An edge holds any angle it is given

The teapot effect is a tension, not a pressure found that a sheet of liquid leaving a lip is held to it by its own surface tension, and lets go at the one speed that tension cannot carry. It found too that the lip’s radius hardly matters. The cures that have lasted — a ridge under the spout, a groove, a downturned lip, a water-repellent coating — all act on the line where the liquid meets the solid at an edge. The essay stopped there, deliberately, because the membrane argument has nothing to say about an edge.

This essay is about the edge. The claim it examines is one that almost every account of wetting states as its starting point: that a liquid meets a solid at a definite angle, the contact angle, set by the three surface tensions between liquid, solid and air. On a smooth face that is true. On an edge it is not, and the way it fails explains the coin that holds thirty drops, the glass standing full above its brim, and why a teapot that dribbles once goes on dribbling.

The angle the chemistry sets #

On a flat, smooth, uniform solid, a contact line settles where the three tensions balance along the surface. Young’s equation says the solid–air tension equals the solid–liquid tension plus the liquid–air tension times the cosine of the angle. Given the three tensions there is one angle, the Young angle θ. It is a real material property, and for water it runs from near zero on clean glass to over a hundred degrees on wax.

The balance is a statement about a line on a face, and it is only the balance along the face. Across the face, the solid pushes back with whatever normal force is needed and asks nothing about the angle. That is why the angle is set only along one direction, and it is what an edge breaks. At an edge the face itself changes direction, so “along the face” has two meanings, one for each side.

What changes on an edge #

Put the line exactly on an edge where the upper face turns down, and give the solid a wedge angle φ at the edge — 180° for no edge at all, 90° for a coin’s square rim, a few degrees for a knife. Measured from the upper face, a line about to move onto that face must make the Young angle with it: θ. Measured from the lower face, a line about to move down onto it must make the Young angle with that face. Seen from above, that is θ plus the angle through which the face has turned, 180° − φ.

Between those two values the line can go nowhere. To retreat it needs an angle as small as θ; to advance it needs one as large as θ + 180° − φ. Any angle between leaves it stuck on the edge. The edge holds every angle in the band θ ≤ ψ ≤ θ + (180° − φ) — Gibbs’s inequality, stated in 1878 and measured on machined edges by Oliver, Huh and Mason in 1977. The band’s width is set by the geometry alone and its position by the chemistry. Inside it, the angle is not a material property at all: it is whatever the rest of the liquid needs it to be.

A square rim therefore gives water on clean glass, at a Young angle of 20°, every angle from 20° to 110°, and a knife edge up to 200°. The band is ninety degrees wide on the rim, where ordinary contact-angle hysteresis on a smooth surface is a few degrees to a few tens.

How high water can stand #

The first consequence is how high a liquid can stand. A puddle wide enough to be flat on top has a height fixed entirely by the angle at its edge. The pressure under the flat top is hydrostatic. Balancing it against the tension pulling in at the edge gives

$$h = 2\ell\,\sin\frac{\psi}{2}, \qquad \ell = \sqrt{\frac{\sigma}{\rho g}},$$

where ℓ is the capillary length, 2.73 mm for water at room temperature. On clean glass at 20° that is 0.95 mm, and a puddle on a flat glass plate is that thin however much water is poured on it. The water spreads rather than rising.

On a rim the same water stands higher, because the edge will hold any angle up to 110°: 4.47 mm. That is the full glass. Filled carefully past the brim, the water rises above it because the line has stopped on the rim’s outer edge. The angle there climbs through the band as more is added, and the surface stands up to 4.5 mm proud before the line runs down the outside. The ceiling for any edge, however sharp, is ψ = 180° and h = 2ℓ = 5.45 mm. There the free surface would have to turn back under itself to go higher.

The number that sets all of this is the one the size a drop is allowed used to decide when a drop is flattened by its weight. The capillary length is the scale at which surface tension and gravity are matched. A puddle is always a couple of capillary lengths high at most, and the angle at its edge decides where in that range it sits.

A coin that holds thirty drops #

The classroom version is a coin and a dropper: count the drops a cent will hold before the water spills. The answer is always surprisingly large — twenty to forty — and it is usually explained as surface tension forming a skin. The skin is real, but the skin is not what decides the count. The edge is.

The calculation is the sessile drop, the same Young–Laplace balance the drop falls at a fold solved for a drop hanging under a tube, with gravity reversed. In capillary lengths, with z measured down from the apex and φ the angle of the drop’s outline from the horizontal, the outline obeys

$$\frac{dx}{d\phi} = \frac{\cos\phi}{\kappa}, \qquad \frac{dz}{d\phi} = \frac{\sin\phi}{\kappa}, \qquad \kappa = \frac{2}{b} + z - \frac{\sin\phi}{x},$$

b being the radius of curvature at the apex. For a coin of radius R and an apparent angle ψ at its rim, the outline is integrated from the apex to φ = ψ. The apex radius is chosen so that it ends at x = R, and the drop’s volume is the integral of πx² down the outline.

