Nematic order on a deformable vesicle: theory and simulation

In membranes with nematic liquid-crystalline order, there is a geometric coupling between the nematic director and the shape: nonuniformity in the director induces curvature, and curvature provides an effective potential acting on the director. For a closed vesicle, there must be a total topological charge of +2, which normally occurs as four defects of charge +1/2 each. Previous research has suggested that these four defects will form a regular tetrahedron, leading to a tetrahedral shape of the vesicle, which may be useful in designing colloidal particles for photonic applications. Here, we use three approaches to investigate the behavior of a nematic vesicle: particle-based simulation, spherical harmonic expansion, and finite-element modeling. When liquid crystal has a purely 2D intrinsic interaction, we find that the perfect tetrahedral shape is stable over a wide range of parameters. However, when it has a 3D intrinsic and extrinsic interaction, the perfect tetrahedral shape is never stable; the vesicle is a distorted tetrahedron for small Frank constant and a highly elongated rectangle for larger Frank constant. These results show the difficulty in designing tetrahedral structures for photonic crystals. We explore the connection between curvature and orientational order in deformable membranes by calculating the shapes of nematic liquid-crystal vesicles.