# Surface tension of membranes depending on the boundary shape

## TL;DR

This paper investigates how the shape of the boundary influences the surface tension of elastic membranes, revealing shape dependence at phase transition points through models inspired by spin systems.

## Contribution

It demonstrates that boundary shape affects membrane surface tension at critical points, extending understanding beyond traditional projected area dependence.

## Key findings

- Boundary shape impacts surface tension at phase transitions.
- Confirmation using Helfrich and deficit angle models.
- Highlights importance of boundary shape in membrane physics.

## Abstract

In this paper, we study the boundary effect on the surface (or frame) tension of elastic membrane surface models. The frame tension generally depends only on the projected area of the boundary over which the surface spans. However, from a spin model analogy, the frame tension is expected to be dependent also on the boundary shape at the continuous transition point. We confirm this expectation using the following fixed-connectivity and tethered surface models: the surface model of Helfrich and Polyakov and a surface model with deficit angle term. We also discuss the reason why this expectation is worthwhile to study.