Phase separation in the presence of spatially periodic forcing

We have investigated theoretically phase separation in the presence of spatially periodic forcing. From the analytic and numeric study of a suitably generalized Cahn-Hilliard equation in one and two dimensions, we find that the forcing amplitude necessary to generate a periodic kink-type state from small random initial conditions depends weakly on wave number. This amplitude is much larger than the one necessary to stabilize the periodic state, i.e., to prevent late-stage coarsening, once it is established. Surprisingly, the destabilizing mode is of long-wave type, which is in contrast to the well-known most rapidly growing coarsening mode in the unforced system. In the Allen-Cahn equation with nonconserved order parameter the relevant modes are always long wave. It appears feasible to observe these effects by imposing a temperature modulation by optical grating which then couples to concentration modulation via the (Ludwig-)Soret effect.