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

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.