The gas sets a squeeze film's loss, not its stiffness

The heat of a squeeze leaves across the gap followed the heat a gas film makes when it is squeezed. Compress gas in a gap twenty micrometres wide and it warms; the warmth leaves through the walls, which are a gap’s width away, and how much of it leaves in a cycle decides whether the film behaves isothermally or adiabatically. One number measures that, the thermal number $\Omega = \omega h^2/\kappa$, the cycle’s rate against the rate heat diffuses across the gap, and between the two limits, near Ω = 13.5, the film loses energy as heat it has no viscous way to lose. For a given device the thermal number is tied to the squeeze number by the geometry.

Every figure there was air at one atmosphere, and the geometry was the only thing that moved the thermal number. The essay ended on the other handle. A gas’s thermal diffusivity is its conductivity over its density and heat capacity, so it falls as the pressure rises and is several times smaller for a heavy gas than for air, while the viscosity barely changes with either. A levitator could be moved onto its thermal loss peak, or off it, without touching the mechanism — though a heavy polyatomic gas also has a ratio of specific heats close to one, which shrinks the whole thermal effect as it moves it. It asked which gas and which pressure put a given levitator where it stores most and wastes least.

One levitator, six gases #

The levitator is the earlier essay’s: a disc of radius ten millimetres oscillating at twenty kilohertz over a twenty-micrometre gap. Its film is the linearised compressible Reynolds film, solved with the gap’s own complex polytropic index — the factor by which the heat crossing the gap makes the gas softer than adiabatic and lossier than isothermal. The six gases are helium, air, argon, carbon dioxide, xenon and sulphur hexafluoride, with their viscosities, conductivities, densities and heat capacities at twenty degrees, and the pressure runs from a tenth of an atmosphere to a hundred.

Two things change with pressure, and they go opposite ways. The squeeze number $\sigma = 12\mu\omega a^2/ph^2$, which decides whether the gas can escape the gap in a cycle, falls as one over the pressure. The thermal number rises in proportion to it, because the density rises and the diffusivity falls. Their ratio goes as the square of the pressure. The film’s force is reported in newtons per unit relative amplitude — the scale $pa^2$ the film’s pressure gives it — so that a film at higher pressure is credited with the force the pressure buys.

Same stiffness, different loss #

At one atmosphere the six films are nearly equally stiff: 26 to 28 newtons per unit relative amplitude. At a squeeze number near seventy the gas is nearly trapped, and a trapped film’s stiffness is the gas’s pressure times its effective exponent times the disc’s area, which differs between gases only through the exponent. The loss is a different matter. The loss tangent, damping over stiffness, runs from 0.19 for helium to 0.36 for xenon, and the difference is almost entirely thermal: helium’s damping is five per cent thermal, air’s 22, argon’s 31, carbon dioxide’s 26, xenon’s 53 and sulphur hexafluoride’s 16.

So the gas in a levitator’s gap is not a matter of its viscosity, which is within a factor of 1.6 across the six. It is a matter of how fast it lets go of its heat. Helium conducts so well that its gap is nearly isothermal at twenty kilohertz and loses almost nothing thermally. Xenon conducts worst of the monatomic gases and sits just below the thermal loss peak, where the heat crosses the gap a quarter-cycle late and is wasted.

Why the gas hardly touches the stiffness #

The near-equality of the six stiffnesses is the trapped film’s arithmetic. A damper that turns into a spring showed that a gas film at a large squeeze number cannot escape the gap within a cycle, so it is compressed and expanded in place like gas in a closed cylinder, and its stiffness is the pressure times the effective exponent over the gap. The pressure is the same for all six, one atmosphere; the effective exponent differs only between the isothermal one and the gas’s own adiabatic γ, and at one atmosphere and twenty kilohertz most of the six are nearer the isothermal end. Their viscosities, which decide how nearly trapped the film is, differ by a factor of 1.55 and the squeeze numbers by about as much, all of them well into the trapped regime.

The loss is not trapped-film arithmetic. It is the small part of the cycle in which the gas either flows or exchanges heat, and both are rates — the gas’s viscosity against its pressure, its diffusivity against the gap — that differ between gases by much more than the exponent does. That is why the gas sets the loss and not the stiffness: the stiffness is a ratio of states, the loss a ratio of rates. It is the same distinction nothing but the shape of the gap drew for a liquid bearing, whose load is set by geometry and whose friction by the liquid.

