The cereal that gathers at the edge of the bowl

How high water will climb balanced a curved surface pulling on a circumference against gravity pulling on a column, and found the capillary length, $\ell_c = \sqrt{\gamma/\rho g}$, the distance over which surface tension and gravity are equal: 2.7 millimetres for water. The angle a liquid makes with what it sits on found the contact angle that decides whether a surface lifts water or pushes it down. Later essays followed capillary forces up tubes, into pores, along corners a liquid never stops climbing, and into the walls a rinse pulls together, where the suction of a liquid bridge bends soft walls into each other. In every one the solid was fixed, or at most bent, and the liquid did the moving.

Turn it round. Let the solids float, free to drift across the surface, and the forces that lifted water up a tube now move the solids instead. Two floating objects near each other drift together or apart, and a floating object near the wall of its container drifts towards it or away from it, for reasons that have nothing to do with any force between the objects themselves. The surface between them has been bent, and each object is sitting on the slope the other made.

A slope to slide on #

A small object floating at a water surface must be held at the surface by something. If it is denser than water, its weight is partly carried by the surface, which is pulled down round it into a dimple, like a sheet with a weight on it. If it is lighter than water, buoyancy pushes it up harder than its weight pulls down, and the surface holds it back, pulled up round it into a little hill. Either way, the object bends the surface round itself, and the bending decays with distance over a few capillary lengths, because beyond that the surface’s weight flattens it.

Now put a second object nearby. It sits, not on a flat surface, but on the sloping flank of the first object’s meniscus. A light object, which is being pushed upward by buoyancy and held down by the surface, will move to where it can rise further: up the slope, towards the first object if that object’s meniscus is a hill. A heavy object will move down the slope, towards the first object if that object’s meniscus is a dimple. So two light objects, both raising the surface, climb towards each other; two heavy ones, both depressing it, slide into each other’s dimples; and a light one beside a heavy one moves apart, each going the wrong way on the other’s slope. Like menisci attract; unlike menisci repel.

The same rule can be read as energy. Two light floaters each lift a hill of liquid round themselves, and lifting liquid costs gravitational energy and stretches the surface. Brought together, their hills overlap and merge, and less liquid in total has to be lifted to hold both: the system’s energy falls as they approach, and a fall in energy with distance is a force. Two floaters with opposite menisci, one lifting and one depressing, have to bend the surface between them into an S, more surface and more displaced liquid than either alone, and the energy rises as they approach. The objects are not attracting each other; the surface is relaxing, and taking them with it.

Vella and Mahadevan called it the Cheerios effect, after the breakfast cereal whose floating rings clump together in a bowl of milk, in a paper of 2005 that worked out the forces in detail; the phenomenon itself had been noticed long before, and the bubble rafts that Bragg and Nye used in 1947 to model the arrangement of atoms in crystals were held together by exactly this attraction between bubbles.

The wall of the bowl #

The same rule decides what happens at the edge of the bowl. The bowl’s wall is a fixed solid that the milk either wets or does not, and it bends the surface beside it.

The exact shape of the surface beside a flat vertical wall can be found by integrating the balance between the surface’s curvature and its height, and it is drawn here. A wall the water wets completely, like clean glass, lifts the surface to $\sqrt{2}\,\ell_c$, 3.8 millimetres, and the meniscus falls back to the level of the rest of the surface within about eight millimetres. A wall the water does not wet pushes it down by the same rule. A cereal ring, light and raising the surface round itself, finds itself on the slope of a wetting wall’s meniscus and climbs it, which is why the last pieces in a bowl end up stuck to the side. A heavy floating object — a needle laid carefully on the surface, a paper clip — sits in its own dimple and slides away from a wetting wall, towards the middle. Fill a glass to the brim, so that its rim no longer lifts the surface but the surface bulges up over the rim, and the meniscus at the edge reverses: a floating cork that clung to the side of a half-full glass drifts to the middle of a full one.

