  A variational model for the interaction between homogenization and phase
separation is considered. The focus is on the regime where the latter happens
at a smaller scale than the former, and when the wells of the double well
potential are allowed to move and to have discontinuities. The zeroth and first
order $\Gamma$-limits are identified. The topology considered for the latter is
that of two-scale, since it encodes more information on the asymptotic local
microstructure. In particular, when the wells are non constant, the first order
$\Gamma$-limit describes the contribution of microscopic phase separation, also
in situations where there is no macroscopic phase separation. As a corollary,
the minimum of the mass constrained minimization problem is characterized, and
it is shown to depend on whether or not the wells are discontinuous. In the
process of proving these results, the theory of inhomogeneous Modica Mortola
functionals is strengthened.
