Phase-field approach to three-dimensional vesicle dynamics

We extend our recent phase-field [T. Biben and C. Misbah, Phys. Rev. E 67, 031908 (2003)] approach to 3D vesicle dynamics. Unlike the boundary-integral formulations, based on the use of the Oseen tensor in the small Reynolds number limit, this method offers several important flexibilities. First, there is no need to track the membrane position; rather this is automatically encoded in dynamics of the phase field to which we assign a finite width representing the membrane extent. Secondly, this method allows naturally for any topology change, like vesicle budding, for example. Thirdly, any non-Newtonian constitutive law, that is generically nonlinear, can be naturally accounted for, a fact which is precluded by the boundary integral formulation. The phase-field approach raises, however, a complication due to the local membrane incompressibility, which, unlike usual interfacial problems, imposes a nontrivial constraint on the membrane. This problem is solved by introducing dynamics of a tension field. The first purpose of this paper is to show how to write adequately the advected-field model for 3D vesicles. We shall then perform a singular expansion of the phase field equation to show that they reduce, in the limit of a vanishing membrane extent, to the sharp boundary equations. Then, we present some results obtained by the phase-field model. We consider two examples; (i) kinetics towards equilibrium shapes and (ii) tanktreading and tumbling. We find a very good agreement between the two methods. We also discuss briefly how effects, such as the membrane shear elasticity and stretching elasticity, and the relative sliding of monolayers, can be accounted for in the phase-field approach.

Figure 1

The final stage of the evolution showing thus the equilibrium shapes for various values of the swelling factor in the prolate regime.

Figure 2

(Color online) The final stage of the evolution showing thus the equilibrium shapes for various values of the swelling factor in the oblate regime.

Figure 3

(Color online) A vesicle under shear flow executing a tanktreading and oriented with an angle (less that $\pi ∕4$) in the flow.

Figure 4

Schematic drawing of the normal vector field inside the vesicle; the two points correspond to curvature centers and the line between then is a line singularity.