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The Helfrich elastic energy of membranes is given by

$F = \int dS (\kappa H^2 + \kappa_G K)$

where $H$ is the mean curvature and $\kappa_G$ is the Gaussian curvature. The derivation in the original paper (W. Helfrich, Z. Naturforsch. 28 c, 693-703, 1973) is rather phenomenological. Could anyone recommend a derivation that starts from Hooke's law (or, the hookian elastic energy) and derives the above expression? In particular, how would you connect $\kappa$ and $\kappa_G$ to the Lamé parameters?

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I believe that what you are asking for is not possible, not from first principles at any rate. The Helfrich free energy describes the curvature free energy of fluid membranes. A Hookean model, which would apply to a 2D solid film, is not appropriate for this physical situation. For a fluid membrane, to begin with, the Lamé shear modulus $\mu$ is identically zero. Such membranes resist stretching, of course, so $\lambda$ does not vanish, but this is not the same as the free energy terms resisting curvature. And curvature of a fluid membrane is not exactly the same as curvature of a solid sheet (see below).

The derivation of Helfrich's expression is almost inevitably going to be phenomenological. Like the Oseen-Frank derivation of the orientational elasticity of nematic liquid crystals, which inspired Helfrich's approach, it relies on general ideas of symmetry and geometry: identifying the independent invariants which characterize the curvature, and postulating that the free energy must be quadratic in these invariants. We can expect this description to apply as long as the curvature deformations have a large length scale (compared with molecular dimensions), and to break down when this condition is not satisfied. Markus Deserno has written a very detailed review article describing this: Chemistry and Physics of Lipids, 185, 11 (2015) (unfortunately, I can't find an open-access version). For a fluid, there will never be a rigorous link between the constants appearing in Helfrich's free energy and a simple microscopic model, such as beads connected by springs.

Having said all that, crude models of lipid bilayers have been proposed, which try to relate the curvature elasticities to properties of the individual layers, including the Lamé constants. Deserno mentions some of these. In Section 3.2, two models of a fluid membrane as a thin solid plate, and a pair of thin solid plates, as analyzed by Landau and Lifshitz, are mentioned, and expressions given for the membrane bending rigidities in terms of Young's modulus $E$ and the Poisson ratio. However the deficiencies of this kind of model are pointed out (e.g. setting $\mu=0$ leads to $E=0$). If this is what you were thinking of, I don't think that it's quite that simple.

However, Deserno mentions a related, and more satisfactory, model, going back to Goetz, Gompper and Lipowsky Phys Rev Lett, 82, 221 (1999), which is generally available here. This involves two fluidlike films with fluidity restricted to within the sheets, so $\mu=0$. Here, a connection is made between $\kappa$ and the area expansion modulus (hence with $\lambda$) of the films, although (according to Deserno) there seems to be no way of computing the Gaussian modulus from this analysis. This is about as close as I can come to what you want.

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  • $\begingroup$ Thank you for the very detailed answer and the excellent references! So far I thought of membranes as elastic plates but this is clearly not the case. Thank you for clarifying this! $\endgroup$
    – Botond
    Oct 5, 2018 at 18:50

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