Toward a thermodynamically consistent picture of the phase-field model of vesicles: curvature energy.

We extend our recent work on phase-field model for vesicles [D. Jamet and C. Misbah, Phys. Rev. E 76, 051907 (2007)]-where only the membrane local incompressibility was treated-to the situation where the bending forces and spontaneous curvature are included. We show how the general phase-field equations can be derived within a thermodynamic consistent picture. We analyze a general form of the bending energy, where the Helfrich bending force is treated as a special case. The dynamical evolution equation derived here for the velocity field allows one to write down a constitutive law of the composite fluid: The ambient fluid plus the membrane. This constitutive law has a viscoelastic form, the viscous part arises from the fluid, while the elastic one represents the action of the membrane. It is shown that the elastic stress tensor is not symmetric, owing to bending torque, inherent to a diffuse membrane model.