Illustrated Physics
physics, drawn
Fluids

The corner a liquid never stops climbing

A narrow tube lifts a liquid to a definite height because it has a smallest width. A corner has none, so the capillary suction it can develop is unbounded — and whether the liquid takes advantage is decided by a single inequality between the contact angle and the corner's own angle.
14 min read 5 figures The shape decidesWhat happens at the edge

Assumes: How high water will climb · The angle a liquid makes with what it sits on

How high water will climb gives Jurin’s law: the rise in a tube is inversely proportional to its radius, so a narrower tube lifts liquid higher. The obvious extrapolation is that the narrowest part of any container fills first and highest.

A corner has no narrowest part. Two flat walls meeting along a line get closer together without limit as the line is approached, and there is no width to put into Jurin’s formula. So the question is not how high but whether — and the answer is a single inequality.

One line decides whether it climbs for ever. The wetting condition for a corner, drawn as a map. The horizontal axis is the corner's half-angle and the vertical axis the liquid's contact angle with the walls; the diagonal is θ + α = 90°. Below it the meniscus in the corner curves into the liquid, the capillary pressure grows without bound as the corner narrows, and the liquid wicks along it indefinitely. Above it the curvature has the other sign and the liquid stays put. The boundary is tested by evaluating the meniscus a millionth of a degree either side of it at seventeen half-angles, and it wicks on one side and not the other every time. water on clean glass, right-angled corner: α = 45°, θ = 5° — wicks; water on glass, a 20° groove: α = 10°, θ = 40° — wicks; water on plastic, right-angled corner: α = 45°, θ = 75° — does not. A right-angled corner needs a contact angle under 45°, which water on clean glass has and water on most plastics does not.
Fig. 1 The wetting condition for a corner as a map: the corner’s half-angle across, the contact angle up, and the diagonal θ + α = 90°. Below it the liquid runs into the corner and keeps going; above it the liquid stays out, however sharp the corner is made. The boundary is tested a millionth of a degree either side at seventeen half-angles.

The condition, and where it comes from

Concus and Finn proved the general form in 1969, and the geometry behind it fits in a sentence.

A meniscus spanning a corner meets both walls at the contact angle θ\theta. If it is a circular arc of radius rr, the geometry fixes where it sits: its nearest point is at a distance rcos⁡(θ+α)/sin⁡αr\cos(\theta+\alpha)/\sin\alpha from the apex, where α\alpha is the corner’s half-angle.

Everything hangs on the sign of that cosine. When θ+α<90∘\theta + \alpha < 90^\circ it is positive, the arc is concave towards the liquid, the pressure under it is below atmospheric, and the liquid is sucked in. When θ+α>90∘\theta + \alpha > 90^\circ it is negative, the curvature reverses, and the liquid is pushed out.

The same statement can be made without any arc at all, which is how Concus and Finn proved it. Ask whether a surface meeting both walls at the contact angle can exist at all near the apex: for a wetting enough liquid there is no such surface with bounded height, so the variational problem has no solution and the liquid must run in. That is a stronger result than the circular-arc argument, since it does not assume the meniscus is any particular shape, and it is why the condition is a theorem rather than a model.

There is no third case and no width to it. The condition is an inequality between two angles, with no length in it at all, which is why it holds for a corner of any size and why sharpening the corner does not help a liquid that fails it.

That is the opposite of the tube result, and the difference is exactly that a tube has a width and a corner does not. Making a tube narrower raises the liquid; making a corner sharper changes nothing about whether the liquid enters, because the criterion is scale-free.

What the column looks like

A column with no top and no width. The width of the liquid filament held in a corner, against how far up the corner it is, for water on clean glass, right-angled corner (α = 45°, θ = 5°) and water on glass, a 20° groove (α = 10°, θ = 40°). The curve is one over the height exactly — checked by differencing the logs on the drawn curve — because the suction holding the liquid up is the hydrostatic head and the curvature supplying it is one over the width. water on clean glass, right-angled corner: 3373.0 µm wide at 2 mm and 112.43 µm at 60 mm; water on glass, a 20° groove: 13735.0 µm wide at 2 mm and 457.83 µm at 60 mm. There is no height at which it stops. A tube has a rise because it has a smallest width; a corner has no smallest width, so it has no rise to compute — only a filament that gets thinner for ever.
Fig. 2 The width of the liquid filament held in a corner, against how far up it is. The curve is one over the height exactly — checked by differencing the logs on the drawn curve — because the suction holding the liquid up is the hydrostatic head and the curvature supplying it is one over the width.

