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

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(kₓ,kz,t) (where kₓ and kz are, respectively, the x and z components of the wave vector) has the generalized dynamical scaling form S(kₓ,kz,t) =lₓ(t)lz(t)F(kₓ{l}ₓ$(t),${k}zlz(t)), where lₓ(t) and lz(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 lz(t){~}t and lₓ(t){~}t1/3. The experimental observability of these numerical results is also discussed.