The gas that flows towards more of itself

The gradient that drives the other thing found that a transport law is never alone. A temperature gradient drives a flow of matter as well as of heat, the cross coefficients are equal, and a system with several ways of being out of equilibrium has a matrix of transport coefficients rather than a list. It named the case where that matrix does something no single law would allow: a mixture of several species, whose off-diagonal diffusion coefficients can be large enough that a species diffuses up its own concentration gradient.

That sounds as though it should be impossible. The equation that only runs forwards derived diffusion from a random walk, and a crowd of random walkers flows from where there are more of them to where there are fewer, as surely as heat flows from hot to cold. The walk is not wrong. What it leaves out is that the walkers of one species collide with the walkers of the others, and in a mixture of three those collisions can push a species where its own walk would never take it.

Three gases, two bulbs and a tube #

The cleanest demonstration is an experiment Duncan and Toor carried out in 1962. Two glass bulbs of about 78 cubic centimetres, joined by a capillary 86 millimetres long and 2 millimetres wide, are filled at the same pressure and temperature: one with nitrogen and carbon dioxide in equal parts, the other with nitrogen and hydrogen in equal parts. Nitrogen starts at almost the same fraction in both, 0.50086 against 0.49879. By Fick’s law, nitrogen has essentially nothing to do.

The calculation that says otherwise treats diffusion as a balance of forces. Each species is pushed along the tube by the gradient of its own mole fraction, and held back by friction against every other species it moves relative to. The friction between a pair is inversely proportional to that pair’s binary diffusivity, the number kinetic theory supplies for two gases alone: 83.8 mm²/s for hydrogen with nitrogen, 68.0 for hydrogen with carbon dioxide, and only 16.8 for nitrogen with carbon dioxide, whose heavy molecules drag hardest on each other. Written for each species, the balance is the Maxwell–Stefan equations, and they can be integrated along the tube at every moment and solved for the fluxes that carry each bulb’s composition to the other’s.

Fick’s law for nitrogen, drawn dashed, lets the two nearly equal fractions drift together and do nothing else. The Maxwell–Stefan solution does something else entirely. Hydrogen, the fast gas, pours out of its bulb; carbon dioxide, the slow one, moves the other way; and nitrogen, gripped far more strongly by the carbon dioxide than by the hydrogen, is carried along with the carbon dioxide into the hydrogen bulb. It keeps flowing into the bulb that has more nitrogen for six and a half hours, until the difference it has built — nearly fifteen per cent of the mixture — is large enough to push back against the drag. Only then does it turn round and relax the way Fick’s law would have predicted from the start.

Duncan and Toor measured nitrogen doing exactly this, and the experiment has been the standard test of multicomponent diffusion theory since. The computed curves come from nothing but the three binary diffusivities and the bulbs’ dimensions.

Why two components are not enough #

In a mixture of only two species, none of this can happen, and seeing why makes the three-species case clearer. With two species at uniform pressure, the mole fractions add to one, so there is only one independent gradient; whatever drags one species along one way drags the other back the same way, and the friction between them has nothing to push against but the gradient itself. The Maxwell–Stefan equations collapse to Fick’s law with the single binary diffusivity, and the flux always runs down the gradient.

A third species adds a second independent gradient, and each flux then depends on both. The nitrogen flux in the bulbs is a sum of a term from nitrogen’s own gradient and a term from hydrogen’s — carbon dioxide’s is fixed by the other two — and the second term has no reason to be small. Working the friction balance at the starting composition gives a coefficient for hydrogen’s gradient nearly twice the size of nitrogen’s own and of the opposite sign, because hydrogen, rushing one way, is balanced by carbon dioxide rushing the other, and carbon dioxide grips nitrogen far more strongly than hydrogen does.

There is one way a binary mixture can defy Fick’s law, and it is thermodynamic rather than frictional. The true driving force is the gradient of chemical potential, and in a mixture whose components prefer their own kind strongly enough, the chemical potential can fall as the concentration rises. Then the effective diffusion coefficient, which carries that slope as a factor, changes sign, and a single species runs up its own gradient with no third party involved. That is the case the limits section below returns to.

