The angle a liquid makes with what it sits on

A drop of water on clean glass spreads until its edge cannot be seen. The same drop on a waxed board sits up as a bead and rolls off at a tilt of a few degrees. Nothing about the water has changed.

Everything below is where that angle comes from, what its cosine decides, and what it costs to treat it as a single number.

Three tensions meeting along a line #

At the rim of a sessile drop, three materials meet along a curve: the solid, the liquid and the vapour. Each pair of them has an interface carrying an energy per unit area — that is what a surface tension is, and it is an energy before it is a force. The rim is therefore a line at which three energies per area are in competition, and the angle is what they settle on.

The settlement follows from moving the line. Push the contact line outward by $\delta$, per unit length of line. An area $\delta$ of dry solid is covered, costing $\gamma_{sl} - \gamma_{sv}$; the liquid–vapour surface, inclined at $\theta$ to the solid, lengthens by $\delta\cos\theta$, costing $\gamma_{lv}\cos\theta$. At equilibrium the total is stationary against that displacement:

$$\gamma_{sv} = \gamma_{sl} + \gamma_{lv}\cos\theta.$$

That is Young’s equation, and the derivation says what kind of statement it is. It is not a force law that happens to be about surfaces; it is the condition that a free energy is at a minimum, applied to the one coordinate a drop still has once its volume is fixed.

The force version gives the same equation and adds something the energy version hides. Each tension pulls along its own interface at the line: $\gamma_{sv}$ outward along the dry solid, $\gamma_{sl}$ inward under the drop, $\gamma_{lv}$ along the liquid surface at angle $\theta$. Resolving those three along the solid — the standard move of choosing axes to suit the problem — reproduces Young exactly. Resolving them perpendicular to it does not balance at all: there is a leftover $\gamma_{lv}\sin\theta$ pulling straight up, 72.8 mN per metre of contact line at 90°, with nothing in the equation to cancel it.

That leftover is taken up by the solid, which deforms by an amount of order $\gamma/E$. On glass, at $E$ near 70 GPa, that is a picometre — a hundredth of an atomic diameter, and unobservable. On a soft silicone gel at 3 kPa it is 24 micrometres, and the ridge the drop raises under its own contact line is plainly visible under a microscope. Young’s equation has the form it has because on ordinary solids the vertical half of the problem costs nothing to ignore.

The angle therefore belongs to the pair of materials, and to the vapour between them, and not to the liquid. Water on clean soda-lime glass sits near 0°; on polyethylene near 96°; on paraffin wax and on PTFE near 110° and 108°. Mercury on glass sits near 140°. One solid can change by tens of degrees between a freshly cleaved surface and one left out for an hour, because a monolayer of airborne hydrocarbon changes what the vapour side of the boundary is.

The sign of one cosine #

Once the angle is known, the previous rung’s result applies unchanged. A liquid in a tube of radius $r$ rises to

$$h = \frac{2\gamma\cos\theta}{\rho g r},$$

and every term in it is positive except the cosine. So the height in a capillary is the sign of $\cos\theta$ multiplied by a magnitude, and the sign changes at exactly 90°, where the cosine is zero and the liquid stands level with the bath outside.

Past 90° the same formula returns a depression. Mercury in a 0.5 mm glass tube at 140° stands 11.24 mm below the reservoir it is dipped in — not held down, but pushed down by the atmosphere, because the convex meniscus puts the liquid beneath it above atmospheric pressure by the Laplace jump and the column sinks until the hydrostatic deficit makes up the difference.

The two liquids in that figure are worth comparing. Mercury’s surface tension of 487 mN/m is six and a half times water’s, but its density of 13,534 kg/m³ is thirteen and a half times water’s, so the depression at 140° comes out 0.38 times water’s rise at 0° in the same bore. Surface tension alone never sets the size of a capillary effect: it is always $\gamma$ against $\rho g$, and the ratio of those two is a length.

Between the extremes the cosine shrinks. A glass tube handled rather than cleaned gives water an angle around 75°, and the same four tubes then lift a quarter of what they would at 0°.

Narrowing the tube changes every height and moves nothing else.

The length below which any of this matters #

That competition — $\gamma$ against $\rho g$ — has a length in it, and the length decides where capillarity is a phenomenon at all. A blob of size $R$ carries a capillary pressure of order $\gamma/R$ and a hydrostatic pressure of order $\rho g R$; setting the two equal gives

$$\ell_c = \sqrt{\frac{\gamma}{\rho g}} = \sqrt{\frac{0.0728}{1000 \times 9.81}} = 2.7\ \text{mm}$$

for water, and 1.9 mm for mercury, whose much larger tension is beaten by its much larger density. Below the capillary length, shape is decided by surface energy and a drop is a spherical cap. Above it, gravity decides and the liquid flattens into a puddle whose depth stops growing: a non-wetting puddle settles at $2\ell_c\sin(\theta/2)$, which is 4.5 mm for water on paraffin wax and 3.6 mm for mercury on glass, whatever volume is poured out.

The pressure a curved surface can hold falls as the radius grows, and the numbers set the scale of everything here. A 3 mm droplet carries 49 Pa, which is exactly the weight of a 5 mm column of water; at 0.02 mm the same law gives 7.3 kPa; at a micron, several atmospheres. So the two effects are comparable at millimetre sizes and the curved surface wins overwhelmingly below that, which is what the capillary length is a statement of.

That is why capillarity looks like a small-scale phenomenon: in a drinking glass 8 cm across the capillary pressure at the meniscus is under a pascal, and in a 10 µm pore it is 15 kPa and dominates everything.

