The drop that will not merge

The skin that is not a skin explains surface tension as an energy: molecules at a surface have fewer neighbours than those inside and cost energy in proportion to the area they occupy, so a liquid pulls itself into the shape of least area. By that account a drop touching a pool of the same liquid should merge with it instantly. The merged state has less surface; the drop’s surface simply disappears into the pool’s, and the energy released pays for the flow. Yet anyone who has watched raindrops on a puddle, or coffee dripping into a cup, has seen drops skate across the surface or bounce before they vanish.

Osborne Reynolds noticed it in 1881 and wrote a short paper about drops floating on water. The explanation he suggested was the right one, and it has nothing to do with any force holding the drop up. Between the drop and the pool is a layer of air, and air takes time to leave a narrow gap. The question is never whether the two liquids want to merge; it is whether the air can get out of the way in time.

The film that thins too slowly #

A drop resting on a pool squeezes the air beneath it out sideways, through a gap that narrows as the air escapes. The flow is governed by the same law that lets an oiled bearing carry a load, the lubrication approximation that Reynolds himself had worked out a few years later for oiled bearings: in a thin gap, the pressure needed to push fluid out sideways grows as the inverse cube of the gap’s thickness.

The consequence is a thinning that slows down as it proceeds. Under a steady squeezing force the film’s thickness falls only as the inverse square root of time once it is thin: a film that halves its thickness in a millisecond takes four milliseconds to halve again, and sixteen for the next halving. The surfaces of drop and pool can merge only when the film is so thin — about a hundred nanometres — that the van der Waals attraction between the two liquids across it can pull them together. For a drop of silicone oil less than a millimetre across pressing on its pool with twice its weight, reaching that thickness takes more than half a second.

The film that goes black before it bursts describes the same race in a soap film, which thins under gravity and suction until it is a few tens of nanometres thick and the forces between its two faces decide whether it survives. Here the film is air between two liquids rather than liquid between two airs, and the timescale is set by how viscous the air is. The air’s viscosity is tiny, fifty times smaller than water’s, and still decisive, because the cube of the gap makes the flow so slow.

The bounce that outruns the air #

A drop that strikes a pool from a height does not rest on the film; it hits it, deforms, and is thrown back up by its own surface tension acting as a spring. The contact lasts a time set by the balance between the drop’s inertia and its surface tension, the capillary time $\sqrt{\rho R^3/\sigma}$, which measured on bouncing drops comes to about two and a half capillary times.

For a drop of radius 0.38 millimetres the contact time is four milliseconds, and the drainage time is 156 times longer. Across the whole range of drop sizes the gap between the two times is two to four orders of magnitude. A drop that bounces is never in contact with the pool for long enough for the air film to drain, and each time it leaves, the gap refills with air. The drop does not merge because it never waits.

The same structure explains a second familiar phenomenon. A drop of water flicked onto a very hot pan skates about on a cushion of its own vapour for a minute before evaporating — the Leidenfrost effect, which the boiling curve that turns back traces through the regimes of boiling. There the cushion is continuously regenerated by evaporation, and the drop can rest on it indefinitely. A drop on a cold pool has no such source of gas. Left at rest it sinks through its film and merges; kept bouncing, it is refreshed every few milliseconds.

When the film finally breaks #

A drop that does eventually reach the pool does not always vanish into it. Watched with a high-speed camera, a drop resting on a pool of the same liquid sits for a fraction of a second on its film, the film ruptures at one point, and the drop begins to empty into the pool — and then, part way through, its neck pinches off and a smaller drop, about half the size of the first, is left behind, sitting on a new film. After a pause it merges again, partly, leaving a still smaller drop. Charles and Mason described this coalescence cascade in 1960, and high-speed imaging since has followed it through five or six generations.