Why the flux turns round when it does #

The barrier, the moment nitrogen’s flux passes through zero while its difference is largest, is not a coincidence of the figure; it is the point at which the two terms cancel. Early on, hydrogen’s gradient is at its steepest and its term wins outright. Hydrogen, the fastest gas, equalises between the bulbs with a characteristic time of about three and a half hours, and as its difference decays, the drag it transmits through carbon dioxide decays with it. Meanwhile nitrogen’s own difference, built up by the drag, pushes back harder the larger it gets.

At 6.6 hours the two are equal. Nitrogen’s difference of 0.1468 is exactly what the remaining hydrogen gradient can hold up, and from then on nitrogen’s own term dominates and the ordinary relaxation begins — slowly, because nitrogen and carbon dioxide, the pair with the smallest diffusivity, set the pace of what remains, and the approach to uniform composition takes the rest of the thirty hours in the figure and beyond. The uphill phase lasts as long as it does because hydrogen is fast and carbon dioxide slow; a mixture in which the drag partner were as quick to equalise as the driving gas would show a smaller, shorter excursion.

Four behaviours on one path #

The strange part of the experiment is not only that nitrogen goes uphill. At different times it does each of the things a single diffusion law forbids.

Plotting nitrogen’s flux against its own difference turns the history into a path, and Fick’s law into a rule about which parts of the plane the path may visit: the two quadrants where flux and difference have the same sign, along a straight line through the origin. The path visits all four regions the multicomponent literature names. At the start it sits almost on the vertical axis, a large flux with no gradient, which is called osmotic diffusion. It then enters a forbidden quadrant, flux from less to more, which is reverse diffusion. It crosses the horizontal axis with a large gradient and no flux, a diffusion barrier. And it finishes in an allowed quadrant, ordinary diffusion, heading home.

None of the four needs anything but friction between the gases. In a matrix form of the same equations, each species’ flux is a sum over every species’ gradient. For nitrogen at the starting composition, the coefficient multiplying the hydrogen gradient is larger than the one multiplying nitrogen’s own, and opposite in sign — so while hydrogen’s gradient lasts, it outvotes nitrogen’s. The barrier is simply the moment when the two contributions cancel.

What the second law actually forbids #

A flow from less to more has the look of a second-law violation, and the second law deserves to be checked rather than trusted.

The rate at which diffusion produces entropy is a sum over species of each flux times the force driving it, the gradient of its chemical potential divided by the temperature. The Maxwell–Stefan equations guarantee that the sum is never negative, because each friction term contributes a square. They guarantee nothing about any single term. For the six and a half hours that nitrogen flows uphill, its share is negative: nitrogen alone is moving in the direction that lowers entropy. Hydrogen and carbon dioxide, running steeply down their own gradients, produce about a hundred times more than nitrogen destroys, and the total never comes near zero.

The second law is a statement about a sum of products of fluxes and forces, not about each product, and that is exactly the room a matrix of transport coefficients needs for its off-diagonal entries. It is the same room the second experiment that cannot disagree found for a thermocouple, where heat can be pumped from cold to hot by a current flowing downhill in voltage. Onsager’s symmetry, which that essay derived from the reversibility of molecular motion, is what makes the multicomponent diffusion matrix consistent with the friction picture: the drag of nitrogen on carbon dioxide and of carbon dioxide on nitrogen are one coefficient, as a mutual inductance is the same both ways.

Carbon that climbs into the richer steel #

Uphill diffusion needs a second species, but not necessarily a second one that moves. In 1949 Lawrence Darken welded together two steels with almost the same carbon content — 0.478 per cent in one, 0.441 per cent in the other — but with 3.8 per cent silicon in the first and almost none in the second, and held the bar at 1050 °C for thirteen days. At that temperature carbon diffuses through iron quickly; silicon barely moves.

The carbon moved from the steel that had more of it into the steel that had less, and it did not stop when the two were equal. It kept going, and at the weld it built a sharp step with far more carbon on the silicon-free side. Silicon makes carbon less welcome in iron — it raises carbon’s chemical potential at any given concentration — so a carbon atom on the silicon side is, thermodynamically, at a higher level than one at the same concentration on the other side. Carbon runs down the gradient of its chemical potential, and across the weld that gradient points the opposite way to the concentration gradient.

The model in the figure contains only that: a carbon flux driven by its chemical potential, silicon fixed in place, and a factor chosen so that 3.8 per cent silicon doubles carbon’s activity. It reproduces the step, conserves the carbon, and heads for a steady state in which the carbon concentration on the two sides differs by exactly that factor of two while nothing flows at all. At equilibrium the concentration is not uniform; the chemical potential is.

