Illustrated Physics
physics, drawn
Concept

Surface tension — where it appears

The energy cost per unit area of a liquid's surface, acting as a force per unit length along it and making a drop behave as though skinned. There is no skin: the cost is there because a molecule at the surface has fewer neighbours than one inside, and the two units are dimensionally the same quantity.

Named by 28 essays across 4 fields — each of them below, with the objects they name alongside it.

Surface at fixed volume. Four shapes of identical volume with their surface areas evaluated. The sphere's is the smallest at 4.836; the flattest shape drawn carries 1.91 times as much. Surface tension is an energy per unit area, so a free drop of liquid has an incentive to be the first of these and none at all to be any of the others.

The skin that is not a skin

A drop of water behaves as though it were wrapped in a stretched membrane, and there is no membrane. What there is instead is an energy cost per unit of surface, and almost everything the apparent skin does follows from a liquid trying to have less of one.

fluids · Surface tension 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 Which wavelength wins. The growth rate of a disturbance on a liquid thread against kR, the circumference divided by the wavelength. Everything to the right of one decays; the maximum sits at kR = 0.697, which is a wavelength of 9.01 radii or 4.51 diameters. That number, and not a property of any particular liquid, is what sets the spacing of the drops a tap breaks into.

The thread that cannot stay a thread

A stream of water from a tap breaks into drops, and it does so at a spacing that is always about four and a half diameters. Nothing chooses that number — it is the wavelength that grows fastest out of a competition between all of them, and it can be computed before any water is poured.

fluids · Surface tension Four tubes, four heights. Water in tubes of radius 0.2, 0.4, 0.8, 1.6 mm, with each meniscus drawn as the spherical cap a 20° contact angle forces and each height computed from it. The narrowest rises 70 mm and the widest 9 mm — in inverse proportion to the radius, with nothing about the glass or the volume of water entering it.

How high water will climb

Water rises up a narrow tube against gravity, and the narrower the tube the higher it goes. The height is set by a curved surface pulling on a circumference while gravity pulls on an area, and the two scale differently — which is the whole of it.

fluids · Capillarity One volume of liquid, several solids. 4 drops of the same 5 µL of liquid, on 4 solids it meets at 20°, 60°, 90°, 140°. Each is the spherical cap that volume and that angle force, so the footprint radius is computed rather than chosen: 20° gives 2.61 mm, 60° gives 1.71 mm, 90° gives 1.34 mm, 140° gives 0.69 mm. At every contact line the three interfacial tensions are drawn to scale, and only their horizontal components balance — γsv = γsl + γlv cos θ. The vertical pull of the liquid surface is taken up by the solid, which is why the angle belongs to three interfaces at once and to no single liquid: change the solid and nothing about the water has changed.

The angle a liquid makes with what it sits on

Everything capillarity does — climbing, beading, wicking, waterproofing — is the sign and size of one cosine, and that cosine belongs to three interfaces at once rather than to the water. Change the solid and nothing about the water has changed, yet the same five microlitres goes from a footprint 2.61 mm across to one of 0.69 mm.

fluids · Capillarity Two ways to weaken a surface. Surface tension is a property of a surface rather than a constant of a liquid, and this shows how far it moves. The steep curve is water with ethanol dissolved in it, against ethanol mole fraction, through nine measured points: two per cent of ethanol takes the tension from 72 to 56.4 mN/m, a fall of 22% for a change of composition small enough to taste and not to see. The initial slope is 837 mN/m per unit of mole fraction, because ethanol collects preferentially at the surface and a little of it covers a great deal of area. The gentle line is pure water against temperature, read on the same horizontal axis as degrees rather than fraction: about 0.15 mN/m per degree, so a difference of ten degrees across a surface is worth about a millinewton per metre. Neither dependence would be interesting if surfaces were uniform. What makes them matter is that a difference in tension across a surface is a force along it, and nothing in the liquid prevents such a difference from existing.

