Undulation amplitude of a fluid membrane surrounded by near-critical binary fluid mixtures

We consider the thermal undulation, or shape fluctuation, of an almost planar fluid membrane surrounded by the same near-critical binary fluid mixtures on both sides. A weak preferential attraction is assumed between the membrane and one component of the mixture. We use the Gaussian free-energy functional to study the equilibrium average of the undulation amplitude within the linear approximation with respect to the amplitude. According to our result given by a simple analytic formula, the ambient near-criticality tends to suppress the undulation of a membrane, and this suppression effect can overwhelm that of the bending rigidity for small wave numbers. Thus, the ambient near-criticality is suggested to prevent a large membrane from becoming floppy even if the lateral tension vanishes at the equilibrium.