A rounded edge spills before its line goes round

An edge holds any angle it is given put a liquid’s contact line on a sharp edge and followed Gibbs’s inequality to its consequences. On a smooth face the line meets the solid at the Young angle, fixed by the chemistry. On an edge the line stops, and the angle it makes with the upper face is free to take any value from the Young angle to the Young angle plus the edge’s turn — 90° more on a square rim. That band is why a coin carries a dome of water nearly three times the volume its chemistry alone allows, why a glass can be filled above its brim, and why a spout that has once dribbled goes on dribbling.

It closed by admitting that the band was drawn for an edge that does not exist. A machined edge is rounded to a few micrometres, a coin’s rim to a fraction of a millimetre, a pottery lip to a millimetre or more. On a rounded edge, it said, the line does not stand still; it slides round the curve, and pinning at one place is the limit of an edge much sharper than anything else in the problem. What it did not say was how much sharper, or what a rounded edge is worth.

The natural suspicion is that rounding destroys the band. The line was pinned because the solid’s face changed direction at a point; if the direction changes smoothly, the line can always find a place where the local face makes the Young angle with the liquid, and it will slide there. On that reading a rounded edge is just a curved surface, and a curved surface holds a liquid at its Young angle like any other. The reading is half right.

A line that slides and an angle that still turns #

Take a plateau whose edge is a quarter-round of radius r, turning down to a vertical face. A line sitting on the curve at an angle φ round it is on a face tilted by φ from the horizontal. Locally it meets that face at the Young angle θ, exactly as the chemistry demands — nothing is pinned. But the face itself has turned, so the meniscus meets the horizontal at θ + φ. By the time the line has slid all the way round, φ is 90° and the apparent angle is θ + 90°, the top of Gibbs’s band. The band has not gone. It has been spread along the curve, one degree of apparent angle per degree of arc.

What rounding adds is a second thing that changes as the line goes round: it drops. On a sharp edge the line stays at the plateau’s level while the angle climbs. On a rounded one it sinks by r(1 − cos φ) as it slides, and a puddle’s height is measured from its line. For a puddle wide enough to be flat on top — the case the size a drop is allowed sets up — the height above the line is 2 sin(ψ/2) capillary lengths for an apparent angle ψ, so the height above the plateau is

$$h(\varphi) = 2\sin\frac{\theta + \varphi}{2} - r\,(1 - \cos\varphi),$$

in capillary lengths — 2.73 mm for water. The first term is Gibbs; the second is the price of the curve.

The figure draws the meniscus on an edge rounded to half a capillary length, about 1.4 mm, at four positions of the line. At 0° it is the ordinary puddle at the Young angle. As the line slides round, the meniscus steepens and the puddle rises, but the line is also going downhill. At 66° the two effects balance, and that is as high as the puddle can stand.

Where the rise stops #

The figure plots h against the line’s position for four roundings. On a sharp edge the height rises all the way round, to the Gibbs value at 90°. On a rounded edge the curve rises, flattens and turns down, and its maximum is the most the edge can hold. The maximum sits where the slope of the first term equals the slope of the second: cos((θ + φ)/2) = r sin φ. Past that point the line would have to go further round for the puddle to grow, and going further round lowers the puddle; there is no equilibrium with more liquid, and the puddle spills over.

This is a statement about stability rather than about force, and it is worth being precise about what spills. Nothing is pushed off the edge. Each position of the line on the curve is an equilibrium, and each holds a definite volume. Up to the maximum, adding liquid moves the line a little further round to a new equilibrium; at the maximum there is none to move to, and the next drop runs down the face. The same thing happens on a sharp edge at the top of the band. On a rounded edge it happens earlier.

How much of the arc a line can use #

The spill condition says how much of the curve is usable. If the rounding radius is less than cos((θ + 90°)/2) capillary lengths — about a half for a Young angle of 30°, a quarter for 60° — the rise never stops before the vertical face, the line uses the whole arc, and the edge holds exactly the angle a sharp edge would, less only the height by which its line sits below the plateau. For larger roundings the line gets only part-way round before the puddle spills, and how far falls as the inverse of the radius. A rim rounded to several capillary lengths is used only in its first few tens of degrees; to the liquid, it is barely an edge at all.