The surface that pulls toward the stronger side

Surface tension is usually treated as a constant of a liquid, and it is not — it depends on temperature and on what is dissolved, and a difference in it along a surface is a force along that surface — which drags the liquid underneath and needs no pressure difference at all.

fluids · Surface tension Where a drying drop loses its liquid. The rate at which liquid leaves the surface of a drying drop, against distance from the centre in units of the drop's radius, for 5 contact angles. The flux is not uniform, and it is not a property of the liquid: it is set by how vapour diffuses away from a lens-shaped object, which is the same boundary-value problem as the field around a charged lens and has the same answer — a power law in the distance from the rim, with an exponent that depends only on the contact angle. at 10° the exponent is 0.471, and the loss has doubled by 87.8 per cent of the way out, at 40° the exponent is 0.357, and the loss has doubled by 92.5 per cent of the way out, at 70° the exponent is 0.182, and the loss has doubled by 98.9 per cent of the way out, at 90° the exponent is 0.000 and the drop dries evenly everywhere, at 120° the exponent is -0.500 and the flux falls toward the rim. Below a right angle the flux diverges at the contact line; at exactly a right angle it is uniform; above it the edge is the slowest-drying part of the drop. Since a pinned edge must be resupplied from the interior, that sign decides which way the liquid inside the drop flows — and therefore whether everything suspended in it ends up in a ring at the rim or in a spot at the centre.

The ring the drop leaves behind

A drop of coffee dries into a ring rather than a disc, and nothing about coffee is responsible. The pattern is produced by a boundary condition — an edge that cannot move — and it survives replacing the coffee with anything else that will stay suspended.

fluids · Surface tension A siphon's pressure, and the 10.09 m it cannot pass. The absolute pressure of the liquid along a siphon, from the upper surface, over the crown, and down to an outlet 1.2 m below the surface, for 4 crown heights. The profile is hydrostatic and depends on nothing but height: the tube's shape, its length and its bore do not appear. At the crown the liquid is below atmospheric pressure by ρg times the lift, and the whole question of how high a siphon can reach is whether that number stays above the liquid's vapour pressure — 2.34 kPa for water at 20 °C, which puts the ceiling at 10.09 m. a 2 m crown sits at 81.7 kPa and holds, a 6 m crown sits at 42.5 kPa and holds, a 9.5 m crown sits at 8.1 kPa and holds, a 11.5 m crown sits at -11.5 kPa and is below the vapour pressure, so it boils. The ceiling is a property of the liquid, not of the mechanism: a degassed liquid that can be pulled into tension has no such limit, and siphons in a vacuum.

The height a siphon cannot pass

A siphon will not lift water more than about ten metres, and the usual explanation for the limit is also given as the explanation for the mechanism. It cannot be both. A siphon runs in a vacuum, with degassed water, over a crown no atmosphere could support.

fluids · Hydrostatics Two straight lines that are not the same line. Half the difference between the liquid and vapour densities, in units of the critical density, against the distance from the critical temperature — both logarithmic, over 5 decades. The van der Waals curve is solved for the coexisting pair at each temperature and its slope in the last decade is 0.500, which is the mean-field ½. Real fluids give 0.326. The two differ by 11 per cent at t = 0.1 and by a factor of 7.9 at the bottom of the axis, which is why an equation with the wrong exponent in it looked right for eighty years.

The point at which the two become one

Heat a sealed tube of carbon dioxide and the meniscus inside it does not boil away — it fades, the two densities converging until there is nothing to separate. Twenty millikelvin before that happens the fluid turns milky, and the exponent describing the last approach is a number van der Waals got wrong and could not have got right.

thermodynamics · Phase change The same block, one of them with no upthrust at all. Two identical blocks 0.8 m tall with their tops 1.2 m under the surface, drawn with the pressure on every wetted face at its true relative size. On the right the block is clear of the floor and the pressure on its underside exceeds that on its top by 7.8 kPa, which is ρgh and is exactly Archimedes' 7.8 kPa. On the left the bedding is perfect and there is no water under it, so nothing pushes up: the resultant is 11.8 kPa downward and the block presses on the floor with more than its own weight. Buoyancy is not something the fluid has. It is what the bottom face is doing, and a face the fluid cannot reach does nothing.

