Internal phase transition induced by external forces in Finsler geometric model for membranes

TL;DR

This study uses Finsler geometry to model membrane shape transformations under external forces, revealing phase transitions in internal vector fields and their effects on membrane tension and elasticity.

Contribution

It introduces a Finsler geometric membrane model incorporating an external vector field, demonstrating force-induced internal phase transitions and their impact on membrane properties.

Findings

Membrane shape transitions depend on the interaction coefficient λ.

Internal vector field σ undergoes phase change influenced by external forces.

String tension exhibits scaling behavior related to membrane size and shape.

Abstract

We numerically study an anisotropic shape transformation of membranes under external forces for two-dimensional triangulated surfaces on the basis of Finsler geometry. The Finsler metric is defined by using a vector field, which is the tangential component of a three dimensional unit vector $\sigma$ corresponding to the tilt or some external macromolecules on the surface of disk topology. The sigma model Hamiltonian is assumed for the tangential component of $\sigma$ with the interaction coefficient $\lambda$. For large (small) $\lambda$, the surface becomes oblong (collapsed) at relatively small bending rigidity. For the intermediate $\lambda$, the surface becomes planar. Conversely, fixing the surface with the boundary of area $A$ or with the two point boundaries of distance $L$, we find that the variable $\sigma$ changes from random to aligned state with increasing of $A$ or $L$ for the intermediate region of $\lambda$. This implies that an internal phase transition for $\sigma$ is triggered not only by the thermal fluctuations but also by external mechanical forces. We also find that the frame (string) tension shows the expected scaling behavior with respect to $A/N$ ($L/N$) at the intermediate region of $A$ ($L$) where the $\sigma$ configuration changes between the disordered and ordered phases. Moreover, we find that the string tension $\gamma$ at sufficiently large $\lambda$ is considerably smaller than that at small $\lambda$. This phenomenon resembles the so-called soft-elasticity in the liquid crystal elastomer, which is deformed by small external tensile forces.