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.

Figure 1

The fluid membrane having a single component fluctuates around the $xy$ plane. The semi-infinite regions on the positive and negative $z$ sides of the membrane are occupied by binary fluid mixtures sharing the same properties.

Figure 2

Logarithmic plots of the terms in the parentheses of Eq. (73). The solid line represents the first term $K^4/2$, while the other curves represent $2{\lambda }^{2}d\left(K\right)$. We use $\lambda =0.1$ and $0.01$ for the dashed curve and for the dash-dotted curve, respectively.