The wet plates that hold on when pulled

A glass of water set down on a wet bar does not stay where it was put. A nudge sends it sliding as if on ice, and a slight slope makes it creep away on its own. It is floating: between its flat base and the bar is a film of water a few tens of micrometres thick, too thin to see and too thick to have been squeezed out by the glass’s weight. The same film works the other way when the glass is lifted. A wet glass slide pressed flat onto another is surprisingly hard to pull straight apart, and two panes of window glass stacked wet can be impossible to separate by pulling, though they slide apart easily sideways.

Josef Stefan measured the second effect in 1874 and found its law; Osborne Reynolds put it in the general theory of lubrication in 1886. The law has an inverse cube in it, and the cube is the whole story — it is why the glass floats, why two wet plates resist parting, and why there is no such thing as squeezing a liquid out of a gap completely.

Liquid that has to leave through a narrowing slot #

Two flat discs of radius $R$, a gap $h$ between them filled with liquid, closing at speed $-\dot h$. Every second, the liquid inside radius $r$ loses volume $\pi r^2(-\dot h)$, and that volume must flow outwards through the slot of height $h$ at radius $r$. Flow through a slot is the flat cousin of the flow through a pipe that the fourth power in a pipe followed: the liquid is stationary against both walls and fastest midway between, and the flow through a slot of height $h$ per unit width is $h^3/12\mu$ times the pressure gradient. The cube appears for the same reason the fourth power appeared in a pipe: narrowing the slot both removes room for flow and slows what is left, because the walls’ grip, momentum going sideways from the walls into the liquid, reaches every part of a thinner layer.

Putting the two together gives the pressure gradient at radius $r$ as $6\mu r(-\dot h)/h^3$, and integrating it from the rim, where the pressure is the surroundings’, gives a parabola:

$$p(r) = \frac{3\mu(-\dot h)}{h^3}\left(R^2 - r^2\right).$$

The pressure is highest in the middle, because the liquid there has furthest to go, and every bit of liquid leaving the centre has to pass through the whole slot on its way out. Adding the pressure over the disc gives the force the liquid exerts:

$$F = \frac{3\pi\mu R^4(-\dot h)}{2h^3}.$$

The force goes as the fourth power of the disc’s radius and the inverse cube of the gap. A disc twice as wide pushes against sixteen times the resistance; a gap half as thick resists eight times as hard.

A flat base that floats #

Pressed with a steady force, two flat plates close their gap at whatever speed makes the liquid’s resistance equal to the force, and since the resistance at a given speed rises as $1/h^3$, the closing slows sharply as the gap narrows — so sharply that, by the classical result of lubrication theory, the gap shrinks only as the inverse square root of the time and never reaches zero. What matters here is what that does to things put down on wet surfaces.

Pressed with five newtons — about the weight of a full half-litre glass — a ten-centimetre base on water reaches ten micrometres of film in half a minute and one micrometre in fifty minutes, each tenfold narrowing taking a hundred times as long as the one before. A glass put down a minute ago is riding on ten micrometres of water, which is a fluid bearing: it supports the glass’s weight on pressure and resists sliding only through the shear of a layer ten micrometres thick, which is very little. That is why a wet glass slides on a bar at a nudge, and why a slight slope sets it creeping. With oil between, every time is ninety times longer for the same gap; with honey, ten thousand times, and a drop of honey squashed between two glass slides is still a measurable film a year later.

The plate that holds itself down #

Reverse the motion and the parabola turns over. Pulling the discs apart at speed $v$ draws liquid into the gap from the rim, and drawing it in needs a pressure at the centre below the surroundings’ by the same amount. The liquid sucks the discs together with the same force, $3\pi\mu R^4v/2h^3$.

For two wet discs ten centimetres across, ten micrometres apart, opened at a micrometre a second, the suction is 29.5 newtons — three times the weight of a litre of water, from a film that weighs less than a tenth of a gram. Opened a hundred times faster, the law demands 2.9 kilonewtons. That cannot happen: the pressure at the centre would have to fall more than seven atmospheres below the surroundings, six below zero absolute, and long before that the water cavitates — bubbles of vapour, or of dissolved air, open in the middle of the film and the suction collapses. The force is capped near the point where the pressure at the centre reaches zero absolute, where the suction is the atmosphere pushing on the outside of the discs with nothing pushing back in the middle.