Needles, striders and the weight a surface carries #

The heavy floaters are the more familiar case. A steel needle laid carefully on water floats although steel is eight times denser, because the surface, bent down into a trough along its length, holds it up, as the skin that is not a skin found for anything smaller than a few capillary lengths. Two such needles laid side by side a few millimetres apart slide together until they touch, each sliding down into the other’s trough, and a handful of pins scattered on water assemble into a raft as surely as cereal does. A floating object’s “charge” is simply the part of its weight, or of its excess buoyancy, that the surface has to carry, and the sign follows from which way the surface is pulled.

A water strider avoids the problem by being water-repellent all over and spreading its weight across long legs, each of which makes a shallow dimple; it is held up by the surface without breaking it. But the dimples are there, and a strider near the bank of a pond, whose meniscus slopes up to a wetting shore, finds itself on the downhill side of the bank’s slope — one reason the insects that need to leave the water have learned to climb it by bending the surface the other way.

A two-dimensional force, screened by gravity #

The force between two floaters can be computed, and its form is familiar from somewhere unexpected. Far from a small floater, where the surface is nearly flat, its height $h$ obeys

$$\nabla^2 h - \frac{h}{\ell_c^2} = 0,$$

the curvature of the surface balancing its height above the level. Without the second term this would be the equation of the electric potential in two dimensions, whose solution round a point charge is a logarithm and whose force between two charged lines falls as one over the distance. The second term, from gravity, cuts the logarithm off beyond the capillary length; the solution round a small floater is $K_0(r/\ell_c)$, a modified Bessel function, which behaves as a logarithm close in and decays exponentially far out. It is the same equation as the potential round a charge in a plasma, where the long-range force that does not reach found the Debye length cutting off the Coulomb field. Gravity screens a meniscus the way mobile charges screen an electric field.

The energy of a second floater sitting in that deformation goes as $K_0$ of the separation, and the force as its derivative, $K_1$. The figure plots it: close in, well inside a capillary length, the force falls as one over the distance, exactly the two-dimensional Coulomb law, and beyond a capillary length it dies exponentially. Each floater’s “charge” is the vertical force it hands to the surface, divided by $2\pi\gamma$ — positive for a floater the surface must hold down, negative for one it must hold up — and the force between two is proportional to the product of their charges, which is the rule of like and unlike in arithmetic form. For water the capillary length is 2.7 millimetres: two cereal pieces a centimetre apart barely feel each other, and at three millimetres they are pulled firmly together.

Rafts #

A crowd of floaters gathers in two stages, and the stages follow from the range of the force. A floater within a capillary length or two of a neighbour is pulled to it quickly. One far from everything feels almost nothing, because the exponential tail of the force is so weak, and waits until a cluster forms near it, whose combined meniscus reaches further. The simulation shows the pattern: small clusters first, scattered across the surface, then slower merging into one raft. It is the pattern of leaves gathering on a still pond, of the bubbles on a cup of coffee drifting into islands and then to the rim, and of froth gathering on the surface of a stirred pot.

Engineers have used it to assemble things. Millimetre-scale parts with chosen faces made to be wetted or not wetted, floated on water, drift together and lock into arrangements set by which faces bend the surface up and which down, a technique of self-assembly demonstrated in the 1990s with plastic tiles that formed ordered arrays by themselves. And some insects use it to climb out of water. The larvae of certain beetles and some water-walking insects reach the edge of a pond by deforming the surface with their bodies or legs so as to make a meniscus of the same sign as the bank’s, and are then drawn up the bank’s meniscus without moving a leg — passive climbing by the Cheerios effect.

Bubble rafts as crystals #

The best-known use of the attraction is older than its name. In 1947 Lawrence Bragg and John Nye, looking for a way to watch the arrangement of atoms in a metal, blew bubbles of identical size, a millimetre or so across, onto the surface of a soap solution through a fine nozzle. Each bubble raises the surface round itself, so every pair attracts; at contact, the bubbles resist being squeezed, because the small bubble blows up the big one — a curved film costs energy to flatten. Attraction at long range and repulsion at contact is the recipe for a crystal, and the bubbles packed into a hexagonal raft with all the defects of a real metal: grain boundaries where rafts of different orientation met, vacancies where a bubble was missing, dislocations that glided through the raft when it was sheared. Bragg and Nye filmed the dislocations moving, at a time when no microscope could see one in a metal, and the films were used to teach how metals bend for decades.