The block the water does not lift

A block bedded flat on the bottom of a tank, with no water underneath it, feels no upthrust at all. It is fully submerged, Archimedes' principle is not suspended, and it presses on the floor with more than its own weight — because buoyancy is not something a fluid has, it is what the bottom face is doing, and a face the water cannot reach does nothing.

fluids · Buoyancy 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 Two angles a film is not allowed to depart from. The two junctions Plateau's laws permit, drawn at the angles a balance of equal tensions requires. A soap film pulls equally in every direction along itself, so where films meet the pulls must sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films can only meet along a line, at 120.0000° to one another, because three equal coplanar vectors sum to zero at 120° and at no other angle. Four such lines can only meet at a point, at 109.4712° — arccos(−1/3), the tetrahedral angle — for the same reason in three dimensions. Both numbers are found here by solving the balance rather than by drawing what is expected, and neither depends on the liquid, the temperature or the size of the foam. A junction of four films along a line, or of three lines at a point, is not merely unusual: the tensions cannot balance there, so it rearranges within milliseconds into the two arrangements drawn.

The angles a film has no choice about

A soap film pulls equally hard in every direction along itself, so wherever films meet the pulls have to sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films meet at a hundred and twenty degrees and four edges at a hundred and nine point four seven, in every foam, of every liquid, at every scale, and nothing about the material appears in either number.

fluids · Surface tension The outward pull on a charged surface. Electrostatic pressure against the field at a conductor's surface. The quantity is ½ε₀E², the energy density of the field itself, and it is outward whatever the sign of the charge — like charges repel, and a charged surface is trying to fly apart. The factor of a half is the interesting part and is where a first attempt goes wrong: the field is σ/ε₀ outside and zero inside, and the layer of charge feels neither of those but their mean, because no charge exerts a force on itself. 0.5 MV/m gives 1.1 Pa, 1 MV/m gives 4.4 Pa, 2 MV/m gives 17.7 Pa, 3 MV/m gives 39.8 Pa, 5 MV/m gives 110.7 Pa. Those are small pressures — three megavolts per metre is the breakdown field of air and pulls with about a hundredth of an atmosphere — which is why electrostatic forces shape soap films and dust and not much that is stiffer, and why the same pressure set against surface tension has a definite size of drop at which it wins.

The pressure a charge puts on its own metal

Charge on a conductor sits on the surface and tries to leave. The outward pull is half epsilon-nought E squared, the half is because a charge exerts no force on itself, and setting that pull against surface tension gives the largest a charged drop is allowed to be — a number Rayleigh wrote down in 1882 and an industry now depends on.

electromagnetism · Conductors Melting curves, and the one that leans the wrong way. Melting temperature against pressure for water, benzene, naphthalene, each measured from its own melting point at one atmosphere, with pressure in bars. The slope of every coexistence line is the latent heat divided by the temperature and the change in volume, and the latent heat of melting is positive for everything — so the sign of the slope is the sign of the volume change, and nothing else. Almost everything expands on melting and its line leans forwards. Water's solid is less dense than its liquid, so its line leans backwards at 135 bars a kelvin: pressing on ice at just below zero melts it, and it takes 135 atmospheres to gain a single degree. The anomaly is not in the thermodynamics; it is in the fact that ice floats.

