  In this work, we study the adhesion of multi-component vesicle membrane to
both flat and curved substrates, based on the conventional Helfrich bending
energy for multi-component vesicles and adhesion potentials of different forms.
A phase field formulation is used to describe the different components of the
vesicle. For the axisymmetric case, a number of representative equilibrium
vesicle shapes are computed and some energy diagrams are presented which reveal
the dependence of the calculated shapes and solution branches on various
parameters including both bending moduli and spontaneous curvatures as well as
the adhesion potential constants. Our computation also confirms a recent
experimental observation that the adhesion effect may promote phase separation
in two-component vesicle membranes.
