The markers that move because atoms are unequal

The equation that only runs forwards derives the diffusion equation from a random walk: particles that each take steps in random directions spread out so that their density obeys Fick’s law, flowing from where there is more of them to where there is less at a rate proportional to the gradient. The derivation is about one kind of particle in an empty space. In a solid there is no empty space. A metal is a lattice of atoms packed as tightly as spheres can be, and an atom in it can move only if something gets out of its way.

For most of the first half of the twentieth century it was not known what that something was. One possibility was that neighbouring atoms swap places directly, or that a ring of three or four atoms rotates together, each moving into the next one’s site. Another was that atoms move by jumping into vacancies, sites in the lattice where an atom is missing, which are always present in small numbers at high temperature. The two mechanisms predict the same Fick’s law for a single species diffusing through its own lattice. They predict different things when two different species diffuse into each other at different rates, and the experiment that decided between them was done with a bar of brass and some wires.

Two metals mixing at unequal speeds #

Ernest Kirkendall had found in the early 1940s that zinc diffuses faster in copper–zinc alloys than copper does. In 1947, with Alice Smigelskas, he made a sample designed to show what that implied. A rectangular bar of seventy-thirty brass was wound with fine molybdenum wires along two opposite faces, then plated with a thick layer of copper, so that the wires lay along the interfaces between brass and copper, embedded in the metal. Molybdenum does not dissolve in either; the wires are inert markers, fixed to the lattice planes they sit in. The bar was heated to 785 °C for fifty-six days, and sections were cut and measured at intervals.

The figure follows a model couple of two species, A on the left diffusing three times as fast as B on the right, through Lawrence Darken’s equations of 1948. The composition profile spreads, as any diffusion does, by a distance growing as the square root of time, and it is lopsided: where A is scarce, deep in B’s side, the mixture’s diffusivity is dominated by A’s own fast motion, so the mixed zone reaches further into B than into A. And the markers do not stay at the original interface. They drift steadily into A’s side.

The brass bar showed the same thing. After fifty-six days the wires on the two faces of the brass were 0.12 millimetres closer together than at the start. The brass between them had lost volume. Zinc had left it faster than copper had entered, and the lattice planes that the departing zinc had occupied had gone with it.

The marker that walks as the root of time #

The rate at which the markers move is a measurement of the mismatch between the two species.

Plotted against the square root of time, the marker’s displacement is a straight line: it moves like a diffusion distance, faster at first and slowing, with the same $\sqrt t$ that sets how far the summer reaches the cellar in December. The slope grows with the ratio of the two species’ intrinsic diffusivities. When they are equal the markers do not move at all, whatever the composition profile does. Darken’s analysis made this exact: the velocity of the lattice planes at any point is the difference between the two intrinsic diffusivities times the composition gradient there. Measuring the marker motion and the composition profile together gives both diffusivities separately — something the composition profile alone cannot give, since it depends only on their weighted sum.

The vacancies that balance the books #

Why the markers move is a matter of bookkeeping. Each species diffuses relative to the lattice with its own intrinsic diffusivity, down its own gradient.

The flux of A towards B’s side is $D_A$ times A’s gradient; the flux of B towards A’s side is $D_B$ times B’s gradient, which is the same gradient with the opposite sign. If $D_A$ is three times $D_B$, three atoms of A cross each plane for every atom of B crossing back. Atoms do not appear or vanish, and every site of the lattice is either occupied or vacant. So the net flow of atoms towards B’s side must be balanced by an equal flow of vacant sites towards A’s side. Vacancies arrive on A’s side, are filled by A atoms leaving for B’s side, and the net effect is that A’s side loses atoms and lattice sites, and B’s side gains them. The lattice itself moves, carrying its planes and any markers embedded in them.

Under a direct-exchange mechanism this could not happen. A swap moves one atom each way, so the fluxes of the two species relative to the lattice would always be equal and opposite and the markers would never move. The Kirkendall experiment therefore ruled out direct exchange and ring mechanisms as the main route for diffusion in copper and its alloys, and made the vacancy mechanism, which had been proposed but not proved, the accepted one. It was not received calmly. Robert Mehl, then the leading figure in American metallurgy, delayed publication of the paper for months because he did not believe it, and accepted it only after the result was repeated. The effect now carries Kirkendall’s name.