The melting curve that leans the wrong way

The slope of any coexistence line is the latent heat divided by the temperature and the change in volume. Latent heat is always positive, so the sign of the slope is the sign of the volume change — and for water the volume change is negative, which is the whole of why ice floats and why the melting curve leans backwards.

thermodynamics · Phase change One line decides whether it climbs for ever. The wetting condition for a corner, drawn as a map. The horizontal axis is the corner's half-angle and the vertical axis the liquid's contact angle with the walls; the diagonal is θ + α = 90°. Below it the meniscus in the corner curves into the liquid, the capillary pressure grows without bound as the corner narrows, and the liquid wicks along it indefinitely. Above it the curvature has the other sign and the liquid stays put. The boundary is tested by evaluating the meniscus a millionth of a degree either side of it at seventeen half-angles, and it wicks on one side and not the other every time. water on clean glass, right-angled corner: α = 45°, θ = 5° — wicks; water on glass, a 20° groove: α = 10°, θ = 40° — wicks; water on plastic, right-angled corner: α = 45°, θ = 75° — does not. A right-angled corner needs a contact angle under 45°, which water on clean glass has and water on most plastics does not.

The corner a liquid never stops climbing

A narrow tube lifts a liquid to a definite height because it has a smallest width. A corner has none, so the capillary suction it can develop is unbounded — and whether the liquid takes advantage is decided by a single inequality between the contact angle and the corner's own angle.

fluids · Surface tension The colours a draining film runs through, and the end of them. The fraction of light a free soap film reflects, against its thickness, at three wavelengths — 450, 550, 620 nm — computed from the sum over multiple reflections rather than from the two-beam approximation. The three curves peak at different thicknesses, which is why a draining film runs through a sequence of colours as it thins. Below about 11 nm every curve is under a tenth of a per cent and the film looks black. At zero thickness the reflectance is exactly zero, checked before the figure is drawn: the two surfaces reflect equally and half a cycle out of step, so a film much thinner than a wavelength cancels itself. That is the whole of why a black film is black. Nothing is absorbing; the film is there and has simply stopped being able to interfere constructively at any visible wavelength — which means the blackness is a measurement, and a film that has gone black is known to be thinner than about a tenth of a wavelength without anything being measured directly.

The film that goes black before it bursts

A soap film drains, runs through every interference colour, and then stops reflecting anything at all. The black patch is not a hole and not a film about to break: it is the thinnest and most stable state the arrangement has, held apart by a pressure between its two surfaces that only exists at distances of nanometres.

fluids · Surface tension The pore size the air decides. The largest pore radius that holds condensed water in equilibrium with air at a given relative humidity, for water at 25 °C, on a logarithmic radius axis. A concave meniscus lowers the vapour pressure over it by exp(−2γVₘ/rRT), so a pore whose meniscus would be tighter than a radius set by the humidity is in equilibrium only when full. The upper curve is the emptying condition, through a hemispherical meniscus with two curvatures; the lower is the filling condition, through the cylindrical film that lines a pore before it closes, with one — so the same pore fills at a higher humidity than it empties at. At 50% humidity a pore empties below 1.51 nm and fills below 0.76 nm; at 90% humidity a pore empties below 9.96 nm and fills below 4.98 nm; at 99% humidity a pore empties below 104 nm and fills below 52 nm. The radii run from molecular at low humidity to a tenth of a micrometre at 99 per cent, and every one was checked by putting it back into the vapour-pressure relation.

The pore that fills from dry air

Water condenses when the air is saturated — on a flat surface. Over a curved one the vapour pressure is different, higher over a drop and lower over a meniscus, and in a pore a few nanometres across it is low enough that the pore fills with liquid from air at half humidity. The water it holds is under a tension of a hundred megapascals, and the pore empties at a lower humidity than it filled at, for a reason that needs no roughness at all.

fluids · Capillarity An angle set by a voltage. The contact angle of water on a fluoropolymer coating with a rest angle of 115°, against the voltage across the coating, for coatings 1 and 3 µm thick with a relative permittivity of 1.93. The Lippmann–Young law cos θ = cos θ₀ + ε₀εᵣV²/2dγ makes the cosine rise as the square of the voltage, so the curve is symmetric about zero volts and steepest where the angle is near 90°. For 1 µm the law reaches complete wetting at 110 V; for 3 µm the law reaches complete wetting at 191 V. It does not get there. In most reported experiments the angle stops falling somewhere in the shaded band and then stays put however high the voltage is raised — contact-angle saturation — and why is still argued: charge trapped in the insulator, ionisation of the air at the sharp edge of the drop, and the thermodynamic stability of the contact line have all been proposed and none accounts for every case.

