The force a contact line holds is a range

A law that is exact as an average finds that the liquid a tube lifts is exactly $2\pi R\gamma\cos\theta/\rho g$ in volume, at every radius: Jurin’s law, which is only approximate as a statement about the height at the centre, is exact as a statement about the mean height of everything lifted. The identity is a force balance on the lifted column — surface tension pulling up round the perimeter against the column’s weight — and it has one contact angle in it.

It ends on the complication every experimenter meets. A real contact line does not sit at one angle. It pins on roughness and on chemical patches, and a liquid front that has been advancing stops at a steeper angle than one that has been receding. Between the receding angle $\theta_r$ and the advancing angle $\theta_a$, the line can sit at any angle at all without moving. This essay follows what that does to two measurements, a column in a tube and a drop on a tilted pane, and finds the same thing in both: the force a contact line holds is not a number but a range.

Static friction for a liquid #

The force a contact line exerts on the liquid, per unit length, is $\gamma\cos\theta$ along the solid, pulling the liquid across it. With one angle it is fixed. With pinning, the angle is whatever it has to be to hold the liquid in equilibrium, provided it stays between the two limits, and the force is anything between $\gamma\cos\theta_a$ and $\gamma\cos\theta_r$. The width of that range,

$$\gamma\,(\cos\theta_r - \cos\theta_a),$$

plays exactly the part of the limiting friction in Amontons’ law: below it, nothing moves, and the line takes whatever force the equilibrium asks for; at it, the line slips. Hysteresis, in this light, is not a nuisance in the measurement of a contact angle but a property of the contact line in its own right, and it makes a liquid on a solid behave in the two measurements below like a block on a rough table.

One tube, two heights #

The lifted volume is $2\pi R\cos\theta$ in capillary units, so the mean height is $2\cos\theta/R$ — and with pinning there are two of each.

The first figure is a glass tube half a capillary length in radius, 1.35 millimetres for water, with the menisci a meniscus solver gives for a column that has risen into place and one that has fallen into place. Clean glass has an advancing angle of about 30 degrees and a receding angle of about 10. The risen column’s mean height is 3.46 capillary lengths, 9.4 millimetres; the fallen column’s is 3.94, 10.7 millimetres; the heights at the wall are 3.62 and 4.19. The two surfaces differ by 14 per cent in the height they report, and a column pinned anywhere between them stays put. Nothing about the liquid or the tube is different. The difference is history.

That is the standard complaint about capillary rise as a way of measuring surface tension: fill a tube from below and it reads low; let it drain to its level from above and it reads high. The usual remedy — to wet the tube well above the final level and let the column fall into place, so that the receding angle, which on clean glass is nearly zero, is the one in play — is the right one, and the arithmetic says why. At a receding angle of 10 degrees $\cos\theta$ is 0.985, and treating it as one costs a per cent and a half. At an advancing angle of 30 degrees it is 0.866, and treating that as one costs thirteen.

The second figure generalises. For a given hysteresis — a fixed number of degrees between the two angles — the height range is $\cos\theta_r/\cos\theta_a - 1$, which depends strongly on where the angles lie. Ten degrees of hysteresis about a mean of 10 degrees moves the height by 3 per cent; about a mean of 60 degrees, by 36. A surface that water wets well hides its hysteresis, because the cosine is flat near zero; a surface it wets poorly amplifies it. The ratio is independent of the tube’s radius, since the lifted volume is exact at every radius, so the same curves hold for a hair-thin capillary and a tube a finger wide.

A patchy wall, against a wall with a past #

The earlier essay’s own question was different: not advancing against receding, but a contact line pinned at a spread of angles round the tube’s circumference, as a wall with a patchy surface would pin it. The force balance on the column is still exact, and it now says the total lift is the perimeter’s average of $\cos\theta$, not the cosine of an average angle.

The third figure computes it for a Gaussian spread about a mean of 20 degrees. The perimeter average of $\cos\theta$ is then $\cos\bar\theta\,e^{-\sigma^2/2}$, checked here against a direct quadrature to a part in $10^9$. A spread of 10 degrees lowers the lift by 1.5 per cent; a spread of 20 degrees, which is ordinary for glass that has not been cleaned with care, by 5.9 per cent. Both are well below the 14 per cent between a risen and a fallen column on the same glass. So of the two ways a real wall departs from the single angle, it is the wall’s memory of which way the liquid last moved, not the patchiness of its surface, that dominates the scatter in a capillary-rise measurement — and the patchiness enters only at second order, as the square of the spread, which is why it is so easily overlooked.

