Critical diffusion in a bounded fluid. I

A mode-mode coupling theory of the critical diffusion of a binary mixture confined between two parallel plates of infinite extent separated by a distance 2L is presented. The effect of boundaries, which we assume constrain the fluid in contact with them to be stationary, modifies the transport properties of the hydrodynamic shear modes and, consequently, the resulting critical relaxation rate. This rate, as measured by light-scattering experiments, is obtained for wave vector q→ parallel to the boundaries by expanding it in powers of (qL${)}^{\mathrm{-}1}$. Under the assumption that none of the components in the fluid has preferential wetting to the boundaries, the term of order (qL${)}^{\mathrm{-}1}$ is explicitly calculated, yielding a suppression to the bulk relaxation rate of the form Γ=${\Gamma }_B$[1-(1/ qL)(lnqL+1.20)], where ${\Gamma }_B$ is the bulk expression in which the weak critical divergent behavior of the viscosity has been taken into account. This result, which is valid in the asymptotic qL≫1 and qξ≫1 region but for ξ/L arbitrary and where ξ is the correlation length, is shown to produce important deviations from the bulk case. In particular, for values of qL∼3, which are within the present experimentally accessible range, it yields approximately a 40% suppression to the bulk value. It is also concluded that the effect of the boundaries completely precludes any crossover to a two-dimensional divergent behavior which would be obtained if physically unrealistic nonsticking boundary conditions are assumed, as has been done in the literature.