The angle a voltage can set

A contact angle is treated as a fact about three materials — a solid, a liquid and the air — fixed the moment they are chosen. Put a voltage across a micrometre of insulator under a drop and the angle falls as the square of the voltage, with nothing about the materials changed. Drops can be steered across a chip with no pump, and a lens can focus with no moving part. The law that describes it is exact in its model, and real surfaces stop obeying it at an angle nobody has fully explained.

fluids · Capillarity Like menisci attract, unlike menisci repel. Cross-sections of a water surface with two small floating bodies, drawn with menisci of the shape K₀(r/ℓc) that a small floater makes, exaggerated vertically. Left: two light bodies, each pulling the surface up round itself as buoyancy pushes it up against the surface; each sits on the other's slope and is drawn towards it, as a bubble rises — the two climb together. Right: a light body that raises the surface beside a heavy one that depresses it; each sits on the other's slope the wrong way and they drift apart. The deformations reach a few capillary lengths, 2.71 mm for water, and the rule is the whole of the Cheerios effect: cereal clumps, and cereal gathers at the wall of a bowl the milk wets.

The cereal that gathers at the edge of the bowl

The last few pieces of cereal in a bowl of milk clump together and drift to the side of the bowl. Nothing is pulling on them but the milk's surface. Each floating piece bends the surface round itself, and a second piece sitting on that bend has somewhere to slide: like menisci attract, unlike menisci repel, and the reach of the force is a few capillary lengths, 2.7 millimetres for water. The arithmetic is the arithmetic of charged lines in two dimensions, screened by gravity, and it says why leaves gather on a pond, why bubbles on coffee drift to the cup's rim, and why particles smaller than about ten micrometres do not clump this way at all.

fluids · Capillarity A floating needle and the water it displaces. A cross-section of a steel darning needle 1.36 mm thick (density 7.85) floating on water, with a contact angle of 100°, its meniscus computed from the balance of surface tension and pressure; one capillary length is 2.73 mm. The needle's centre sits 1.43 mm below the undisturbed surface, and the water meets it 117° from the top, sloping at 37°. Two regions of water have been displaced. Below the level, under the needle, a region whose weight — 21 per cent of the support — is what pressure on the needle's underside supplies. Beside it, the two dents in the surface, whose weight — the other 79 per cent — is exactly the vertical pull of surface tension along the two contact lines. Together they weigh what the needle weighs: Archimedes, counting the dent.

The water the dent displaces

A steel needle laid carefully on water floats, and the usual explanation is that surface tension holds it up instead of buoyancy, as though the two were rival mechanisms. They are not rivals. The needle presses a dent into the surface, and the upward pull of surface tension along its edges is exactly the weight of the water missing from that dent. Archimedes' principle holds without amendment: a floating needle displaces its own weight of water. Most of that water is simply not underneath it.

fluids · Buoyancy Six cells of a two-dimensional foam, by number of sides. Cells of a two-dimensional soap froth with three to eight sides, each drawn with its films meeting three at a time at 120°, as surface tension requires. A hexagon can do that with straight sides. A cell with fewer than six sides cannot: to make its corners 120° its sides must bulge outward, so its gas is at higher pressure than its neighbours' and leaks out. A cell with more than six must have sides curving inward, lower pressure, and gains gas. The rate of change of area, in units of πκγ/3 where κ is the films' permeability, is n − 6: −3 for a triangle, +2 for an octagon.

