Illustrated Physics
physics, drawn
Concept

Laplace pressure — where it appears

The excess pressure inside a curved interface, equal to the surface tension times the curvature, so the smaller the radius the larger it is. That inverse dependence is why a small bubble connected to a large one empties into it, and why nucleating a new bubble is hard.

Named by 9 essays across 2 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 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 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 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 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 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 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

Named alongside it

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

Surface tensionCapillarityContact angleWettingCapillary lengthCavitationCurvatureInstabilityMeniscusNucleationScalingSurface energy

All concepts