The cap is half the atmosphere’s push on the disc’s area, not all of it, because the pressure is a parabola: when the centre reaches zero absolute, the pressure is still atmospheric at the rim, and the average suction over the disc is half the peak. It is 398 newtons for a ten-centimetre disc, more than forty kilograms’ weight. Pull faster and the force cannot rise; it sits at the cap until the gap has widened enough for the viscous suction to fall below it. Water can in principle sustain tension — the column that is pulled, not pushed found sap held at minus tens of atmospheres in a tree — and the height a siphon cannot pass found the same water breaking in a siphon at the first nucleus available. A film between glass plates has dissolved air and microscopic dust and breaks near zero, as a siphon does.

This is the force that makes two wet panes of glass hard to separate by pulling. It is not surface tension. The meniscus at the rim of a wet joint does pull the plates together, by a capillary pressure of about the surface tension over the gap — the force the walls a rinse pulls together found strong enough to collapse microscopic structures on drying — and for a ten-micrometre water film that is a few kilopascals spread over the meniscus. But the large, rate-dependent resistance that makes the panes stick when pulled briskly and part easily when slid or peeled is viscous, and it has the signature of viscosity: it depends on how fast they are pulled, and on nothing at all once they are still.

The same law with a ball, and with a slide #

A ball pressed onto a wet floor meets the same difficulty in a far milder form, and the difference is the shape. The drag a distant wall decides found that a sphere moving slowly through a liquid has a drag that depends on how far away the walls are; brought towards a wall head-on, its drag grows without bound as the gap closes. The liquid in the narrow region between the sphere’s nearest point and the wall has to escape through a slot that is thin only over a small patch, of radius about $\sqrt{ah}$ for a sphere of radius $a$ at gap $h$, and putting that patch into the disc’s law gives a force that grows as $a^2/h$ rather than as $1/h^3$. The sphere’s resistance rises more gently than the disc’s, because its curvature keeps the thin region small — but it still diverges, and a sphere settling under its weight onto a wall in a viscous liquid approaches it exponentially slowly rather than as a power, and also never arrives. A ball bearing dropped into a jar of glycerine settles onto the bottom over minutes, closing the last micrometres far more slowly than it fell the centimetres before them.

The ratio between the two is $R^2/4h^2$, and it is enormous at small gaps: at a micrometre, a flat disc ten centimetres across resists six hundred million times as hard as a ball of the same radius. That is the whole reason a ball settles onto a wet floor while a flat plate floats on it, and why every design that must touch down through a liquid — a tyre’s tread, a frog’s toe pad, the foot of a beetle that walks on wet leaves — is cut into small patches with channels between: each patch behaves like a small flat disc, whose resistance falls as the fourth power of its size, and the channels give the liquid somewhere near to go.

Reynolds’s general theory of lubrication, of which the squeeze film is one case, adds sliding. If one surface slides over the other while the gap narrows in the direction of sliding, the liquid dragged into the converging gap piles up and raises the pressure, exactly as the squeezed liquid did, without either surface approaching the other. That converging wedge is how every oil-lubricated journal bearing in an engine floats its shaft: the shaft sits slightly off-centre in its bearing, the gap converges on one side, and the rotation drags oil into it hard enough to hold the load on a film a few micrometres thick. The squeeze film is the same equation with the sliding switched off and the approach switched on, and in a bearing under a pulsing load — a connecting rod’s big end, carrying a force that reverses twice a revolution — the two act together: when the wedge fails momentarily, the inverse cube of the squeeze film buys the time until it returns. The oil that is a glass for a quarter of a millisecond followed that film into the gear and rolling-element contacts where the pressure is so high that the oil’s viscosity rises a hundred-millionfold, which is how a film survives where the squeeze alone would not hold it.

Why size matters so much #

The fourth power of the radius makes the effect enormously sensitive to size, and in a way that favours the large. A plate pressed with a given pressure — a load proportional to its area — closes a film in a time that grows as the square of its radius: the load grows as $R^2$, the resistance as $R^4$.

A small foot touches down: a water film under a millimetre of contact clears in milliseconds. A large flat base floats: under a sheet of glass a metre across it takes two hours to reach ten micrometres. That asymmetry runs through biology and engineering. A tyre’s tread is cut into narrow blocks with grooves between them so that no part of the contact is wide enough to float on the water it meets at speed — aquaplaning is a squeeze film that does not clear in the time a tyre’s contact patch spends on the road. A tree frog’s toe pad is divided into hexagonal cells separated by channels, which drain the film under each cell quickly and let the pad touch down, while the meniscus and the remaining thin film give it a hold that resists sliding.

