The small bubble blows up the big one

Blow two soap bubbles on the ends of a tube, one small and one large, and open the tap between them. Almost everybody predicts that they will equalise. What actually happens is that the small one shrinks away to nothing while the large one grows, and it does so faster and faster as it goes.

The previous rung established that a liquid surface costs energy per unit area. This one is the immediate consequence of that for a curved surface, and it is a consequence with a great deal in it.

Where the pressure jump comes from #

Take a spherical drop of radius $R$ and imagine growing it by $\mathrm{d}R$. Two things change. The surface area grows by $8\pi R\,\mathrm{d}R$, costing $\gamma \cdot 8\pi R\,\mathrm{d}R$ of surface energy. The volume grows by $4\pi R^2 \mathrm{d}R$, and if there is a pressure difference $\Delta p$ across the surface, that expansion does $\Delta p \cdot 4\pi R^2\,\mathrm{d}R$ of work.

For the drop to be in equilibrium the two must balance:

$$\Delta p \cdot 4\pi R^2 = \gamma \cdot 8\pi R, \qquad \Delta p = \frac{2\gamma}{R}.$$

The inside is at higher pressure than the outside, by an amount inversely proportional to the radius. This is the Young–Laplace relation in its simplest form, and the derivation is worth noticing for what it does not use: no force balance across a hemisphere, no arrows, no cutting anything in half. It is an energy argument, which is why it generalises to surfaces that are not spheres by replacing $2/R$ with the sum of the two principal curvatures.

The factor of two nobody remembers #

A soap bubble is not a drop. It is a thin film of liquid with air on both sides, so it has two surfaces, each costing $\gamma$ per unit area. Repeat the derivation and the area term doubles:

$$\Delta p = \frac{4\gamma}{R}.$$

Drawn on logarithmic axes both laws are straight lines of slope minus one, exactly a factor of two apart at every radius — and the separation is not a property of soap. It is the number of surfaces: a droplet has one and a bubble has two, so a half-millimetre droplet carries 291 pascals where a soap bubble of the same size carries 582. Nothing about the film’s chemistry enters, which is why the factor is exactly two rather than approximately two.

Getting this wrong by a factor of two is the standard error in the subject, and it survives because both answers are the same shape. The way to keep it straight is not to memorise the two but to remember what is being counted: one interface or two.

Why there is no equilibrium #

Now the two-bubble experiment. Both are connected to the same air, so if they were in equilibrium their internal pressures would have to be equal, which by $\Delta p = 4\gamma/R$ requires equal radii. Any other configuration has a pressure difference, and air flows down it.

The direction is the surprising part and it follows immediately: the smaller bubble has the higher pressure. So the small one drives air into the large one. And as it shrinks, its radius falls, so its pressure rises, so the driving difference increases. The configuration is not merely out of equilibrium — it is unstable in the specific sense that departing from equality accelerates the departure.

That is a familiar shape once it is stated as a landscape. Equal radii is a maximum rather than a minimum, and a system balanced at a maximum leaves it under any disturbance at all.

The symmetric configuration sits at the top of the energy curve rather than at the bottom, so the arrangement everybody’s intuition proposes as the resting state is the one place the system cannot stay. Equal bubbles are an equilibrium and an unstable one, which is not a contradiction but a statement about curvature: the first derivative vanishes and the second has the wrong sign. The same distinction settles whether a floating hull rights itself, by exactly this reasoning about the second derivative at the equilibrium.

The total surface energy confirms it. Two bubbles of equal volume-sum have more surface than one bubble holding the whole lot, by the same cube-root-of-two arithmetic that makes coalescing drops release energy. Everything about the process runs downhill in surface energy, and the end state is one bubble.

What the number actually is #

It is worth putting sizes to the relation, because the $1/R$ makes it span an enormous range and the two ends behave like different subjects.

At the scale of things that can be seen, the pressure is trivial. A three-millimetre raindrop carries about 49 pascals — five thousandths of an atmosphere, and about the same as a five-millimetre head of water. Nothing about a raindrop is dominated by it.

