Interfaces, Surfaces and Membranes

Instructor: Yao Zhenwei
中文版：界面、表面与膜
Back to: Soft Matter Physics Course Notes

Core Picture

Interfaces cost free energy because molecules at an interface have a different environment from molecules in the bulk. Droplets, foams, emulsions, wetting, capillarity, and biological membrane shapes are controlled by surface energy, curvature, and bending energy.

Surface Tension

Surface tension gamma is free energy per unit area and also acts as a contracting force along the interface.

surface F=$F_s=\gamma A$

work=$dF=\gamma dA$

capillary=$l_c=\sqrt{\gamma/(\rho g)}$

At small scales, surface tension dominates gravity. At large scales, gravity flattens interfaces.

Young-Laplace and Wetting

A curved interface has a pressure jump across it. Larger curvature gives larger capillary pressure.

Laplace=$\Delta p=\gamma\left(\frac{1}{R_1}+\frac{1}{R_2}\right)$

sphere=$\Delta p=2\gamma/R$

droplet=smaller R gives larger pressure

The contact angle comes from force balance among three interfacial tensions.

Young=$\gamma_{SV}=\gamma_{SL}+\gamma_{LV}\cos\theta$

wetting=small theta means good wetting

Helfrich Membrane Energy

A lipid membrane is a two-dimensional fluid surface. Molecules can move in the membrane, but bending the whole membrane costs energy.

mean=$2H=\frac{1}{R_1}+\frac{1}{R_2}$

Gaussian=$K_G=1/(R_1R_2)$

Helfrich=$F=\int[\frac{\kappa}{2}(2H-C_0)^2+\bar\kappa K_G]dA$

kappa is bending rigidity, C_0 is spontaneous curvature, and kappa_bar controls the Gaussian curvature term.

Small-Slope Approximation and Fluctuations

Write the membrane as z=h(x,y). If the slope is small, curvature simplifies.

small slope=$|\nabla h|\ll1$

curvature=$2H\approx\nabla^2h$

bending=$F_b=\frac{\kappa}{2}\int(\nabla^2h)^2d^2r$

tension=$F_\gamma=\frac{\gamma}{2}\int|\nabla h|^2d^2r$

Fourier-mode fluctuations follow from equipartition.

mode=$F_q=\frac12(\kappa q^4+\gamma q^2)|h_q|^2$

fluctuation=$\langle|h_q|^2\rangle=\frac{k_BT}{\kappa q^4+\gamma q^2}$

Exam Cue

For interfaces, write F=gamma A. For curved pressure, write Young-Laplace. For wetting, write Young equation. For membrane shape, write Helfrich. For small fluctuations, use h(x,y) and Fourier modes.