n -Atic Order and Continuous Shape Changes of Deformable Surfaces of Genus Zero

Abstract

We consider in mean-field theory the continuous development below a secondorder phase transition of n-atic tangent plane order on a deformable surface of genus zero with order parameter ψ = e inθ . Tangent plane order expels Gaussian curvature. In addition, the total vorticity of orientational order on a surface of genus zero is two. Thus, the ordered phase of an n-atic on such a surface will have 2n vortices of strength 1/n, 2n zeros in its order parameter, and a nonspherical equilibrium shape. Our calculations are based on a phenomenological model with a gauge-like coupling between ψ and curvature, and our analysis follows closely the Abrikosov treatment of a type II superconductor just below H c2 .

Numerical insights into the phase diagram of p-atic membranes with spherical topology

“…reflected in the noncommutative nature of ∇ q and it causes a geometrical frustration of the ordered state on curved membranes inducing or leading to coupling between in-plane ordering and membrane conformations (see appendix A for description of the properties of ∇ q ). Equation (2) is relevant for the more general class of p-atic membranes with in-plane anisotropic elasticity, which is invariant under rotation of its components by the angle 2π p in the plane [10,11]. For the cases p = 1 (polar) and p = 2 (nematic) a more specific energy expression holds:…”

Theoretical study of fluid membranes of spherical topology with internal degrees of freedom

“…The lowest eigenfunctions of this operator and similar operators for other gauges, are presented in different forms in [14], [15] and [11], and are re-expressed here as…”

“…al. [11], embodied in equations (1.4) and (1.5), has been reduced to the much simpler form of equation (2.5) (2.7), and integrals of the form…”

“…So far, the model and formalism are identical to that used by Park, Lubensky and MacKintosh in their mean-field treatment of n-atic fluid membranes of genus-zero (vesicles) [11]. They noted that, not only is n-atic ordering partly frustrated by curvature of its twodimensional space, making parallelism impossible to achieve, but it is further frustrated by the spherical boundary conditions.…”

Textures and shapes in nematic elastomers under the action of dopant concentration gradients

“…The equilibrium lowest-energy state in a disk with either normal or tangential anchoring on the edge contains a pair of defects with the charge +1/2, while defects with the charge −1/2 are obtained in the pretzel topology [9]. On a spherical shell, devoid of boundaries, topology requires the total charge equal to two [11,12], which is accommodated by four defects with the charge +1/2 [13,14], though their configuration depends on the ratio of splay to bend nematic elasticity [15]. These patterns lead to a very limited variety of shapes, which all contain singularities at defects locations, smoothed out by bending rigidity of the film.…”