Thermal membrane undulation

A thermal membrane undulation is a shape fluctuation excited by thermal energy. In a planar membrane, a Fourier mode has stiffness $k_c{q}^4+\sigma q^2$, so tension suppresses long-wavelength excess area while bending suppresses short wavelengths.

In a membrane-mechanics experiment, a narrow glass pipette applies a controlled pressure difference and draws a tongue of membrane into its opening. Imaging the aspirated length and vesicle geometry estimates the change in apparent membrane area. At low tension, much of that change can flatten thermal undulations; at higher tension it can measure actual area dilation. For ideal spherical caps with pipette radius $R_p$ and vesicle radius $R_v>R_p$, the Young–Laplace equation gives $\Delta p=2\sigma (1/R_p-1/R_v)$. Bending at the rim, adhesion and a membrane-associated cytoskeleton can require corrections to this simple geometric inference.

This coefficient multiplies the squared mean-curvature term $\frac{\kappa }{2}\int (2H-C_0{)}^{2}dA$ in the Helfrich membrane energy. It has units of energy, unlike a beam's flexural rigidity. For small height modes of a nearly flat membrane of area $A$, with Fourier normalization $h_q=A^{-1}\int h(\mathbf{x})e^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$, quadratic bending and tension energies give $⟨|h_q{|}^2⟩=k_{B}T/[A(\kappa q^4+\sigma q^2)]$ by the equipartition theorem. Thermal-flicker measurements can therefore separate short-wavelength bending from longer-wavelength tension effects.

For small-slope height fluctuations of a nearly planar membrane, $A$ is projected area, $\sigma$ is membrane tension, $\kappa$ is membrane bending modulus, and $q$ is a wavenumber. With a Fourier convention in which the mode amplitude $h_{\mathbf{q}}$ has units of length, the quadratic Helfrich membrane energy and the equipartition theorem give the displayed thermal variance. Bending dominates for $q\gg \sqrt{\sigma /\kappa }$, giving a $q^{-4}$ spectrum. Tension dominates at smaller $q$, giving $q^{-2}$. The corresponding membrane elastic length is $\sqrt{\kappa /\sigma }$ when length is measured as $1/q$; a wavelength convention multiplies it by $2\pi$.