Illustrated Physics
physics, drawn
Series

Diffusion — the series

17 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A spike, spreading. The solution of the diffusion equation at three times, with a seeded random walk histogrammed behind it. The area under every curve is the same because nothing is lost; only the width changes, and it grows as the square root of the time.

    The equation that only runs forwards, and the walk underneath it

    A drop of ink spreads and never gathers. The equation describing it is one of the few in physics that is not reversible — and underneath it is nothing but a coin being tossed.

    part 1 · thermodynamics
  2. Seven walks from one point. 7 random walks of 400 steps each, all starting at the same place. None of them goes anywhere in particular and none of them stays put; the typical distance reached after n steps is the square root of n, so quadrupling the time doubles the spread. The tracks are seeded, so this is a property of the figure rather than of any one run.

    The jiggle that proved atoms

    A pollen grain in still water never stops moving. For eighty years that was a curiosity with no explanation; then it became the measurement that settled whether matter is made of particles, by turning a microscope and a stopwatch into a count of how many molecules are in a mole.

    part 2 · thermodynamics
  3. The fraction of walks that have come home, against how long they have walked. The proportion of 1,600 lattice random walks that have returned to their starting point at least once, against the number of steps taken, on a logarithmic horizontal axis, in one, two and three dimensions. In one dimension almost every walk is home almost at once and the fraction climbs toward one. In two it climbs more slowly — the return is still certain, but only logarithmically, so a two-dimensional walk that has not come back after a thousand steps is unremarkable. In three the curve flattens: it reaches 0.3481 and stops, against the exact value 0.3405, drawn as a line. That number is Watson's integral, and it is the probability that a three-dimensional walk ever comes home at all. The difference between the cases is not one of degree. In one and two dimensions the expected number of returns is infinite and a diffusing particle visits every site eventually; in three it is finite, and a molecule released in a room will, with probability two-thirds, never pass through its starting point again. The same statement runs the other way round: a reaction that needs two diffusing partners to meet is a very different problem on a membrane from what it is in a cell.

    The walk that comes home

    A particle wandering at random on a line returns to where it started, with certainty. On a plane it returns, with certainty. In three dimensions the probability is 0.3405 — so two out of three molecules released in a room never pass through their starting point again, and the difference between the cases is not a matter of degree.

    part 3 · thermodynamics
  4. Two experiments, two computations, one coefficient. The Seebeck coefficient of a resonant conductor against where its resonance sits relative to the chemical potential, together with the Peltier coefficient divided by the temperature. The first is obtained by applying a temperature difference and finding the voltage that stops the current; the second by applying a voltage at uniform temperature and taking the ratio of the heat flow to the current. Different driving, different measurement, different integral — and the two curves agree to 3.8e-7 of the sweep's own scale across the whole of where they pass through zero and change sign. That equality is Kelvin's relation Π = ST, guessed in 1854 from an argument its author knew was not sound and proved by Onsager in 1931 from microscopic reversibility. It is not a property of this conductor; it holds for every one.

    The second experiment that cannot disagree

    Heat one end of a wire and a voltage appears across it. Pass a current through the same wire at uniform temperature and it carries heat. Those are two different experiments with two different apparatus, and the coefficient in front is the same number in both — not approximately, and not for some materials. The reason is that the equations of motion underneath look the same run backwards.

    part 4 · thermodynamics
  5. The year, arriving underground at four different times. Temperature against depth in soil of diffusivity 0.5 mm²/s, at four times of year, measured from the annual mean. The surface swings by ±12 K; the swing underground is smaller and later, and both are governed by one length, δ = √(2D/ω) = 2.24 m. The amplitude falls as e^(−z/δ) — the dashed envelope — and the phase lags by z/δ radians, so the curves lean over as they go down and cross the axis at different depths. At 7.0 m the lag is half a year: the ground there is at its coldest in August and its warmest in February, in antiphase with the sky, with a swing of 0.52 K left. That is a cellar, and it is also why a water main below about a metre and a half does not know that it froze last week.

    The summer that reaches the cellar in December

    Drive the diffusion equation at its boundary instead of releasing something into it and the solution is a decaying, lagging oscillation with a single length in it. That length governs both the shrinking and the delay, which is why the depth at which the ground is coldest in August is fixed by the same number as the depth at which the seasons stop being felt at all.

    part 5 · thermodynamics
  6. Effusion rate against molecular mass. The rate at which a gas escapes through a small hole, against its molar mass, normalised to hydrogen. The rate is a quarter of the number density times the mean speed times the area, and the mean speed goes as the inverse square root of the mass — so the rate does too, which is Graham's law of 1848. Hydrogen escapes four times faster than oxygen and 13.3 times faster than uranium hexafluoride. The practical consequence is isotope separation, and its difficulty is on this chart. The two uranium hexafluorides differ by 3 out of 352 in mass, so a single stage enriches by a factor of only 1.00429 — four parts in a thousand. Reaching 90 per cent from natural uranium's 0.72 per cent therefore takes about 1665 ideal stages, and a real cascade needs more because each stage is imperfect. That number is why gaseous-diffusion plants were among the largest industrial structures ever built, and why centrifuges — which separate by mass directly rather than by the square root of it — replaced them.

