The thickness that goes both ways

Viscosity is defined the same way for everything that flows. Shear a fluid and it resists; the resistance divided by the rate of shearing is the viscosity, and the definition says nothing whatever about what the fluid is made of or how it carries the resistance.

Because the definition is neutral, it is easy to assume the mechanism is too. It is not. Everything anybody knows about how honey behaves when it is warmed is wrong for air, and the failure is not a detail: the sign of the temperature dependence is opposite, and the size differs by a factor of thirty.

What a gas is doing #

Viscosity in a gas is momentum diffusing sideways. Two layers of gas sliding past one another exchange molecules; a molecule arriving from the faster layer brings extra forward momentum with it, and one leaving takes some away. The net transfer across the surface is a shear stress, and nothing rubs against anything.

That picture gives an expression with three factors — the density, the mean speed and the mean free path — and the celebrated consequence that the density cancels against the free path, so the viscosity of a gas does not depend on how much gas there is. What is left is the mean speed, which goes as the square root of the temperature.

So a hotter gas should be more viscous, and it is. Air at 900 kelvin is more than half again as viscous as air at 250. That is the wrong way round for anybody whose intuition comes from honey, and it is the strongest single piece of evidence that a gas’s viscosity is not friction: friction between layers would fall as the layers were shaken apart, and this rises.

It also settles an old question about what viscosity is. Newton’s own account was of layers dragging on one another, and for a liquid that is a serviceable metaphor. For a gas it predicts the wrong sign, and it predicted the wrong dependence on pressure as well — Maxwell’s calculation that a gas’s viscosity does not depend on its density was received as evidence against the kinetic theory until he went and measured it. A theory whose most implausible prediction turns out to be right is in a strong position, and the temperature dependence is the second such prediction from the same argument.

The rise is also the reason a hot gas is harder to pump and why the drag on a body in a hot exhaust is larger than the same body in cold air at the same density. It has been measured since Maxwell, who predicted it before it was seen and then measured it himself.

The exponent that is not a half #

The hard-sphere prediction is that viscosity goes as the square root of the temperature. Measurement gives something steeper, and the discrepancy is informative rather than embarrassing.

Real molecules attract one another. A pair passing at a distance is bent together a little, which increases the chance of a collision — so the effective cross-section is larger than the geometric one. A fast pair is bent less than a slow one, because the deflection depends on the ratio of the interaction energy to the kinetic energy. So heating a gas shrinks its molecules’ effective size, and the viscosity rises for that reason on top of the mean speed rising.

That is Sutherland’s correction, and its one adjustable constant is a temperature: the temperature at which the kinetic energy is comparable with the depth of the attraction. Below it the attraction matters; above it, it does not. Carbon dioxide, whose molecules attract most strongly of the three drawn, has the largest constant and the steepest curve; helium, which barely attracts at all, is the closest to the hard-sphere line.

By four thousand kelvin the exponent has fallen to 0.503 and the hard sphere is back. That is a satisfying thing to be able to say about an approximation: it is not merely a simplification, it is the high-temperature limit of the real behaviour, and the departure from it is a measurement of the intermolecular potential.

What a liquid is doing #

In a liquid the molecules are permanently in contact. There is no free flight, no mean free path worth the name, and nothing to carry momentum across a plane in the way a gas does. What has to happen instead for one layer to slide past another is that individual molecules squeeze past their neighbours, and squeezing past costs energy.

If getting past costs a fixed energy, the rate at which a molecule manages it carries a Boltzmann factor, and the viscosity — which is a rate turned upside down — carries the reciprocal. That gives a straight line of $\log\eta$ against $1/T$, whose slope is the barrier height. Glycerol’s barrier is nearly four times water’s, which is why its viscosity falls by more than a factor of five over eighty kelvin where water’s falls by two.

Heating therefore helps: it does not add carriers, it removes an obstacle. That is the opposite of the gas mechanism in the most direct possible sense, and it is why the two families in the opening figure slope in opposite directions.