When the liquid does wick, what it forms is not a column with a top.

At any height, the filament in the corner has whatever width makes its capillary suction equal to the hydrostatic head at that height. Higher means more head, which means more suction, which means a narrower filament. The relation is exact: width times height is a constant, and the constant is the capillary length squared times a factor of geometry.

So the filament thins for ever and never stops. At two millimetres it is tens of micrometres across; at sixty it is a couple; at a metre it would be a fraction of one. There is no height at which the argument fails and no height at which the liquid runs out of suction.

A tube has a rise and a corner has a profile. Asking “how high does it go” of a corner is asking for a number that does not exist, and the useful question is how wide the filament is at a height one cares about.

The same filament, followed further

A column with no top and no width. The width of the liquid filament held in a corner, against how far up the corner it is, for water on clean glass, right-angled corner (α = 45°, θ = 5°) and water on glass, a 20° groove (α = 10°, θ = 40°). The curve is one over the height exactly — checked by differencing the logs on the drawn curve — because the suction holding the liquid up is the hydrostatic head and the curvature supplying it is one over the width. water on clean glass, right-angled corner: 1349.2 µm wide at 5 mm and 33.73 µm at 200 mm; water on glass, a 20° groove: 5494.0 µm wide at 5 mm and 137.35 µm at 200 mm. There is no height at which it stops. A tube has a rise because it has a smallest width; a corner has no smallest width, so it has no rise to compute — only a filament that gets thinner for ever.
Fig. 3 The same corners followed from five millimetres to twenty centimetres. The curve does not change shape, it only continues: at two hundred millimetres the filament is under a micrometre across, and there is still nothing in the calculation that stops it. Every decade of height is a decade of width.

Extending the plot is worth doing because a curve that is one over the height looks the same at every scale, and that sameness is the result.

There is no feature anywhere on it. No height at which the suction runs out, no width at which the balance changes, no crossover to another regime. That is what a scale-free criterion produces downstream: having no length in the condition, the solution has no length in it either, and the profile is a pure power.

The practical reading is a table rather than a limit. A corner at two millimetres holds a filament tens of micrometres wide; at two hundred it holds one under a micrometre. Whether that counts as “the liquid climbs” depends entirely on what is being asked — a fuel tank cares about the flow it can carry, an assay cares about whether any liquid arrives ahead of the front, and a contact-angle measurement cares about whether it perturbs the reading.

A power law with no cutoff is a statement that the physics ran out of scales, not that the effect is unlimited in practice. What limits it is always something outside the model: the rounding of the corner, the evaporation of a filament a molecule or two thick, the time the flow would take.

An infinite column with a finite volume

Everything above this height, and it is not much. The volume of liquid held in the corner above a given height, for water on clean glass, right-angled corner and water on glass, a 20° groove. The filament's cross-section goes as the square of its width and so as one over the square of the height, and that integral converges — the total above any height is finite and equals the cross-section there times the height, which the figure checks against a sum over four thousand times the starting height. water on clean glass, right-angled corner: 22.754 µL above 2 mm; water on glass, a 20° groove: 66.529 µL above 2 mm. So the column is infinitely tall and holds a microlitre. That combination is what makes corner wicking useful rather than merely strange: it moves liquid a long way and commits almost none of it to being in transit.
Fig. 4 The volume of liquid held in the corner above a given height. The cross-section falls as the square of the width and so as one over the square of the height, so the integral converges — the total above any height is that height times the cross-section there, checked against a sum over four thousand times the starting height.

The obvious objection to an unbounded column is that it would need an unbounded amount of liquid, and it does not.

