  We consider a class of Cahn-Hilliard equation that models phase separation
process of binary mixtures involving nontrivial boundary interactions in a
bounded domain with non-permeable wall. The system is characterized by certain
dynamic type boundary conditions and the total mass, in the bulk and on the
boundary, is conserved for all time. For the case with physically relevant
singular (e.g., logarithmic) potential, global regularity of weak solutions is
established. In particular, when the spatial dimension is two, we show the
instantaneous strict separation property such that for arbitrary positive time
any weak solution stays away from the pure phases +1 and -1, while in the three
dimensional case, an eventual separation property for large time is obtained.
As a consequence, we prove that every global weak solution converges to a
single equilibrium as the time goes to infinity, by the usage of an extended
Lojasiewicz-Simon inequality.
