The oil that is a glass for a quarter of a millisecond

Every essay on viscosity so far has varied the temperature. The third is the sharpest of them: heat a liquid and it thins, heat a gas and it thickens, and the word viscosity names one measurement made on two mechanisms with almost nothing in common.

Pressure is the other variable, it is rarely taught, and over the range a machine reaches it is by far the larger effect. Every liquid thickens under pressure, they all do it in the same direction, and they all do it exponentially.

Barus’s rule, from 1893, is $\eta = \eta_0 e^{\alpha p}$. The coefficient $\alpha$ is between ten and thirty inverse gigapascals for an ordinary oil and around half of one for water, and that spread of a factor of thirty between water and an oil is the whole of why one of them is a lubricant.

The mechanism is not mysterious and it is worth naming before the consequences, because it also says where the rule fails. A liquid flows by molecules changing places — which is how a liquid conducts momentum sideways in the first place — a molecule can only change places if there is somewhere to go, and squeezing a liquid removes the somewhere. The free volume falls roughly in proportion to the pressure, the rate of rearrangement falls exponentially in the inverse of the free volume — an exponential that decides everything once it is in an exponent — and an exponential in the pressure comes out over the range where the free volume has not fallen too far. It is the same free-volume argument that accounts for a liquid’s approach to a glass on cooling, with pressure doing what temperature otherwise does.

Inside a gear tooth #

The pressures in the previous figure sound extreme and they are entirely ordinary. A rolling-element bearing or a gear tooth concentrates its load onto a contact a fraction of a millimetre across, and what the pressure there is can be computed rather than guessed: it is the Hertz solution for two elastic bodies pressed together, which for a line contact gives a peak pressure of $\sqrt{w'E'/\pi R}$ and a contact half-width of $\sqrt{4w'R/\pi E'}$.

The pressure is 1.32 gigapascals, which is thirteen thousand atmospheres, and it is generated by nothing more dramatic than a load of five hundred newtons per millimetre of contact length — on a roller of two centimetres’ radius. What turns that load into a pressure is the area it acts over, and the area two curved bodies genuinely touch over is far smaller than they look. There is no hydraulics involved. The pressure is simply what an ordinary load comes to once elasticity has concentrated it onto a contact that small.

What the oil experiences is therefore a journey. It enters the contact at atmospheric pressure as a liquid of a tenth of a pascal-second — about the consistency of a thin syrup. Within a quarter of a millimetre the pressure has risen to a gigapascal and its viscosity has risen by twelve decades, which takes it to within a factor of three of what a glass has. It crosses the contact in that state, and on the far side the pressure falls and it is a thin syrup again.

The transit takes 241 microseconds at two metres a second, and the contact is passed several thousand times a second for the life of the machine. That is the quarter of a millisecond in the title: a liquid, a solid, and a liquid again, several thousand times a second, for years.

The film that would not otherwise exist #

The point of all of this is a number that can be computed both ways.

A film between two surfaces sliding past one another is the classical problem of hydrodynamic lubrication, and the answer there is old and straightforward: the surfaces drag fluid into a converging gap, the fluid has to squeeze out sideways, and the pressure that builds up holds them apart. The thickness depends on the speed, the viscosity and the load, and the arithmetic is the same slow-flow calculation a pipe poses.

In a gear or a bearing the classical answer is wrong by two orders of magnitude, and it is wrong in the direction that matters. Two things it leaves out both act to increase the film. The surfaces deform elastically under the pressure, flattening the gap and making the converging region far longer than the undeformed geometry suggests; and the oil stiffens.

The subject is written in three dimensionless groups, and the one that carries the pressure dependence is the materials parameter $G = \alpha E'$ — the viscosity’s pressure coefficient multiplied by the effective elastic modulus of the two solids. For steel on steel with an ordinary oil it is about five thousand. Setting it to one, which is what removing the pressure dependence amounts to, divides the film by $5000^{0.6}$, which is a factor of a hundred and sixty.

That factor is the difference between a working machine and a seizing one. The realistic film is nine hundred nanometres, which is several times the roughness of a ground steel surface; the other is five nanometres, which is a fortieth of the roughness, and a calculation returning it is describing two surfaces in contact.

So the whole of rolling-element engineering rests on a property of liquids that has nothing to do with lubrication and was measured by Barus for its own sake. The pressure that would crush the film is the same pressure that thickens the oil enough to survive it, and the two scale together in a way that makes the arrangement work over decades of load.

How the coefficient is got at, and how the film is #

Both of the numbers this argument turns on are measured, and neither measurement is obvious.

The pressure coefficient comes from a high-pressure viscometer, and the usual design is a weight falling down a tube of the fluid inside a pressure vessel. The fall is slow, Stokes’s drag law applies, and timing the fall gives the viscosity — the same arrangement that measures an ordinary liquid, put inside a vessel that holds a gigapascal. What limits it is the range: a viscosity of $10^{10}$ pascal-seconds means a body falling a centimetre in a year, so the measurement stops several decades below the glass and the crossing is reached by extrapolation from both sides rather than by observation.

