Viscoelasticity of two-dimensional fluid membranes under oscillatory loadings

Biomembranes consisting of two opposing phospholipid monolayers, which comprise the so-called lipid bilayer, are largely responsible for the dual solid-fluid behavior of individual cells. Quantifying the mechanical characteristics of biomembrane, including the dynamics of their in-plane fluidity, can provide insight not only into active or passive cell behaviors but also into vesicle design for drug delivery systems. Despite numerous studies on the mechanics of biomembranes, their dynamical viscoelastic properties have not yet been fully described. We thus quantify their viscoelasticity based on a two-dimensional (2D) fluid membrane model, and investigate this viscoelasticity under small amplitude oscillatory loadings in micron-scale membrane area. We use hydrodynamic equations of bilayer membranes, obtained by Onsager's variational principle, wherein the fluid membrane is assumed to be an almost planar bilayer membrane. Simulations are performed for a wide range of oscillatory frequencies f and membrane tensions. Our numerical results show that as frequencies increase, membrane characteristics shift from an elastic-dominant to viscous-dominant state. Furthermore, such state transitions obtained with a 1-$\mu$m-wide loading profile appear with frequencies between $O(f) = 10^1-10^2$ Hz, and almost independently of surface tensions. The transition shifts to a lower frequency range as the width of the loading profile increases. We discuss the formation mechanism of the viscous- or elastic-dominant transition based on relaxation rates that correspond to the eigenvalues of the dynamical matrix in the governing equations.