The cross-section of the filament goes as the square of its half-width, and the half-width goes as one over the height, so the area falls as 1/h21/h^2. That integral converges. The total volume above any height is finite, and it works out to be the cross-section at that height multiplied by the height itself — a fraction of a microlitre for a millimetre-scale corner.

That combination is what makes corner wicking useful rather than merely a curiosity. It moves liquid a long way and commits almost none of the liquid to being in transit, which is precisely what a transport channel should do.

It is also why the effect is easy to overlook. A filament a few micrometres wide, carrying nanolitres, is invisible unless somebody looks for it — and the first careful observations were made in spacecraft, where the absence of gravity removes the hydrostatic head and the filament fills the whole corner.

Where else the corners are

Once the criterion is in hand, corners turn up in places that were not described as having any.

A porous medium is mostly corners. Sand, paper, fabric and soil are packings whose pore space is a network of angular gaps between grains, and a wetting liquid entering one does not fill the pores in order of size — it runs along the corners first and fills the bodies afterwards. That is why a drop on a paper towel spreads faster than any single-pore calculation predicts, and why the wetted region has a ragged edge rather than a sharp one.

A fibre bundle — a wick, a rope, a tendon, a paintbrush — is a set of long grooves between round fibres, and the half-angle at the contact line between two touching cylinders is zero. So a wetting liquid always wicks along a fibre bundle, whatever its contact angle up to ninety degrees, which is why a lamp wick works and why a wet rope is hard to dry. The same geometry sets the drying of a fibre, since a filament retreating along a groove leaves the ring the drop leaves behind along its length rather than at a single edge.

A crack in a solid is a corner of very small half-angle, so it wicks for almost any wetting liquid — and the tension the liquid can then develop in it is the one the column that is pulled not pushed computes for a tree. That is one route by which water reaches the tip of a crack in stone or concrete, where freezing then does the damage.

And a plant’s leaf axil or an insect’s grooved cuticle is a corner shaped by selection. Several desert beetles and lizards move water along grooves to their mouths by exactly this mechanism, with no muscular pumping at all, and the grooves are cut at half-angles well inside the criterion.

The condition being about angles rather than sizes is what lets one criterion cover all of those. A tube result would have needed a different number for each.

The square tube

The square tube that beats the round one. A square tube 1 mm on a side and a round tube of the same cross-section, against the liquid's contact angle. The round tube's rise is Jurin's height and is finite for every angle under 90°: 26.2 mm at 5°, 24.7 mm at 20°, 20.2 mm at 40°, 9.0 mm at 70°. The square tube's corners are right angles, so their half-angle is 45° and they wick without limit whenever the contact angle is under 45°. The curve drawn for them is not a rise but the height at which the filament has narrowed to 1 µm — a stand-in for how far it can be followed — and it outruns the bulk column by a factor of tens at every wetting angle. Above 45° the corners do nothing at all and the square tube behaves like the round one.
Fig. 5 A square tube one millimetre on a side against a round tube of the same cross-section, on a logarithmic height axis. The round tube’s rise is Jurin’s height for every angle under ninety degrees. The square tube’s corners wick without limit below forty-five, and the drawn curve is only where the filament has narrowed to a micrometre.

The practical version of all this is that container shape matters in a way Jurin’s law does not anticipate.

A square tube has right-angled corners, so α=45∘\alpha = 45^\circ, and water wicks into them whenever the contact angle is under forty-five degrees. Clean glass and water manage that easily; most plastics do not. Above forty-five degrees the corners are empty and the square tube behaves like a round one of the same area.

Below it the two are not comparable at all. The round tube stops at a height; the square tube’s corners carry liquid past that height by a factor of tens before the filament becomes thin enough to argue about, and in principle without limit. Anyone measuring a contact angle by capillary rise in a non-circular tube is measuring two effects at once.

That is exploited deliberately, and for the same reason that a small bubble beats a large one in the small bubble blows up the big one: a smaller radius of curvature wins. Propellant management devices in spacecraft use vanes and corners rather than tubes, because a corner keeps working when there is no gravity to define a height and no pressure difference to push against — and because it delivers liquid to a specific place rather than merely holding it. Microfluidic channels of rectangular cross-section carry corner flow ahead of the bulk meniscus, which is a known source of error in timed assays and a known tool for passive pumping.

