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Towards a thermodynamically consistent picture of the phase-field model of vesicles: local membrane incompressibility.

A phase-field model for vesicles including hydrodynamics was presented in two and three dimensions [T. Biben and C. Misbah, Phys. Rev. E 67, 031908 (2003); T. Biben, K. Kassner, and C. Misbah, Phys. Rev. E 72, 041921 (2005)]. A particularly important feature for vesicles is that their membrane is locally incompressible. In these works a tension field defined everywhere in the bulk was introduced in order to fulfill local membrane inextensibility. Here we reconsider the original model by treating the phase field as a thermodynamic variable and develop a picture which is consistent with the second law of thermodynamics. This enables us to write the phase-field evolution equations in terms of a thermodynamical potential. This potential acquires, at global equilibrium, a Lyapunov functional character. The goal of this paper is twofold: (i) The first and primary goal is purely conceptual, in that we can write down a first and second principle for membranes, from which the evolution equations follow, thanks to the evaluation of the entropy production and the use of concepts of irreversible thermodynamics. (ii) Due to the monotonous character of the evolution of the functional (at global equilibrium), we expect this formulation to be more appropriate for numerical studies. The formalism developed to account for the local incompressibility of the membrane is believed to offer a systematic framework in order to include naturally other physical ingredients, as briefly discussed here and demonstrated in future works.