Illustrated Physics
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Capillarity — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · fluids
  2. 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.

    part 2 · fluids
  3. 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.

    part 3 · fluids
  4. 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.

    part 4 · fluids
  5. 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.

    part 5 · fluids
  6. The column that overshoots its own height. The height of water in a vertical glass tube, as a fraction of Jurin's equilibrium height 2γ/ρgr, against time for tubes of radius 0.2, 0.4, 1, 2 mm, integrated from the capillary pull, the weight and the Poiseuille drag with the inertia of the rising column. The 0.2 mm tube (Jurin height 74.4 mm) creeps up without overshooting; the 0.4 mm tube (Jurin height 37.2 mm) creeps up without overshooting; the 1 mm tube (Jurin height 14.9 mm) overshoots to 1.33 of it; the 2 mm tube (Jurin height 7.4 mm) overshoots to 1.46 of it. The linearised motion about the equilibrium is underdamped when the radius exceeds (32η²γ/ρ³g²)^(1/5) = 0.48 mm for water: wider tubes ring, narrower ones do not.

    The pore that lifts highest fills slowest

    Jurin's law says how high water climbs in a tube and nothing about how long it takes. The rise has three laws in succession — the column's inertia, then its viscous drag, then its weight — and the same narrowness that lifts water highest throttles it hardest, so for any deadline there is a best pore. A wide enough tube does something Jurin's law cannot: it overshoots and rings. And whatever the route, exactly half the energy the surface releases is lost.

    part 6 · fluids
  7. The pull of a meniscus against the stiffness of a wall. For two neighbouring lines of a wet pattern leaning towards each other, the net Laplace pressure pulling them together (heavy curve) and the bending stiffness pushing them back (straight lines), against the lean as a fraction of the half-gap. The pull grows faster than linearly, because narrowing a gap raises its suction, and it diverges as the gap closes. The stiffness is a straight line whose slope falls as the fourth power of the line's height; the lines drawn are for 0.8 times, 1 times, 1.25 times the critical height. At 0.8 of the critical height the straight state is stable and an unstable balance lies at a lean of 0.77 of the half-gap, a barrier the lines must be pushed over; at the critical height the two start with the same slope and the straight state is marginal; at 1.25 times the straight state is unstable and the pair collapses.

    The walls a rinse pulls together

    Every result about capillary rise assumes the walls hold still. Make them thin enough and they do not. The water between two walls is below atmospheric pressure, so it pulls them together, and as the gap narrows the suction grows. Beyond a critical height or length the walls cannot push back and they snap shut. It is why wet hair clumps, why a paintbrush comes out of water as a point, and why a chip factory cannot rinse lines of resist taller than about three times their width without laying them flat.

    part 7 · fluids
  8. 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.

    part 8 · fluids
  9. 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.

    part 9 · fluids

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