Hydrodynamic liquid crystal models for lipid bilayers

Coarse-grained continuous descriptions for lipid bilayers are typically based on minimizing the Helfrich energy. Such models consider the fluid properties of these structures only implicitly and have been shown to nicely reproduce equilibrium properties. Model extensions that also address the dynamics of these structures are surface (Navier--)Stokes--Helfrich models. They explicitly account for membrane viscosity. However, these models also usually treat the lipid bilayer as a homogeneous continuum, neglecting the molecular degrees of freedom of the lipids. Here, we derive refined models which consider in addition a scalar order parameter representing the molecular alignment of the lipids along the surface normal. Starting from hydrodynamic surface liquid crystal models, we obtain a hydrodynamic surface Landau--Helfrich model for asymmetric lipid bilayers and a surface Beris--Edwards model for symmetric lipid bilayers. The fully ordered case for both models leads to the known surface (Navier--)Stokes--Helfrich models. Besides more detailed continuous models for lipid bilayers, we therefore also provide an alternative derivation of surface (Navier--)Stokes--Helfrich models.