The straight line is a fit and has a range. Water is measurably non-Arrhenius over a wide interval — its structure changes with temperature in ways a single barrier cannot represent — and the numbers quoted here reproduce it between 280 and 360 kelvin and nowhere else. Naming the range a model holds over is the whole obligation, and this is a model with a narrow one.

Where neither picture works #

Between the two there is a region where neither description applies, and it is not a small one.

Neither picture works near a critical point, and the reason is that both rest on a separation of scales that has gone. There the mean free path is comparable with the molecular spacing, the fluid is neither a gas nor a liquid, and there is no sense in which momentum is either carried by flying molecules or passed along by neighbours in contact. The viscosity is finite and measurable and neither mechanism computes it.

A dense gas near its critical point has a mean free path comparable with the distance between molecules. The kinetic derivation needs a molecule to travel many diameters between collisions and it does not; the liquid derivation needs a well-defined cage of neighbours and there is not one. Measured viscosities in that region do neither thing: they rise with density along an isotherm in a way that no simple expression captures, and correlating them is an empirical business.

The same is true of a supercritical fluid used industrially. Supercritical carbon dioxide is used as a solvent precisely because it has a gas’s viscosity and a liquid’s density, and neither of the two mechanisms in this essay predicts its transport properties to better than tens of per cent.

The liquid that runs out of the argument #

Push a liquid the other way, towards cold, and the Arrhenius picture fails differently and more spectacularly.

Cooling a liquid toward its glass transition stretches the timescale on which it responds to a steady stress out of the laboratory’s reach. Somewhere along that stretch the distinction between a very slow liquid and a solid stops being operational — not because the material changes character at a point, but because the answer to “does it flow?” becomes a statement about how long anyone is prepared to watch. Window glass does not sag; the timescale is a factor of $10^{20}$ too long.

A glass-forming liquid cooled towards its transition does not follow a straight line on the Arrhenius plot. It curves upward, so steeply that the viscosity rises through twelve or thirteen decades over a temperature interval in which nothing structural changes that a diffraction pattern can detect. The apparent activation energy is not constant; it grows as the liquid cools.

The conventional definition of the glass transition is a viscosity — the temperature at which it reaches $10^{12}$ pascal seconds — which is an admission that no property changes discontinuously there. What is being named is the point at which the material stops flowing on a laboratory timescale, and that is a statement about the observer as much as about the substance.

Whether there is a genuine thermodynamic transition underneath the kinetic one is an open question, and it is the reason the glass problem is still worked on. What is clear is that the single-barrier picture has no room for it: one barrier gives one straight line, and the data are not one straight line.

Two mechanisms, two instruments #

The measurement itself is a good place to see that these are different subjects wearing one name.

Momentum diffusing away from a suddenly moved surface spreads to a depth that grows as the square root of the time, and the constant in that square root is the viscosity divided by the density. That combination — the kinematic viscosity — is what a transient measurement actually returns, which is worth knowing before comparing two numbers: a dense oil and a light one with the same dynamic viscosity diffuse momentum at quite different rates.

A liquid is measured by making it flow: down a capillary, past a falling ball, between a rotating cone and a plate. All of these need a steady flow, a measurable pressure drop or torque, and a known geometry, and they work because a liquid’s viscosity is large enough for the forces to be comfortable.

A gas is a thousand times less viscous and cannot be measured that way at ordinary pressures without either an impractically fine capillary or an impractically sensitive transducer. The classical instrument is instead an oscillating disc, whose damping is read off over many cycles, and the modern one is a vibrating wire or a torsional crystal. What those return is not quite the same quantity: they measure the momentum that diffuses out during one oscillation, so what is being determined is the kinematic viscosity — the viscosity over the density — and the density has to be supplied separately.

That distinction is worth carrying, because the kinematic viscosity of air is fifteen times that of water. By the measure that decides how far momentum spreads in a given time, air is the thicker of the two, and the everyday ordering is reversed. Which of the two quantities matters depends on the question: a force on a surface is set by the dynamic viscosity, and a spreading or a settling time by the kinematic one.