Where each gas sits on the thermal number #

The thermal number shows why. At one atmosphere helium’s is 0.29, deep in the isothermal regime; air’s is 2.4; xenon’s 7.9, approaching the peak; and sulphur hexafluoride’s 15.6, just past it. Raising the pressure slides every gas to the right in proportion. A sixfold rise in pressure puts air on the loss peak; a fiftyfold rise puts helium there.

That is the handle the earlier essay asked about, and it is a strong one: a factor of four in pressure moves the thermal number by four, where moving it by the geometry would mean changing the gap by two. But the pressure also changes the squeeze number the other way, so the handle cannot be turned without changing the film’s mechanics too.

Sulphur hexafluoride on the peak #

Sulphur hexafluoride is the earlier essay’s cautionary case. Its thermal number at one atmosphere is on the peak, where the gap’s thermal loss is largest, yet its film loses only sixteen per cent of its damping thermally. The reason is its ratio of specific heats, 1.098. A gas whose compression barely heats it has little heat to lose: the gap’s largest thermal loss — the peak value of the imaginary part of its polytropic index — grows from zero in proportion to $\gamma - 1$. It is 0.040 for sulphur hexafluoride, 0.116 for carbon dioxide, 0.157 for air and 0.251 for the monatomic gases. A heavy polyatomic gas moves the film onto its loss peak and shrinks the peak by a factor of six at the same time.

The polytropic index shows the two gases side by side. Xenon’s effective exponent climbs from one at low pressure towards 5/3 at high, and its loss peaks on the way. Sulphur hexafluoride’s exponent can only rise to 1.098, and its loss peak is a sixth of xenon’s, though it arrives at a lower pressure. A gas that does not heat when squeezed is a gas whose film is nearly the same at every thermal number.

Stiffness has a peak #

Raising the pressure stiffens every film, up to a point. At low pressure the film is trapped and its stiffness is proportional to the pressure. At high pressure the squeeze number falls below one, the gas flows out of the gap and back each cycle, and the film becomes a damper with little stiffness at all. Between, each gas peaks near ten atmospheres: carbon dioxide at 99 newtons, sulphur hexafluoride at 109, air at 116, helium at 127, argon at 136 and xenon at 149. The monatomic gases peak highest, because their adiabatic exponent is largest and a trapped film’s stiffness scales with it; xenon, whose film is the least isothermal of the monatomic gases at the peak’s pressure, gets most of that exponent.

What a film can store for what it wastes #

The question the earlier essay posed has an answer in this plane. Each gas, traced over pressure, is a curve: higher pressure buys stiffness and costs loss. Up to about a hundred newtons per unit relative amplitude, helium’s curve lies lowest — at fifty newtons its loss tangent is 0.29, against air’s 0.40 and xenon’s 0.46 — so for any stiffness in that range, helium at the pressure that gives it is the least wasteful fill. Beyond a hundred newtons helium is approaching its own peak of 127, its loss climbs steeply, and the heavy monatomic gases take over: at 120 newtons xenon loses 0.73 to helium’s 0.78, and above 127 only argon and xenon can be used at all.

So there are two answers, as a design question usually has. For a levitator that needs modest stiffness and little loss, fill it with helium near or below an atmosphere. For one that needs the most stiffness the geometry allows, fill it with xenon at about ten atmospheres and accept that half its damping will be heat. Sulphur hexafluoride loses least to heat but buys the least stiffness, and never wins.

A levitator’s power bill #

The loss tangents become watts with the levitator’s own numbers. Oscillating the disc by a tenth of its gap — two micrometres — the film dissipates, each cycle, π times its stiffness times its loss tangent times the square of the relative amplitude times the gap: about three microjoules for helium at one atmosphere and six for xenon. At twenty thousand cycles a second that is 0.06 watts for helium and 0.12 for xenon, drawn continuously from the transducer and delivered as heat to the disc and its stage. For a levitator holding a wafer whose position is measured in nanometres, where the heat delivered to the stage sets its thermal drift, the fill alone halves or doubles that heat without touching the mechanics.

The pressure is the larger lever on the same bill. Taking the helium fill from one atmosphere down to a third cuts the loss tangent from 0.19 to 0.10 and the stiffness from 26 to under 10 newtons; since the dissipated power goes as stiffness times loss tangent, it falls fivefold, for a film a third as stiff. Raising it to three atmospheres raises the stiffness two-and-a-half-fold and the power nearly fivefold. Every fill has its own version of that trade, and the frontier figure is the whole of it: a designer picks a stiffness, reads off which gas reaches it with the least loss tangent, and the pressure follows.