The effect is not a laboratory curiosity. Where a low-alloy steel is welded to a high-chromium one — in the pipework of power stations and refineries — carbon migrates across the joint during years at operating temperature, leaving a carbon-starved, weakened band on one side and a hard, brittle one on the other, and joints of that kind are designed around it.

Where species climb outside a laboratory #

The welded-steel case has relatives wherever one element changes another’s chemical potential and one of them is much more mobile. In silicon devices, dopant atoms carry charge, and the electric field set up by a heavily doped region pushes other charged dopants around; measured profiles routinely show one dopant piling up against a gradient where another is concentrated, and process models have to include the coupling to predict where junctions end up. In geology, minerals and silicate melts preserve compositional profiles from the diffusion that happened as they cooled, and some of those profiles run uphill for one element because of the gradients of others, which matters because the widths of such profiles are used as clocks for how long a magma sat before it erupted.

What these cases share with the two bulbs is that a single-species law, fitted to an ordinary profile, would give a diffusivity that is negative across part of the sample. That is the practical signature of coupling: a fitted coefficient that comes out with the wrong sign is not a bad measurement, it is a second gradient left out of the model.

Where the picture stops #

The gases were ideal and the capillary quasi-steady. The chemical potentials were those of an ideal-gas mixture, and the tube, holding a few tenths of a per cent of the gas, was taken to adjust instantly to the bulbs. Both are excellent for Duncan and Toor’s gases; neither would be for a liquid, where the diffusivities depend strongly on composition and the chemical potentials depend on how the molecules interact.

The binary diffusivities were constants. In a gas they are nearly independent of composition, which is why three numbers suffice. In liquids and solids the Maxwell–Stefan coefficients vary with composition and must be measured across it.

Temperature and pressure were uniform. A capillary narrow enough for molecules to strike its walls more often than each other changes the friction terms, and a temperature gradient adds the thermal diffusion of the previous kind. Neither was included.

The steel model had one parameter for all of chemistry. A single exponential factor stood for silicon’s effect on carbon, silicon was frozen, and the diffusivity was held constant across the weld. The real bar’s carbon profile depends on how those vary, and the figure reproduces the phenomenon rather than Darken’s measured numbers.

And the diagonal terms stayed positive. In every case here a species’ own coefficient was positive and the uphill flow came from its neighbours. If a mixture’s own diffusion coefficient becomes negative — which happens inside the unstable region of a mixture about to separate, the spinodal of a phase diagram — a species diffuses up its own gradient with no help at all, and small fluctuations grow into domains. That is spinodal decomposition, and it is uphill diffusion with nobody else pushing.

What the figures leave out #

The two-bulb figure shows nitrogen only. Hydrogen and carbon dioxide are what drive the whole effect, and their curves — hydrogen equalising within a few hours, carbon dioxide over a day — are the context that makes nitrogen’s behaviour intelligible; they were left out to keep the one surprising curve legible.

The steel figure draws the silicon as a sharp step. In the real bar silicon also diffuses, a thousand times more slowly than carbon, and over thirteen days its step softens by a fraction of a millimetre; over years at service temperature that softening is part of how a dissimilar weld ages.

Still open: predicting the coefficients of a liquid #

For gases, the three binary diffusivities in the figures come from the kinetic theory that proved atoms and its descendants, and a mixture of any number of gases can be predicted from its pairs. For liquids there is no such theory. The Maxwell–Stefan coefficients of a liquid mixture depend strongly on composition, the chemical potentials depend on molecular interactions that are themselves hard to compute, and near a composition at which the mixture would separate, the thermodynamic factors that convert one into the other go to zero.

Molecular simulation can now compute multicomponent diffusion coefficients for simple liquids, and the results often agree with experiment to tens of per cent; for mixtures of molecules that associate, for electrolytes, and for silicate melts in which uphill diffusion has been observed in geological samples, predictions from first principles remain unreliable, and the coefficients that decide whether a species goes uphill are still measured one system at a time.

The habit worth carrying away is to ask what a gradient is a gradient of. Matter flows down the slope of its chemical potential, and the concentration is only a proxy for that slope when nothing else in the mixture changes it — a proxy that fails the moment there are three species in a tube or two elements in a steel.