Where the vacancies pile up #

The vacancy flux is not uniform, and where it converges vacancies accumulate.

On A’s side, vacancies arrive faster than they leave; on B’s side, they leave faster than they arrive. A crystal can absorb or emit vacancies only at certain places — dislocations, which can climb by absorbing a row of vacancies, grain boundaries and free surfaces — and those are what allow the lattice planes on A’s side to be destroyed and those on B’s side to be created, which is what moves the markers. If the sinks cannot keep up, the excess vacancies on A’s side condense together into voids, each growing by collecting the vacancies that wander onto it, as the cluster that grows where the walkers arrive grows by catching diffusing particles. The resulting band of porosity on the fast diffuser’s side of a joint is called Kirkendall porosity.

It is a practical hazard. Gold wires bonded to aluminium pads on microchips form intermetallic compounds at the joint over years of operation at elevated temperature; the gold diffuses into the aluminium much faster than the reverse, and Kirkendall voids on the gold side weaken the bond until it fails — a failure mode called “purple plague”, after the colour of one of the compounds. Solder joints between tin-based solders and copper develop similar voids, and the reliability of electronics under heat depends partly on the vacancy flux. In other contexts the voids are made on purpose: nanoparticles of one metal coated with another and heated hollow themselves out from the inside, as the core diffuses outward faster than the shell inward, leaving a void at the centre. Hollow nanospheres for catalysis are made this way.

Atoms that move only where there is a hole #

The vacancy mechanism can be seen working directly in a lattice model with no forces in it at all.

Each move in the simulation picks a vacancy at random and a neighbouring site at random; if the neighbour is an A atom, they swap; if it is a B atom, they swap one time in three. No atom is pushed, no gradient is imposed, and no rule mentions the boundary. Yet after six hundred thousand moves the vacancies have collected on A’s side, fifty-six of them against thirty-eight on B’s, and the mixing is lopsided in the way the continuum model describes. The reason is simply that a vacancy surrounded by fast neighbours moves more often than one surrounded by slow ones, and a crowd of vacancies wandering through a mixture spends its time preferentially where its moves are frequent, in the A-rich region, and pushes A atoms out of that region as it goes.

This is the statistical content of the jiggle that proved atoms, applied inside a crystal. Einstein’s explanation of Brownian motion turned a visible random motion into evidence for invisible molecules; the Kirkendall experiment turned a visible drift of wires into evidence for invisible holes. In both cases the argument was quantitative: the size of the effect could only be explained by the mechanism proposed.

The walk a vacancy takes back #

The vacancy mechanism has a consequence for single atoms that makes their diffusion slower than a free random walk would be. An atom that has just jumped into a vacancy has, by construction, a vacancy right behind it, in the site it just left. Its next jump is therefore more likely than chance to be straight back, undoing the first. The walk that comes home shows that a random walker in one or two dimensions returns to its starting point with certainty and in three dimensions with a definite probability; a vacancy is such a walker, and each of its returns reverses an atom’s step. The atom’s steps are correlated, and its diffusion is reduced by a correlation factor, 0.78 for the face-centred cubic lattice of copper, computed by John Bardeen and Conyers Herring in 1951.

In a two-species alloy the correlation is different for the two species, because a vacancy beside a fast atom is likely to exchange with it again rather than with a slow neighbour, and Darken’s simple equations, which ignore correlation, are corrected by terms that depend on it. The corrections change the predicted marker shift by tens of per cent in some alloys, and measuring the shift precisely became, after 1947, one of the ways of measuring the correlation itself.

The same markers on a growing oxide #

Markers turned out to answer a question in an entirely different field. When a metal oxidises at high temperature, a scale of oxide grows on its surface, thickening as the square root of time, because the reaction requires atoms to diffuse through the scale that has already formed — the parabolic law that the ice that grows more slowly the thicker it gets derives for ice on a pond, where heat rather than atoms must cross the layer already grown. Which atoms cross? Either metal ions diffuse outward through the scale to meet oxygen at its outer surface, or oxygen diffuses inward to meet the metal underneath.