The analogy worked because the force between bubbles, like the force between atoms, has a range of a few diameters and a hard core, and because it acts in two dimensions, where a raft can be watched whole. The capillary length sets the range: bubbles much larger than it make menisci that reach only a fraction of their own size, and rafts of large bubbles barely hold together, which is why the method used bubbles of a millimetre or less.

A drop of soap scatters the raft #

Touch the middle of a floating raft with a sliver of soap and it flies apart. Soap lowers the surface tension where it lands, and the surface, pulled harder by the clean water round it than by the soapy water in the middle, flows outward and carries every floater with it — the effect the surface that pulls toward the stronger side found driving tears of wine. That flow, set by differences of tension, is far stronger than the attraction set by gravity’s slight bending of the surface, and it overrides it entirely. In a real bowl of milk, whose surface is crowded with natural surfactants from the milk’s proteins and fats, gradients of tension stir the floating cereal too, and the clumping is the slower effect that wins once the surface has settled.

Too small to clump #

The weight of a floater is what makes its meniscus, and weight scales steeply with size.

A floating sphere’s “charge” is set by the part of its buoyancy or weight that the surface must balance, which grows as the cube of its radius, and the energy with which two of them attract grows roughly as the sixth power. Two millimetre spheres of half water’s density hold each other with tens of billions of times the thermal energy, which is to say absolutely. Two spheres of a hundred micrometres still hold with a hundred thousand times it. At ten micrometres the binding has fallen below the thermal energy, and spheres that size, jostled by the random motion of the molecules around them, are pulled apart as fast as their menisci draw them together. The weight-driven Cheerios effect switches off near a radius of a dozen micrometres.

That would seem to make it irrelevant to colloids, the micrometre particles that are trapped in great numbers at oil–water and air–water interfaces in emulsions and foams. It does not make interfacial clumping irrelevant: such particles do clump, often strongly. But they do it by other means. Their contact lines are never perfectly smooth — the line where the interface meets a real particle undulates with the particle’s roughness and shape — and an undulating contact line bends the interface into a pattern with no net height but with lobes up and lobes down, which attract one another between particles as quadrupoles do. Ellipsoidal particles, whose contact lines are forced to undulate by their shape, attract much more strongly than spheres, end to end or side by side, and assemble into chains. The physics is the same equation; what changes is which term of the multipole expansion carries the force when the monopole, the weight, is negligible.

What the pictures cannot show #

The force and energy figures use the linear theory, valid when the surface’s slopes are small and the floaters are far apart compared with their own size; close to contact and for heavy floaters that deform the surface steeply, nonlinear effects change the numbers. The meniscus charge of a real floater depends on its shape, its density and the contact angle at its surface, which the size figure collapses into a single model of light spheres half submerged. The raft simulation treats the floaters as points with a short-range repulsion, ignores the hydrodynamic interactions between moving floaters, and uses a drag simply proportional to speed. The menisci in the first figure are exaggerated vertically to be visible. And no figure includes the thin layers of surfactant that cover most real surfaces, including milk’s, which change the surface tension locally and can drive floaters by gradients of tension instead.

Still open: shaping a crowd by bending its surface #

If the force between floaters is set by how each one bends the surface, then floaters can be designed to assemble into chosen patterns by designing their menisci — giving them shapes, edges or chemistry that make the surface rise here and fall there. Directing that assembly further, by imposing a curvature on the whole interface so that particles migrate to where the curvature suits their own distortions, has been demonstrated for small particles at curved oil–water interfaces. How rich a set of structures can be built this way, how to avoid the jammed, disordered clusters that form when attraction is strong and arrival is fast, and whether such interfacial assembly can be scaled up to make materials, are being explored.

The habit worth carrying away is to ask what an object does to its surroundings before asking what force it feels. Each floater bends the surface round itself into a hill or a dimple, and a second floater sitting on that bend slides along it: like menisci attract and unlike repel, with a force ∝ $K_1(\ell/\ell_c)$ — a two-dimensional Coulomb law screened by gravity at 2.7 mm — strong for cereal and gone, against thermal motion, below about twelve micrometres. The milk is the medium and the message.