At the scale of a millimetre it starts to matter: a soap bubble that size is at 291 pascals, which is why a bubble resists being deformed and springs back. At ten microns a droplet is at 15 kilopascals, a seventh of an atmosphere. At a hundred nanometres it is 1.5 megapascals — fifteen atmospheres, inside a droplet of water sitting in ordinary air.

That last number is the one to carry forward, because it says the small end of this law is not a small correction to anything. A nanoscale interface is a high-pressure environment, and the pressure is supplied by geometry rather than by any pump. Chemistry inside a small droplet happens at a pressure nothing external is applying.

What makes the Laplace pressure severe is not that it is large at ordinary sizes — it is not, 49 Pa across a three-millimetre drop — but that it has no floor. Every halving of the radius doubles it, without limit, all the way down to where the continuum description of a surface stops meaning anything at a few molecular diameters. An inverse first power is gentle over the range anybody measures and violent at the end nobody can reach.

The lung, which cannot work this way #

The instability above is a genuine problem for anything built out of small connected bubbles, and the human lung is exactly that: some three hundred million alveoli, connected through a shared airway, with radii from about 0.1 to 0.3 millimetres.

Run the arithmetic with water’s surface tension. An alveolus of 0.1 mm radius carries $2\gamma/R \approx 1{,}400$ pascals of excess pressure — about 1.4 per cent of an atmosphere, which is a substantial fraction of the pressure difference driving breathing — and is comparable with the whole hydrostatic variation across a standing human. Worse, the two-bubble instability applies: small alveoli should empty into large ones and the lung should collapse into a few big cavities. It does not.

The reason is that the fluid lining the alveoli is not water. It is a surfactant, and its distinguishing property is not merely a low surface tension but a surface tension that falls as the surface is compressed. As an alveolus shrinks, the surfactant molecules crowd together, $\gamma$ drops, and $2\gamma/R$ falls even though $R$ is falling too. The instability is not merely reduced; the sign of the effect is reversed.

Infants born prematurely often lack this surfactant, and the resulting condition — stiff lungs that collapse between breaths — was a leading cause of neonatal death until artificial surfactant became available in the 1980s. That treatment is one equation on this page, read as an engineering requirement.

Why a clean liquid will not boil #

The other consequence of $1/R$ is what happens as $R$ goes to zero, and it is severe.

To boil, a liquid must form a bubble of vapour. That bubble starts small, and at small radius the Laplace pressure is enormous: at one micron, $2\gamma/R$ for water is about 1.5 atmospheres; at a hundred nanometres it is fifteen. So the vapour inside a newly forming bubble would have to be at many atmospheres to push the surface outward — which requires a temperature far above the nominal boiling point.

The consequence is that a scrupulously clean liquid in a scrupulously smooth vessel can be heated well past its boiling point without boiling. Water can be superheated to around 280 °C under the right conditions. When such a liquid does finally nucleate, it does so explosively, because the whole superheat is released at once. Superheating in a microwave oven is a modest domestic version of this, and it is a real hazard.

What makes ordinary kettles boil at 100 °C is dirt and roughness. A crevice in the metal holds a pocket of trapped gas whose radius is set by the crevice rather than by the vapour, and a pocket a tenth of a millimetre across needs a negligible overpressure to grow. Boiling begins at nucleation sites, which is why bubbles in a pan come from particular spots on the base, in streams, rather than from the body of the liquid.

Everything in this essay is a rearrangement of one statement: surface costs energy per unit area. A curved surface can lower its energy by moving, and the pressure difference is the rate at which it would like to — so a bubble’s pressure, a nucleus’s barrier, and a suspension’s coarsening are three readings of one quantity with three different geometries in front of it.

A boiling curve says a liquid boils at a temperature and contains no statement whatever about how a bubble gets started. That is what the barrier above supplies, and it is why a clean liquid can sit well to the right of the plateau with nothing happening: a bubble small enough to appear by chance is a bubble the Laplace pressure crushes, and one large enough to survive is one nothing produces.

