The rings a reaction lays down in a gel

The equation that only runs forwards found that diffusion smooths everything it touches: a drop of ink spreads, sharp edges blur, and the concentration of anything diffusing tends towards uniformity. The heap that sets itself alight followed diffusion of heat competing with a reaction that makes heat, and found a threshold beyond which the heap runs away. This essay follows diffusion competing with a reaction that makes a solid, and finds something that seems to contradict the first essay entirely: diffusion, together with a reaction and a threshold, producing a sharp, regular pattern out of a perfectly uniform start.

In 1896 the German chemist Raphael Liesegang put a drop of silver nitrate on a layer of gelatin that contained potassium dichromate. As the silver diffused outward into the gel and met the dichromate, red silver dichromate precipitated — not as a spreading red stain but as a set of concentric rings, separated by clear gel. The same thing happens in a test tube of gel with one salt soaked in and the other poured on top: bands form down the tube, close at the top and further and further apart below. The pattern is found in nature in banded agates and some iron-stained sandstones, in kidney stones, and in thin films during the manufacture of semiconductors, where it is usually a nuisance.

Bands where a smear was expected #

The figures here come from a model of the experiment computed directly. One electrolyte, A, is held at a fixed concentration at the entrance of a long tube of gel and diffuses in. The gel already holds a uniform concentration of a second, B. Where they meet they react to form a product, C, which is soluble and diffuses too. And C becomes solid, D, only under a condition that is the heart of the matter: a new solid starts only where the concentration of C passes a nucleation threshold $c^*$, well above its solubility $c_s$; but once a grain of solid is present, C deposits on it wherever its concentration exceeds the lower value $c_s$.

The result is a set of sharp bands. Close to the entrance they are crowded; down the tube they spread out. Each band holds more solid than the one before, roughly in proportion to its distance. Between them the gel is clear: the reaction happened there too, the product C was made there, but no solid formed — the C made between bands ended up in the bands on either side.

A solid that will not start #

The two thresholds are not an invention of the model. The barrier a new phase has to climb found that a new solid cannot form at its solubility, because the first tiny cluster of it has a large surface for its volume and the surface costs energy; it forms only when the solution is supersaturated enough to pay for the surface. Once a solid surface exists, the cost has been paid, and growth onto it needs only ordinary saturation. That gap between starting and growing is what turns a smear into bands.

As A advances into the gel, the reaction zone where A meets fresh B moves forward and produces C. Ahead of the existing bands, C accumulates, its concentration rising until somewhere it passes the threshold, and a new band nucleates. From then on the new band is a sink. C diffuses into it from both sides and deposits, so the concentration of C in a whole region around the band is pulled down to near the solubility. The reaction zone keeps moving forward, keeps making C, and as long as it is close to the band, that C diffuses back and feeds the band instead of reaching the threshold. Only when the reaction zone has moved far enough away that the band’s pull no longer reaches it can C accumulate again, pass the threshold, and nucleate the next band.

The gap between bands is therefore the reach of the last band’s drain at the moment the next one forms. It is not a property of the gel or the reaction alone, and it is not set once: it grows as the pattern develops, because the reaction front slows as it moves further from the source of A.

The time law #

The bands appear where the reaction front is when they appear, and the front moves by diffusion. A substance diffusing from a fixed source into a medium that consumes it advances as the square root of time — the same law that the ice that grows more slowly the thicker it gets found for the thickness of ice on a pond and the summer that reaches the cellar in December for heat soaking into the ground. So each band’s position, plotted against the square root of the time it formed, falls on one straight line. Harmon Morse and George Pierce stated the law in 1903. It is the easier of the two laws of Liesegang patterns to explain, and it says nothing about where the bands are, only how fast the place where they can form moves along.