Inert markers placed on the original metal surface decide it. If the metal moves outward, new oxide forms on top of the markers and they end up buried at the bottom of the scale; if oxygen moves inward, new oxide forms beneath them and they ride on the top. Experiments of this kind, in the 1950s, showed that iron, copper and nickel oxidise mainly by outward diffusion of metal through vacancies in the oxide’s metal sublattice, while titanium and zirconium oxidise mainly by inward diffusion of oxygen. The design of oxidation-resistant alloys for turbine blades rests on that distinction, and the method is Kirkendall’s: put an inert marker where something is changing, and see which way the lattice carries it.

The gradient that is not a force #

There is a subtler lesson in the bookkeeping. Diffusion is often described as driven by a concentration gradient, as if the gradient pushed atoms down it. In a two-species lattice the fluxes depend on the gradients, but the gradients drive nothing: the atoms make random jumps into vacancies, and the flux is the statistical excess of jumps in one direction over the other, set by how many atoms and vacancies are on each side. A flux of vacancies arises not because anything pushes vacancies, but because the A atoms’ greater jump rate makes vacancies move through A-rich regions faster, and the net result of random jumps is a flow towards where they move most readily.

Thermodynamics generalises this. The gradient that drives the other thing shows that in a mixture a gradient of one quantity drives flows of others, with coefficients tied together by Onsager’s reciprocity, and the Kirkendall effect is one of those cross-couplings: a composition gradient produces a flow of lattice sites. The gas that flows towards more of itself shows the same kind of thing in a gas mixture, where a component can flow up its own gradient for a time because the others’ fluxes carry it along. In both cases the intuition that each species simply flows down its own gradient independently of the others fails, because the species share a space — a lattice of sites, or a vessel of fixed volume — and the space keeps its own accounts.

Measuring a hole that is not there #

The vacancy concentration in a metal near its melting point is about one site in ten thousand, and it falls exponentially as the metal cools, with an energy of formation of about one electronvolt. It can be measured in several ways. Heating a metal bar and measuring both its length and the spacing of its lattice planes by X-ray diffraction gives two different expansions: the bar lengthens slightly more than the lattice, because new sites are being created as vacancies at the surface. Simmons and Balluffi did this for aluminium in 1960 and obtained the vacancy concentration directly from the difference. Positrons implanted into a metal annihilate more slowly when they are trapped in vacancies, where there are fewer electrons, and their lifetimes count the holes. And quenching a metal rapidly from high temperature freezes vacancies in, raising its electrical resistance until annealing lets them escape.

The Kirkendall experiment measured something different: not the number of vacancies but their flow. It is the only one of these that shows vacancies doing what the theory of diffusion needs them to do, which is to move atoms.

Constant diffusivities and perfect sinks #

The continuum model takes the two intrinsic diffusivities as constants, independent of composition, and the molar volume as the same for both species; in real alloys both diffusivities change with composition, often by orders of magnitude, and a difference in atomic size changes the sample’s volume as the mixing proceeds. It assumes that vacancy sinks and sources are everywhere and perfectly efficient, so that the vacancy concentration stays at equilibrium and the lattice planes move freely; the accumulation figure shows where the sinks must work hardest, not where voids would actually form. The marker in the model sits exactly at the original interface, which is also the Matano plane only when the diffusivities are equal. The lattice simulation is two-dimensional, with a vacancy fraction hundreds of times larger than any real metal’s so that the effect is visible in a small picture, and its swap probabilities stand in for the ratio of jump rates.

Still open: what the voids decide in a joint #

Whether Kirkendall voids form in a given joint, and where, depends on the competition between the vacancy flux and the efficiency of the sinks that absorb it, which depends in turn on the joint’s microstructure: its grain size, its dislocation density, its impurities and the stresses that develop as one side shrinks and the other swells. Predicting the life of a solder joint or a wire bond from its composition and temperature history requires modelling all of this together, and the models are still largely empirical. How the stress generated by the moving lattice feeds back on the diffusion itself — compressive on one side, tensile on the other, each altering the vacancy concentration — is an active question in the reliability of the ever-smaller joints in modern electronics.

The habit worth carrying away is to ask what has to move for something to move. When a fast-diffusing metal and a slow one mix, more atoms cross one way than the other, so lattice sites must flow back as vacancies, and inert markers at the weld drift into the fast side as $\sqrt t$ — the 0.12 mm by which Kirkendall’s wires closed up in brass after 56 days was the proof that diffusion in metals is atoms jumping into holes. A wire that moves is a ledger that does not balance unless something invisible is moving the other way.