Chapter One - Continuum Theory for the Edge of an Open Lipid Bilayer

Abstract

Abstract Using a variational approach, the Euler–Lagrange equations of an open lipid bilayer subject to forces and couples distributed on its surface and edge are derived. Both constant and geometry-dependent edge-energy densities are considered. For the second of these alternatives, the edge-energy density is a general function of the normal and geodesic curvatures and geodesic torsion of the edge. Focusing on a generic segment of the edge, the global forms of the force and moment balances and the free-energy imbalance are stated and their local counterparts are derived. While the force and moment balances lead to the governing equations of the edge element under internal and external loads, the free-energy imbalance provides a mechanism for ensuring the thermodynamic compatibility of constitutive relations. Inspired by various experimental and theoretical studies showing the importance of dissipative mechanisms at the edge of an open lipid bilayer, the internal force and moment are decomposed into elastic and viscous parts. Considering the geometry-dependent edge-energy density and following the Coleman–Noll procedure, constitutive relations for the elastic contributions to the internal moment and tangential component of the internal force are derived. Additionally, the constitutive relations for the viscous contributions to the internal force and moment are restricted by a reduced dissipation inequality. In the purely elastic regime, it is shown that the governing equations for the edge arising from augmenting the force and moment balances with thermodynamically compatible constitutive relations reduce to the Euler–Lagrange equations previously obtained on variational grounds.