Phase field calculus, curvature-dependent energies, and vesicle membranes

Abstract

Lipid bilayer vesicles are membranes formed spontaneously in aqueous environments under suitable conditions. They have been extensively studied in literature as a simple model of cell membranes. In this work, we discuss some recent developments in the phase field modeling and simulations of vesicles. We describe how the equilibrium and dynamic properties of the vesicles can be effectively described by the phase field calculus which is a natural translation of the energy variations with respect to the vesicle geometry to the variations of the corresponding diffuse interfacial energy with respect to the phase field functions. We also illustrate how phase field techniques can be generalized to model a multi-component vesicle, a vesicle interacting with a surrounding fluid and the adhesion of a vesicle to a substrate.

Keywords:

• diffuse interface
• phase field
• curvature
• vesicle deformation
• vesicle fluid interaction
• vesicle substrate adhesion

Acknowledgements

This research is supported in part by NSF-DMS 0712744 and NIH NCI-125707. I would like to thank a number of people, including Roy Nicolaides and Morton Gurtin of Carnegie Mellon who introduced the Cahn–Hilliard theory to me more than two decades ago; Max Gunzburger of Florida State who brought to my attention the Ginzburg–Landau models which led to many joint works on the subject, together with other collaborators; and Long-Qing Chen who shared with me his knowledge and insight on the various aspects of phase field modeling in materials sciences at Penn State. On the phase field modeling and simulations of vesicles, I am particularly grateful to Chun Liu and Xiangqiang Wang for many stimulating discussions and collaborations that resulted in a number of works discussed here. I also thank the contributions made by other collaborators that include Rolf Ryham, Jian Zhang, Sova Das, Liyong Zhu, Manlin Li and Yanxiang Zhao. In addition, I thank the authors of 76 and 85 who permitted the use of their experimental pictures in 21 for comparison purposes; some of them are again shown in this paper.