Illustrated Physics
physics, drawn
Concept

Capillary length — where it appears

The size at which surface tension and gravity balance in a liquid, below which drops are spheres and above which they are puddles. It is about 2.7 millimetres for water, which is why raindrops are never much larger than that and why a spilled drop flattens.

Named by 9 essays across one field — 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 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 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 A closed cabin accelerating at 3 m/s², and everything in it leaning the same way. A closed box accelerating to the right at 3 m/s², with water in the bottom and air above. Everything that can hang or float lines up with the effective gravity g − a, tilted 17.0° from the vertical towards the back: the water's surface and the isobars below it are perpendicular to it, a floating block stands along it, a plumb bob hangs along it — and a helium balloon on a string leans the other way along the same line, forward, into the acceleration. The balloon is not doing anything unusual. Buoyancy always points opposite to gravity, and in the cabin the effective gravity has a horizontal part, so the air's pressure gradient does too, and it pushes the balloon along it.

The balloon that leans the wrong way

When a car pulls away, everything loose in it swings back — except a helium balloon on a string, which swings forward. Nothing strange is acting on it. Buoyancy points against gravity, and inside an accelerating cabin gravity has a sideways part. Read that way, a lift cannot change how deep a boat floats, free fall abolishes floating altogether, and a centrifuge is Archimedes' principle with the word "up" pointing at the axis.

fluids · Buoyancy 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 One height inside the bottle where the pressure is the air's. Gauge pressure against height inside a sealed bottle with an open tube reaching down to 10 cm above its outlet, while it is bubbling, at three water levels: 30.0, 22.0, 13.0 cm. Each line rises at the same hydrostatic slope, ρg, from the air pocket at the top, which is below atmospheric, to the outlet at the foot. All three cross atmospheric pressure at the tube's tip — 7.4 mm below it, the extra a bubble needs to leave a 2 mm tube — and so all three reach the outlet at the same 906 Pa above atmospheric, whatever the level. Dashed: an open bottle at the same levels, whose pressure at the outlet is 2937, 2154, 1273 Pa.

The bottle that pours at one rate however full it is

Water running from a hole in the foot of an open bottle slows as the bottle empties, because the pressure at the hole is the depth of water above it. Seal the bottle and push an open tube through the stopper, down into the water, and it stops slowing: the flow holds steady from full until the level reaches the bottom of the tube. Nothing regulates it. The tube's tip is the one point inside where the pressure is the atmosphere's, the water above it hangs from an air pocket below atmospheric, and the outlet sees only the height of the tip.

fluids · Hydrostatics 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

Named alongside it

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

Surface tensionContact angleHydrostatic pressureLaplace pressureCapillarityMeniscusWettingArchimedes principleBuoyancyIntermolecular forcesScalingSurface energy

All concepts