The film whose thinnest point sets the flow

The fountain a lamp can drive mentioned in passing the most photographed thing superfluid helium does: a beaker of it, lifted out of the bath, drains itself, the liquid creeping up the inside wall, over the rim and dripping from the bottom. That essay was about the two fluids and the heat they carry, and it left the film with a sentence: it flows at a speed set by gravity and by the critical velocity rather than by drag. The speed below which nothing can be made then found where a critical velocity comes from — the speed above which a moving superfluid can shed energy into excitations, and so the speed below which it cannot be slowed at all.

Put the two together and the film turns out to be a stranger flow than “frictionless liquid running uphill” suggests. It is a flow whose rate is fixed not by the force driving it but by a speed limit, and the limit bites at one place only. Everything about how the beaker empties follows from where that place is.

A film every liquid makes #

Liquid helium wets everything. Its atoms attract one another so weakly — helium boils at 4.2 kelvin — that almost any solid attracts them more strongly than they attract each other, and a solid surface above a pool of liquid helium is coated with a film of it. The film is held by the van der Waals attraction between the wall and the helium, the weak pull between induced dipoles that the attraction that needs no charge found neutral atoms exert on one another, summed over the wall’s atoms; it falls as the cube of the distance from the wall.

A film at height $h$ above the bath is in equilibrium with the bath when lifting a little more helium to that height costs as much as the wall’s pull gives back. Gravity costs $gh$ per unit mass; the wall’s pull at the film’s surface is $\alpha/d^3$ per unit mass, for a film of thickness $d$. Setting them equal:

$$d = \left(\frac{\alpha}{gh}\right)^{1/3}.$$

The constant $\alpha$ depends on the wall’s material, and measured films on glass are about thirty nanometres thick a centimetre above the bath. The thickness has been measured by the change in polarisation of light reflected from the coated wall, and by weighing the film on a vibrating quartz crystal whose frequency falls as helium condenses on it, both of which see a few atomic layers clearly. The film thins only slowly with height, as the cube root, so it is still a few nanometres thick a metre up: every surface inside a helium cryostat that is cold enough is wet.

None of this has anything to do with superfluidity. Ordinary liquid helium above 2.177 kelvin coats the wall in exactly the same way and to exactly the same thickness. The same balance of a surface force against a pressure holds up the film that goes black before it bursts in a soap bubble, and water forms such films too, a few molecules thick, on any surface above it in saturated air. The difference below the λ point is not that the film exists. It is that the film can move.

A flow that does not care how hard it is pushed #

An ordinary liquid film thirty nanometres thick cannot flow along a wall in any useful amount. The no-slip condition holds the liquid at the wall still, and viscous drag across so thin a layer is enormous; the film is effectively frozen in place, and it stops wherever it has climbed. A superfluid film has no such drag on its superfluid part, which slides along the wall freely. So once the film reaches the rim of a beaker and continues down the outside to a bath at a lower level, there is a continuous path of frictionless liquid from the higher surface to the lower, and liquid flows along it.

How fast? For a frictionless flow the honest answer is: as fast as anything allows, and what allows it is the critical velocity. Below some speed the superfluid in the film cannot lose energy to anything; above it, it can, by creating excitations or vortices, and the excess flow is dissipated. The flow therefore settles at the critical speed, and the volume carried per second is that speed times the film’s thickness times the superfluid fraction — independent of the difference of levels that drives it.

John Daunt and Kurt Mendelssohn measured it in Oxford in 1938 and 1939, timing the levels in small beakers raised above or lowered into a bath. The transfer rate, about $7.5 \times 10^{-5}$ cubic centimetres per second for every centimetre of rim near 1.4 kelvin, did not depend on how far apart the levels were, or on how high the rim was within a few centimetres. A beaker drained at a constant rate, and its level reached the bath’s in a straight line rather than in the slowing curve of any drain driven by pressure. Bernard Rollin and Francis Simon had proposed the film in 1939 to explain why helium seemed to escape from containers and why heat leaked along walls faster than conduction allowed; Kamerlingh Onnes had seen levels in nested vessels equalise in 1922 and put it down to evaporation.

The straight line is the signature. A siphon, or water draining through a pipe, slows as the head shrinks because the flow is proportional to the push. The film’s flow does not respond to the push at all, within wide limits, because it is not limited by friction but by a speed beyond which the superfluid stops being superfluid.

