  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).
