A Leidenfrost drop floats on a bearing it feeds itself

Nothing but the shape of the gap and the gap that carries the most described a bearing. A film of liquid between a shaft and a pad holds a large load, because a viscous fluid in a thin gap must be pushed through it, and that takes a pressure that goes as the inverse cube of the gap. So it needs relative motion, a converging wedge and a supply of oil, or else two surfaces being pressed together. Take all three away and there would seem to be nothing left to carry a load. A levitator showed one way round this, with a surface in motion. There’s another, with nothing moving at all.

Flick some water onto a hot frying pan. If the pan is at about 130 °C, each drop sizzles and has gone in seconds. At 250 °C, each one skates about, lasts minutes and doesn’t seem to touch anything. Johann Leidenfrost described this in 1756, in a tract on the properties of water, and Boerhaave had noticed it earlier. Every cook has used it as a thermometer. The reason it’s a remarkable fact isn’t that the hot plate gives slower evaporation than the moderate one, although it does, but that the drop’s own vapour is carrying its weight, in a thin film, and the film is a real lubrication problem with a clean answer.

The balance #

A drop flattened by its own weight into a puddle of radius $R$ has a thickness of twice the capillary length, $2\ell_c$, about 5 millimetres in water, and so an excess pressure at its base of $\rho_lg\cdot2\ell_c$. That’s all the load is: about 47 pascals, spread evenly over the base. Below the puddle sits a film of steam of thickness $e$, at a mean temperature between the plate’s and the boiling point.

Heat conducts through it, at a rate $k\Delta T/e$ per unit area, and this evaporates the bottom of the drop. So there’s a vertical velocity of new vapour entering the film at the top, at $w=k\Delta T/(\rho_v Le)$. It has to leave by flowing sideways. Conservation of mass in a radial film with a uniform source gives a radial flux per unit width of $wr/2$. And a viscous thin film flows under a pressure gradient by Reynolds’s lubrication law, so

$$\frac{wr}{2}=-\frac{e^3}{12\mu_v}\frac{dp}{dr},\qquad \Delta p=\frac{3\mu_vwR^2}{e^3}.$$

Setting that pressure drop equal to the load gives the whole thing:

$$e^4=\frac{3\,\mu_v k\,\Delta T\,R^2}{\rho_vL\,\rho_l\,g\,h}.$$

Biance, Clanet and Quéré wrote it down in 2003, with measurements. The fourth power is unusual. It comes from the cube of the lubrication law and one more factor, from the evaporation’s supply falling as the film thickens. The film’s rule is thus the same one as a viscous bearing’s, with one extra power of $e$ because the supply of lubricant depends on how thick the film is.

That is the bearing’s whole trick. A normal bearing has its oil supplied from outside at some rate, and if the load rises, the gap narrows and the pressure rises, in the inverse cube. In this one the gap narrows and the supply rises as well, so the pressure rises by the inverse fourth power. So it’s very stiff. A tenfold increase in load changes $e$ by a factor of less than two. The film isn’t delicate at all, as its survival under a skating drop suggests: push a drop down and the film thins, evaporation speeds up, and the extra vapour pushes back.

A doubling of the plate’s excess temperature raises the film by $2^{1/4}$, or 19 per cent, if the steam’s properties are held fixed. With the steam’s actual viscosity, conductivity and density, which all vary with temperature in the film, it rises 38 per cent. The result is a thickness of about a tenth of a millimetre over the whole range of temperatures anyone is likely to meet, a thickness that’s small and very insensitive to everything. Measurements give about that: Biance measured them from the diffraction pattern that a laser makes passing through the gap, and more recent interferometry has confirmed the size. Burton and colleagues in 2012, using a laser under the drop, have shown the film’s detailed shape to be a shallow dome with a thin neck at the rim, not a flat one, a refinement on the flat film used here.

A film that would be gone in a millisecond #

How much of a surprise it is that this bearing holds can be measured against what the film would do on its own. Take away the evaporation, and the drop is simply squeezing a thin film of gas out from under itself, as a lubricated plate does when pressed against another. The exact theory a thin gap allows supplies Stefan’s formula of 1874 for that squeeze, which gives a time of order $\mu_vR^2/(\Delta p\,e^2)$. For a tenth of a millimetre of steam, under 47 pascals and across 5 millimetres, it’s about half a millisecond.