    The gradient that drives the other thing

    A concentration gradient drives a flow of matter and a temperature gradient drives a flow of heat. Each also drives the other, by coefficients that are equal — a relation nobody could have guessed and which follows from the fact that the underlying motion runs the same forwards and backwards in time.

    part 6 · thermodynamics
  7. Nitrogen that runs the wrong way. The nitrogen mole fraction in each of two bulbs joined by a capillary, as in Duncan and Toor's experiment: one bulb starts with 0.50086 nitrogen and the rest carbon dioxide, the other with 0.49879 nitrogen and the rest hydrogen, at 35 °C. Solid curves: the capillary solved at each instant from the Maxwell–Stefan equations with the three pairs' diffusivities, 83.8, 68.0 and 16.8 mm²/s. Dashed: Fick's law for nitrogen alone, which can only let the two start values relax together. At the start the nitrogen gradient is 0.00207, yet nitrogen flows at 309 times the rate that gradient would drive. From 0.1 to 6.5 hours it flows from the bulb with less nitrogen into the bulb with more, opening a difference of 0.1468 at 6.5 hours; at 6.6 hours its flux passes through zero with a difference of 0.1468 still in place. Each gas is conserved to a part in 10⁹. The carbon dioxide moving out of the first bulb drags nitrogen with it, because the nitrogen–carbon dioxide pair has by far the smallest diffusivity and so the strongest friction.

    The gas that flows towards more of itself

    Fick's law says a substance diffuses from where there is more of it to where there is less. In a mixture of three gases, nitrogen can do the opposite for hours — flowing into the bulb that already holds more nitrogen, and then stopping while a difference remains — and in a welded bar of steel, carbon crosses into the side that is already richer. Nothing is wrong with the second law. Diffusion flattens chemical potential, and with more than two components, or a second element changing it, that is not the same as flattening concentration.

    part 7 · thermodynamics
  8. The same rate for every size of molecule. The diffusion-limited rate constant for two equal spherical molecules meeting in water at 25 °C, against their radius from 0.1 to 100 nm: 8kT/3η = 7.43·10⁹ per molar per second, flat. The dashed line is what a larger target would give if its partner kept the mobility of a 0.3 nm molecule — 7.4·10¹¹ at 30 nm — the intuition that a bigger molecule is easier to hit. Stokes and Einstein's law takes it away again: a molecule twice as big is twice as easy to hit and diffuses half as fast, and the two cancel exactly. Every reaction whose partners react at the first touch runs at close to this one number, whether the partners are ions, small organic molecules or proteins.

    The speed limit that does not care how big the molecules are

    Two molecules in water cannot react faster than they can find each other, and how fast they find each other is set by diffusion. A bigger molecule is an easier target, so it seems a reaction between big molecules should be faster. It is not: a molecule twice as big is twice as easy to hit and wanders half as fast, and the two cancel exactly. Every reaction that happens at the first touch runs at the same rate, about seven billion per molar per second in water, whether the partners are ions or proteins — a speed limit written by the viscosity of water and the temperature, and nothing else.

    part 8 · thermodynamics
  9. What one trajectory says, and what the crowd says. For the walk with α = 0.5: the time-averaged mean-square displacement of each of 12 single trajectories, each watched for 10000 time units — the average of the squared displacement over every window of a given lag along that one path — against the lag, on logarithmic axes (thin lines), and the mean-square displacement of an ensemble of 800 walkers at the same times (thick). The ensemble spreads as t^0.5. Every single trajectory's time average grows nearly in proportion to the lag — mean slope 0.82, where the ensemble's is 0.5 — as if it were ordinary diffusion, and the trajectories disagree with one another about the diffusion coefficient by more than a factor of ten. A time average along one path and an average over many paths, which for ordinary diffusion are the same thing, here give different laws. That is weak ergodicity breaking, and it is what a single-molecule tracking experiment has to be read against.