The film is measured in the machine, which is much less obvious. A steel ball is loaded against a glass or sapphire disc coated so that part of the light reflects at the coating and part at the ball; the two reflections interfere; and the colour of the interference fringe reports the gap. It is the same measurement a soap film makes as it thins, read deliberately, and with a spectrometer rather than an eye it gives the film to a nanometre over a contact half a millimetre wide, while the contact is running.

That technique is what established the shape the formulas fit. The nearly parallel central region, the constriction at the exit, the pressure spike that goes with it — all of it was predicted numerically in the 1960s and photographed afterwards, and the agreement between a numerical solution of a coupled elasticity-and-lubrication problem and a colour photograph of an oil film is one of the cleaner confirmations in the subject.

It also fixes what the Dowson–Higginson expression is. It is a fit to numerical solutions, checked against those measurements, accurate to some tens of per cent across the range it was fitted over, and it is not a derivation. Every film thickness in this essay carries that accuracy and no more.

Temperature and pressure turn out to be one variable #

The two dependencies have been treated here as separate facts, and there is good evidence that they are not.

The free-volume argument says what they have in common. Both heating and decompressing make room; both cooling and compressing take it away; and what a molecule needs in order to move past its neighbours is room. If that is the whole story then the viscosity should not depend on temperature and density separately but on some single combination of them.

The combination turns out to be $T\rho^{-\gamma}$, with $\gamma$ a number between about one and six that belongs to the liquid. Measured viscosities of a great many molecular liquids, taken over wide ranges of temperature and pressure, collapse onto a single curve when plotted against that one variable — hundreds of separate measurements, spanning fourteen decades of viscosity, falling on one line per substance with one fitted exponent. The result is known as thermodynamic scaling and it has been established since the early 2000s.

What makes it more than a curve fit is that $\gamma$ can be predicted. For a liquid whose molecules repel as an inverse power $r^{-n}$ of their separation, a straightforward scaling argument gives $\gamma = n/3$, and the measured exponents for simple liquids agree with the repulsive parts of their known potentials. So the exponent is a statement about the shape of the intermolecular repulsion, extracted from a viscometer.

That is the surprising connection here. A viscosity measured under pressure reports the steepness of the repulsion between two molecules — which is otherwise got at by scattering something off them — and it does so because the only thing squeezing a liquid does is push molecules up their mutual repulsive wall.

It also explains something about the earlier essays that looked like a coincidence. The opposite signs of a gas and a liquid finds a liquid’s viscosity falling roughly exponentially in the inverse temperature; the Barus rule above finds it rising roughly exponentially in the pressure; and if both are functions of one variable then neither exponential is fundamental and both are local approximations to the same curve seen along different paths. Which is exactly what is observed: neither the Arrhenius fit nor Barus’s rule holds over a wide range, and they fail together.

Which regime a machine is in, and why it is the starting that wears it #

A film thickness is only meaningful next to a roughness.

The ratio is conventionally called $\Lambda$ and the two thresholds are conventional too, but the shape they describe is not. Below $\Lambda \approx 1$ the high points of the two surfaces are touching through the film, the load is shared between asperity contacts and fluid pressure, and the surfaces wear. Above $\Lambda \approx 3$ the film separates them completely and, in the absence of contaminants and fatigue, nothing wears at all.

Because the film grows as roughly the seven-tenths power of the speed, a bearing at rest is at $\Lambda = 0$ and passes through the whole mixed regime every time the machine starts. Most of the wear in a bearing’s life happens in those few seconds, which is why the interesting engineering questions are about starting and stopping rather than about running, and why a machine that runs continuously outlasts one of the same design that is switched on and off.

It also says exactly what polishing buys. Halving the roughness moves the safe speed down by the same factor raised to the reciprocal of seven-tenths — about a factor of two and a half — so the surface finish and the speed trade against each other on a fixed exponent. That is a designer’s curve rather than a rule of thumb, and it is computed from the same three groups.

The glass that is sold on purpose #

A vitrified film sounds like a failure mode and it is a product.

A traction drive transmits torque between two rollers that do not touch. There is a film of fluid between them, it is a solid while it is in the contact, and it shears — a solid can be sheared — carrying a stress from one roller to the other. The ratio of that shear stress to the pressure holding the rollers together is the traction coefficient, it is between about five and nine per cent for a fluid designed for the purpose, and it is the number a continuously variable transmission is designed around.

What makes it useful is that it is nearly independent of speed over a wide range, which is what a friction coefficient is usually assumed to be and what a tyre’s grip conspicuously is not. The reason is the limiting stress: once the film is vitrified it does not shear faster when pushed harder, it shears at a stress fixed by the pressure, so the transmitted torque is set by the normal load and not by the slip. A drive built on that behaves like a friction clutch with no wear surface.

