Fluctuation tension and shape transition of vesicles: renormalisation calculations and Monte Carlo simulations

Abstract

It has been known for long that the fluctuation surface tension of membranes $r$, computed from the height fluctuation spectrum, is not equal to the bare surface tension $\sigma$ introduced in the Helfrich theory. In this work we relate these two surface tensions both analytically and numerically and compare them to the Laplace tension $\gamma$, and the mechanical frame tension $\tau$. Using one-loop renormalisation calculations, we obtain, in addition to the effective bending modulus ${\kappa }_{\mathrm{eff}}$, a new expression for the effective surface tension ${\sigma }_{\mathrm{eff}}=\sigma -\epsilon k_{\mathrm{B}}T/(2a_p)$ where $a_p$ the projected cut-off area, and $\epsilon =3$ or 1 according to the allowed configurations. Moreover we show that the crumpling transition for an infinite planar membrane occurs for ${\sigma }_{\mathrm{eff}}=0$, and also that it coincides with vanishing Laplace and frame tensions. Using extensive Monte Carlo (MC) simulations, triangulated membranes of vesicles made of $N=100-2500$ vertices are simulated. No local constraint is applied. It is shown that the numerical $r$ is equal to ${\sigma }_{\mathrm{eff}}$ both with radial MC moves ($\epsilon =3$) and with corrected MC moves locally normal to the fluctuating membrane ($\epsilon =1$). For finite vesicles of typical size $R$, two different regimes are defined: a tension regime for ${\hat{\sigma }}_{\mathrm{eff}}={\sigma }_{\mathrm{eff}}{R}^2/{\kappa }_{\mathrm{eff}}>0$ and a bending one for $-1<{\hat{\sigma }}_{\mathrm{eff}}<0$. A shape transition from a quasi-spherical shape imposed by the large surface energy, to more deformed shapes only controlled by the bending energy, is observed numerically at ${\hat{\sigma }}_{\mathrm{eff}}\simeq 0$. We propose that the buckling transition, observed for planar supported membranes in the literature, occurs for ${\hat{\sigma }}_{\mathrm{eff}}\simeq -1$, the associated negative frame tension playing the role of a compressive force.

to be published in Soft Matter