Long-time behavior of velocity correlation of a Brownian particle in a near-critical binary fluid mixture

We consider a rigid spherical particle suspended in a binary fluid mixture in the one-phase region near the demixing critical point, assuming the critical composition far from the particle. The particle surface is assumed to interact with mixture components via short-range forces, which can generate the adsorption layer rich in the preferred component and can influence the force exerted on the particle. Applying hydrodynamics based on a coarse-grained free-energy density, we calculate the linear response of the particle velocity to an oscillating external force. We devise a procedure for calculating the first four terms in the low-frequency expansion of the dissipative part of the hydrodynamic force exerted on the particle. We numerically show that the preferential adsorption (PA) changes the coefficients of the four terms, which determine the leading and the next-to-leading terms in the long-time expansion of the particle's velocity correlation function at equilibrium. The temporal exponents of these terms are not changed by PA. According to our numerical results, the back-flow effects on the first two terms in the low-frequency expansion cancel each other out and PA does not affect the coefficient of the leading term, which is consistent with the previous results obtained in the framework of the Gaussian model. The coefficient of the next-to-leading term can be changed by PA. This change is in the direction of suppressing the correlation for a neutrally buoyant particle.