Nematic membranes: shape instabilities of closed achiral vesicles

We consider the coupling between the local curvature tensor of a membrane and the local two-dimensional nematic order parameter, deriving it from a quasi-microscopic argument. This coupling makes the nematic director aligned along the lowest curvature eigenvector in a local metric. Local bending of a membrane may then generate nematic ordering. Alternatively, emerging nematic order leads to shape instabilities of closed vesicles. The theory is applied to a spherical isotropic vesicle, which turns into a prolate shape with two +1 disclinations on its poles as the nematic order sets in the membrane, described within the Landau-de Gennes continuum model.