Foam Drainage: How Bubbles Slowly Lose the Fight Against Gravity

A foam is mostly channels, not films

Look closely at a soap foam and the flat films between neighbouring bubbles are almost incidental — the liquid mostly lives in the thin curved channels that run along every edge where films meet. Joseph Plateau observed in the 19th century that these edges always meet in threes, at a fixed angle of exactly 120°, a purely geometric consequence of surface tension trying to minimise total film area at every junction. The channel this produces — a curved triangular tube running along the edge — is called a Plateau border, and it is where drainage physics actually happens.

Laplace pressure drives the flow

A curved liquid surface under tension sustains a pressure difference across it — the smaller the radius of curvature, the bigger the jump. A Plateau border's cross-section is sharply curved, so the liquid pressure just inside it is lower than in the flat, nearly uncurved films nearby:

ΔP = γ · (1/r1 + 1/r2)      Laplace pressure across a curved interface

γ  = surface tension of the soap solution
r1 = radius of curvature of the Plateau border cross-section
r2 = second principal radius (often much larger, along the border's length)

This suction pulls liquid out of the flat films and into the borders, which is why films thin from the middle outward even before gravity gets involved — capillary suction alone is enough to start draining a horizontal foam. Add gravity and the borders themselves drain downward, with liquid content decreasing steadily up the height of the foam column: a fresh foam is wet at the bottom and dry — and much more fragile — near the top.

The drainage equation

Treating the network of Plateau borders as a porous medium that liquid flows through under gravity and capillary suction gives a nonlinear diffusion-advection equation for the liquid fraction ε:

∂ε/∂t = ∂/∂z [ C·ε^2 · (∂ε/∂z) ] - ∂/∂z [ K·ε ]

first term: capillary-driven diffusion (borders pull liquid from wetter regions)
second term: gravity-driven advection (liquid falls under its own weight)
C, K = constants set by border geometry, surface tension and viscosity

Because the coefficients scale with powers of ε, the equation is genuinely nonlinear: dry regions (small ε) drain much more slowly than wet ones, since the Plateau border channels themselves shrink as they lose liquid, which increases their flow resistance. This self-limiting feedback is why a foam's drainage rate slows dramatically over time rather than emptying at a constant rate — the classic observed curve is fast initial drainage followed by a long, slow tail.

Disjoining pressure: what keeps a thin film from vanishing

If capillary suction pulled every last drop out of a film, it would rupture instantly. In practice, films thin down to a stable equilibrium of tens of nanometres and can persist there for a long time. What stops the drainage is disjoining pressure — a short-range repulsion between the two surfactant-coated interfaces of the film, arising mainly from electrostatic double-layer repulsion (both surfaces carry the same sign of charge from the adsorbed surfactant) and steric crowding of the surfactant molecules once the two interfaces get close enough to interact directly. As the film thins, disjoining pressure rises sharply and eventually balances the capillary suction pulling liquid toward the neighbouring Plateau border, pinning the film at a stable thickness — until evaporation, mechanical disturbance or coalescence pushes it thin enough to lose that balance and rupture.

Coarsening and the endgame

Drainage is only half the story of a foam's decline. Gas also diffuses between bubbles — small bubbles have higher internal Laplace pressure than large ones (the same ΔP = 2γ/r relationship, now applied to the whole bubble rather than a border), so gas slowly diffuses from small bubbles into large ones through the intervening film. This process, called coarsening or Ostwald ripening, shrinks small bubbles until they vanish and grows large ones, steadily reducing bubble count and total film area even in a foam that never loses a drop of liquid. Combine coarsening with drainage-thinned, disjoining-pressure-exhausted films, and the two effects compound: thinner films rupture more easily, and each rupture merges bubbles and accelerates further coarsening — the runaway collapse familiar from watching foam on a drink slowly disappear.

Frequently asked questions

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