How slow a real tube is #

The model’s units hide how slow the real experiment is. Small ions in water diffuse with coefficients of about $10^{-9}$ square metres per second, and a gel slows them only a little, because its fibres occupy a small fraction of its volume. A front that advances as $\sqrt{Dt}$ therefore covers a millimetre in about a quarter of an hour, a centimetre in about a day and ten centimetres in about four months. The first bands in a test tube appear within hours, close together near the top; the later ones take weeks, and the last bands in a long tube can take a year. The jiggle that proved atoms found the same diffusion coefficients in the wandering of a pollen grain, and the slowness is the same slowness: every molecule reaches a centimetre only by taking an astronomical number of steps a fraction of a nanometre long.

That slowness is also why the gel is needed. In plain water, the solution near the reaction zone changes its density as salts are consumed and product forms, and density differences drive convection that stirs the tube far faster than diffusion could; precipitate grains, being denser than water, settle as they form. A gel suppresses both. Its tangled fibres stop the liquid flowing on scales larger than their spacing while leaving the ions free to diffuse between them, and they hold each grain of precipitate where it forms. The gel turns a stirred, settling suspension into a medium in which diffusion is the only transport, which is the condition for the laws drawn here. Liesegang bands have been grown without gels in thin capillaries and in microgravity, where convection is suppressed in other ways, and they follow the same laws.

The spacing law #

The second law is the remarkable one.

After the first few bands, the ratio of each band’s distance from the entrance to the previous band’s settles to a constant. The positions form a geometric progression: $x_{n+1} = (1+p)\,x_n$, with the spacing coefficient $p$ the same for every band. Kazimierz Jabłczyński found this rule in 1923, and it is followed by real Liesegang patterns in experiment after experiment, typically with $p$ between about 0.05 and 0.4.

It can be understood from the mechanism. If every length in the problem grows in proportion as the pattern advances — the distance of the front from the entrance, the reach of a band’s drain, the region over which C has to accumulate — then each band forms at a position that is a fixed multiple of the one before. Diffusion has no length of its own; it converts times into lengths as their square roots, and a process controlled by diffusion from a fixed source has a structure that looks the same at every scale, magnified. The pattern is self-similar, and a geometric progression is what a self-similar sequence of points looks like.

The same arithmetic explains why the bands grow heavier. Each band collects the product made in the stretch of tube between it and the previous band, and that stretch grows in proportion to the band’s distance. A band twice as far down the tube holds twice the solid.

The threshold sets the scale #

The spacing depends on how far the product must be supersaturated before it nucleates. A higher threshold means the reaction zone has to move further ahead of the last band before C can build up enough, and a band that finally nucleates at a higher concentration drains a wider region. The bands come further apart and fewer of them fit in the tube. That dependence — the Matalon–Packter law, relating the spacing coefficient to the concentrations and to the threshold — is how the same two salts can give coarse rings in one gel and fine ones in another. Anything that changes how hard nucleation is changes the pattern: the gel’s density, impurities that seed crystals, the concentration of the inner salt.

The sensitivity has a practical use. Because the spacing is so dependent on nucleation, Liesegang patterns have been proposed and used as a way of making regular microstructures without lithography: a reaction that lays down conducting or optical material in bands, spaced by a law that can be tuned by concentration rather than drawn by a machine. Experiments in which an electric field is applied across the gel move the bands and change their spacing, which has been used to make patterns with spacings chosen in advance.

A rhythm with no clock #

It is worth being precise about what is surprising. A periodic pattern usually needs something periodic to produce it: a wave, a vibration, a clock, a template. A Liesegang tube has none. Its source holds A at a constant concentration; the gel and its B are uniform; the reaction and the diffusion are smooth in time. The only discontinuous element is nucleation, which switches on sharply at one concentration and then, through the drain it creates, switches itself off for a while. A threshold followed by depletion is the recipe for a relaxation oscillation — the same recipe as a dripping tap, which fills until a drop breaks free and then must fill again, the rhythm the map a dripping tap turns out to be followed from regular dripping into chaos — and here the oscillation is laid out in space by the moving front instead of in time.

The analogy can be pushed one step further. A dripping tap’s rhythm is set by how long the tap takes to refill after a drop leaves; the Liesegang rhythm is set by how long the front takes to escape the last band’s drain. Because the front slows as it goes, the “refill” takes longer each time, and the rhythm slows geometrically — which is the spacing law seen as a clock running ever more slowly.

