Morphology of fluctuating spherical vesicles with internal bond-orientational order

We investigate the tangent-plane n-atic bond-orientational order on a deformable spherical vesicle to explore continuous shape changes accompanied by the development of quasi-long-range order below the critical temperature. The n-atic order parameter ψ = ψ₀e^inΘ, in which Θ denotes a local bond orientation, describes vector, nematic and hexatic orders for n = 1, 2 and 6 respectively. Since the total vorticity of the local order parameter on a surface of genus zero is constrained to 2 by the Gauss-Bonnet theorem, the ordered phase on a spherical surface should have 2n topological vortices of minimum strength 1/n. Using the phenomenological model including a gauge coupling between the n-atic order and the curvature, we find that vortices tend to be separated as far as possible at the cost of local bending, resulting in a non-spherical equilibrium shape, although the tangent-plane n-atic order expels the local curvature deviation from the spherical surface in the ordered phase. Thus the spherical surface above the transition temperature transforms into ellipsoidal, tetrahedral, octahedral, icosahedral and dodecahedral surfaces along with the development of the n-atic order below the transition temperature for n = 1, 2, 3, 6 and 10 respectively.