We have investigated the concentration fluctuations at the interface between two liquid phases. The sample investigated is a binary mixture prepared at its critical consolution concentration, and kept below its critical temperature, so that it is separated into two phases. The experimental technique used is small angle static light scattering, an unusual technique to study interfacial fluctuations. We have investigated both the equilibrium interfacial fluctuations at a fixed temperature, and the nonequilibrium ones during the transient following a sudden increase of the temperature. The nonequilibrium condition is obtained by suddenly raising the temperature below its critical value, so that the binary mixture is still separated into two phases, and a very slow partial diffusive remixing of the two phases occurs. The scattered light contains two contributions. One is due to the interfacial fluctuations, and the other is due to the fluctuations in the bulk phases. At equilibrium light is mainly scattered from the capillary waves at the interface between the two phases, and the well known q^(-2) spectrum is observed. We show that during the transient the capillary waves contribution to the scattered light is superimposed to a q^(-4) term due to nonequilibrium fluctuations in the bulk phases. The light scattering data allow us to show for the first time that the interface is preserved during the diffusive remixing of the bulk phases. Moreover, the surface tension and concentration difference across the interface attain almost instantaneously their equilibrium values. We also show for the first time that giant fluctuations are present in the bulk phases during the transient following the fast rearrangement of the interface, and their contribution to the scattered light is even larger than that associated to capillary waves. During the diffusive remixing, the presence of a macroscopic concentration gradient makes spontaneous velocity fluctuations of the fluid in the vertical direction give rise to nonequilibrium concentration fluctuations. We have developed a fluctuating hydrodynamics treatment of the fluctuations which, together with a mean field description of the macroscopic state of the system, provides a satisfactory description of the experimental results.

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