Fluids are formulated for it, and formulated against the ordinary requirement. A lubricant is wanted to have a small traction coefficient, because its traction is the drag that heats a gearbox; a traction fluid is wanted to have a large one. Both are made by choosing the molecular shape — compact, rigid, cyclic molecules vitrify at lower pressure and hold a higher limiting stress; long flexible chains do neither — and the two products sit at opposite ends of one design axis.

That the same physics is sold twice, once as a virtue and once as a defect, is the clearest evidence that it is real rather than an artefact of how bearings are analysed.

Barus’s rule is a fit extrapolated a hundredfold past its own range #

Barus’s rule is a fit made far below the pressures it is being used at. It was established over a few tens of megapascals and it is applied here over a few thousand. Extrapolated to a gigapascal it over-predicts — badly for a fluid with a large coefficient, where it returns numbers no material has — and the Roelands equation, which is the usual repair, under-predicts at the same place. The honest statement is that the crossing into a glass happens at roughly a gigapascal for an ordinary lubricant, that both rules bracket it, and that neither gives a viscosity beyond it. The figures cut off at the glass value for that reason.

And past the glass there is no viscosity to give. A vitrified film does not shear at a rate proportional to the stress; it shears at a limiting stress, like a soft solid, and the limiting stress is roughly a twentieth of the pressure. That is why traction drives work at all — a fluid whose shear stress saturates can transmit torque through a film a micron thick — and it is a different constitutive law with a different variable in it. Everything here describes the pressure that gets the film to that point and not what happens once it is there.

The oil in the contact is not at room temperature. Shearing a fluid of $10^{11}$ pascal-seconds at $10^6$ per second dissipates an enormous power in a very small volume, and the film heats by tens of degrees in the time it takes to cross. Since viscosity falls steeply with temperature, the heating partly cancels the pressure thickening, and a calculation that ignores it over-predicts the film at high speed. The heat has to leave through the solids, and how fast it does is a diffusion problem with its own length rather than a matter of how hot the film gets. The curves here are isothermal and the discrepancy grows to the right.

And the load has been treated as steady. A gear tooth’s load rises and falls within a single mesh, a bearing’s rolling element is loaded and unloaded once per revolution, and the film does not have time to reach the steady value the formula gives. What survives from one pass to the next is a squeeze-film effect that keeps the gap open longer than a steady calculation allows, which is a help rather than a hazard and is not in the figures.

The constriction at the exit, and a coefficient that is not one number #

They cannot show the shape of the gap. The film in an elastohydrodynamic contact is not uniform: it is nearly parallel across most of the contact and has a sharp constriction at the exit, where the gap narrows to about three quarters of the central value and the pressure spikes above the Hertzian value. That constriction is the minimum film thickness the formula reports, and it exists because the elasticity and the lubrication problem are coupled — the pressure deforms the surfaces and the deformed surfaces set the pressure. Drawing it requires solving that coupled problem rather than evaluating a fit to it.

Nor can they show that $\alpha$ is not a constant. A pressure coefficient quoted for an oil is a slope taken over some range, and the slope falls as the pressure rises. Different laboratories quote different values for the same fluid because they used different ranges, and a difference of twenty per cent in $\alpha$ moves the film by twelve per cent through the three-fifths power — which is inside the scatter of the fit anyway, and is worth knowing before a film thickness is quoted to three figures.

And they cannot show what the fluid does to a rough surface at the scale of the roughness itself. The film here is a mean thickness, and a real gap is a landscape of asperities each locally either touching or not. The ratio $\Lambda$ is a summary statistic of that landscape, it is the thing that correlates with life, and it is not a description of what is happening at any point.

Still open: what the limiting stress actually is #

The limiting shear stress of a vitrified lubricant is the number that decides how much torque a traction drive transmits and how much heat a bearing generates, and it is measured rather than derived. It is roughly proportional to the pressure with a coefficient of a few per cent, it depends on the molecular structure of the fluid in ways that are used empirically — and a fluid that has one at all is one that remembers, rather than merely resisting — and whether it is a genuine yield stress of a glass or a shear-induced flow instability at very high rate has been argued for forty years.

The difficulty is experimental as much as conceptual. Measuring a stress inside a contact half a millimetre wide, at a gigapascal, while the material is passing through in a quarter of a millisecond, is hard; measuring the same fluid in a high-pressure viscometer takes the pressure but not the shear rate, which is six orders of magnitude lower. Whether the two measurements are of the same quantity is not settled, and the machines are designed around correlations rather than around a law.

The habit worth carrying away concerns extrapolated exponentials. An exponential fitted over one decade of its argument and used over three is not a measurement; it is a statement about where the substance stops being what it was. Barus’s rule is extremely useful and it is useful for locating a transition rather than for computing what lies past one — and reading an exponential past the point where its mechanism holds is how a figure comes to report $10^{16}$ pascal-seconds for a liquid.