A cent is 19.05 mm across, 3.49 capillary lengths in radius. At 60° it holds 0.52 mL, at 90° 0.82, at 120° 1.13 and at 150° 1.44. None of these drops is a spherical cap: the tops are flattened by their weight into puddles with rounded edges, 2.8 to 5.5 mm high. What fixes which one appears is the Young angle and the rim together. Water on a copper surface with a Young angle of 60° fills the coin until the line reaches the rim at 60°. It then holds at the rim while the angle climbs to 150°, and spills there, carrying 1.44 mL — twenty-nine drops of a twentieth of a millilitre.

What the rim is worth #

The same coin with no edge — a disc of the same size inked on a large flat plate of the same material — holds only what the Young angle allows before the line moves outward. At 60° that is 0.52 mL, and at 30° only 0.25 mL. The rim holds 1.44 and 1.13. The edge multiplies what the coin holds by 2.7 and 4.4. The more wetting the liquid, the more the edge is worth, because a wetting liquid would otherwise spread and the rim is the only thing stopping it. At 90° the gain is 2.1, and the drop at the top of the band is nearly a whole sphere of water resting on its rim, 5.7 mm high.

The count of drops therefore measures the edge first and the chemistry second: 23 drops at a Young angle of 30°, 29 at 60°, 34 at 90°. A worn coin with a rounded rim holds fewer — the rounding lets the line creep round the curve instead of stopping at it, and the band of held angles shrinks. A greasy coin holds more, because grease raises the Young angle and moves the whole band up.

Filling and emptying are different journeys #

The band has a second consequence, and it is the one the teapot left open. Fill the coin from a small drop and the line spreads at the Young angle until it reaches the rim. It then stops, while the angle climbs through the band. Draw water off again from a full coin and the line does not retreat when the angle passes back through the value it had on the way up. It stays on the rim until the angle has fallen all the way to the Young angle, and only then moves inward. Every drop between 0.52 and 1.44 mL can be reached on the way up and on the way down, and the rim holds the line through all of them.

That is hysteresis made by geometry. On a smooth surface the difference between the angle at which a line advances and the angle at which it recedes is a property of the surface’s roughness and chemistry. It is usually a few degrees, sometimes a few tens. What a plate takes with it and the drop a no-slip wall would never let spread both treat a line that is free to move, and the dynamics of its motion. On an edge the difference is 180° − φ, fixed by the shape and independent of the surface’s finish: 90° on a square rim.

This is the teapot’s hysteresis. A spout starting to pour has a dry lip. The line sits on the lip’s outer edge, and the sheet leaves at whatever angle the flow needs, provided it lies in the band. A slow start — a pour that begins at a trickle — lets the line run over the edge and down the outside. The outside of the spout is then wet, and it has no second edge to stop anything. Speeding the pour up moves the sheet, but the line on the outside has nowhere to be held. It has to be dragged all the way back to the lip before the band applies again, so a spout that has once dribbled goes on dribbling at speeds that would have left it clean from a fast start.

The cures the teapot essay listed now read as edge engineering. A ridge or a groove under the spout gives the liquid a second edge to be held on. A thin downturned lip makes the edge sharper and widens the band. A water-repellent coating raises the Young angle, which lifts the whole band. The sheet can then leave at angles that on a wetting spout would already have sent the line over. The 2010 experiments of Duez and colleagues, in which a spout made hydrophobic stopped dribbling, measured exactly this.

The same band, holding liquid off a surface #

The surprising application of Gibbs’s band is not holding liquid on a surface but holding it off one. A lotus leaf, and every engineered water-repellent surface, carries a forest of tiny posts. A drop sits on their tops with air trapped underneath, touching only a few per cent of the solid. What stops the water falling into the gaps between them is the post edges. The water’s surface bridges each gap and meets each post’s top edge at an angle that has to lie in that edge’s band, and for a square-topped post the band reaches 90° above the Young angle.

That is why a liquid that wets the solid material can still be held off it. A post with an overhanging, re-entrant cap has a wedge angle below 90°, and its band reaches past 180° above the Young angle. Oils, whose Young angle on almost anything is well below 90°, can then be suspended on air as water is on a lotus leaf. The surfaces called omniphobic — Tuteja and colleagues’ of 2007, and the cuticle of springtails that live in wet soil — work on this edge-pinning. None of it would be possible if the contact angle were a material property.