What the sign is really about #

The two mechanisms can be told apart by an argument that does not need either formula.

In a gas, the carriers move and the transport is limited by how fast they move. Heat supplies more of what limits it, so the transport goes up. Every transport coefficient of a dilute gas behaves this way — thermal conductivity and diffusivity rise with temperature too, and by comparable powers, which is why the ratio of viscosity to conductivity is nearly a constant across gases.

In a liquid, the carriers are stuck and the transport is limited by how often one gets free. Heat supplies more of what limits that, so the resistance goes down. Diffusion in a liquid rises with temperature for the same reason viscosity falls, and the product of the two is nearly temperature-independent — the Stokes–Einstein relation, which holds surprisingly well and fails, informatively, near the glass transition.

So the sign of $\mathrm{d}\eta/\mathrm{d}T$ is a diagnostic. A substance whose viscosity rises with temperature is transporting momentum by free flight, and one whose viscosity falls is transporting it by activated hopping — whatever it is called and whatever phase diagram it appears on.

The solid that is measured by waiting #

The liquids in the figures have viscosities that a bench instrument can measure in minutes. There is a range beyond them where the only usable instrument is patience, and two of the measurements in it are worth describing because they are made on materials nobody would call a fluid.

The pitch-drop experiment was set up at the University of Queensland in 1927. Pitch — the tar-like residue that shatters into fragments if struck with a hammer — was heated, poured into a funnel and left for three years to settle before the funnel’s stem was cut open. It has been dripping ever since.

Nine drops have fallen: the first in 1938 and the ninth in 2014. From the drop interval and the funnel’s geometry the viscosity comes out around $2\times10^8$ pascal seconds, some two hundred thousand million times water’s. Nobody saw a drop fall for eighty-six years; the custodian who looked after the experiment for fifty-two of them died in 2013, having missed all three that fell on his watch, and the first one ever recorded fell at a similar experiment in Dublin later that year.

The experiment has an incidental result that belongs to this essay. Air conditioning was installed in the building in 1988, and the interval between drops lengthened. A temperature change of a few degrees, measured over decades, moved the viscosity by a factor visible in a drop rate — which is the Arrhenius slope, determined by an experiment nobody was running.

The other measurement is on a scale where no apparatus is possible at all. The Earth’s mantle transmits shear waves from earthquakes, which is what a solid does, and it also flows. Its viscosity is about $10^{21}$ pascal seconds, and it is known because a natural experiment was performed on it.

Ice sheets several kilometres thick loaded Scandinavia and northern Canada until about ten thousand years ago, pressing the crust down. When the ice went, the load was removed, and the land began to rise. It is still rising — a centimetre a year around Hudson Bay and the Gulf of Bothnia — and the rate of that recovery is a relaxation time, from which the viscosity follows. Haskell did the calculation for Fennoscandia in 1935 and got a figure that has survived.

The number that reconciles the two behaviours is a ratio. Dividing the mantle’s viscosity by its shear modulus gives a time of a few hundred years: shorter than that, the mantle is a solid, and longer than that, it is a fluid. An earthquake wave, which crosses in minutes, sees a rock; an ice sheet, which sits for ten thousand years, sees a liquid. Same material, one number deciding which, and the number is a ratio of the material’s own time to the observation’s.

The window that did not flow #

The pitch drop invites a well-known claim about a much stiffer material, and the claim is false.

The story is that the windows of medieval cathedrals are thicker at the bottom than the top because the glass has flowed downward over the centuries, and that this proves glass is a liquid.

The arithmetic refuses it, and by a very wide margin. This essay’s own account of the glass transition defines it as the temperature at which the viscosity reaches $10^{12}$ pascal seconds. Room temperature is hundreds of degrees below that for a window glass, and the Arrhenius extrapolation — which understates the rise, since the curve steepens as it cools — puts the viscosity somewhere above $10^{18}$ and plausibly far higher. At those values the flow of a pane under its own weight over a thousand years is a small fraction of the diameter of one atom.

