  In this paper, phase field models are developed for multi-component vesicle
membranes with different lipid compositions and membranes with free boundary.
These models are used to simulate the deformation of membranes under the
elastic bending energy and the line tension energy with prescribed volume and
surface area constraints. By comparing our numerical simulations with recent
experiments, it is demonstrated that the phase field models can capture the
rich phenomena associated with the membrane transformation, thus it offers
great functionality in the simulation and modeling of multicomponent membranes.
