Illustrated Physics
physics, drawn
Concept

Nucleation — where it appears

The formation of the first small bubble or droplet of a new phase, which has an energy barrier to pay before the transition can proceed. The barrier exists because a small nucleus is nearly all surface, so the surface cost beats the volume gain until a critical radius is reached.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

The small bubble empties into the large one. Two soap bubbles of radius 4 mm and 12 mm joined by an open tube. The excess pressure inside each is 4γ/R — 72.8 Pa and 24.3 Pa — so the 4 mm bubble is at the higher pressure and blows itself into the other. The smaller a bubble gets the harder it pushes, so the process runs away rather than settling: there is no equilibrium anywhere except both bubbles equal.

The small bubble blows up the big one

Connect two soap bubbles of different size and the small one empties into the large one. Everybody expects the opposite, and the reason it happens is one equation with a radius in the denominator — which also means the process runs away rather than settling.

fluids · Surface tension The phase boundary of water, from one equation. Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

A boiling point is a pressure, not a temperature

Nothing about water names one hundred degrees; the air does. Where a liquid turns to vapour in its bulk is fixed by what pushes on it, which makes the familiar figure a coordinate on a curve — 71 °C on Everest, 119.5 °C in a sealed pot — and one latent heat draws the whole curve.

thermodynamics · Phase change The part of the curve no fluid follows. Van der Waals' isotherms in reduced units, at 5 temperatures either side of the critical one, so that nothing about any particular substance appears. Above the critical temperature the pressure falls monotonically as the volume grows, which is what a fluid does. Below it the curve develops a loop with a rising section in the middle, and that section says the pressure increases as the substance expands — a material with negative compressibility, which cannot exist, because any fluctuation would run away. What happens instead is drawn as the horizontal line: the substance separates into two phases at one pressure, and the volume moves along the line as the proportions change. Its height, 0.6470 of the critical pressure, is fixed by requiring the two areas the line cuts off to be equal, which is the condition that the two phases have the same Gibbs energy. It meets the curve at volumes 0.603 and 2.349, a ratio of 3.9, and those are the densities of the liquid and its vapour. The two turning points of the loop, at 0.72 and 1.53, bound the part that is not merely unobserved but impossible; between them and the construction the substance can be made to sit, superheated or supercooled, until something nucleates.

The part of the curve no fluid follows

One equation for a real gas produces isotherms with a rising middle section, which says a substance would expand as the pressure on it grows. Nothing does that. What replaces it is a horizontal line whose height is fixed by making two areas equal, and the condition is not a convenience.

thermodynamics · Phase change A pore that lifts 100 m has to be 0.1 µm or finer. Capillary rise against pore radius, both logarithmic, for a liquid of surface tension 72.8 mN/m at a contact angle of 20°. The relation is a straight line of slope −1 — halve the pore and double the rise — and the two horizontal marks are the height in question, 100 m, and the 10.3 m that one atmosphere supports. 0.01 µm lifts 1394.7 m; 0.1 µm lifts 139.5 m; 1 µm lifts 13.9 m; 5 µm lifts 2.8 m; 20 µm lifts 69.7 cm; 50 µm lifts 27.9 cm. The conducting vessels of a tree are tens of microns across and lift under a metre; the pores in the membranes between them are tens of nanometres and would lift kilometres. Those are the same expression at two scales, and only one of them is a pipe.

The column that is pulled, not pushed

A capillary fine enough to lift a hundred metres is far too fine to carry any flow, and one wide enough to carry the flow lifts under a metre. Neither is how the water gets up a tree. The column is under tension — an absolute pressure of −0.88 MPa at the top, which a gas cannot have — held together by cohesion and prevented from tearing by pores a few tens of nanometres across.