    The diffusion that slows the longer it is watched

    A random walker whose steps come at irregular times still diffuses normally, provided the waits have a finite average. Let the waits have no average — mostly short, occasionally enormous, as they are for a molecule that keeps getting stuck — and three things change at once. The spread grows more slowly than time. A diffusion coefficient measured by following one particle depends on how long it was followed. And following one particle for a long time no longer tells what many particles do: averages along a path and averages over a crowd give different laws. Tracking single molecules inside living cells runs straight into all three.

    part 9 · thermodynamics
  10. The temperature through growing lake ice. Temperature against depth below the ice surface in a lake under air 10 K below freezing, after 1, 4 and 16 days, from Stefan's law with the ice's top held at the air's temperature. The ice is 11.1, 22.3, 44.6 cm thick; within it the temperature rises in a straight line from −10 °C at the top to 0 °C at the bottom, and below it the water is at 0 °C throughout. Ice forms only at the bottom, where water meets ice at the freezing point, and the latent heat it releases there has to be carried up through all the ice above by conduction. The straight lines get shallower as the ice thickens: the same temperature difference drives heat across a longer path, so less heat escapes and the ice grows more slowly. Sixteen times as long gives four times the thickness.

    The ice that grows more slowly the thicker it gets

    A lake under air ten degrees below freezing grows eleven centimetres of ice on the first day and only another eighteen in the following six. The ice forms at its underside, and every gram of it releases latent heat that must be carried up through the ice already there before more can form: the ice is its own insulation. The thickness therefore grows as the square root of time, and the rule, Stefan's, depends on temperature and time only through their product — the freezing degree-days that lake-ice forecasters count. A few centimetres of snow on top slows it more than any amount of cold speeds it up.

    part 10 · thermodynamics
  11. A cluster grown by random walkers. 3000 particles, each released far away to wander at random on a square grid until it first touches the cluster and sticks there, starting from one seed at the centre. Colour marks the order of arrival: the first third (violet), the second (blue), the last (red). The cluster is 220 grid steps across, branched and open, and almost every late arrival is at the outer tips: walkers wander into the outer branches long before they can find their way into the gaps between them. The gaps never fill.

    The cluster that grows where the walkers arrive

    Let particles wander in one at a time from far away, each sticking where it first touches a growing clump. The rule has nothing in it about shape, and the clump that results is a lacy, branched thing whose mass grows more slowly than its area, so it gets emptier the bigger it gets. The reason is that a random walker reaches the outermost tips long before it can find its way into a gap between branches. The same arithmetic — the probability that a walker arrives is a solution of Laplace's equation — shapes electrodeposits, lightning-like discharges in plastic, fingers of water pushed into oil, and colonies of bacteria short of food.

    part 11 · thermodynamics
  12. Where skin and the thing it touches meet. Temperature across the contact between skin at 33 °C (left) and a block at 20 °C (right), against distance from the contact in millimetres, at 0.1, 1 and 10 s after touching, for steel (blue) and pine wood (red). Each pair of curves is two error functions meeting at the contact, and the meeting temperature does not change with time: 21.0 °C against steel and 30.3 °C against wood. The nerve endings a fraction of a millimetre under the skin report something close to that interface temperature, so the steel feels like an object at 21 °C and the wood like one at 30 — although both are at 20. The profiles spread as the square root of time, much further into the steel than into the wood.

    The metal that feels colder than the wood

    A steel railing and a wooden bench in the same shade are at the same temperature, and the railing feels much colder. The usual explanation, that metal conducts heat better, is half the answer, and it gets water wrong. When two bodies touch, the surface between them jumps at once to a temperature that then stays fixed, a weighted average of the two in which each body's weight is the square root of its conductivity times its heat capacity per volume. Skin at 33 °C touching steel at 20 °C meets it at 21; touching wood, at 30. The nerves report the meeting temperature, not the object's.

    part 12 · thermodynamics
  13. Insulating a thin wire makes it lose more heat. The heat lost per metre from a wire 2 mm across held 40 K above the air, against the outer radius of the PVC insulation round it (k = 0.19 W/m K), with the surface losing 10 W/m²K to still air by convection and radiation, on a logarithmic radius axis; dashed, the bare wire. The first layers of insulation increase the loss, from 2.51 to a maximum of 12.11 W/m at an outer radius of 19 mm — the critical radius k/h — because they add surface faster than they add resistance. Beyond the critical radius the loss falls again, but so slowly — the resistance of the insulation grows only as the logarithm of its radius — that it does not get back down to the bare wire's until the PVC is 178 km thick. Any sheath a cable could carry makes it lose more heat, not less.

