Vesicle Model with Bending Energy Revisited
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Vesicle Model with Bending Energy Revisited Henri Gouin
Received: 7 November 2013 / Accepted: 23 February 2014 © Springer Science+Business Media Dordrecht 2014
Abstract The equations governing the conditions of mechanical equilibrium in fluid membranes subject to bending are revisited thanks to the principle of virtual work. The note proposes systematic tools to obtain the shape equation and the line condition instead of Christoffel symbols and the complex calculations they entail. The method seems adequate to investigate all problems involving surface energies. Keywords Vesicles · Surface energy · Differential geometry of surfaces · Shape equation Mathematics Subject Classification (2010) 74K15 · 76Z99 · 92C37 1 Introduction Lipid molecules dissolved in water spontaneously form bilayer membranes, with properties very similar to those of biological membranes and vesicles [1]. The knowledge of the mechanics of vesicles started more than thirty years ago when both experimental and theoretical studies of amphiphilic bilayers engaged the attention of physicists and the interest of mathematicians [2]. In a viscous fluid, vesicles are drops a few tens of micrometers wide, bounded by impermeable lipid membranes a few nanometers thick. The membranes are homogeneous down to molecular dimensions; consequently, it is possible to represent the vesicle as a two-dimensional smooth surface in three-dimensional Euclidean space. Depending on the cases, the bilayers may be considered as liquid or solid. When liquid, the lipid molecules form a two-dimensional lattice and the membranes are described with an effective energy that does not penalize tangential displacements. Their mechanical properties permit a continuous mechanical description; such a deformable object is characterized by a flexion-governed membrane rigidity resulting from the curvature energy. The general
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H. Gouin ( ) Aix-Marseille Université, CNRS, Centrale Marseille, M2P2 UMR 7340, 13451 Marseille, France e-mail: [email protected] H. Gouin e-mail: [email protected]
H. Gouin
theory accounts for surface strain, director extension, and director tilt associated with the misalignment of the surface normal. Galilean invariance is tantamount to the invariance of the energy under arbitrary two-dimensional orthogonal transformations and regarded as a function of a symmetric two-dimensional tensor. In this respect, the Helfrich Hamiltonian, which is quadratic in the curvature eigenvalues, provides a good description of lipid membranes and associates bending with an energy penalty [3–5]. Equilibrium configurations of the membrane satisfy a single normal ‘shape’ equation corresponding to the extrema of the Hamiltonian. Fournier used a variational method to study fluid membranes [6]. Contrary wise, in this note, we revisit the mechanical behaviour of vesicle membranes by using the principle of virtual work [7–9] together with lemmas from the intrinsic differential geometry of surfaces; consequently, we do not need to use coordinate lines and Christoffel symbols
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