Undulation amplitude of a fluid membrane in a near-critical binary fluid mixture calculated beyond the Gaussian model supposing weak preferential attraction

We calculate the mean square amplitude of the shape fluctuation - an equal-time correlation - of an almost planar fluid membrane immersed in a near-critical binary fluid mixture. One fluid component is usually preferentially attracted by the membrane, and becomes more concentrated around it because of the near criticality. This generates osmotic pressure, which influences the amplitude. The amplitude is also affected by the reversible dynamics of the mixture, which moves with the membrane. By assuming the Gaussian free-energy functional and weak preferential attraction, the author previously showed that a new term is added to the restoring force of the membrane and tends to suppress the amplitude. Not assuming both of them, but still focusing on modes with wavelength longer than the correlation length, we here calculate the amplitude of a tensionless membrane. First, within the Gaussian model, we solve the governing equations to show that, for long wavelength, the additional term becomes predominant, although decreased hydrodynamic effects make its numerical factor approximately half that of the previous result. The change in the term turns out not to be monotonic with the wavelength, which is mainly caused by the change in the induced mass. Second, assuming the critical composition far from the membrane, we calculate the amplitude beyond the regime of the Gaussian model. The result coincides roughly with the corresponding result in the Gaussian model if the correlation length is interpreted as one close to the membrane.