In this paper we derive thermodynamically consistent higher order
phase �eld models for the dynamics of vesicle membranes in incompressible
viscous
uids. We start with basic conservation laws and an appropriate version
of the second law of thermodynamics and obtain generalizations of models
introduced by Du, Li and Liu [5] and Jamet and Misbah [13]. In particular we
derive a stress tensor involving higher order derivatives of the phase �eld and
generalize the classical Korteweg capillarity tensor.