A drop on a tilted pane #

The same range decides how large a drop a tilted surface can hold. A drop on a plate tilted at an angle $\alpha$ is pulled downhill by the component of its weight along the plate. As the tilt grows, its downhill edge steepens towards the advancing angle and its uphill edge flattens towards the receding one; when both reach their limits, it slides. For a ridge of liquid — a drop long across the slope, whose cross-section is the same everywhere — the force balance along the plate at that moment is exact, per unit length:

$$\rho g A \sin\alpha = \gamma\,(\cos\theta_r - \cos\theta_a),$$

with $A$ the cross-sectional area. In capillary units, the largest ridge the plate holds has an area $(\cos\theta_r - \cos\theta_a)/\sin\alpha$.

That balance says nothing about the drop’s shape, and the shape is the check. Young–Laplace along the surface, with the hydrostatic pressure of a tilted layer, was integrated from the uphill contact point at the receding angle, with the pressure there chosen by bisection so that the surface comes back to the plate at the advancing angle. The area enclosed was then compared with the force balance, which the integration never used.

The fourth figure shows three of the shot drops, on a plastic with an advancing angle of 90 degrees and a receding one of 70. Each leans downhill, its downhill edge standing at 90 degrees and its uphill edge at 70, and the steeper the plate, the smaller the ridge it can hold: 1.97, 0.68 and 0.39 square capillary lengths at 10, 30 and 60 degrees, which for water is 14.5, 5.0 and 2.9 square millimetres of cross-section. The force balance gives 1.970, 0.684 and 0.395, and across nine cases on three surfaces the shot areas match it to six parts in $10^8$.

What a pane holds, and why glass holds least #

The fifth figure sets three surfaces side by side, and the order is the opposite of what intuition expects. Clean glass, which water wets readily, holds the smallest drops: on a plate tilted 30 degrees, a ridge of at most 0.24 square capillary lengths. A plastic that water wets far less holds 0.68, and a water-repellent coating with the same 20 degrees of hysteresis, 115 degrees advancing and 95 receding, holds 0.67. What holds a drop is not how much the liquid likes the surface but how different its two angles are — and, through the cosine, where they lie. On clean glass the two cosines, 0.985 and 0.866, differ by only 0.12; on the plastic, 0.342 and 0, by 0.34.

That is why a window that has just been cleaned sheds rain in thin, fast-running films and a dirty one holds beads. It is also the whole design brief of a surface meant to shed water: what matters is not a large contact angle but a small hysteresis. A coating that raises both angles together, as the third surface does, sheds no better than the plastic. The surfaces that shed drops at a tilt of a degree or two are the ones engineered to have almost no hysteresis at all.

A drop of finite width is harder, because its contact line is curved and the angle varies round it. The same balance holds with the drop’s width across the slope in place of a unit length and with a shape factor of order one that depends on how the angle is distributed round the line; it has been measured on many surfaces and is usually close to one. The two-dimensional ridge is the case in which the factor is exactly one, and in which the shape can be solved and checked.

A windscreen’s worth of water #

The numbers say what a tilted pane of glass actually keeps. On clean glass at 30 degrees, the largest ridge that stays put is 5.6 millimetres from its uphill edge to its downhill one and half a millimetre deep — a thin lens of water, 1.7 square millimetres in cross-section. At 60 degrees, the rake of a car’s windscreen, it is 4.2 millimetres long and 0.4 deep. The same plate coated with the plastic holds a ridge 4.2 millimetres long and 1.6 deep at 30 degrees, and 3.1 by 1.25 at 60: a bead rather than a lens, and three times the water.

The shapes explain the difference as well as the sizes. On glass the ridge at the onset of sliding is nearly flat, because both its angles are small; most of its length is at a depth of a fraction of a millimetre, and a small push — the air flowing over a moving car — is enough to start it. On the plastic the ridge stands up, its downhill edge vertical, and it has to grow to three times the volume before its weight matches what its two contact lines can hold. A rain-repellent treatment for a windscreen that raised the contact angles without narrowing the gap between them would give exactly this: taller, stickier beads that the wipers have to move.

