Stresses in curved nematic membranes

Area of Science:

• Soft Matter Physics
• Theoretical Physics
• Materials Science

Background:

• Curved membranes with ordered director fields, like those found in biological systems or liquid crystals, experience intrinsic stresses.
• Understanding these stresses is crucial for predicting membrane behavior and stability.

Purpose of the Study:

• To develop a theoretical framework for calculating stress and torque induced by nematic ordering on curved membranes.
• To investigate the nature of forces between defects and their interaction with Gaussian curvature.
• To derive modified mechanical laws for curved membranes, such as spherical vesicles.

Main Methods:

• Utilizing a variational principle and energy invariance under Euclidean motions.
• Deriving Euler-Lagrange equations and boundary conditions for the membrane.
• Employing the Green function of the Laplace-Beltrami operator to mediate forces.

Main Results:

• The stress tensor includes attraction-repulsion forces between defects, influenced by Gaussian curvature.
• Non-isotropic forces involving derivatives of the Green function and Gaussian curvature are identified.
• A modified Young-Laplace law is derived for spherical vesicles due to nematic texture.
• A repulsive force from a defect at the north pole of a spherical cap indicates an unstable equilibrium.

Conclusions:

• The theoretical framework successfully quantifies stress and torque from nematic ordering on curved membranes.
• Defect interactions and curvature-mediated forces play a significant role in membrane mechanics.
• The findings provide insights into the stability and behavior of curved membranes with ordered structures.