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

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 $\kappa_{\rm eff}$, a new expression for the effective surface tension $\sigma_{\rm eff}=\sigma - \epsilon k_{\rm B}T/(2a_p)$ 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 $\sigma_{\rm eff}=0$, 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 $\sigma_{\rm eff}$ 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 $\hat \sigma_{\rm eff}=\sigma_{\rm eff}R^2/\kappa_{\rm eff}>0$ and a bending one for $-1<\hat \sigma_{\rm eff}<0$. 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 $\hat \sigma_{\rm eff}\simeq 0$. We propose that the buckling transition, observed for planar supported membranes in the literature, occurs for $\hat \sigma_{\rm eff}\simeq-1$, the associated negative frame tension playing the role of a compressive force.