fluids · Capillarity The barrier a new phase has to climb. The free energy of a droplet against its radius, at four supersaturations. Two terms compete: the volume term is a gain and goes as r³, the surface term is a cost and goes as r². At small radius the surface wins, so a droplet that forms by chance is more expensive than the vapour it came from and evaporates again; past a critical radius the volume wins and the droplet grows without limit. The maximum between them is the barrier. At S = 1.5 the critical radius is 2.66 nm and the barrier 533.8 kT, S = 2 the critical radius is 1.56 nm and the barrier 182.6 kT, S = 3 the critical radius is 0.98 nm and the barrier 72.7 kT, S = 5 the critical radius is 0.67 nm and the barrier 33.9 kT. The critical radius contains a few hundred molecules at low supersaturation and a handful at high, which is the first sign that a theory built on a surface tension and a bulk free energy is being applied outside its comfort. Both the critical radius and the barrier are located here by searching the drawn curve and checked against the closed forms 2γ/|Δg| and 16πγ³/3Δg², which agree to a part in a thousand.

The barrier a new phase has to climb

Water vapour three times supersaturated is thermodynamically desperate to condense and will sit there indefinitely if it is clean enough. The obstacle is that a droplet has to start small, and a small droplet is nearly all surface — so the first nanometre of every phase transition costs energy rather than releasing it, and what decides whether anything happens is the height of that cost divided by kT.

thermodynamics · Phase change The boiling curve, which turns back. Heat flux from a horizontal surface into water boiling at one atmosphere, on a logarithmic axis, against how far the surface is above the boiling point, also logarithmic, from standard correlations. At a few degrees heat leaves by convection; from about 5 K bubbles nucleate and the flux climbs as the cube of the superheat, to the critical heat flux of 1.11 MW/m² at 20 K. Beyond it the flux FALLS as the surface gets hotter — the transition regime, drawn dashed as the interpolation it is — down to 19 kW/m² at 67 K, the Leidenfrost point, where a continuous film of vapour insulates the surface. Past that, film boiling and radiation let the flux rise again, slowly: 310 kW/m² at 1000 K above boiling. Between the crisis and the Leidenfrost point, a surface 3 times hotter passes 58 times less heat.

The boiling curve that turns back

Heat a surface under water and more heat flows from it the hotter it gets — up to a point. Past about twenty degrees above boiling the bubbles crowd together, merge, and begin to blanket the surface with vapour, and from then on a hotter surface passes less heat, not more, until by a few hundred degrees above boiling it is wrapped in an insulating film and passes fifty times less than it did at its best. Shigeo Nukiyama found the turn in 1934 by melting wires. It is why a water drop skates for a minute on a very hot pan but vanishes in two seconds on a merely hot one, and why every boiler, steam generator and water-cooled reactor is designed around a number called the critical heat flux.

thermodynamics · Phase change How much of supercooled water freezes in an instant. The fraction of a body of supercooled water that freezes at once when it is nucleated, against how far below 0 °C it had been cooled: the ice that forms releases latent heat until the whole of it is back at 0 °C, and freezing stops there until heat is taken away. Solid: with water's measured heat capacity, which rises steeply below about −20 °C; dashed: with the heat capacity it has at 0 °C. At −15 °C, 19.1 per cent freezes at once; at −20 °C, 25.7 per cent; at −38 °C, the coldest liquid water reaches before it freezes by itself, 51 per cent. The rest stays liquid at 0 °C. Water would have to be cooled about 80 degrees for all of it to freeze at once, and no liquid water survives that long.

The supercooled water that only partly freezes

Water can be cooled well below 0 °C and stay liquid, and when a crystal finally starts, the whole bottle seems to turn to ice at once. It does not. Freezing releases heat, and with nowhere for that heat to go in the instant of freezing, it warms everything back to 0 °C — at which point freezing must stop. Water cooled to −15 °C freezes about a fifth at once; to −38 °C, the coldest liquid water gets, about half. The rest has to wait for the surroundings to take the heat away, and a hand warmer is the same arithmetic run above body temperature.