Pulled apart rather than heated #

The same barrier can be crossed from the other side. A liquid can be brought to vapour by lowering the pressure at constant temperature instead of raising the temperature at constant pressure, and where that happens fast — at a propeller tip, in a pump’s suction, in the wake of a rapidly moving surface — it is called cavitation.

Cavitation is destructive out of proportion to the energies involved, and the reason is the collapse rather than the growth. A vapour cavity that finds itself back in high pressure implodes, and the implosion is driven by the surrounding liquid’s inertia rather than limited by anything; local pressures reach thousands of atmospheres and temperatures thousands of kelvin over a region a few microns across. It erodes bronze, and it does so on ship propellers, pump impellers and turbine blades — components made of metal, worn away by water at room temperature, through a mechanism whose entire driving term is the curvature of a bubble a few microns across.

There is a related and stranger fact worth stating, because it is the extreme case of everything on this page: a liquid can be put under genuine negative pressure, meaning it is in tension rather than compression. Water in a sealed capillary has been held at tens of atmospheres of tension without cavitating. Tall trees exploit this — the water column in a redwood is under tension, pulled from above by evaporation rather than pushed from below by a pump, and the tension exceeds what any hydrostatic argument would permit for a pushed column.

What curvature actually means here #

The spherical case has one radius, and a general surface has two. The full relation is

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

with $R_1$ and $R_2$ the principal radii of curvature, counted positive when the surface curves away from the higher-pressure side. Three consequences follow that the sphere hides.

A cylinder is half a sphere’s jump. One principal radius is the cylinder’s, the other is infinite, so $\Delta p = \gamma/R$. That asymmetry is what makes a liquid thread unstable, and it is the next rung.

A saddle can have no jump at all. If the two curvatures are equal and opposite the sum is zero, and there is no pressure difference across the surface. Soap films spanning a wire frame, with air at the same pressure on both sides, are exactly this: every point is a saddle with equal and opposite curvatures. That is what a minimal surface is, expressed as a pressure statement rather than a geometric one.

Concave means the inside is at lower pressure. A meniscus curving down into a tube has its centre of curvature above the liquid, so the pressure just under the surface is below atmospheric — which is what pulls a column of water up a narrow tube and is the subject of the capillarity ladder.

The same law, sorting a suspension #

There is a consequence of $1/R$ that acts slowly and is responsible for a surprising amount of the world going stale, and it is the two-bubble result with the bubbles replaced by something solid.

Take a suspension of small crystals in a solution, of assorted sizes. The solubility of a particle depends on its curvature for exactly the reason a bubble’s pressure does: the surface costs energy, and a smaller particle carries more surface per unit of material, so it is slightly more soluble. So the solution is supersaturated with respect to the large particles and undersaturated with respect to the small ones. Material dissolves off the small and deposits on the large.

The result is Ostwald ripening: the small particles vanish, the large ones grow, and the mean size drifts upward for as long as the system is left alone. It is not a chemical change — the composition is fixed throughout — it is a redistribution driven entirely by curvature.

Ice cream going gritty is this. So is a foam’s bubbles growing coarse while the total gas is unchanged, and an emulsion’s droplets doing the same. Anything that must stay finely divided has to be defended against it, usually by a surfactant that makes the interface cheap enough for the driving difference to be negligible.

Nothing here reverses, and the reason is that every step of ripening lowers the total surface. There is no configuration of many small particles the system prefers to one large one at the same volume, so the direction is the one a count of arrangements always gives — and the process stops only when there is one particle left, or when something intervenes to stop it.

The instability, sold as an instrument #

The two-bubble result says that a bubble whose radius is falling is a bubble whose pressure is rising, and that no equilibrium exists between two of them. Turned round, that gives one of the standard ways of measuring a surface tension.

Push gas slowly through a capillary of radius $r$ submerged in the liquid. As gas enters, a bubble forms at the tip, and its radius of curvature starts very large — the surface is almost flat — and falls as the bubble grows, because the bubble is a spherical cap on a fixed circular mouth. So the pressure needed to push more gas in rises as the bubble grows, which is the opposite of the intuition and is the same inversion as the two-bubble experiment.

