consider rodlike particles that align to form a so-called nematic phase in the plane of the membrane. We call any sheet with such embedded orientational order a nematic membrane. Nematic membranes can occur in biological cells, liquid crystal films, manufactured materials, and other soft matter systems. By formulating the free energy of nematic films using tensor contractions from differential geometry, we elucidate the elastic terms allowed by symmetry, and indicate differences from hexatic membranes. We find that topological defects in the orientation field can cause the membrane to buckle over a size set by the competition between surface tension and in-plane elasticity. In the absence of bending rigidity the resulting shape is universal, known as a parabolic pseudosphere or a revolved tractrix. This buckling is the two-dimensional analog of the bent cores of line defects that are frequently observed in bulk nematic liquid crystals. Bending costs oppose such buckling and modify the shape in a predictable manner. In particular, the anisotropic rigidities of nematic membranes lead to different shapes for aster and vortex defects, in principle enabling measurement of couplings specific to nematic membranes.