For a Young angle of 90° or more the limit is zero: the rise stops before 90° however small the rounding, because the Gibbs term’s slope cos((θ + φ)/2) has already reached zero at the face. On a sharp edge that makes no difference — the line cannot slide and the puddle spills at the face anyway — but it means a water-repellent edge is never used to its full turn once it has any rounding at all.

What an edge is worth #

The useful number is how much of a sharp edge’s advantage a rounded one keeps: the extra height it holds over a flat plate, as a fraction of the extra height a sharp edge holds. For water on a surface with a Young angle of 60°, rounding the edge by a tenth of a millimetre keeps 96 per cent of the advantage; a third of a millimetre, 88; a millimetre, 62; two millimetres, 40; three millimetres — about a capillary length — 30. Far beyond, the advantage decays as $\cos^2(\theta/2)/2r$, the inverse of the radius.

So the scale that decides whether an edge is sharp is the capillary length, not anything microscopic. An edge is sharp, for this purpose, when its rounding is small against 2.7 mm. A machined edge at a few micrometres is sharp by three orders; a coin’s milled rim at a tenth of a millimetre is sharp to within a few per cent; a glass’s brim rounded to a millimetre is worth about two-thirds of a knife edge; a thick earthenware lip rounded to three or four millimetres keeps under a third. The static picture of the previous essay is right for anything much finer than a millimetre and wrong, by a factor that this calculation supplies, for anything coarser.

The coin again #

The same calculation on a coin the size of a US cent, 19.05 mm across, with a Young angle of 60° and its drop computed as a surface of revolution rather than a wide puddle: a sharp rim holds 1.44 mL, the figure the previous essay found. Rounded by a tenth of a millimetre it holds 1.41; by a millimetre, 1.16; by three millimetres, 0.73 — still well above the 0.52 mL the surface’s chemistry allows with no edge at all. On the coin the line uses the whole arc for roundings up to about a millimetre and a half, because a finite drop’s hoop curvature adds pressure that a wide puddle does not have, so the loss on a slightly rounded rim is almost all the line sitting lower, below the coin’s face, and the volume of rim that sits inside the drop.

A glass filled above its brim is the same calculation turned inside out: the liquid stands inside a ring rather than on a disc, and a brim a couple of millimetres thick is rounded on both sides. For a clean glass, with a Young angle of perhaps 20° to 30°, a brim rounded to a millimetre still lets the water stand three and a half millimetres or so above it before it spills — about four-fifths of what a knife edge would allow — which is why the trick works on ordinary glassware and fails on a thick, rolled rim.

The coin-and-dropper demonstration, in which pupils count how many drops a coin will hold before it spills, is therefore mostly measuring the coin’s milling and wear as well as the water. A new coin with a crisp rim and an old one whose rim has been rounded by handling can differ by several per cent before any difference in cleanliness is counted.

Where the scale is not millimetres #

The capillary length sets the scale here because gravity is what a puddle’s height works against. Where gravity does not matter, the scale that replaces it is whatever else sets the meniscus’s curvature, and the conclusion changes accordingly.

In a microfluidic channel the meniscus is curved by the channel itself, tens of micrometres across, and the pressure it holds is set by that curvature — the pressure a law that is exact as an average computes for a meniscus in a tube. A capillary valve, a sudden widening that stops the advancing liquid until the pressure behind it has bent the meniscus through the band, works on the same geometry as the puddle: the line slides round a rounded corner, drops back relative to the widening as it goes, and bursts early if the rounding is comparable with the channel. There the rounding that matters is a fraction of the channel’s width, and an etched corner rounded by a couple of micrometres in a twenty-micrometre channel gives away a noticeable part of the burst pressure. The same edge would be perfectly sharp on a coin.