Why the criterion is about a sum

The form of the condition — a sum of two angles rather than a comparison between them — is worth a moment, because it is what makes the result general.

Both angles are measured from the same reference: the contact angle from the wall, and the half-angle from the corner’s bisector. Their sum is the angle between the liquid surface and the bisector, and the criterion says that surface must lean towards the apex rather than away from it. Written that way it is almost a tautology, and the content is in the geometry that turns “leaning towards the apex” into “the curvature has the right sign”.

The generality follows immediately. Nothing in the argument requires the corner to be straight, the walls to be flat, or the liquid to be under gravity. Concus and Finn’s theorem covers a corner between curved surfaces and a container of any cross-section, and what it says is that the criterion is applied at each point of the edge with the local half-angle. A groove that tapers can therefore wick along part of its length and not the rest, and the crossover happens where the local half-angle passes through 90∘−θ90^\circ - \theta.

That has been used deliberately: a tapered groove is a one-way valve for a wetting liquid, because the liquid runs into the sharpening end and cannot run back out of the widening one. The condition being local is what makes a geometry into a mechanism, and it is the same move that makes a ratchet out of an asymmetric surface.

Where the model stops

The corner is perfectly sharp. Any real corner has a radius of curvature at its apex, and below that radius the geometry is a tube rather than a wedge. So the filament stops narrowing at a height set by the apex radius, and the unbounded column is unbounded only in the idealisation. For a machined corner with a ten-micrometre radius, that is around a metre.

Gravity is the only body force. In microgravity the head vanishes, the balance that set the width goes with it, and the liquid fills the corners entirely and travels along them at a speed set by viscosity rather than stopping at a shape. That is the regime the spacecraft applications live in and it is a different calculation.

The contact angle is treated as one number. Real surfaces have advancing and receding angles that differ by tens of degrees — the angle a liquid makes is that subject — and a corner near its criterion can wick when the liquid advances and not when it recedes. The map’s boundary is a band on any real material.

And nothing here is dynamic. The profiles are equilibrium shapes. How long a filament takes to reach a given height is a viscous problem whose answer goes as the square root of time, and for a filament a few micrometres wide the times are long — minutes to travel centimetres, not the fraction of a second the equilibrium picture suggests.

The liquid is assumed to wet the walls uniformly. A corner between two different materials has two contact angles, the criterion becomes an inequality involving both, and the meniscus is asymmetric — a case that arises constantly in a device made by bonding a lid onto an etched channel, where the floor and the ceiling are different materials.

And the pressure in the gas is taken as uniform. For a filament a micrometre across in a closed container, evaporation from the thin end and condensation at the thick end move liquid faster than the flow does, and the equilibrium shape is set by the vapour rather than by the hydrostatics. That is the mechanism by which a heat pipe works, and it is why a corner-flow device in a sealed system behaves differently from one open to the air.

What the pictures cannot show

The criterion map draws a sharp line and the physics near it is delicate. Within a degree or so of the boundary the meniscus is nearly flat, the filament is enormous, and the equilibrium is reached slowly and is easily disturbed. A drawing of a boundary cannot convey that the interesting behaviour is a band around it rather than the line itself.

The profile figures draw a two-dimensional filament and the real object is three-dimensional, joining a bulk meniscus at the bottom and tapering into nothing above. The transition region where the corner filament meets the bulk is where most of the liquid is, and it is off the bottom of every plot here.

A third thing outside the figures is what happens at the very top. The filament does not simply become thin; below a few tens of nanometres it stops being a liquid governed by surface tension at all, because disjoining pressure — the direct interaction between the two walls across the film — takes over from curvature. The film then has a thickness set by that interaction rather than by the head, and it is a wetting film rather than a filament. Where the crossover happens depends on the material and not on the geometry, and it is the true top of the column in any real corner. None of the curves drawn here knows about it, and all of them run past it.