The sliding threshold is a Bond number #

In capillary units the ridge’s cross-section is itself a Bond number — its area over the square of the capillary length, the ratio of a weight to a tension force that the size a drop is allowed puts at the centre of every problem in this family. The force balance then reads as a threshold on that number: a ridge slides when

$$\mathrm{Bo}\,\sin\alpha > \cos\theta_r - \cos\theta_a,$$

and the hysteresis sets the critical Bond number directly. That puts the tilted pane beside two thresholds already found for a drop’s size: the pendant drop that falls at a fold, where the weight outgrows the largest force a tube’s rim can supply, and the falling drop that no speed can break below a Weber number of its own. In each, a drop’s fate is decided by comparing its weight with a capillary force that has a ceiling, and the ceiling is set by the geometry of the contact line: the rim’s circumference for the pendant drop, the difference of two cosines for the pane.

Read the other way, the threshold is a measurement. A drop of known size on a plate tilted slowly until it slides gives $\cos\theta_r - \cos\theta_a$ directly, from the tilt at which it moves, without either angle being measured on its own. For a ridge of half a square capillary length the sliding tilt is 13.7 degrees on clean glass and 43 degrees on the plastic, and one of a fifth of a square capillary length slides on glass at 36 degrees and never slides at all on the plastic, even held vertically. This is the tilting-plate method, and the ridge solved here is the case in which it is exact rather than approximate.

The edge that holds any angle #

There is one place where hysteresis becomes unlimited. An edge holds any angle it is given: at a sharp corner in the solid, a contact line can sit at any angle within a band as wide as the corner’s own angle, the inequality Gibbs wrote down, and the range of force it holds widens to match. A drop on a plate with a raised rim, or a column in a tube whose top edge it has reached, is pinned far more strongly than any chemistry could pin it — which is why a glass filled to its brim can be overfilled into a dome, and why the meniscus of a siphon’s column can be held at the lip of a tube.

What was checked #

The sixth figure lists them. The shot ridges were checked against the exact force balance on three surfaces at three tilts. The tube’s two mean heights were computed by the meniscus solver and checked against $2\cos\theta/R$, the exact lifted volume, to four parts in $10^7$. And the effect of a spread of angles was computed by quadrature over the perimeter and checked against its closed form.

What the picture cannot show #

Where the angles come from. The advancing and receding angles are stated for each surface, as representative values, not derived from any roughness or chemistry. How a given surface’s texture sets them is a separate problem.

Motion. Once the contact line slips, the angle depends on its speed, and a sliding drop’s shape and speed are a dynamic problem; a plate drawn out of a liquid is the steady version of it.

A two-dimensional drop. The shot shapes are ridges. A real drop’s contact line is curved and its shape must be solved in three dimensions.

Evaporation and contamination. A contact line pinned for long enough can move by evaporation alone, and a contaminated surface’s angles drift.

The convention the numbers depend on #

Contact angles are measured through the liquid. Lengths are in capillary lengths, $\sqrt{\gamma/\rho g}$, which for water at room temperature is 2.71 millimetres; areas in its square, 7.34 square millimetres. The “mean height” of a column is its lifted volume over the tube’s cross-section. The tilt $\alpha$ is the plate’s angle to the horizontal. The Bond number that gives this family of problems its name is the square of a length in capillary lengths.

Who found it, and when #

Contact-angle hysteresis was noted in the nineteenth century and studied systematically in the twentieth; the balance for a drop on a tilted plate is due to C. G. L. Furmidge, who published it in 1962 for the retention of spray droplets on leaves. Gibbs gave the inequality for a contact line at an edge in his work on heterogeneous equilibrium in the 1870s. Jurin’s law dates from 1718, and the exact volume identity behind it is Laplace’s.

Still open: the drop that has started to slide #

The balance above says when a drop starts to move and nothing about what it does next. A sliding drop’s contact angles depend on its speed through the dissipation near the moving contact line, which grows logarithmically as the line is approached and has to be cut off at a microscopic length. The calculation that follows would give the ridge a speed, apply the relation between the dynamic angle and the speed that comes out of the viscous corner flow, and ask for the steady speed at each tilt above the threshold — and whether a drop that has just started to slide can stop again, which is the question of whether a contact line has the equivalent of a lower kinetic friction than its static one.