Microfluidic devices use the same inequality to stop flow. A channel that opens suddenly into a wider chamber presents the advancing liquid with an edge. The line holds there until the pressure behind it has bent the meniscus through the whole band, which makes a capillary valve with no moving parts, whose burst pressure is set by the edge’s geometry. A law that is exact as an average computes the pressure a meniscus carries in a tube, and the valve is that pressure with an edge deciding the angle.

What the picture cannot show #

Real edges are round. A machined edge has a radius of a few micrometres, a coin’s rim more. On a rounded edge the line is not held at one place. It slides round the curve to the point where the local face makes the Young angle with the surface, and the angle measured from the upper face changes as it goes. The band is the same, but the line moves through it rather than standing still. Pinning at one place is the limit of an edge much sharper than anything else in the problem.

The surface is clean and uniform. Every face here has one Young angle, with no hysteresis of its own. A real coin has an advancing angle and a receding one on its flat face, and both widen the loop further. Dust on the rim is a local edge of its own, and a line can catch on it before it reaches the rim.

The drops are static. A drop fed by a dropper is jolted at each addition, and near the top of the band the equilibrium is barely held. A disturbance can push the line over the edge early, so a coin in a real classroom spills a drop or two before the calculation says it must. The loop is drawn for infinitely slow filling.

The coin is level. Tilt it and the angle round the rim varies. The line goes over the edge first on the downhill side, where the angle reaches the top of the band soonest, and the capacity falls.

The convention the numbers depend on #

The Young angle θ is measured through the liquid on a smooth face of the material. The apparent angle ψ at an edge is measured through the liquid from the upper face. The wedge angle φ is the solid’s interior angle at the edge: 180° for none, 90° for a square rim. Water is taken at 20 °C, σ = 0.0728 N/m and ρ = 998 kg/m³, giving ℓ = 2.73 mm. The top of a square rim’s band for θ = 90° is taken as 179° rather than 180°, where the outline would close into a sphere touching the coin at a point. A US cent is 19.05 mm across. A “drop” in the count is 0.05 mL.

How each number was checked #

Three checks bear on the drops. The first is exact: the weight of every drop must equal the pressure on the coin under it, less the pull of the surface round the rim, V = πR²(2/b + z) − 2πR sin ψ in capillary units. It holds to 4 × 10⁻¹⁰ across nine drops of three sizes and three angles. The second is a limit: a small enough coin carries a spherical cap, since gravity cannot reach it. The departure from a cap must fall as the square of the coin’s size, and halving the coin divides it by 4.00, 4.00 and 3.95 at 30°, 90° and 150°. The third is the other limit: a wide coin carries a puddle whose height tends to 2ℓ sin(ψ/2). What is left over is the rim’s curvature round the coin, which must fall as the inverse of its size, and doubling the coin divides it by 1.96, 2.01 and 2.00. The calculation refuses a Young angle outside 0° to 180°, a wedge wider than flat, a coin of zero size, and a force-balance tolerance of zero.

Who found it, and when #

Young stated the angle in 1805 and Gibbs the inequality for an edge in 1878, in the same memoir that made the thermodynamics of surfaces a subject. Oliver, Huh and Mason measured it on machined edges in 1977, and showed that the line holds across the whole band. The re-entrant surfaces that use it to repel oils are Tuteja, Choi, McKinley, Cohen and Rubner’s of 2007. The teapot’s hysteresis was put in these terms by Kistler and Scriven in the 1990s, and Duez and colleagues’ experiments of 2010 showed that the spout’s wettability decides whether it dribbles.

This field keeps finding the pattern that the effect that explains nothing and the effect that is real, and where it stops began with: a real observation, a real mechanism, and a textbook quantity promoted into an explanation it cannot give. Here the quantity is the contact angle. It is a property of the material on a face, and on an edge it is a property of the shape.

Still open: the sheet that leaves an edge at speed #

The static band says which angles an edge can hold. It does not say which angle a moving sheet asks for. That is the calculation the teapot actually needs: a sheet of given thickness and speed, turned by a lip of given radius, arriving at the lip’s edge at an angle set by the flow — the membrane balance of the teapot essay carried to the edge — and compared with the band there. Where the flow’s angle lies inside the band the pour is clean; where it falls outside it, the line goes over and the dribble begins. Mapping that boundary in speed and Young angle would give the coating the teapot needs for a given pour, and it would say how much a ridge under the spout is worth.

Beside it is the fate of the sheet once it has left cleanly, which is to thin, flap and break — where a jet stops being a jet — and the rounded-edge version of the band, in which the line slides rather than holds. That second one is the version every real rim is, and it decides how sharp an edge has to be before the static picture here is the right one.