So the film has to be remade all the time, and the arithmetic bears that out. The vapour enters it at about seven centimetres a second, so the film’s whole contents are replaced in about one and a half milliseconds. Its steam is gone almost as soon as it’s made. The drop isn’t resting on a cushion of vapour, as it’s often pictured; it’s resting on a stream, one being renewed about seven hundred times a second, and that’s what makes the phenomenon steady rather than transient. At each instant the drop is falling onto its film at a speed matched by the film’s growth, and at no point has it got anywhere.

On the size of the drop #

The formula’s $R^2$ applies to a puddle. A drop much smaller than the capillary length is nearly spherical, and it rests on a flattened patch whose radius is set by the balance of its weight and its surface tension: $r_c\sim R^2/\ell_c$. That patch supports the weight, on a pressure that’s Laplace’s, so the same balance has different inputs and gives

$$e\propto R^{5/4},$$

which is the steeper line on the left of the first figure, computed from the same four-power law with those inputs. A half-millimetre drop has a film of about 9 micrometres, which is why very small drops seem to touch the plate and, in fact, are near to doing so. Mahadevan and Pomeau’s rolling droplets of 1999 use the same balance for a drop on an unwetted surface.

At the other end, there’s a limit. A puddle larger than about four capillary lengths in radius, about a centimetre, can’t sustain a flat film: the vapour collects in a pocket under the drop and rises through it as a bubble, a chimney of vapour in the centre. That’s a Rayleigh–Taylor instability, the one a ceiling’s drip has, with the heavy water on top of a light vapour. Its wavelength is $2\pi\sqrt3\,\ell_c$, about 2.7 centimetres, which sets the largest flat base. Larger puddles have holes in them, and they live by a different balance.

The lifetime’s surprise #

So far nothing distinguishes a hot plate from a very hot one. The practical consequence shows up in how long a drop lasts.

A boiling drop in contact takes its heat directly from the metal, through a very thin layer, by rapid nucleate boiling. At 30 degrees above boiling, the flux can be a megawatt per square metre. A 100-micrometre layer of steam, with a conductivity of 0.03 watts a metre per kelvin, passes $0.03\times200/10^{-4}$ or 60 kilowatts per square metre, at an excess nearly seven times as large. It’s a film insulator. That is what lets a drop survive so long, and the ratio of the two lifetimes can be a factor of a hundred. It’s the same physics that makes a film of steam on a quenched steel bar so disastrous for cooling, and it’s why the worst heat-transfer regime in nuclear reactor safety and in the heat treatment of metal is called “film boiling”. It’s the regime where the surface is too hot to be wetted, and where the critical heat flux has just been passed.

Between the branches, the drop’s life jumps at the Leidenfrost point. That temperature is not predicted by this calculation. It marks where the film can no longer be sustained, and for a real plate, it’s controlled by a good deal more than this balance: the roughness of the plate, its thermal conductivity and how much it cools locally when a drop touches, the liquid’s own properties and how the drop arrived. On a polished plate, water’s point is about 200 °C. On a rough plate it can be 300 or more, because the film must rise above the roughness to be a film, and rough asperities conduct and nucleate. Vakarelski and colleagues showed in 2012 that a water-repellent textured surface suppresses contact altogether, so that a sphere cooled in water keeps a stable vapour film all the way down until the plate is nearly at the boiling point. Thus the Leidenfrost point is a property of the surface as well as the liquid. That is what a chemical engineer designing a quench, or a fuel injection designer, spends time on.

The heat budget #

The drop on its film has one more source of heat, and it depends much more strongly on temperature.

Radiation from the plate reaches the drop across a film that’s transparent to it, and water is close to a perfect absorber in the infrared. Its share rises as $T^4$, and by a plate at 600 °C it’s a considerable part of the total. On a small drop it matters even more, since its contact patch shrinks faster than its projected area. That’s why the model’s smallest drops have their lifetimes set by radiation and by the diffusion of vapour off the top.

The shape of a drop’s life comes from this budget.

A puddle’s height is fixed at $2\ell_c$, and it shrinks by losing only base area, so its radius falls fairly slowly and evenly at the start, on a time determined by that height and the flux. Below the capillary length, it’s a sphere. Then its volume falls faster than its contact area, so the shrinkage first accelerates. As the sphere gets small, its contact patch falls rapidly, as $R^2$, while the volume goes as $R^3$. The last stages take a surprisingly long time for such a small drop, and that’s why a very small drop on a hot pan is the last to go. The result is a much slower end than a naive extrapolation suggests. It’s also what makes the tiny drops skittle about: each is hardly touching anything.

