The gradient that drives the other thing

The transport laws come one to a subject. A concentration gradient drives a flux of matter, with a diffusion coefficient. A temperature gradient drives a flux of heat, with a thermal conductivity. A velocity gradient drives a flux of momentum, with a viscosity. Each has its own name, its own coefficient, its own chapter.

They are not independent. A temperature gradient drives a flux of matter as well as of heat, and a concentration gradient drives a flux of heat as well as of matter — and the two cross coefficients are equal.

The first of those is the Soret effect, noticed by Ludwig in 1856 and measured by Soret in 1879, and it is easy to miss because it is small. A few per cent of separation across a hundred degrees is what a liquid gives, and a gas gives rather more.

The steady state is a balance, not a stillness #

The figure is worth reading carefully because it shows a state that is often described wrongly.

There is no flow anywhere in it. The mixture has come to rest and its composition will not change again. But the reason is not that nothing is driving anything: the temperature gradient is still there, still pushing one component toward the cold end, and the concentration gradient it has built up is still pushing back. The two cancel exactly, at every point, and the profile is where that happens.

That is a different kind of steady state from equilibrium, and the difference matters. At true equilibrium every process and its reverse balance individually, which is detailed balance. Here they do not: heat is flowing steadily through the cell from hot to cold, entropy is being produced continuously, and the composition is stationary only because two matter fluxes cancel. The system is dissipating and unchanging at the same time.

Distinguishing the two is worth doing because they behave differently. An equilibrium state can be reasoned about with free energies; a non-equilibrium steady state cannot, and the whole apparatus of irreversible thermodynamics exists because so much of the world is in one.

The relation nobody could have guessed #

Write the two fluxes as linear in the two gradients and there are four coefficients: the diffusion coefficient, the thermal conductivity, and two cross terms. The cross terms describe different experiments and involve different measurements, and there is no obvious reason for them to be related.

Onsager showed in 1931 that they are equal.

The argument is one of the more surprising in the subject, because it reaches a statement about macroscopic transport from a statement about microscopic reversibility. The underlying equations of motion run the same forwards and backwards in time, so a fluctuation in one variable is correlated with a later fluctuation in another in exactly the way the second is correlated with a later fluctuation in the first. Those correlations are what the transport coefficients are, by the fluctuation relation — and the symmetry of the correlations becomes a symmetry of the coefficients.

What that gives is a genuine economy. A system with $n$ transport processes has $n^2$ coefficients, and the relations reduce the count to $n(n+1)/2$ — for two processes, four coefficients become three. Every one of those savings is an experiment that does not have to be done, and a check on one that has.

The relations were the first result of non-equilibrium thermodynamics and remain the most useful, and they have been confirmed in a dozen unrelated systems: heat and matter in a mixture, heat and charge in a thermocouple, matter and charge in an electrolyte, and the coupling between diffusion of different species in a multicomponent solution.

Where the coupling is not small #

A separation of a few per cent is not useful and a cascade of them is. What makes the cascade possible is a piece of engineering rather than of physics, and it is worth describing because it is so economical.

Put the hot wall and the cold wall close together and make the whole thing tall. The Soret effect separates the mixture across the narrow gap; buoyancy drives the warm fluid up the hot face and the cool fluid down the cold face; and because the rising fluid is enriched in one component and the falling fluid in the other, the convection carries one component to the top of the column and the other to the bottom. Every effective stage compounds what the one below achieved.

Clusius and Dickel built the first in 1938 — a vertical glass tube with a heated wire down the middle — and separated the isotopes of chlorine, which nothing else at the time could do. The apparatus is a tube, a wire and a power supply, and the mass difference it exploits is three parts in seventy.

The method’s limitation is what it costs in time and heat rather than in complexity. A column takes days to reach its steady state, because the transport along it is slow, and the whole of the heat put in is dissipated. That is why it lost to the centrifuge for uranium and remains standard for separating isotopes of light gases in quantities of grams.

The cross effect nobody expects #

A related effect belongs here because it is the same failure of an intuition.

Join two containers at different temperatures with a wide tube and they come to equal pressure, as anybody would expect. Join them with a hole smaller than the mean free path and they do not: they come to rest at pressures in the ratio of the square roots of their temperatures, because what crosses a small hole is a sample of the molecules weighted by their speed rather than a flow of bulk gas.

