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.