    The insulation that makes a wire lose more heat

    Wrap a hot water pipe in glass wool and it loses less heat. Wrap a thin electrical wire in PVC and it loses more — four times more, at the best thickness — and runs cooler for it. The insulation adds a resistance to heat flowing through it, but it also adds surface for the heat to leave from, and around a thin enough cylinder the surface wins. The crossover is a radius, k/h, set by the insulation and the air; below it, more insulation means more heat lost, and a small enough ball cannot be insulated by anything wrapped round it.

    part 13 · thermodynamics
  14. The fold beyond which no heap is steady. The centre's steady temperature rise, in units of RTₐ²/E, against the Frank-Kamenetskii parameter δ, for a slab, a long cylinder and a sphere, found by integrating the steady heat equation outward from the centre: each shape has two steady states for every δ below a critical value — a cool, stable one and a hot, unstable one — and none above it. The fold is at δ = 0.878, 2.000, 3.322 for the slab, cylinder and sphere: a sphere tolerates nearly four times a slab's δ at the same half-width, because its surface is larger in proportion to its volume and heat from its centre has more ways out. A real heap, which is neither, has a critical δ between, and fixing it for a given shape is the first step in calculating the size at which a store becomes unsafe.

    The heap that sets itself alight

    A haystack, a coal stockpile, a bin of oily rags or a silo of damp grain can catch fire with no spark at all. A slow reaction inside makes a little heat; the heap holds it in; the warmer middle reacts faster and makes more. Whether that ends in a warm heap or a burning one is decided by a single number built from the heap's size, the air's temperature and the reaction's speed — and past one value of it there is no steady temperature at all. The critical size shrinks steeply as the weather warms, and the heap gives almost no warning before it goes.

    part 14 · thermodynamics
  15. Bands of precipitate laid down in a tube of gel. A tube of gel holding one salt, B, with a second salt, A, diffusing in from its left end, computed at the end of the run: the shading shows where solid precipitate has formed, darker for more. The product of their reaction precipitates only where its concentration first passes a nucleation threshold, 0.5 in the units used, and then wherever solid is already present. The solid is not spread smoothly. It forms in 23 sharp bands, close together near the entrance and further and further apart down the tube — the last two are 26.0 units apart, against 0.8 between the first two. Each band holds more precipitate than the last, in proportion to its distance. Nothing in the gel was patterned to begin with.

    The rings a reaction lays down in a gel

    Let one salt soak into a tube of jelly that holds another, and where they meet a solid should form, spreading smoothly behind the advancing front. It does not. It forms in sharp, separate bands, close together near the entrance and further and further apart, each one a fixed multiple of the distance of the last. Raphael Liesegang saw the rings in 1896 and they have been found since in rocks, in agate, in gallstones and in microchips. Nothing in the jelly is patterned to begin with. The pattern comes from a solid that will not start until the solution is overloaded, and that then empties the solution around it.

    part 15 · thermodynamics
  16. Two metals mixing at unequal speeds. The fraction of the fast-diffusing species A against position across a welded couple, A on the left and B on the right, at three times in the ratio 1 : 4 : 16, when A diffuses 3 times as fast as B, from Darken's equations, in units in which B's diffusivity and the middle time are both one; each profile's marker — an inert wire laid at the original weld — as a dot on it. The profiles spread as the square root of time, and they are lopsided: A pours into B's side faster than B into A's, so the mixed zone extends further into B. The markers do not stay at the weld. They move into A's side — by 0.39, 0.80 and 1.63 units at the three times — because the lattice planes of A's side are being emptied faster than they are filled. That drift, first seen in copper and brass in 1947, is the Kirkendall effect.

    The markers that move because atoms are unequal

    In 1947 Ernest Kirkendall and Alice Smigelskas wound fine molybdenum wires around a bar of brass, plated it thickly with copper, and heated it for fifty-six days. The wires, which do not dissolve in either metal, ended closer together: the brass inside them had shrunk. Zinc had diffused out of the brass faster than copper diffused in, and the lattice planes of the brass had been taken away along with the zinc. If atoms in a metal moved by swapping places with each other, the two flows would have had to be equal. They were not, and the only way the books balance is if atoms move by jumping into vacancies — empty lattice sites — that flow the other way. The wires were the first direct evidence that diffusion in metals runs on holes.

    part 16 · thermodynamics
  17. A cooling planet, read from the top. Temperature against depth in a body cooling by conduction from a uniform start at 3870 °C, with a surface held at nought, at four ages: the exact error-function solution for a half-space. Cooling reaches downward as the square root of time, so after a hundred million years the upper few hundred kilometres have cooled and the interior knows nothing about it. The gradient at the surface is what a mine measures, and it falls as the inverse root of time. A thermal gradient therefore gives an age, and this is what Kelvin saw in 1862. He assumed the Earth was a solid conductor from the start, on a good basis, as everyone believed then, and these curves are the mathematical consequence of that belief.

    The age a cooling Earth gives

    In 1862 William Thomson, later Lord Kelvin, took a thermometer reading from the bottom of a mine and a piece of mathematics from Fourier and computed the age of the Earth: about a hundred million years. Geologists, needing far longer, had no physics with which to reply. The arithmetic was right. The usual story blames radioactivity for overturning it, but half the heat is not radioactive, and the error lay in assuming the inside of the planet stays still.

    part 17 · thermodynamics

All series