A million times more than a viscous film #

How different the frictionless flow is can be put in a single comparison. If the same film, thirty nanometres thick, were an ordinary liquid with the viscosity of helium just above the λ point, gravity pulling it down a vertical wall would drive it at a rate set by its viscosity and by the cube of its thickness, as the fourth power in a pipe found for a liquid in a tube: a film’s flow per unit width is $gd^3/3\nu$. For helium’s kinematic viscosity, about $2 \times 10^{-8}$ square metres a second, that is a few times $10^{-11}$ cubic centimetres a second per centimetre. The superfluid film carries $7.5 \times 10^{-5}$ — nearly two million times more.

The comparison also shows why the superfluid flow is not simply “a viscous flow with the viscosity set to zero”. Set the viscosity to zero in $gd^3/3\nu$ and the rate is infinite: with nothing to resist it, gravity would accelerate the film without limit. A real superfluid film accelerates until it meets the critical speed and then stops accelerating, because beyond that speed the energy gravity feeds in goes into making vortices rather than into flow. The film’s rate is finite because superfluidity is not perfect at any speed, only below one.

That is the general lesson the film teaches about superflow. The liquid that will not slow down found that a superfluid set flowing in a ring keeps flowing for as long as anyone has watched. The film shows the converse: a superfluid that is pushed cannot be made to go faster than its critical speed, however hard the push. A flow with no friction below a speed and copious dissipation above it is neither of the textbook idealisations, and it behaves like a current through a device that passes any amount up to a fixed limit and no more.

Switched off at the λ point #

Only the superfluid part of the film moves, since the normal part is held to the wall by its viscosity, and the superfluid fraction falls from nearly one at a kelvin to zero at the λ point. The transfer rate follows it down. At 2.1 kelvin, with less than a quarter of the liquid superfluid, the film carries a fifth as much as at low temperature, and at 2.177 kelvin it stops. The film is still there, as thick as before, and the beaker still holds its liquid indefinitely.

That switch-off is one of the cleanest demonstrations that the film’s flow is superfluidity and not some surface peculiarity. Daunt and Mendelssohn’s temperature dependence tracked the superfluid fraction that the two-fluid model had been built to explain from quite different measurements — the heat that arrives as a wave and the fountain effect — and fitted it.

One place decides #

The film’s thickness varies along its path, from tens of nanometres near the liquid surfaces to the thinnest part at the rim, the highest point. The same volume must pass every point of the film each second — liquid cannot pile up anywhere along a steady flow — so where the film is thin it must flow fast.

So the film runs fastest at the rim, and only at the rim does it reach the critical speed. Everywhere else it flows below the limit, without dissipation of any kind. The rim is a bottleneck in the strict sense: the whole flow is fixed by what the film can carry there, and the rest of the film simply passes on what it is given. The figure’s numbers are the ones the measured rate implies — a critical speed of about 27 centimetres a second in a film thirty nanometres thick.

The bottleneck explains the experiments’ details. The rate depends on the perimeter of the rim, not on the beaker’s area or the depth of liquid, because a longer rim is a wider bottleneck. It depends only weakly on the height of the rim above the liquids, through the thinning of the film there, which in this simple picture goes as the inverse cube root of the height. And it depends sensitively on the rim’s condition: a scratch, a speck of dust or a film of frozen air on the rim changes the thickness there and with it the whole flow, which made the early measurements frustratingly irreproducible until clean surfaces were used.

The same reasoning lets the film be stopped. Low-temperature physicists who do not want helium to creep out of a cell up a pumping line put a sharp knife-edge or a small orifice in its path, a place where the film must be very thin; the bottleneck there limits the creep to a trickle. Without one, film creeping up a warm tube reaches a height where it evaporates, and the evaporation is a steady heat load on the cell — the reason superfluid helium is so hard to keep in a vessel with an opening.

The heat that a creeping film carries away is not small. A film climbing the inside of a tube a centimetre across carries about $3 \times 10^{-5}$ grams of helium a second; if it evaporates where the tube warms, it takes its latent heat with it, about three-quarters of a milliwatt — a large load for a refrigerator working at 1.4 kelvin, where every milliwatt must be paid for by pumping away vapour at a fraction of a millibar. Rollin’s first clue to the film, in 1936, was exactly such an anomaly: heat seemed to travel up the walls of his apparatus far faster than the walls could conduct it, and the explanation was helium travelling up and boiling away.

