Phase Separation Dynamics in a Concentration Gradient

Phase separation dynamics with an initially non-uniform concentration are studied. Critical and off-critical behavior is observed simultaneously. A mechanism for an expanding phase separated region is demonstrated and the time dependence of the concentration is determined. The final equilibrium state consists of a planar interface separating one phase from the other. The evolution to this state is characterized by an experimentally observable flux, $j$, crossing this interface. We find that $j \sim t^{-2/3}$ if patterns are formed in the bulk and $j \sim t^{-1/2}$ if the bulk remains homogeneous. The results are explained in terms of scaling arguments which are confirmed numerically. (postScript figures appended to end of lateX file).