What a drop does when it moves #

The film offers almost no resistance to lateral motion. That’s the most visible property of the phenomenon, and it’s what makes the skating drop so fast. A film that thin and that low in viscosity transmits a shear stress of $\mu_vV/e$, which for a drop moving at ten centimetres a second over a film of 100 micrometres comes to about 0.012 pascals. For a puddle one centimetre across, it’s a force of about a micronewton, against a weight of about four millinewtons. The result is a sliding friction coefficient of a few ten-thousandths, better than a well-lubricated bearing’s, in a device with no oil.

Such low friction means that very small asymmetries in the film can propel it. Linke and his colleagues showed in 2006 that a Leidenfrost drop on a sawtooth surface climbs a slope, driven by the viscous drag of its own vapour flowing preferentially along the teeth. Lagubeau later showed a herringbone pattern doing it. There’s no pump in this system, and no moving part in it: the plate is at one temperature, and the drop travels. A Leidenfrost drop on a ratchet is a heat engine with a working fluid that happens to be the object being moved.

Compare it with the film that heats itself. There, a bearing’s oil warms by shear, and that reduces its viscosity. Here the film is cooled from the drop’s side and heated from the plate’s, and it’s the vapour’s supply that depends on the film’s own thickness, without any motion. Both are films that alter their own driving term. Between them they cover the two self-acting modes such a thin layer has.

What the picture cannot show #

Inertia in the film. The film’s Reynolds number at its edge is about 4.5 for a 5 mm puddle at this excess, so lubrication is only barely justified at the rim. The vapour speeds up as it leaves, and a modest inertial correction applies at the edge. It’s a matter of a few per cent in the thickness, because of the fourth root.

A flat base. The film’s shape is taken to be uniform. In reality the base is a shallow dome, and the film is thinner near its rim, with a pressure profile that isn’t the parabola of a uniform film. That changes the coefficient and not the exponents.

Everything over the top of the drop. Evaporation from the top surface by diffusion and the drop’s internal convection are neglected. The model’s lifetimes are consequently too long: a drop of a millimetre’s radius lives 336 seconds at this excess, in the model, and measurements give nearer a hundred seconds at a comparable temperature. So the lifetime figures should be read as being within a factor of three, and the ratio across the Leidenfrost point as being of the right order.

The approach to the Leidenfrost point. The film-boiling branch is drawn down to the lowest temperature at which it exists, and the contact branch up. In between, a real drop splutters in a transition regime, sometimes touching, and the sharp vertical drop shown between the branches is really a range, tens of degrees wide, in which contact is intermittent. Neither the contact flux’s simple cube nor the film’s formula applies there.

The convention #

Water at 100 °C has a density of 958, surface tension of 0.059 newtons a metre and a latent heat of 2.26 megajoules per kilogram, so its capillary length is 2.51 millimetres. Steam’s properties are taken at the film’s mean temperature, with viscosity going as the 0.8 power of temperature and density as its inverse. A puddle above the capillary length is modelled as a disc of height $2\ell_c$, a sphere below; the two regimes are joined at $R=\ell_c$ without any smoothing, which accounts for the kink in the first figure. Contact boiling uses a cube-law flux scaled to reach a megawatt per square metre at 30 K, capped at 1.2 megawatts. The plate’s emissivity is 0.96 with the drop treated as a black absorber.

How it was checked #

The formula’s powers are checked directly, by changing one input at a time and reading the ratio. The film’s absolute size agrees with the order measured by Biance and colleagues. The model refuses a drop of no radius, a plate at or below the boiling point and a lifetime from nothing.

Who found it, and when #

Boerhaave’s remark is from 1732, and Leidenfrost’s tract from 1756. Gottfried’s measurements of drop lifetimes date from 1966. Biance, Clanet and Quéré’s scaling is from 2003, Linke’s ratchet from 2006, Burton’s measurement of the film from 2012 and Vakarelski’s stabilised film from 2012.

Still open: the stiffness that makes it ring #

The film’s stiffness has a consequence nobody has yet put numbers to here. A soft drop on a stiff gas spring has a natural frequency, and Leidenfrost drops at millimetre sizes are observed to vibrate on their own: a star-shaped oscillation of the puddle’s rim at a frequency set by its size. The next calculation would linearise the same lubrication equation about the steady film, with a time-varying thickness and the evaporative source’s lag, as a spring in series with the drop’s mass, and find whether its damping can go negative, as the source’s lag behind the thickness suggests it might. That would turn the observed stars into a self-excited vibration, with a threshold in the plate’s temperature that could be checked.