The radius reaches its minimum when the cap is a hemisphere, at which point it equals the capillary’s own radius and the pressure is at a maximum of $2\gamma/r$ above the hydrostatic head. Grow the bubble any further and the radius increases again, the required pressure falls, and the bubble runs away and detaches.

So the whole measurement is: record the peak pressure, subtract the depth term, and multiply by $r/2$. Nothing has to be seen, which is the method’s virtue — it works in an opaque melt, a slurry or a hot liquid metal, where an optical technique has nothing to look at. And varying the rate at which bubbles are produced varies the age of the surface being measured, from a few milliseconds to seconds, which turns the same apparatus into a way of watching a surfactant arrive at a fresh interface.

A bubble is a resonator, and it is what water sounds like #

A bubble in a liquid has a stiffness and a mass, so it has a natural frequency, and it is worth working out because the answer explains a sound everybody knows.

The stiffness comes from the gas: squeeze the bubble and the pressure rises, adiabatically, so the restoring force is set by $\kappa P_0$ with $\kappa$ the ratio of specific heats. The mass is not the gas, which is negligible, but the liquid around the bubble that has to be pushed out of the way as it expands. Assembling the two gives Minnaert’s result of 1933,

$$f_0 = \frac{1}{2\pi R}\sqrt{\frac{3\kappa P_0}{\rho}},$$

which for an air bubble in water is about 3.3 kilohertz divided by the radius in millimetres. A one-millimetre bubble rings at 3.3 kHz; a tenth-of-a-millimetre bubble at 33 kHz.

That is the sound of running water. A stream, a tap filling a sink and rain on a puddle are all audible because entrained bubbles are pinched off and ring at their own natural frequency, each one for a few milliseconds, and the pitch of the noise is a direct readout of the sizes being produced. Oceanographers count bubbles under breaking waves this way, by listening.

The surface tension appears in this as a correction rather than as the restoring force, because the Laplace pressure adds to the gas pressure inside. At ordinary sizes it is negligible: $2\gamma/R$ for a millimetre bubble is 145 pascals against an atmosphere. It takes over below about a micrometre, where $2\gamma/R$ exceeds atmospheric pressure and the bubble’s stiffness is being supplied by its own surface — which is the same crossover, at the same radius, as everything else on this page.

Where the model stops #

The interface is treated as a mathematical surface. At a radius of a few nanometres the surface is a substantial fraction of the drop, $\gamma$ itself becomes size-dependent, and the Tolman correction enters. Nucleation theory is built on the relation above and is known to be quantitatively unreliable for exactly this reason.

Equilibrium is assumed. A surface being created rapidly has not had time for surfactant to reach it, so the effective $\gamma$ is higher than the equilibrium value — which matters enormously in the lung and in every spraying process.

The film has uniform thickness. A real soap film drains under gravity, thins at the top and eventually ruptures. Nothing in $4\gamma/R$ knows about that, and the lifetime of a bubble is a drainage problem rather than a curvature one.

The surface tension is a constant. It is not, wherever a surfactant is present, and the lung is the case where that variation is the whole point rather than a correction. A relation written with $\gamma$ outside the bracket quietly assumes the interface does not care how much it has been compressed.

Gravity is neglected. The relation holds pointwise, and a large drop has different curvature at its top and bottom because the hydrostatic pressure differs. The shape that results is what a pendant-drop measurement fits, and it is only tractable numerically.

The ladder from here #

Later rungs on this anchor: the instability of a liquid thread, and the drop spacing it selects. Minimal surfaces, where the curvature sum is zero everywhere. Nucleation and the critical radius, which is the barrier above turned into a rate. Ostwald ripening, in which the same $1/R$ makes large particles in a suspension grow at the expense of small ones — the two-bubble result with the bubbles replaced by crystals. Foam coarsening, which is the same process again. And the surfactant story properly told, as molecules with one end that likes water and one that does not.

The neighbouring ladder is capillarity, where the pressure jump computed here meets a solid wall and decides how far water will climb.