The bubble that grows by counting its sides

A froth of soap bubbles squeezed between two sheets of glass coarsens over a few days: small cells vanish and large ones swell. The obvious guess is that size decides which cells win, as it does for two bubbles joined by a tube, where the small one always empties into the large one. In a froth it is wrong. John von Neumann showed in 1952 that a cell's area changes at a rate fixed entirely by how many sides it has — shrinking if fewer than six, growing if more — and not at all by how big it is. The proof is a single sum of angles.

fluids · Surface tension Water's surface tension, all the way to the critical point. The surface tension of water against its liquid vapour, from the international standard correlation, against temperature from 0 °C to the critical point at 373.946 °C. 20 °C: 72.7 mN/m; 100 °C: 58.9 mN/m; 200 °C: 37.7 mN/m; 300 °C: 14.4 mN/m; 370 °C: 0.4 mN/m. It falls almost in a straight line for most of the range and then bends down to reach zero exactly at the critical point, where liquid and vapour stop being two things. The bend is a power law, the gap to the critical temperature raised to 1.256.

The tension that vanishes where the phases meet

Surface tension is usually treated as a property of a liquid, like its density. It is a property of a boundary between two things, and it can only exist while there are two things. Heat water in a sealed tube and its surface tension falls steadily, then faster, and reaches exactly zero at 374 °C, where liquid and vapour stop being different. On the way the surface stops being a surface: the step in density spreads from a molecule's width to thousands, and the rate at which the tension dies turns out to be a number the whole family of critical points shares and the simplest theory gets wrong.

fluids · Surface tension A drop in a tapered tube. A section through a glass tube tapering from 1.2 mm to 0.2 mm in radius over 60 mm, holding a 2 μL plug of water (contact angle 20°), not to scale across. Both menisci curve into the empty tube, so the water just inside each is below atmospheric pressure — by 269 Pa at the narrow end, radius 0.50 mm, and by 252 Pa at the wide end, radius 0.54 mm. The pressure is lower at the narrow end, so the water flows towards it: a net push of 17 Pa, with no gravity, no pump and no motion of the tube.

The drop that runs to the narrow end

Put a drop of water into a glass tube that narrows towards one end and it moves, on its own, towards the narrow end — horizontally, with nothing pushing it. Francis Hauksbee reported it to the Royal Society in 1712, and the explanation is two curved surfaces pulling with different strengths: each meniscus lowers the pressure of the water behind it by an amount that grows as the tube narrows, so the end in the narrower part sucks harder. A drop of mercury, which does not wet glass, goes the other way. Which way a drop goes depends on the cosine of one angle, and, for a short drop, on its length.

fluids · Capillarity Which waves a moving object can make. The phase speed of waves on deep water against wavelength (red), with horizontal lines at the speeds of an object moving steadily across the surface: 15.0 cm/s, 23.1 cm/s, 30.0 cm/s, 50.0 cm/s. A steady object can only feed waves that keep pace with it, whose phase speed equals its own, and those are where its line meets the curve. At 15 cm/s the line misses the curve entirely: no wave can keep up, and the object leaves no wake. At 23.1 cm/s it touches the curve once. Above that it meets it twice: at 30 cm/s a ripple of 5.6 mm and a gravity wave of 5.2 cm; at 50 cm/s a ripple of 1.9 mm and a gravity wave of 15.8 cm.