The two situations differ only in the size of a hole, and the intuition that pressure equalises is a statement about the first. It has no standing in the second, and the failure is complete rather than a correction: the pressure ratio is a factor of three at a ten-fold temperature ratio, which nothing about the wide-tube reasoning would tolerate.

The practical consequence is a nuisance in every vacuum measurement made at a temperature different from the gauge’s, and it has to be corrected for. It is also, run deliberately, a pump with no moving parts, which is how a Knudsen compressor works.

Comparing the two temperature differences says which knob is worth turning, and the answer is not the obvious one.

The composition profile is a logistic in the temperature, so raising the temperature difference moves the ends along a curve that flattens: the first forty degrees are worth much more than the second forty. Doubling the temperature difference does not double the separation, and there is a ceiling that no temperature difference reaches.

Stacking stages is different in kind. Each stage acts on what the previous one produced, so the effect compounds rather than saturating — the logit of the composition grows in proportion to the number of stages, without limit, and the separation approaches complete because the composition does. That is why the practical apparatus is tall rather than hot.

The distinction between an effect that saturates and one that compounds is worth carrying beyond this. A single-stage effect with a ceiling becomes an unbounded one as soon as it can be cascaded, and finding a way to cascade it is usually worth more than improving it — which is the whole of what a distillation column, an isotope cascade and an amplifier chain have in common.

What a coupled flux is, generally #

The pattern is worth stating in the abstract because it applies far beyond mixtures.

Wherever a system has several ways of being out of equilibrium — a temperature difference, a concentration difference, a voltage, a pressure — each of them drives every flux, not only its own. The coefficients form a matrix; the diagonal entries are the familiar transport coefficients; the off-diagonal ones are the cross effects; and the matrix is symmetric.

The most familiar instance is thermoelectric. A temperature difference across a conductor drives a current, which is the Seebeck effect and is what a thermocouple is. A current drives a heat flow, which is the Peltier effect and is what a solid-state cooler is. The two coefficients are related by exactly Onsager’s relation — the Kelvin relation, found empirically by Thomson in 1854 and unexplained for seventy-seven years — and every thermoelectric device is designed with both of them.

That is a good illustration of what a general principle buys. Thomson’s relation was a puzzle: two coefficients from two quite different measurements, agreeing to within experiment, for no reason anybody could give. Onsager’s derivation did not improve the number. It showed that the agreement was compulsory, that it would hold in systems nobody had looked at, and that it followed from time-reversal symmetry — which is a statement about the microscopic world and not about thermocouples at all.

The separation that is measured to find something else #

Effusion and thermal diffusion are the two ways of separating a mixture without a chemical difference between its components, and putting them beside each other explains why the second is now used more as a measurement than as a method.

Effusion’s dependence on mass is exact and simple: the enrichment is the square root of the mass ratio, and it depends on nothing else about the molecules. That makes it a good separation method and an uninformative measurement, since it reports a mass anybody already knows.

Thermal diffusion depends on the interaction between the molecules as well as on their masses, in a way that no simple theory captures, and its coefficient can even change sign as the temperature or the composition is varied. That makes it a poor separation method for anything where a centrifuge will do — and a genuinely informative measurement, since the coefficient is sensitive to a part of the intermolecular potential that little else reaches.

So the modern use of the Soret effect is in reverse. It is measured in polymer solutions, colloids and biological mixtures to learn about the interactions, and it is used as a way of moving small amounts of material about in a microfluidic device, where a temperature gradient is easy to make and a pump is not.

Where the model stops #

Everything here is linear in the gradients. The fluxes are taken proportional to the driving forces, which is excellent for small departures from equilibrium and fails for large ones. Far from equilibrium the coefficients themselves depend on the gradients, the reciprocal relations do not hold, and there is no general theory in their place.

The Onsager relations require the microscopic dynamics to be time-reversible. In a magnetic field it is not, quite: reversing the motion also requires reversing the field, and the relations acquire a sign — the coefficient at field $B$ equals the transposed coefficient at $-B$. That modification is not a technicality; it is what allows the Hall effect and its thermal counterparts to exist at all.