thermodynamics · Phase change The tilt that starts the reversal, in a well of soft crystal. A planar defect three wall parameters wide (three times √(A/K)) in which the crystal's anisotropy falls to zero, drawn as the anisotropy K(x) across it; the curve is the shape of the first tilt of the magnetisation to become unstable as a reverse field is raised, the lowest mode of −A d²/dx² + K(x), and the dashed line is its eigenvalue. The equation is Schrödinger's, with the exchange stiffness A in place of ħ²/2m and the anisotropy as the potential, and the reversal begins in its ground state. Here the ground state lies at 0.373 K, so the magnet reverses at 37 per cent of the anisotropy field. The mode reaches out of the well a distance of about √(A/K) on each side, because exchange will not let the tilt change abruptly.

The magnet that reverses where its crystal is weakest

A perfect crystal of neodymium-iron-boron should hold its magnetisation against a reverse field of 7.7 tesla. The best magnets made of it give way at one to three. The shortfall, known for eighty years as Brown's paradox, is not a failure of the crystal but of its edges: reversal begins wherever the crystal is weakest, in a layer a few nanometres thick. Working out how weak that layer can afford to be turns out to be a problem already solved in quantum mechanics — the nucleation field is the ground-state energy of a particle in a well shaped like the defect.

electromagnetism · Magnetisation 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.

thermodynamics · Diffusion Salt that takes in water from the air at three-quarters humidity. The water held by a crystal of common salt, in grams per gram of salt, against the relative humidity of the air around it at 25 °C, with the humidity rising (solid) and falling (dashed). Rising, the crystal stays dry until 75.2 per cent and then dissolves at once into a drop holding 2.79 g of water per gram — the saturated solution — which goes on drinking as the air gets damper: 4.24 g at 85 per cent, 11.5 g at 95. Falling, the drop does not crystallise at 75 per cent. It sheds water along the same curve, becomes supersaturated, and stays liquid down to about 45 per cent before a crystal finally nucleates and it dries out at once.

The salt that dissolves in the air's own water

Leave a dish of salt in a bathroom and it does not just clump; on a humid enough day it turns into a puddle, though no water was poured on it. Above 75 per cent humidity, common salt pulls water out of the air until it has dissolved itself. The number is not a property of salt's attraction for water in any vague sense. It is the vapour pressure of a saturated salt solution, read as a fraction of pure water's: wherever the air holds more water than that, the solution is the state with the lower chemical potential. Bring the humidity back down, and the puddle does not turn back into salt until the air is far drier — about 45 per cent.

thermodynamics · Chemical potential The boiling point in a column of water, and the depth heating reaches it first. The temperature at which water boils against depth in a geyser's conduit 2240 m above sea level, where the air's pressure is 77.6 kPa (solid): 92.7 °C at the surface, 103.6 °C at 4 m, 115.5 °C at 10 m and 131.8 °C at 22 m — rising steeply near the top and more gently below, because each extra metre adds the same pressure to water whose boiling point responds less as it gets hotter. Dashed: the water as it heats from beneath, cooled to 80 °C at the top by the pool and below that warming by only 0.8 °C per metre, mixed by convection; 6 °C and 3 °C short of the curve at 4 m, and then touching it. It touches at 4 m, not at the bottom: there the water is still 13.7 °C below boiling.

The column that boils from the middle

Water at the bottom of a geyser's tube is hotter than water anywhere else in it, and it is not the water that boils first. The weight of the column raises the boiling point with depth faster than the heat from below raises the temperature, so the column reaches its boiling curve partway down, under a cooler surface layer. A little boiling there lifts the water above. Everything beneath is then at a lower pressure than the one it was heated under, so it is superheated and flashes, and the eruption feeds itself: past about fourteen centimetres of water thrown out, each slug of steam makes more room than it was given.

thermodynamics · Phase change

Named alongside it

The objects these essays reach for when they reach for this one.

Latent heatMetastabilitySupersaturationSurface tensionThe Boltzmann factorCavitationFree energyHydrostatic pressureHysteresisInstabilityLaplace pressurePressure

All concepts