The film is also a filter. Only the superfluid moves along it, and the superfluid is the part of the liquid in its lowest quantum state, so anything that cannot join that state is left behind: atoms of helium-3 dissolved in helium-4, which are fermions and cannot condense with it, travel with the normal part and stay where they are. A film, or a plug of fine powder whose pores are so narrow that only superfluid can pass, lets pure helium-4 through and holds back the rest. Such superleaks are used to purify helium-4 and to separate the two isotopes, and they work for the same reason the beaker empties: the film’s flow is the superfluid’s, and nothing else is admitted.

Waves on a film #

A film that slides freely also carries waves. If the film is made a little thicker in one place, the extra liquid feels a weaker pull from the wall than the film around it, and the superfluid slides away from the bulge to thinner regions, overshoots, and comes back: a ripple in the film’s thickness, restored by the van der Waals force, carried by the superfluid alone while the normal part stays put.

Kenneth Atkins predicted these waves in 1959 and with colleagues measured them soon after, calling them third sound, after the ordinary first sound of a pressure wave and the second sound of the heat wave. Their speed is set by how stiffly the wall’s pull resists a change of thickness, and since the pull goes as the inverse cube of the thickness, its rate of change goes as the inverse fourth power, and a thin film is far stiffer than a thick one: third sound runs at half a metre a second on a film thirty nanometres thick and at sixteen metres a second on one of three nanometres.

Because only the superfluid moves in a third-sound wave, its speed is a direct measure of the superfluid fraction of a film too thin for any other measurement. That made third sound the instrument for the thinnest films, a few atomic layers, where the film’s superfluidity behaves differently from the bulk liquid’s: in a film that is effectively two-dimensional, superfluidity appears not gradually but in a sudden jump, at a temperature where bound pairs of vortices come apart — the transition with nothing to order — and measurements of film flow and of third sound by David Bishop and John Reppy in 1978 confirmed the jump’s predicted size.

What the flat-wall picture leaves out #

The figures treat the film as a uniform layer on a flat, smooth, vertical wall, thickened only by the van der Waals balance, with a single critical speed and a superfluid fraction taken from an approximate formula. Real films on real walls depart from each of these. Near a rim the wall curves, and surface tension at the film’s free surface thickens the film over the curve; real surfaces are rough on the scale of the film, so the “thickness” at a scratch is not the formula’s; and the van der Waals pull is weakened at distances beyond tens of nanometres by the finite speed of light, which makes thick films thinner than the cube-root law. The critical speed itself is not a single number: it depends on the film’s thickness and on how the measurement is made, and its theory, in terms of vortices crossing the film, gives the trend but not the value.

The temperature dependence uses the empirical rule that the normal fraction grows as the 5.6th power of the temperature over the λ point’s, which is good to some per cent below 2 kelvin and poor right at the transition. And the beaker figure assumes the film’s flow switches on immediately, with no time for the film to form; a beaker raised from a bath is already coated, but one filled from below takes time to grow its film.

The domain of the argument is a superfluid film thick enough to behave like the bulk liquid, on clean surfaces, flowing slowly enough that the critical speed is reached only at its thinnest point. Within it, the film’s flow is set by a speed limit at one place, and the push that drives it does not matter.

Still open: what sets the critical speed in a film #

The critical speed of about thirty centimetres a second in a thirty-nanometre film is a measurement, not a calculation. Landau’s criterion, which gave the speed below which a superfluid cannot create excitations, predicts a critical speed of tens of metres a second, a hundred times too high; real superfluids reach their limit much sooner, by creating quantised vortices, and in a film the vortices form at the rough surface of the wall and are pinned and released by its imperfections. A theory of vortex nucleation in a film gives a critical speed falling roughly as the inverse of the thickness, in agreement with the trend, but whether the nucleation is thermally activated, quantum tunnelling or triggered by specific defects, and so what the speed would be on a perfectly smooth wall, is not settled.

What is settled is the shape of the flow. A helium film is held to the wall by van der Waals attraction, 30 nm thick a centimetre above the bath and as thick above the λ point as below it; below it the superfluid in the film flows without friction up to a critical speed, reached first at the rim where the film is thinnest, so the film carries about 7.5 × 10⁻⁵ cm³/s per centimetre of rim whatever the difference of levels — and a beaker empties in a straight line rather than a slowing curve. The most famous trick of superfluid helium is a flow limited by a speed and decided at a single edge.