The puddle that could dry in a second

Pressure is a rate of arrival counted the molecules of a gas striking a wall and found the gas law in the count. The gas that leaves is not the gas inside cut a hole in the wall and found that the same count, a quarter of the number density times the mean speed, is the rate at which gas escapes through it. And how far a molecule gets measured the distance a molecule of air travels between collisions, about seventy nanometres. This essay points the same count at the surface of a liquid, and finds that it sets a ceiling on how fast anything can evaporate — a ceiling so high that almost nothing on Earth comes near it, which turns out to be the more interesting fact.

The ceiling exists because of a balance. Put water in a closed jar and leave it. Molecules leave the surface, the vapour above it grows denser, molecules from the vapour strike the surface and are captured, and eventually the two rates are equal: the vapour has reached its saturation pressure, and the liquid neither gains nor loses. That equality is the whole of the argument. In equilibrium the rate of leaving equals the rate of arriving, and the rate of arriving is a kinetic-theory count that can be done on the back of an envelope.

Leaving as fast as arriving #

The molecules of a gas at pressure $p$ and temperature $T$ strike each square metre of any surface at a rate $p/\sqrt{2\pi m k T}$, where $m$ is the mass of one molecule; multiplied by $m$, that is a mass flux $p\sqrt{M/2\pi RT}$ for a molar mass $M$. At saturation, if every molecule striking the liquid is captured, the liquid must be emitting molecules at exactly the same rate, or the vapour above it would not stay saturated. This is detailed balance: in equilibrium each microscopic process runs as fast as its reverse. It is the same argument by which the glow that says nothing about the surface tied a body’s emission of light to its absorption — a surface that absorbs well must emit well, or it could not stay in equilibrium with the radiation around it — applied to molecules instead of photons.

The rate of emission is a property of the liquid surface and its temperature. It does not depend on whether any vapour happens to be present above it. So if the vapour is removed, the liquid keeps emitting at the same rate, and nothing comes back. The emission rate at saturation is therefore the most the liquid can lose by evaporation, at that temperature, into anything:

$$J_{\max} = \alpha\, p_{\text{sat}}(T)\,\sqrt{\frac{M}{2\pi R T}},$$

where $\alpha$, between zero and one, is the fraction of molecules striking the surface that stick. Heinrich Hertz wrote down the first version of this in 1882, measuring the evaporation of mercury into an evacuated vessel and finding it far below his estimate; Martin Knudsen in 1915 showed that with a truly clean mercury surface the rate matched the formula with $\alpha$ close to one, and that Hertz’s surfaces had been contaminated. The formula is now called the Hertz–Knudsen relation and the ceiling it gives the kinetic limit.

Ten thousand times below its ceiling #

Saturation pressure climbs steeply with temperature, as a boiling point is a pressure traced, and the ceiling climbs with it. For water at 20 °C the saturation pressure is 2.34 kilopascals and the ceiling is 2.54 kilograms per square metre per second. Water is a kilogram per litre, so that is a layer 2.5 millimetres deep every second: nine metres an hour, a lake drained in a working day. At the boiling point the ceiling is near a hundred kilograms per square metre per second. Nothing about a glass of water left on a table suggests any of this. A shallow pan outdoors in summer loses five or ten millimetres a day, which is about a hundred thousand times less than its ceiling.

The reason is the air. A molecule leaving the surface of a puddle does not fly off into space, although it leaves at the hundreds of metres a second that the speeds in a still room found for every molecule of a gas. It travels some tens of nanometres before it hits a molecule of air, and is as likely to be knocked back toward the water as onward. The air immediately above the surface fills with vapour until it is almost saturated, and once it is, molecules return to the liquid nearly as fast as they leave. Net evaporation goes on only as fast as the vapour can be carried away from that saturated layer, by diffusing through still air and then being swept off by convection or wind. It is the same bottleneck that set the pace in the ice that grows by stealing from the droplets, where an ice crystal in a cloud grew only as fast as vapour could diffuse to it, whatever its surface could have taken.

The two lower curves on the figure compute that, with the vapour diffusing across a layer of still air half a millimetre or five millimetres thick into air at half humidity. A few millimetres is about the thickness of the stagnant layer over a still puddle; a breeze thins it to a fraction of a millimetre, and the evaporation rises in proportion. Both curves rise with temperature at the same steep rate as the ceiling, because both are driven by the same saturation pressure, but they lie four to five decades below it. Every puddle, wet towel and lake on Earth evaporates at a rate set not by its liquid, but by its air.

Two resistances in a row #

The kinetic ceiling and the diffusion through air act in series, like two resistors carrying one current. The vapour first has to leave the surface, which it can do at most at the kinetic rate; then it has to cross the diffusion layer, at a rate proportional to the diffusion coefficient and inversely to the layer’s thickness. Written as resistances to the flow of vapour, they add.

The figure plots evaporation against the thickness of the layer the vapour must diffuse through. For very thin layers the rate sits on the ceiling, because diffusion removes the vapour faster than the surface can supply it. For thick layers it falls in inverse proportion to thickness, as plain diffusion would. The two meet, with each resistance equal, at a thickness of 0.17 micrometres in air at one atmosphere.