The speed below which nothing on water leaves a wake

Every wave on clean water travels at 23 centimetres a second or faster: long waves are fast because gravity drives them, short ones because surface tension does, and in between there is a slowest wave. A boat, a duck or a fishing line can only make waves that keep pace with it, so anything crossing the water more slowly than 23 centimetres a second makes no waves at all. Faster, it makes two kinds at once — short ripples that run ahead of it and long waves that trail behind — and the threshold between nothing and both is set by gravity, surface tension and density alone.

waves · Wave motion A surface whose thickness depends on how much of it is looked at. The rms roughness of a water surface at 20 °C, in nanometres, against the size of the patch over which it is measured, from the thermal ripples that fit inside the patch — with gravity (solid) and without it (dashed) — for ripples down to a molecular wavelength of 0.6 nm. Over 10 nm the surface is 0.16 nm rough; over a micrometre, 0.26; over a millimetre, 0.36; and beyond a few millimetres gravity flattens the longest ripples and the roughness stops at 0.390 nm. In free fall it would keep growing as the square root of the logarithm: 0.43 nm over a metre. An X-ray reflectivity measurement sees the ripples smaller than its own coherence length, typically some micrometres, and reports a thickness of about a third of a nanometre — a number that belongs to the instrument as much as to the water.

The surface whose thickness depends on the view

The surface of a glass of water looks like a perfect plane, and over most distances it nearly is. But it is at room temperature, and every ripple it could carry — from a few molecules long to a few millimetres — is shaken by heat with its own share of thermal energy. The ripples are tiny, a fraction of a nanometre high, and they add up in an odd way: every factor of ten in wavelength contributes the same amount of roughness. So the thickness of a water surface has no single value. It depends logarithmically on how large a patch is looked at, until gravity stops it at a few millimetres, and the X-ray measurements that report a third of a nanometre are reporting a property of their own resolution as much as of the water.

fluids · Surface tension The film that thins too slowly. The thickness of the air film under a silicone-oil drop of radius 0.38 mm pressed onto its pool, against time, on logarithmic axes, by Reynolds's law: dh/dt ∝ −h³, so the film thins quickly at first and then ever more slowly, its thickness falling only as the inverse square root of time once it is thin. The shaded band is one bounce, 4.1 ms, in which the film falls from 10 µm to 1240 nm; the dashed line is the 100 nm at which van der Waals forces pull the surfaces together, reached after 0.65 s. The air does not have to be held up; it has only to be slow to leave, and the cube of the thickness makes it slower and slower the thinner it gets. A vibrating bath that lifts the drop off every few milliseconds refills the film each time.

The drop that will not merge

A drop of oil let fall onto a pool of the same oil ought to vanish into it at once: merging releases surface energy, and nothing holds the two apart. Instead it bounces. A film of air is trapped between drop and pool, and air, though it offers no strength at all, is slow to leave a gap a few micrometres thick: the film takes the better part of a second to drain to the hundred nanometres at which the surfaces can touch, while a bounce lasts four milliseconds. Shake the pool and the drop can bounce for ever, in rhythms that double as the shaking grows — and, near the pool's own instability, it starts to walk across the surface on the waves of its own making.

fluids · Surface tension The force that does not count the water. The force of a water bridge between two touching spheres of radius R, in units of 2πRγ, against the bridge's volume on a logarithmic scale running over six decades, for three contact angles. Each curve is nearly flat: from a millionth of R³ to a thousandth, the force on a wetting grain falls only from 0.99 to 0.92 of 2πRγ, and it stays close to cos θ for the others. A droplet between two grains of sand pulls almost as hard as one a thousand times larger. This is the reason sand wet with one per cent of water and sand wet with ten per cent hold together with about the same strength. The curves fall where a bridge is large enough for its neck to stop being small compared with the grain, and bridges of that size merge with their neighbours.

The sandcastle a spoonful of water holds

A dry heap of sand stands at thirty degrees and no steeper. Add a spoonful of water to a bucketful, and the same grains will stand in a vertical wall and carry a person's weight. Add ten times as much water and the wall is no stronger, because a bridge of water between two grains pulls with almost the same force whatever its size. Its strength belongs to the grain and the surface tension, and the amount of water hardly enters.

fluids · Granular matter

Named alongside it

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

Laplace pressureWettingCapillarityContact angleCapillary lengthEquilibriumMeniscusCavitationCurvatureMetastabilityPressureSurface energy

All concepts