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We calculate the mean square amplitude of the shape fluctuation -- anequal-time correlation -- of an almost planar fluid membrane immersed in anear-critical binary fluid mixture. One fluid component is usuallypreferentially attracted by the membrane, and becomes more concentrated aroundit because of the near criticality. This generates osmotic pressure, whichinfluences the amplitude. The amplitude is also affected by the reversibledynamics of the mixture, which moves with the membrane. By assuming theGaussian free-energy functional and weak preferential attraction, the authorpreviously showed that a new term is added to the restoring force of themembrane and tends to suppress the amplitude. Not assuming both of them, butstill focusing on modes with wavelength longer than the correlation length, wehere calculate the amplitude of a tensionless membrane. First, within theGaussian model, we solve the governing equations to show that, for longwavelength, the additional term becomes predominant, although decreasedhydrodynamic effects make its numerical factor approximately half that of theprevious result. The change in the term turns out not to be monotonic with thewavelength, which is mainly caused by the change in the induced mass. Second,assuming the critical composition far from the membrane, we calculate theamplitude beyond the regime of the Gaussian model. The result coincides roughlywith the corresponding result in the Gaussian model if the correlation lengthis interpreted as one close to the membrane.

journal of the physical society of japan

Undulation Amplitude of a Fluid Membrane in a Near-Critical Binary Fluid  Mixture Calculated beyond the Gaussian Model Supposing Weak Preferential  Attraction