Coarsening dynamics of a one-dimensional driven Cahn-Hilliard system

We study the one-dimensional Cahn-Hilliard equation with an additional driving term representing, say, the effect of gravity. We find that the driving field E has an asymmetric effect on the solution for a single stationary domain wall (or ''kink''), the direction of the field determining whether the analytic solutions found by Leung [J. Stat. Phys, 61, 345 (1990)] are unique. The dynamics of a kink-antikink pair (''bubble'') is then studied. The behavior of a bubble is dependent on the relative sizes of a characteristic length scale E(-1), where E is the driving field, and the separation L or the interfaces, For EL much greater than 1 the velocities of the interfaces are negligible, while in the opposite limit a traveling-wave solution is found with a velocity upsilon proportional to E/L. For this latter case (EL much less than 1) a set of reduced equations, describing the evolution of the domain lengths, is obtained for a system with a large number of identifies and implies a characteristic length scale growing as (Et)(1/2). Numerical results for the domain-size distribution and structure factor confirm this behavior, and show that the system exhibits dynamical scaling from very early times.