The cascade is driven by the pressure inside the drop. A drop’s surface tension compresses its interior by $2\sigma/R$, larger for a smaller drop, as the small bubble blows up the big one shows for two connected bubbles: once the film breaks, the higher pressure in the drop drives its liquid down into the pool. But the liquid moves as a column, which narrows as it drains, and a liquid column is unstable to pinching for the reason the thread that cannot stay a thread gives: a column longer than its circumference lowers its surface area by breaking into drops. The draining drop becomes a column, and the column pinches before the drop has emptied, leaving a daughter. The cascade ends when the drops are so small that viscosity damps the flow before the column can pinch, which for water is at a few tens of micrometres.

So a drop on a still pool merges in steps, each step a race between the film draining and the neck pinching, and each daughter drop again a race between its own film and its own bounce. In a rain shower on a pond, the tiny drops seen skittering across the surface for a moment are often the last generations of such cascades from larger drops.

What a bounce costs and how shaking repays it #

A drop bouncing on a still pool loses energy at each impact: to viscous flow inside the drop, to the waves it makes in the pool, and to the air squeezed out of the gap. The bounces that add up to a stop shows what happens to a ball that loses a fixed fraction of its speed at each bounce: the bounces get lower and closer together, and the series of their durations has a finite sum, so the ball comes to rest after a definite time. A drop on a still pool does the same, coming to rest on its film within a few bounces — after which the film drains and it merges.

A shaken pool repays the loss. At each impact the surface is moving up to meet the drop and returns more energy than the drop arrived with, and in a steady bouncing state the energy lost and the energy gained balance exactly. The drop’s rhythm settles where they do. That balance is what fixes the landing phase in the figures below: land too early in the cycle and the surface is still accelerating upward, giving a large kick and a long flight that carries the drop to a later landing; land too late and the opposite happens. The phase at which kick and loss balance is stable, and it is where the drop’s landings settle.

Shaking the pool #

The way to keep a drop bouncing is to shake the pool up and down. If the pool’s surface accelerates downward faster than gravity during part of each cycle, a drop resting on it is left behind and launched; it then lands again on a surface that is coming up to meet it, and is thrown again. Yves Couder and his colleagues in Paris showed in 2005 that drops of silicone oil on a bath shaken at around eighty hertz can be kept bouncing for days.

The figure follows a model of the drop as a mass on a damped spring — the spring standing for the restoring force of the drop’s deformed surface — that pushes only while the drop presses into the bath. The bath’s peak acceleration is measured in units of gravity, Γ. Below about Γ = 0.7 the drop never leaves the bath. A rigid ball would first be thrown at exactly Γ = 1, when the bath first accelerates downward faster than gravity; the springy drop leaves a little earlier, because the shaking excites its own bounce. Between about 0.7 and 2 it lands once a cycle at a fixed phase of the shaking. Higher, the rhythm changes.

The rhythm a shaken drop settles into #

Following the phase at which the drop lands, cycle after cycle, once it has settled, gives a picture of all its possible rhythms at once.

The single landing phase drifts as the shaking grows, and near Γ = 2 it splits into two: the drop alternates a high bounce and a low one, each landing at its own phase. Near Γ = 3 it lands only once every two cycles, sailing over the intermediate crest of the bath’s motion. Beyond, the landings scatter over bands of phase and no short rhythm repeats. The same cascade of doublings appears in a ball bouncing on a vibrating table, a classic model of chaos, and in the dripping tap that the map a dripping tap turns out to be reduces to a one-dimensional rule. Real drops show it too, with thresholds that depend on their size and viscosity and on details of their deformation that the spring model does not capture; what the model gets right is the first threshold, set almost entirely by gravity, and the existence of the cascade.

The once-in-two-cycles rhythm turns out to be the interesting one, because it is the rhythm at which the bath itself would respond to the drop.

The wave a bouncing drop rides #

Shaken hard enough, a bath of liquid becomes unstable by itself. Above a threshold acceleration its flat surface breaks into a pattern of standing waves oscillating at half the shaking frequency — the Faraday instability, whose mechanism the ripples that answer at half the shaking traces to parametric resonance, the same effect by which the swing that is pumped, not pushed is driven by standing up and squatting at twice its own frequency.