Where the ladder goes next

The surface-tension ladder began with the skin that is not a skin, passed through Laplace’s pressure and the instability of a thread, and reached the angles a film has no choice about, where three films meeting must do so at one hundred and twenty degrees. This rung asks what a liquid does at a corner and finds a condition with no length in it. The rungs after it: the dynamics of corner flow, where the filament advances as the square root of time; the wicking of a porous medium, which is a network of corners rather than of tubes; and the shapes a liquid takes in a container with no gravity at all, which is where the criterion was proved and where it is used.

The habit worth carrying away is that a scale-free condition is a different kind of result from a scale-dependent one. Jurin’s law answers how much and Concus–Finn answers whether, and a criterion with no length in it applies to a corner of any size — including one too small to see and one large enough to walk into.

Part 7 of 8

This essay is one argument about Surface tension. The others:

Two angles a film is not allowed to depart from. The two junctions Plateau's laws permit, drawn at the angles a balance of equal tensions requires. A soap film pulls equally in every direction along itself, so where films meet the pulls must sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films can only meet along a line, at 120.0000° to one another, because three equal coplanar vectors sum to zero at 120° and at no other angle. Four such lines can only meet at a point, at 109.4712° — arccos(−1/3), the tetrahedral angle — for the same reason in three dimensions. Both numbers are found here by solving the balance rather than by drawing what is expected, and neither depends on the liquid, the temperature or the size of the foam. A junction of four films along a line, or of three lines at a point, is not merely unusual: the tensions cannot balance there, so it rearranges within milliseconds into the two arrangements drawn. The angles a film has no choice about Part 6 — A soap film pulls equally hard in every direction along itself, so wherever films meet the pulls have to sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films meet at a hundred and twenty degrees and four edges at a hundred and nine point four seven, in every foam, of every liquid, at every scale, and nothing about the material appears in either number. The colours a draining film runs through, and the end of them. The fraction of light a free soap film reflects, against its thickness, at three wavelengths — 450, 550, 620 nm — computed from the sum over multiple reflections rather than from the two-beam approximation. The three curves peak at different thicknesses, which is why a draining film runs through a sequence of colours as it thins. Below about 11 nm every curve is under a tenth of a per cent and the film looks black. At zero thickness the reflectance is exactly zero, checked before the figure is drawn: the two surfaces reflect equally and half a cycle out of step, so a film much thinner than a wavelength cancels itself. That is the whole of why a black film is black. Nothing is absorbing; the film is there and has simply stopped being able to interfere constructively at any visible wavelength — which means the blackness is a measurement, and a film that has gone black is known to be thinner than about a tenth of a wavelength without anything being measured directly. The film that goes black before it bursts Part 8 — A soap film drains, runs through every interference colour, and then stops reflecting anything at all. The black patch is not a hole and not a film about to break: it is the thinnest and most stable state the arrangement has, held apart by a pressure between its two surfaces that only exists at distances of nanometres.

Where this goes next

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. 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. Two angles a film is not allowed to depart from. The two junctions Plateau's laws permit, drawn at the angles a balance of equal tensions requires. A soap film pulls equally in every direction along itself, so where films meet the pulls must sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films can only meet along a line, at 120.0000° to one another, because three equal coplanar vectors sum to zero at 120° and at no other angle. Four such lines can only meet at a point, at 109.4712° — arccos(−1/3), the tetrahedral angle — for the same reason in three dimensions. Both numbers are found here by solving the balance rather than by drawing what is expected, and neither depends on the liquid, the temperature or the size of the foam. A junction of four films along a line, or of three lines at a point, is not merely unusual: the tensions cannot balance there, so it rearranges within milliseconds into the two arrangements drawn. The angles a film has no choice about A soap film pulls equally hard in every direction along itself, so wherever films meet the pulls have to sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films meet at a hundred and twenty degrees and four edges at a hundred and nine point four seven, in every foam, of every liquid, at every scale, and nothing about the material appears in either number.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

CapillarityContact angleCurvatureGeometryHydrostaticsLaplace pressureMeniscusScalingSurface tensionWetting