The same law runs the other way in joints. The cartilage surfaces of a knee are pressed together hundreds of times a day under several times body weight, and the synovial fluid between them is squeezed out on every step. The cube buys time: under a brief load the film cannot get much thinner before the load is lifted, and the surfaces, at least in the part of the cycle where the squeeze film carries the load, are kept apart by liquid that has nowhere quick to go.

An air film that damps a machine #

Air is a fluid too, with a viscosity fifty times less than water’s — and, as the viscosity that does not care how much gas there is found, almost the same at any pressure down to a fraction of an atmosphere. In a gap of a few micrometres, the inverse cube makes it formidable.

The accelerometers in a phone are silicon plates a fraction of a millimetre across, a few micrometres thick, hung on silicon springs, with fixed electrodes a few micrometres away that sense the plate’s motion by capacitance. Air in the gap between plate and electrode is squeezed every time the plate moves, and the force it exerts is the squeeze-film force, proportional to the plate’s speed. It is a damper whose strength is set by the gap.

A plate 0.6 millimetres across on a 2-kilohertz spring is critically damped at a gap of 24 micrometres, and at a gap of 2 micrometres it would be overdamped by a factor of nearly two thousand — it would creep back from a disturbance instead of responding to it. The quality factor, which the width that is a lifetime related to how long a resonator rings, falls a thousandfold for every tenfold narrowing of the gap. Designers use that sensitivity: an accelerometer must be close to critically damped, to settle quickly without ringing, and its damping is set by etching the gap to the right width and by cutting holes through the plate, which give the air a short path out and replace one large squeeze film with many small ones. A gyroscope or a resonant sensor, which must ring for as long as possible, is sealed in vacuum instead, because no amount of geometry makes a micrometre air gap gentle — the noise a high Q moves out of the way found why such devices want their resonance as sharp as it can be.

Where the law stops #

Every figure assumes a liquid that sticks to both surfaces, flows as a Newtonian fluid at low Reynolds number, and fills a gap whose surfaces are perfectly flat and parallel. Each assumption fails somewhere near the bottom of the curves.

Real surfaces are rough. Polished glass has bumps of a few nanometres, ground metal of a micrometre, and when the film has thinned to the height of the bumps the plates touch at their high points and the liquid in the valleys has a network of channels to escape through. The squeeze then ends in contact at a finite time, set by the roughness. Two gauge blocks — steel blocks lapped flat to a few tens of nanometres — can be wrung together until they hold each other’s weight, and the film between them is a few nanometres thick, where the liquid is no longer a continuum and molecular attraction does part of the work.

Real liquids are not always Newtonian. In a film a few molecules thick, liquids form layers against the walls and resist being squeezed in steps rather than smoothly, as the force measured between mica sheets in surface-force apparatus shows. And air in micrometre gaps begins to slip at the walls when the gap is a few mean free paths — at atmospheric pressure the mean free path is 68 nanometres — which reduces the damping below the formula in the narrowest MEMS gaps and at low pressure.

The domain of the drawings is a Newtonian liquid, flat parallel discs, gaps from a micrometre to a millimetre, and closing speeds slow enough that the liquid’s inertia is negligible. Within it, the law has been confirmed to the precision of the measurements since Stefan’s.

Still open: how a film finally breaks #

What happens at the very end — how a thinning film between two surfaces actually ruptures and lets them touch — is not fully settled, because it depends on forces that act only at the last few nanometres and on the shape the film takes as it thins. Squeezed films often do not stay flat; the pressure peak in the middle can elastically dent the surfaces, trapping a dimple of liquid at the centre while the rim thins first, an effect seen in interferometry of drops approaching a wall and of bubbles approaching each other. When a drop approaches a surface, whether it bounces, coalesces or rests on a trapped film of air for seconds depends on a competition between this squeeze-film resistance and the forces that break thin films, and drop impact, coalescence in emulsions and the wetting of a surface by a falling drop all turn on it. Which molecular force wins at the last nanometre, and how the dimple and the roughness steer the rupture, are questions answered case by case.

The law above the last micrometre is exact. Liquid squeezed from a gap $h$ between discs of radius $R$ resists with $3\pi\mu R^4\dot h/2h^3$, so under a steady load the gap falls as $1/\sqrt t$ and never closes — ten centimetres of glass on water reach one micrometre in fifty minutes — while pulling them apart draws a suction capped only by cavitation at half the atmosphere’s push, 398 N; and a micrometre of air critically damps a machine a millimetre wide. Two flat things meeting through a liquid are separated by a cube, and the cube has no last step.