That number is not arbitrary. The diffusion coefficient of a gas is roughly a third of its mean free path times its mean molecular speed — the same kinetic argument that gave the viscosity that does not care how much gas there is. The kinetic flux, divided by the vapour’s density, is a speed equal to a quarter of the mean molecular speed. Their ratio, the crossover thickness, is therefore about four-thirds of a mean free path, and with real numbers for water vapour in air it comes to two and a half mean free paths. The kinetic ceiling binds only when vapour can get away from the surface within a few free flights; past that distance the collisions take over, and diffusion sets the pace.

At lower pressure both the mean free path and the diffusion coefficient grow in inverse proportion to pressure, and the crossover moves out with them: 1.7 micrometres at a tenth of an atmosphere. Evaporation into thin air is faster for the same reason the note too high for thin air gets lower — fewer collisions — but the ceiling stays where it was, because the liquid knows nothing about the gas above it. Only in a vacuum, or across a gap shorter than a mean free path, does the liquid itself set the rate.

The ratio of the gap to the mean free path is the Knudsen number, and it is the quantity that separates two worlds here as it does wherever molecules meet a boundary. At large Knudsen number the vapour flies off in straight lines and the kinetic count holds; at small Knudsen number the vapour is a continuum, moving by diffusion and flow. Ordinary evaporation sits at Knudsen numbers of a ten-thousandth or less.

Droplets small enough to feel it #

A droplet is the one place in ordinary air where the vapour’s escape path can be short enough. Vapour leaving a sphere diffuses outward into a volume that widens with distance, so the effective thickness of the diffusion layer around a droplet is its radius. A drop of drizzle a millimetre across has a diffusion layer of a millimetre; an aerosol particle a tenth of a micrometre across has one a tenth of a micrometre thick, comparable with the crossover.

The lifetime of an evaporating droplet in still, dry air shows the crossover as a change of slope. For large drops the diffusion resistance dominates, the flux per unit area falls as the radius grows, and the lifetime rises as the radius squared: a millimetre drop lasts about nineteen minutes and a ten-micrometre cloud droplet a tenth of a second. For drops below about a third of a micrometre the kinetic resistance dominates, the flux per unit area is pinned at the ceiling, and the lifetime rises only in proportion to the radius.

This is not a laboratory curiosity. Atmospheric aerosols, the particles on which clouds form, are mostly in this size range, and the rate at which water condenses on them — the mirror image of evaporation, governed by the same two resistances — decides how many of them become cloud droplets when air rises and cools. Models of cloud formation must therefore know the sticking fraction $\alpha$ for water, because in the kinetic regime the rate is directly proportional to it. A cloud with a low $\alpha$ forms more, smaller droplets than one with a high value, and reflects more sunlight. The same coefficient that Hertz could not pin down for mercury enters climate models through the brightness of clouds.

A surface that freezes itself #

Evaporation carries away energy: each kilogram of water that leaves takes its latent heat, about two and a half megajoules, with it. At the kinetic ceiling, water at 20 °C would need 6.2 megawatts per square metre of heat supplied to its surface to keep going — some six thousand times the full noon sunlight falling on the same square metre. Nothing supplies that for long, so water exposed to a vacuum is never at its ceiling for more than an instant. Its surface cools, its vapour pressure falls steeply with temperature, and the ceiling drops with it, until the evaporation the surface can afford matches the heat that reaches it.

The figure solves that balance for a steady heat supply, from a tenth of a watt to ten megawatts per square metre. The surface temperature climbs only slowly across eight decades of heat, because the vapour pressure rises so steeply with temperature that a modest warming multiplies the evaporation many times. Under the full, unreflected sunlight that reaches Earth’s distance from the Sun, ice in vacuum settles near −68 °C, subliming about half a gram per square metre per second. That is close to the measured temperatures of the active, sunlit surfaces of comets near the Sun, which are cooled by the same balance. If only one molecule in ten sticks, the surface has to run warmer to lose the same mass, and the dashed curve lies some fifteen to twenty degrees above.

This is why a beaker of water in a vacuum chamber boils violently, then crusts over with ice, and then sits there subliming slowly — a demonstration often presented as the triple point, which it passes through on the way down. It is also the principle of freeze-drying, where food or a vaccine is frozen and held under vacuum, and the ice sublimes away at a rate set by the heat supplied to it through the shelf. The process is deliberately slow; the ceiling would let it go millions of times faster, if the heat could be got in.

How a sticking fraction was argued over #

Whether the coefficient $\alpha$ for water is close to one or much smaller was argued for most of a century. Early experiments on evaporation from still surfaces gave values from about 0.01 to 0.04. Later ones gave values near one. The difficulty is the cooling just described: a surface evaporating anywhere near its ceiling chills itself, the temperature right at the surface is very hard to measure, and a surface a few degrees colder than assumed has a lower vapour pressure, which looks exactly like a lower $\alpha$. Many of the low values were surfaces cooler than their experimenters believed, or contaminated with traces of oil.

