Elastic Bending Energy: A Variational Approach

Abstract

Geometric continuum models for fluid lipid membranes are considered using classical field theory, within a covariant variational approach. The approach is cast as a higher-derivative Lagrangian formulation of continuum classical field theory, and it can be seen as a covariant version of the field theoretical variational approach that uses the height representation. This novel Lagrangian formulation is presented first for a generic reparametrization invariant geometric model, deriving its equilibrium condition equation, or shape equation, and its linear and angular stress tensors, using the classical Canham-Helfrich elastic bending energy for illustration. The robustness of the formulation is established by extending it to the presence of external forces, and to the case of heterogenous lipid membranes, breaking reparametrization invariance. In addition, a useful and compact general expression for the second variation of the free energy is obtained within the Lagrangian formulation, as a first step towards the study of the stability of membrane configurations. The simple structure of the expressions derived for the basic entities that appear in the mechanics of a lipid membrane is a direct consequence of the well established power of a Lagrangian variational approach. The paper is self-contained, and it is meant to provide, besides a new framework, also a convenient introduction to the mechanics of lipid membranes.

A simultaneous variational principle for elementary excitations of fluid lipid membranes

“…In the first half of the proof, the shape functions are kept fixed, and the deviation vector is varied. The result is exactly like the one given above for a soap bubble (16), that does not depend on the specific form of the soap bubble energy. For the second half of the proof, the general expression for the stress tensor in terms of partial derivatives of the energy density (7) is needed.…”

“…where the second equality follows from recognizing that the integrand coincides with the variational derivative of the energy density along the deviation vector field Z [16]. This observation is crucial in exhibiting that the simultaneous variational functional is a first variation in a not so veiled disguise, and this explains the discount mentioned earlier in jest.…”

“…The energy density can be written as  = g f , including the surface Jacobian, for convenience. In this, and other conventions, this note follows the general treatment of [16], where the free energy curvature model is cast as a higher derivative Lagrangian classical field theory. For the sake of simplicity, in this note the surface is assumed to be closed, modelling a lipid vesicle.…”

“…In this free energy, the two independent fields are the surface shape functions x ( ) X a , through the stress tensor, and a deviation spatial vector field x ( ) Z a . The linear stress tensor is a vector density of weight one, as emphasized by the tilde, that can be obtained in a manifestly covariant way from the energy density of a given geometric model with [16]…”

“…Here   ( )is the Euler-Lagrange derivative of the energy density  , that is given the divergence of the stress tensor [27], and that can be written in an explicitly covariant way as [16]…”