The Soret coefficient is not predicted by any of this. Onsager’s relation says the two cross coefficients are equal and says nothing about how large either is, or even which way a given component moves. Predicting the sign of a Soret coefficient from a molecular model remains genuinely difficult, and for many mixtures it changes sign with temperature or composition for reasons that are argued about.

And the column’s performance is a transport calculation the essay does not do. The separation depends on the gap width, the viscosity, the diffusion coefficient and the temperature difference in a combination with a sharp optimum, and building one that works needs that calculation rather than the exponential drawn here.

How the relation is checked #

An exact relation between two coefficients invites a measurement, and the way such a check is actually made is worth describing because it is harder than it sounds.

The two experiments are not symmetric in difficulty. Measuring the Soret coefficient means imposing a temperature difference and waiting for a composition to settle, which takes hours and needs a composition measurement good to a fraction of a per cent. Measuring the Dufour coefficient — the heat carried by a concentration gradient — means imposing a composition difference and detecting the temperature difference it produces, which is a few millikelvin and disappears within seconds as the concentrations diffuse.

So the check has usually been made in gases, where the Dufour effect is large enough to see and the timescales are seconds rather than hours. Waldmann’s measurements in the 1940s established the equality in several gas pairs to a few per cent, which is about the accuracy the harder of the two measurements allows.

That asymmetry is the ordinary situation with a reciprocal relation and it is why they are so valuable. The point of the relation is not that both coefficients can be measured; it is that only one of them needs to be, and it is the easier one. A relation whose two sides were equally easy to measure would be a curiosity; one whose sides differ by orders of magnitude in difficulty is a tool.

What the pictures cannot show #

The concentration profile is drawn at steady state and hides the time it takes to get there. The approach is diffusive, so the time goes as the square of the cell’s width divided by the diffusion coefficient — hours for a centimetre of liquid, and days for a column. Every measurement of a Soret coefficient is a slow measurement, and separating the true steady state from a partially relaxed one is most of the experimental difficulty.

Nor does anything here show the entropy being produced. A cell with a temperature gradient across it is dissipating steadily, at a rate set by the heat flowing and the temperature difference, and the separation it maintains is being paid for continuously. A figure of the entropy production would show that the steady state is not free, which the composition profile makes it look.

What the effect looks like in a gas #

Gases are where the mechanism can be said out loud, and it is worth doing because the liquid case cannot.

In a gas the Soret effect comes from the fact that a molecule’s mean free path depends on its speed, and a molecule arriving at a point from the hot side arrives faster than one arriving from the cold side. If the two species differ in mass or in the way they scatter, the imbalance is not the same for both, and one species drifts. Chapman worked the calculation out in 1917 from the same kinetic theory that gives viscosity and thermal conductivity, before anybody had measured the effect — and Dootson found it within the year.

Two features of that story are worth noting. It was predicted from a theory built for something else, which is the strongest kind of confirmation a theory gets. And the prediction was quantitative: the coefficient depends on the exact form of the intermolecular force, so measuring it tests a molecular model at a level of detail that viscosity does not reach.

In a liquid none of that survives. There is no mean free path, molecules are always in contact, and no comparably clean calculation exists — which is why the sign of a liquid’s Soret coefficient still cannot be predicted reliably and why the same mixture can separate one way at one temperature and the other way at another.

Where the ladder goes next #

The diffusion ladder began with the equation that only runs forwards, went through the jiggle that proved atoms, the walk that comes home, the second experiment that cannot disagree and the summer that reaches the cellar in December. This rung asks what else a gradient drives. The rungs after it: the entropy production rate, which is the quantity all of irreversible thermodynamics is organised around and which is a sum of flux times force; the minimum entropy production principle and its limits; and diffusion in a multicomponent mixture, where the matrix has off-diagonal terms large enough that a species can diffuse up its own concentration gradient.

The habit worth carrying away is to count the couplings. A system with several ways of being out of equilibrium has a matrix of transport coefficients rather than a list, the off-diagonal entries are real effects rather than corrections, and they are related to one another by a symmetry that costs nothing to assume and saves half the measurements.