Stresses in curved nematic membranes

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.

To appear in PRE

References in corpus (7)

• Nematic textures in spherical shells
• Topological Defects in Spherical Nematics
• Crystallography on Curved Surfaces
• Defects in nematic membranes can buckle into pseudospheres
• Vesicle shape, molecular tilt, and the suppression of necks
• Hydrodynamic theory for nematic shells: the interplay among curvature, flow and alignment
• Hyperbolic Interfaces

Cited by in corpus (9)

• Theory of defect-mediated morphogenesis
• Theory of nematic and polar active fluid surfaces
• Morphodynamics of Active Nematic Fluid Surfaces
• Microdomains and Stress Distributions in Bacterial Monolayers on Curved Interfaces
• Membrane stress and torque induced by Frank's nematic textures: A geometric perspective using surface-based constraints
• Mechanics of nematic membranes: Euler-Lagrange equations, Noether charges, stress, torque and boundary conditions of the surface Frank nematic field
• Passive defect driven morphogenesis in nematic membranes
• Rotational mobility in spherical membranes: The interplay between Saffman-Delbrück length and inclusion size
• Induced stresses in quasi-spherical elastic vesicles: local and global Laplace-Young law