Stresses in curved nematic membranes

TL;DR

This paper develops a theoretical framework to analyze stresses and defect interactions in curved nematic membranes, revealing how nematic ordering influences membrane mechanics and defect stability.

Contribution

It introduces a variational approach to calculate stress tensors and defect interactions on curved membranes, including effects of Gaussian curvature and Green functions.

Findings

Defects attract each other via Green function-mediated forces.

Defects are attracted to regions with similar Gaussian curvature signs.

Spherical vesicles exhibit a modified Young-Laplace law due to nematic textures.

Abstract

Ordering configurations of a director field on a curved membrane induce stress. In this work, we present a theoretical framework to calculate the stress tensor and the torque as a consequence of the nematic ordering; we use the variational principle and invariance of the energy under Euclidean motions. Euler-Lagrange equations of the membrane as well as the corresponding boundary conditions also appear as natural results. The stress tensor found includes attraction-repulsion forces between defects; likewise, defects are attracted to patches with the same sign in gaussian curvature. These forces are mediated by the Green function of Laplace-Beltrami operator of the surface. In addition, we find non-isotropic forces that involve derivatives of the Green function and the gaussian curvature, even in the normal direction to the membrane. We examine the case of axial membranes to analyze the spherical one. For spherical vesicles we find the modified Young-Laplace law as a consequence of the nematic texture. In the case of spherical cap with defect at the north pole, we find that the force is repulsive respect to the north pole, indicating that it is an unstable equilibrium point.