Surface tension and Laplace pressure in triangulated surface models for membranes without fixed boundary

TL;DR

This Monte Carlo study investigates the surface tension of spherical membrane models without fixed boundaries, comparing it with frame tension and analyzing their relationship across different rigidity and size conditions.

Contribution

The paper introduces a method to evaluate surface tension directly from the surface area in triangulated membrane models and compares it with frame tension derived from Laplace pressure.

Findings

Reasonable consistency between surface tension and frame tension at high bending rigidity.

Frame tension becomes constant as the surface area per vertex increases.

Surface tension evaluation is feasible without fixed boundary conditions.

Abstract

A Monte Carlo (MC) study is performed to evaluate the surface tension $\gamma $ of spherical membranes that may be regarded as the models of the lipid layers. We use the canonical surface model defined on the self-avoiding triangulated lattices. The surface tension $\gamma $ is calculated by keeping the total surface area $A$ constant during the MC simulations. In the evaluation of $\gamma $, we use $A$ instead of the projected area $A_p$, which is unknown due to the fluctuation of the spherical surface without boundary. The pressure difference ${\it\Delta}p $ between the inner and the outer sides of the surface is also calculated by maintaining the enclosed volume constant. Using ${\it\Delta}p $ and the Laplace formula, we obtain the tension, which is considered to be equal to the frame tension $\tau$ conjugate to $A_p$, and check whether or not $\gamma $ is consistent with $\tau$. We find reasonable consistency between $\gamma$ and $\tau$ in the region of sufficiently large bending rigidity $\kappa$ or sufficiently large $A/N$. It is also found that $\tau$ becomes constant in the limit of $A/N\to \infty$ both in the tethered and fluid surfaces.