Below the threshold the flat surface is stable, but only just: a disturbance excites Faraday waves that die away, and the closer the shaking is to threshold, the more slowly they die. A drop bouncing once every two cycles strikes the bath in step with those waves, and each landing leaves a little circular patch of them, of wavelength 4.75 millimetres at eighty hertz, that lingers for many cycles. When the drop next lands, it lands on the slope of the waves its earlier landings left behind, and the slope gives it a small sideways kick. Above a further threshold of shaking the kicks reinforce one another, and the drop sets off across the bath at a steady speed of a few millimetres a second, guided by the wave it keeps renewing. Couder’s group called such drops walkers.

A particle guided by its own wave #

A walker is a particle accompanied by a wave that it creates and that steers it — a picture strikingly like the one Louis de Broglie proposed in the 1920s for electrons, the pilot wave, whose wavelength the wave a row of moving clocks leaves on an instant derives from relativity. The resemblance produced a decade of experiments. Walkers confined to a rotating bath settle into orbits of only certain radii, set by the wavelength, as a charged particle in a magnetic field occupies only certain Landau orbits. Walkers in a small corral wander in a way that, averaged over a long time, builds up a probability distribution with the pattern of the corral’s wave modes. Walkers approaching a submerged barrier sometimes cross it, unpredictably, in a classical analogue of tunnelling. And early experiments reported that walkers passing one at a time through one of two slits built up an interference pattern.

The last claim did not survive. Careful repetitions in 2015 and 2018 found that the walkers’ deflections through a slit depended on details of how they entered it and did not build up the reported pattern. The other effects are real, but they are not quantum mechanics. The walker’s wave lives in ordinary three-dimensional space, around one drop, whereas a quantum wavefunction of two particles lives in a six-dimensional space of both their positions together. The walker can always be watched without disturbing it. And no classical wave, however guided, can reproduce the correlations between measurements on two separated entangled particles, which the entangled pair a list of answers can imitate shows exceed what any local hidden mechanism allows. The walker is a beautiful demonstration that a particle with a memory, steered by its own past, can show statistics that look quantum. It is not a demonstration that quantum mechanics is such a thing.

What the models leave out #

The drainage figure uses Reynolds’s lubrication law for a flat disc pressed with a constant force, with the contact radius taken as half the drop’s radius and the force as twice its weight; a real film is dimpled, thinnest at a rim rather than at the centre, and drains faster at first and slower at the end, and the hundred-nanometre threshold for merging is a representative figure, not a constant. The contact time is the measured scaling for bouncing drops with a prefactor of 2.6. The bounce model replaces the drop’s deforming surface with a linear spring and a damper, which is enough to show the thresholds and the cascade but not the precise accelerations at which real drops change rhythm, which depend on drop size, viscosity and the shape of the deformation. The bath’s waves are not included in the bounce model at all, and so it does not walk. The Faraday wavelength assumes deep liquid; real baths are a few millimetres deep and the wavelength shifts slightly.

Still open: what decides when a drop finally merges #

Even a well-bouncing drop eventually merges, from a dust particle, a vibration of the room, or a slow loss of the air film’s integrity, and the life of a drop on a shaken bath is a few hours to days in practice. What exactly ends it — whether the film ruptures at its thinnest point under a fluctuation, or the drop’s own evaporation changes its size until it falls out of its bouncing window — has not been fully settled. The answer matters for applications of the same physics, in which drops are kept separate on purpose: in microfluidic devices that manipulate droplets as tiny reaction vessels, in fuel sprays where colliding droplets must or must not merge, and in raindrops in clouds, whose growth by collision depends on whether they bounce apart or coalesce.

The habit worth carrying away is to ask whether a process is forbidden or merely slow. Two drops of the same liquid lower their surface energy by merging and nothing pushes them apart — but the air between them must drain through a gap whose resistance grows as the cube of its narrowness, which takes 0.65 s for a drop that bounces off in 4 ms, so a drop kept bouncing never merges at all. A liquid that will not mix with itself is obeying the clock of the air, not a law of its own.