  We prove the convergence of phase-field approximations of the Gibbs-Thomson
law. This establishes a relation between the first variation of the
Van-der-Waals-Cahn-Hilliard energy and the first variation of the area
functional. We allow for folding of diffuse interfaces in the limit and the
occurrence of higher-multiplicities of the limit energy measures. We show that
the multiplicity does not affect the Gibbs-Thomson law and that the mean
curvature vanishes where diffuse interfaces have collided.
  We apply our results to prove the convergence of stationary points of the
Cahn-Hilliard equation to constant mean curvature surfaces and the convergence
of stationary points of an energy functional that was proposed by Ohta-Kawasaki
as a model for micro-phase separation in block-copolymers.