The evidence agrees with the arithmetic. Roman glass vessels two thousand years old show no thickening. Optical components stored for a century hold their figure to a fraction of a wavelength, which is a sensitive test indeed. And the medieval panes themselves are not consistently thicker at the bottom: a sizeable fraction are thicker at the top or at one side.

The actual explanation is manufacturing. Crown glass was made by spinning a blob into a disc, which comes out thicker near the centre and thinner near the rim, so every pane cut from it has a thickness that varies across it. A glazier fitting such a pane put the heavy edge downward, most of the time, because it is more stable that way — and sometimes did not.

The general point is the one the glass-transition section makes and this myth inverts. Calling a glass a liquid is a statement that no sharp thermodynamic transition separates it from one; it is not a prediction that it flows on any timescale a building lasts. A material’s classification and its behaviour are answers to different questions, and the distance between them here is thirty orders of magnitude.

What this costs an engineer #

The practical consequences are large and run in opposite directions.

Flow in a pipe carries the consequence. The throughput goes as the fourth power of the radius and inversely as the viscosity, so a temperature change of twenty kelvin — which moves an oil’s viscosity by a factor of two — moves the pumping power by the same factor. That is why a pipeline’s operating cost is a function of the weather, and why heating a crude oil line is worth doing even at the price of the heat.

A lubricating oil thins as an engine warms, which is why viscosity index — the flatness of the curve, not its height — is the property an oil is graded on, and why a multigrade oil contains polymers whose coiling counteracts the thinning. A hot oil that is too thin loses the film that separates the surfaces, and the contact then behaves quite differently.

A gas turbine’s compressor works harder on a hot day partly for this reason, and a vacuum system’s pumping speed in the viscous regime shifts with room temperature by a measurable amount.

And the sign matters for stability. A liquid whose viscosity falls with temperature can run away: a sheared film heats itself, thins, shears faster, heats more. That feedback is why lubricated contacts can fail suddenly rather than gradually, and it has no counterpart in a gas, where the same feedback is negative and self-limiting.

What the pictures cannot show #

Both formulas are fits with named ranges. Sutherland’s is good to a per cent or so for simple gases between about a fifth and twice its own constant, and poor outside that; the Arrhenius fits here reproduce their liquids between 280 and 360 kelvin and were not tested elsewhere.

Building a viscosity out of a free path is the construction that gives the square-root dependence, and everything in this essay about the exponent coming out nearer three quarters is a statement about what that construction leaves out. It assumes hard spheres with no attraction, a single collision scale, and a distribution unperturbed by the shear — and each of those, corrected, moves the exponent in the same direction.

Pressure is absent from both. Gas viscosity is nearly independent of pressure over an enormous range and is not at high density; liquid viscosity rises with pressure, by about a factor of two per hundred megapascals, which is what makes elastohydrodynamic lubrication work at all.

Nothing here is non-Newtonian. Every curve assumes the stress is proportional to the shear rate, so there is one viscosity to plot. Many fluids do not oblige, and for those the whole quantity being drawn depends on how hard it is being measured.

And the exponent chart runs to four thousand kelvin, where a real gas is dissociating and ionising and the molecules whose cross-section the chart is about have stopped existing. The curve there is the formula extrapolated, drawn to show where the hard sphere is recovered, not a prediction about hot air.

The ladder from here #

Later rungs on this anchor: the Chapman–Enskog solution of the Boltzmann equation, which gives the numerical prefactor the elementary derivation cannot; the Stokes–Einstein relation between viscosity and diffusion, and where it breaks; the pressure dependence of liquid viscosity and the elastohydrodynamic films it makes possible; and the quantum lower bound on the ratio of viscosity to entropy density, which is saturated by a quark–gluon plasma and by nothing else known.

The neighbouring ladders are momentum going sideways, where the gas mechanism is built, the viscosity that does not care how much gas there is, which is the same construction’s other surprise, and the liquid that remembers, where a single viscosity stops being enough to describe the response at all.