Patterns from a uniform start #

Liesegang rings are among the oldest known examples of a pattern forming spontaneously in a chemical system, and they are often grouped with the reaction–diffusion patterns Alan Turing predicted in 1952 — spots and stripes arising from two diffusing chemicals that react with each other. The mechanism is different in an important way. Turing’s patterns form in a steady state from an instability of a uniform mixture, with a spacing set by the diffusion constants and reaction rates, and they would form the same way anywhere in an endless medium. Liesegang bands form behind a moving front, one at a time, with a spacing that keeps growing, and they need a sharp threshold — nucleation — that Turing’s mechanism does not. They are a record of the front’s history, written down as it passed, rather than a standing wave of chemistry.

The closer relative may be the branching growth that the cluster that grows where the walkers arrive followed: a solid growing by collecting particles that diffuse to it, where existing growth starves the region around it. In diffusion-limited aggregation the starving happens sideways and makes branches; in a Liesegang tube it happens along the direction of the front and makes bands. In both, a structure that has already formed shapes where the next part can form, by drinking from the diffusing supply around it.

Agates show the result on a geological scale. Their concentric bands of silica, coloured by traces of iron and other elements, have been explained in several ways, and one of them is a Liesegang-like mechanism: silica-rich fluid diffusing into a cavity in volcanic rock and precipitating in rhythm. Whether the bands in a particular agate record that, or episodes of changing fluid, or something else, is argued; the laboratory patterns show that a rhythm needs no external clock.

One dimension, two thresholds and no ripening #

The model is one-dimensional: a straight tube, with every quantity varying only along its length. Real gels have Liesegang patterns that are rings, spirals and helices in three dimensions, and the transition from bands to spirals, which is seen experimentally, needs a second dimension the model lacks. The precipitation rule is the simplest that produces bands — a sharp nucleation threshold and growth above solubility — and it ignores the size of the grains, whereas real precipitates coarsen after they form, small grains dissolving in favour of large ones in the ripening that the small bubble blows up the big one described for bubbles. Ripening can split or move bands after they form. The gel is treated as a passive medium in which everything diffuses freely, and the electrolytes as ideal, with no charge effects. The reaction is fast enough to be confined to a thin zone, which real precipitation reactions usually are, but a slower reaction would smear the zone and with it the bands. The parameters are in dimensionless units chosen to give clear bands, not fitted to a particular pair of salts, and on the grid on which the tube is computed each band occupies a single cell, so the figures resolve the bands’ positions and the amount of solid in each, but not their widths. The domain is the regime in which bands form one at a time behind a diffusion-controlled front, well away from the entrance and from the end of the tube.

Still open: why the width law, and why sometimes the bands run backwards #

The time law and the spacing law are explained, in the sense that simple models reproduce them. A third regularity, that the width of each band is proportional to its position, is reproduced by some models and not others, and depends on what happens to precipitate after it forms. More puzzling are the patterns that break the rules. Some systems produce “revert” spacing, with bands closer together the further they are from the entrance; others produce bands that move after forming, or that dissolve and reform downstream. These depend on the precipitate redissolving when more of the outer electrolyte arrives, on complexes forming in solution, or on the gel itself responding to the reaction, and no single model accounts for all of them. Whether there is a common account of the irregular Liesegang patterns, or only a collection of special cases, is not settled.

The regular pattern is settled. When a diffusing reactant makes a product that becomes solid only above a nucleation threshold $c^*$ but grows on existing solid above its lower solubility, each band drains the product around it, and the next band forms only once the reaction front has outrun that drain; the front moves as $\sqrt t$, so the bands obey $x_n \propto \sqrt{t_n}$, and the problem has no length of its own, so their positions form a geometric progression $x_{n+1}/x_n \to 1 + p$ — 1.09 or 1.19 for thresholds of 0.35 and 0.5 here — with $p$ rising as nucleation gets harder. A uniform jelly and a steady supply produce rhythm, because a solid that is hard to start is greedy once it has.