= Solution
A <synthetic fluid membrane> is a thin, laterally fluid sheet whose components can rearrange in its plane while the sheet resists changes of <area> and <curvature>. Synthetic <amphiphiles> can assemble into two opposed layers with solvent-compatible groups facing outward. Their fluid in-plane response gives no equilibrium static <shear modulus>, but does not imply zero <membrane bending rigidity> or zero area-compression modulus. A thin membrane can therefore bend easily while remaining difficult to stretch. These notes cover generic synthetic interfaces.
TODO: biological, biofluid and biomolecular membrane applications are user-deferred.
A convenient continuum description is the <Helfrich energy>, augmented by <spontaneous curvature> and the Gaussian term:
$$
F=\int\left[\frac\kappa2(2H-C_0)^2+\bar\kappa K+\sigma\right]dA.
$$
Here $H$ and $K$ are <mean curvature> and <Gaussian curvature>, $C_0$ is preferred <curvature>, $\kappa$ and $\bar\kappa$ have <energy> units, and $\sigma$ is <membrane tension> with units $\mathrm{N\,m^{-1}}$. Leaflet asymmetry can favor nonzero $C_0$. At fixed topology the integrated Gaussian term is constant for a closed surface by the <Gauss-Bonnet theorem>. A symmetric tensionless spherical sheet has mean-curvature bending <energy> $8\pi\kappa$, independent of its <radius> in this ideal model. Edge <energy> can favor closure, while fixed <area> and enclosed <volume> constrain the resulting shapes.
For a nearly flat sheet, its <small-slope elastic membrane energy> becomes
$$
F=\frac12\int[\kappa(\nabla^2h)^2+\sigma|\nabla h|^2]d^2x.
$$
With $h_{\mathbf q}=A^{-1}\int h(\mathbf x)e^{-i\mathbf q\cdot\mathbf x}d^2x$, the <equipartition theorem> gives the <membrane fluctuation spectrum under tension>:
$$
\boxed{\langle|h_{\mathbf q}|^2\rangle=\frac{k_BT}{A(\kappa q^4+\sigma q^2)}.}
$$
Bending suppresses short wavelengths, whereas positive tension suppresses the longest undulations. Their crossover is $q\sim\sqrt{\sigma/\kappa}$. Measuring the spectrum can separate these two contributions, provided finite image resolution, projected versus true <area>, and external confinement are accounted for. The drawn modes below are synthetic calculations, not an image of a sample.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2012/paper-1-membrane-spectrum.png]
{title=Membrane mode stiffness}
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