Experiments since the early 2000s have used liquid microjets — streams of water a few micrometres across, fired into vacuum, which evaporate so quickly from such a small volume that their cooling can be calculated from the measured rate. They gave values around 0.6, which are now widely used. Molecular simulations of a water surface find values close to one. Measurements of condensation onto growing cloud droplets in large cloud chambers are consistent with values above about 0.3. The disagreement has narrowed to a factor of two or so, but the number that controls the smallest droplets in every cloud is still not known as well as most quantities in this subject.

A lamp that boils away #

The ceiling’s most consequential application was the light bulb. A tungsten filament runs at temperatures where its vapour pressure, though tiny, is not zero, and it evaporates atom by atom, thinning until a weak spot overheats and the filament fails. In a vacuum bulb nothing stands between the filament and the glass; every atom that leaves the filament is lost, and the kinetic ceiling is the actual rate.

Irving Langmuir at General Electric turned this around in 1913. He measured how fast filaments lost weight in vacuum at known temperatures, used the Hertz–Knudsen formula backwards to read off tungsten’s vapour pressure, and so produced the first reliable values of the vapour pressure of a metal at three thousand kelvin, too low to measure in any other way. The figure uses a modern fit to the same quantity. The rate climbs fiercely with temperature, by a factor of about four hundred between 2400 and 2800 kelvin, which is why a vacuum lamp had to run cool and reddish to last.

Langmuir’s other result that year was the gas-filled lamp. Filling the bulb with an inert gas at about atmospheric pressure puts a diffusion layer around the filament, exactly as the air puts one over a puddle: an evaporating tungsten atom now collides within a fraction of a micrometre, and many are knocked straight back onto the filament. In the simple series model of the figure, a millimetre of argon cuts the loss about seventy-fold, and the same filament can run nearly three hundred kelvin hotter for the same life. Because a hotter filament radiates a much larger share of its output as visible light, the gas-filled lamp was markedly more efficient, and it became the standard incandescent lamp for the rest of the century. Halogen lamps later went further, adding a chemical cycle that carries tungsten from the glass back to the filament.

The surprising connection is that the reason a puddle dries slowly and the reason a light bulb lasts are the same reason. In both, a gas in the way turns a molecular flight that would never return into a random walk that usually does.

Where the picture stops #

The figures take $\alpha$ as one unless stated, and treat the kinetic and diffusive resistances as simply adding. That is accurate when the net evaporation is small compared with the one-way flux. Near the ceiling it is not: when evaporation is strong, the departing vapour itself moves away as a wind, the distribution of molecular velocities just above the surface is far from Maxwell’s, and a layer a few mean free paths thick — the Knudsen layer — has to be treated with the full kinetic theory of gases. The standard correction raises the maximum net flux somewhat for a given $\alpha$ and changes the details of the crossover without moving it by more than a factor of order one.

The diffusion figures assume still air and a fixed diffusion layer, when real evaporation is usually governed by convection, by the wind, and by the buoyancy of humid air, which is lighter than dry. The droplet figure holds the drop at the air’s temperature, when in fact an evaporating droplet cools toward the wet-bulb temperature and evaporates two or three times more slowly in dry air; and it ignores the higher vapour pressure over a strongly curved surface, which the pore that fills from dry air computed and which speeds the evaporation of the smallest droplets. The vacuum figure ignores radiation from the surface, which matters at the low-heat end. The filament figure uses a single effective diffusion coefficient for a layer whose temperature actually falls from three thousand kelvin at the filament to a few hundred at the glass.

The domain of the Hertz–Knudsen ceiling itself is any condensed phase below its critical point evaporating into a gas whose molecules are free over the distance that matters. It is not a ceiling on boiling, where vapour forms inside the liquid and the surface area is not fixed, nor on the ejection of material by a laser or a shock, where the surface is driven far out of equilibrium.

Still open: how close to the ceiling a surface can be run #

Engineers who need to remove heat from a small area — from a processor, a laser diode or a power transistor — would like to use evaporation, because the latent heat of water is enormous, and the ceiling says that the limit is megawatts per square metre. Ordinary boiling and evaporation reach a small fraction of that, held back by the vapour’s escape. Membranes with pores tens of nanometres across, from which vapour leaves into a gap shorter than a few mean free paths, have pushed evaporative heat fluxes far beyond what open surfaces reach, and the question of how close a practical device can come to the kinetic limit, and what then limits it — the supply of liquid through the pores, the sticking fraction, the Knudsen layer — is an active one. So is the sticking fraction of water itself, which a factor-of-two uncertainty leaves open in the smallest droplets of every cloud.

The habit worth carrying away is to ask, when something happens slowly, whether it is slow at its source or slow in its escape. A liquid emits molecules as fast as its saturated vapour would strike it, $p_{\text{sat}}\sqrt{M/2\pi RT}$ — 2.54 kilograms per square metre per second for water at 20 °C — and evaporation is tens of thousands of times slower only because the air returns almost everything within a fraction of a micrometre. The ceiling binds when the escape is shorter than a few mean free paths: in vacuum, where the surface freezes itself; in droplets below a third of a micrometre; and on a lamp filament, where a gas in the way buys its life.