The same heat in other places #

The gap’s complex polytropic index is not special to squeeze films. The same calculation — heat of compression diffusing to a boundary a set distance away in the time of one cycle — gives the complex exponent of the gas in an oscillating bubble that above their resonance, bubbles make water faster used, where the boundary is the bubble’s own wall and the distance its radius. It is the relaxation behind a gas that has not finished being shocked, with heat conduction in place of molecular vibration, and the bulk loss that the viscosity nobody uses adds to a compression. In every case the loss peaks where the relaxation time equals the period, and in every case the peak’s height is set by how far the fast and slow responses differ — here, by γ − 1.

A liquid film has none of this, which is why the last of the oil and a torn film still pulls could treat their films as incompressible and isothermal without loss of anything. The thermal handle belongs to gases.

Checks on the gas-filled film #

The two numbers must scale exactly with pressure, and do: quadrupling the pressure multiplies the thermal number by four and divides the squeeze number by four, for all six gases, to rounding. Air at one atmosphere must be the earlier essay’s levitator, and its squeeze number is 67.343 to the last digit. And the gap’s peak thermal loss must vanish in proportion to $\gamma - 1$: doubling $\gamma - 1$ from 0.01 to 0.02 multiplies it by 1.997. The tests also refuse a pressure of zero, a negative one, and a gas whose ratio of specific heats is one.

What the film model assumes #

A dilute gas. Viscosity and conductivity are taken independent of pressure, as kinetic theory gives. At a hundred atmospheres xenon and sulphur hexafluoride are far from dilute — sulphur hexafluoride liquefies near twenty atmospheres at room temperature — and the upper end of the pressure range is a statement about an ideal gas, not about those substances. The comparison up to ten atmospheres is safer for the light gases than the heavy.

Walls at a fixed temperature. The heat that crosses the gap is assumed to be swallowed by walls that do not warm. Metal walls at these frequencies nearly are: the thermal wave a twenty-kilohertz cycle drives into aluminium penetrates a few tens of micrometres and carries the heat away faster than the gas delivers it. A glass or polymer disc would not be so obliging, and its surface would warm and cool with the cycle, returning part of the heat to the gas and moving the film back towards adiabatic.

Small amplitude. The film is linearised, which holds while the oscillation is a small fraction of the gap; at a tenth, the second-order pressure is already of order a tenth of the first. A levitator lifts its load by the second-order, time-averaged part of the pressure, which this calculation does not compute; the stiffness and loss are the linear film’s.

No rarefaction. At a tenth of an atmosphere air’s mean free path is two-thirds of a micrometre, a thirtieth of the gap, and slip at the walls begins to soften the film; helium’s longer mean free path makes it the first to need the correction.

The convention: force per unit relative amplitude, and the loss tangent #

Stiffness and damping are the film force’s in-phase and out-of-phase parts for a gap oscillation of unit relative amplitude, in newtons, from the dimensionless film force times $pa^2$. The loss tangent is the ratio of damping to stiffness, the fraction of the energy stored per radian that is dissipated. The thermal share is the damping beyond what an isothermal film of the same stiffness would have, over the whole damping. One atmosphere is 101,325 pascals.

Who filled the gap #

Squeeze-film levitation was demonstrated by Salbu in 1964 and its isothermal theory is Langlois’s, from 1962. The thermal correction for the heat crossing a thin gas film was worked out for acoustic transducers and micro-resonators, where the same complex polytropic index appears, and filling a gap with a heavy or light gas to tune a device’s damping is standard practice in hard-disc drives, which are filled with helium to cut the gas’s drag and the flutter it drives on the spinning platters — a different loss, turned by the same handle. The comparison of six gases across three decades of pressure for one levitator, the stiffness peak near ten atmospheres and the crossover between helium and xenon are what this calculation adds.

Still open: the levitation force itself #

The levitator lifts its load by the mean of the film’s pressure, a second-order effect of the oscillation, and the linear film here gives only stiffness and loss. The next calculation carries the film to second order in the amplitude with the thermal polytrope included — the mean pressure a gas trapped at a complex exponent builds over a cycle — and asks whether the gas that wastes least also lifts most, or whether xenon’s larger exponent buys more lift than helium’s isothermal gap saves, which would turn the stiffness frontier above into a load frontier.