On the water-repellent surfaces that hold a liquid off a forest of posts, the posts’ top edges are what pin the meniscus bridging each gap, and the gaps are micrometres wide. A post whose cap is rounded to a fraction of the gap cannot turn the meniscus through its full band before the liquid wets down the post’s side, which is one reason the re-entrant caps that let such surfaces hold oils are made with overhangs sharp on the scale of the spacing, not merely on the scale of a drop.

In every case the rule is the one the puddle gives: an edge is sharp when its rounding is small against the length over which the meniscus it holds is curved. For a drop standing on a coin that length is the capillary length; for a meniscus in a channel it is the channel. The effect that is real, and where it stops found the same structure for a jet on a curved wall — a real effect, and a ratio that says where it ends.

The teapot’s lip #

The teapot effect is decided at a lip, and the cures for it are edges. A spout that dribbles is one where the line has run over the lip onto the outside, and the previous essay read the standard cures — a thin, sharp, downturned lip, a groove under the spout, a water-repellent coating — as ways of widening or relocating Gibbs’s band. This calculation adds the missing condition on the first cure. A thin lip helps only because a thin lip is also a sharp one. A potter’s spout is thrown and glazed, and glaze rounds the edges it covers, often to the order of a millimetre; on the numbers above that gives away a third or more of the band’s worth before the flow has had any say. The lip that works is one turned or ground to a fraction of a millimetre after glazing, and the same is true of the ridge under a spout, which is an edge and is worth its sharpness like any other.

It is the same move the effect that explains nothing made about Coandă: the name given to a behaviour — here “pinning” — stands in for a calculation, and the calculation says how much of the behaviour is there. A rounded lip still pins, in the sense that its apparent angle is not the Young angle; it pins less, by an amount set by one ratio.

What was checked #

With the rounding taken to zero the largest puddle must be Gibbs’s 2 sin((θ + 90°)/2), and it is to a part in a billion. With a rounding of four hundred capillary lengths its extra height over a flat plate must approach $\cos^2(\theta/2)/2r$, and it does to 0.06 per cent. The spill position must satisfy the stationarity condition, and does to rounding error. On the coin, a rim rounded to a quarter of a micrometre must hold what the sharp coin of the previous essay holds, 1.44 mL, and does to five parts in a hundred thousand; the sharp coin itself was checked there against the force balance at its rim. The tests also refuse a negative rounding, a Young angle of zero and a tolerance of zero.

What the model leaves out #

Contact-angle hysteresis. Every face here has one Young angle. A real surface advances at a larger angle than it recedes, so a line on a rounded edge sticks until the local angle has moved by the hysteresis before it slides. That makes a rounded edge behave a little more like a sharp one, by the hysteresis angle, and it restores some of the loop between filling and emptying that an ideal rounded edge does not have.

A quarter-round. Real roundings are not circles, and a chamfer — two small sharp edges — would hold differently: each of its corners has a Gibbs band of its own, and a line on a 45° chamfer can be held at either. What a plate takes with it treats the line in motion, and none of this does.

Static liquid. The puddle and the coin are filled infinitely slowly. A pour is not, and whether a moving sheet can use the band at all is the question that remains.

Who worked it out #

Gibbs’s inequality is from 1878, and Oliver, Huh and Mason measured it on machined edges in 1977, noting that any real edge has a finite radius below which the inequality is only an idealisation. The treatment of a puddle’s height as a function of the line’s position is the textbook capillary balance of de Gennes, Brochard-Wyart and Quéré; what is done here is only to put the rounding into it and find where the maximum falls.

Still open: the sheet that leaves at speed #

Everything so far is static, and the teapot is not. A sheet of liquid arriving at a lip with speed U and thickness h carries a momentum flux $\rho U^2 h$ that the static band knows nothing about, and whether its lower surface leaves the edge at an angle the band can hold is set by the flow as well as the geometry. The next calculation carries the membrane balance of the teapot essay onto a rounded lip — the sheet’s effective tension $2\sigma - \rho U^2 h$ pulling it round the curve, the line on the curve sliding at the Young angle — and asks at what speed, for a given rounding and Young angle, the line is carried over the lip onto the outside, which is the dribble’s threshold drawn in speed, rounding and wettability together.