Dynamical scaling in anisotropic phase-separating systems in a gravitational field

Abstract

We derive a generalization of the Cahn-Hilliard equation for phase separation in the presence of a field that varies linearly in (say) the z direction, e.g., gravity. The starting point of our derivation is the master-equation description for a spin-exchange kinetic Ising model in the presence of such a field. We neglect thermal fluctuations and the effect of hydrodynamic interactions. We also mptivate the concept of generalized dynamical scaling as a means of characterizing anisotropic domain growth. We present numerical results from a two-dimensional (2D) simulation of this equation. Our results indicate that the 2D time-dependent structure factor S(k_x,k_z,t) (where k_x and k_z are, respectively, the x and z components of the wave vector) has the generalized dynamical scaling form S(k_x,k_z,t) =l_x(t)l_z(t)F(k_xl_x(t),k_zl_z(t)), where l_x(t) and l_z(t) are time-dependent length scales in the x direction (perpendicular to direction of gravity) and in the z direction (in the direction of gravity), respectively; and F(x,y) is a universal function of its arguments. The temporal behavior of these length scales is l_z(t)~t and l_x(t)~t^1/3. The experimental observability of these numerical results is also discussed.