The theory of irreversible thermodynamics for arbitrarily curved lipid membranes is presented here. The coupling between elastic bending and irreversible processes such as intra-membrane lipid flow, intra-membrane phase transitions and protein binding and diffusion is studied. Employing the linear irreversible thermodynamic framework, the governing equations of motion along with appropriate boundary conditions are provided. Next,an arbitrary Lagrangian-Eulerian (ALE) finite-element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity is independent of the in-plane material velocity and which can be specified arbitrarily. A finite-element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf-sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks, including the lid-driven cavity problem. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are numerically and analytically found to be unstable with respect to long-wavelength perturbations when their length exceeds their circumference. It is then shown that cylindrical lipid membrane tubes, which in the limit of no bending modulus are identical to fluid films, can be stabilized when there is nonzero bending cost. Such tubes are analytically and numerically shown to have a nontrivial stability phase diagram.
University of California, Berkeley
Lipid Membranes: Continuum Theory and Simulation
Area of Study