What capillarity is competing against is depth, and depth is all a column of liquid knows. Three metres of water is 29.4 kPa of gauge pressure — six hundred times the 49 Pa a 3 mm meniscus can hold — so at that scale the curved surface is a rounding error and the shape of the free surface is flat. The same 49 Pa is worth 5 mm of column, and 5 mm is where a liquid surface starts deciding shapes instead of merely lying on them.

Not one angle, but a band #

Everything so far has treated $\theta$ as a number belonging to a pair of materials. It is not one, and the failure is not small.

Advance a drop across a dry solid and its edge sits at one angle. Pull the same drop back over the solid it has just wetted and its edge sits at a smaller one. The two are the advancing and receding angles, their difference is the contact-angle hysteresis, and for water on ordinary glass it is 20° to 40°. A carefully prepared surface can be brought under 5°; nothing real reaches zero.

The cause is that a contact line does not slide, it sticks and jumps: roughness gives it ridges to hang on and chemical patchiness gives it patches of higher and lower energy. Both are facts about the history of the surface rather than about its chemistry, which is why the hysteresis of a real sample cannot be looked up. It is the same behaviour as a static friction that takes whatever is asked of it up to a limit: an interface pinned on defects supplies any force in a range without moving.

The width of that band decides whether a drop moves. Furmidge’s balance sets the drop’s weight along a slope against the retention its contact line supplies:

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

with $w$ the width of the contact line. For water on a windscreen at 85° advancing and 55° receding, $\gamma(\cos\theta_r - \cos\theta_a)$ is 35.4 mN per metre. A 10 µL drop 4 mm across then weighs 101 µN and is held by 142 µN, so it stays put on a vertical window; a 35 µL drop 6 mm across weighs 339 µN against 212 µN of retention, and runs. The rain that clings to a car window and the rain that streaks down it are on either side of that arithmetic, and both drops are the same water at the same angle.

The number that decides whether a drop moves is therefore the difference between two angles rather than either of them. A windscreen treated to shed water is engineered for low hysteresis, not for a large angle — and a surface can be given a spectacular 150° and still hold every drop that lands on it if its two angles are far apart.

Roughness, used on purpose #

If roughness is what pins a contact line, it can also be built deliberately, and two different things happen depending on which side of 90° the flat surface sits.

When the liquid follows the texture into every crevice, the true area exceeds the projected area by a roughness ratio $r$, and Wenzel’s relation gives an apparent angle $\cos\theta^* = r\cos\theta$. Roughness amplifies whatever the flat material already did: at $r = 1.5$, a surface at 110° reads 121° and a surface at 60° reads 41°. The amplification works both ways and pivots on 90°, where the cosine is zero and roughness changes nothing.

Push the wetting case far enough and it stops being an angle. Once $r\cos\theta$ exceeds one — at $r = 1.5$, once $\theta$ drops below 48° — the texture wicks: liquid runs ahead of the drop through the grooves and the drop sits on a film of itself. That is what a paper towel and a sintered heat-pipe wick do, and why texturing a wetting surface makes it dramatically more wetting rather than slightly.

The other case is the one that gets photographed. A rough hydrophobic surface traps air in its texture, leaving the drop on a composite of solid tops and air pockets. With a solid fraction $f$, Cassie and Baxter’s relation gives $\cos\theta^* = f(\cos\theta + 1) - 1$: at $f = 0.1$ on a material whose flat angle is 110°, the apparent angle is 159°. The contact with the solid is a scatter of tiny patches and the hysteresis collapses with it, which is the real reason a lotus leaf sheds water and takes the dirt with it, and why the roll-off angle on a good superhydrophobic surface is under 5°.

That air is held only by the menisci spanning the gaps, which survive a pressure of order $2\gamma|\cos\theta|/d$ with $d$ the gap width — 50 kPa at a micron and 50 Pa at a millimetre — and a texture that loses it becomes more adhesive than the flat material, not less.

What it costs #

A windscreen is specified in hysteresis. The rain-repellent treatments sold for glass are fluorosilanes that raise the angle to around 105°, but the property being bought is the 5°–10° hysteresis, which is what lets 3 mm drops leave at highway airspeeds instead of accumulating.

A waterproof fabric is a pore size and an angle, not a barrier. Breakthrough comes when the water pressure exceeds $2\gamma|\cos\theta|/r_{\text{pore}}$: at 110° and a 10 µm pore that is 5 kPa, half a metre of water, and the “hydrostatic head” ratings on outdoor gear, up to 10,000 mm, correspond to sub-micron pores. Vapour still passes, having no contact line.

A solder joint is a contact angle inspection. Molten solder on clean copper wets at well under 30°, and flux exists to strip the oxide and raise $\gamma_{sv}$ so that it does. An angle above 90° is a cold joint: mechanically attached, electrically unreliable, and rejected on sight because the angle is visible and the intermetallic layer is not.

An entire mineral industry runs on it. Froth flotation makes the wanted mineral non-wetting with an adsorbed collector and blows bubbles through the slurry: the non-wetting particles ride up on the bubbles and the wetted rock sinks. Billions of tonnes a year are sorted this way, and the reagent chemistry is all about moving $\gamma_{sl}$.

Where the model stops #

Young’s equation assumes five things, and real surfaces break all five. It wants a rigid solid, a smooth one, a chemically uniform one, a non-reactive one, and an equilibrium that has actually been reached. A soft solid deforms by $\gamma/E$ — 24 µm on a 3 kPa gel, which changes the angle itself. A rough or patchy one gives a band 20° to 40° wide instead of a value. A reactive one moves its own $\gamma_{sl}$ while it is being measured.

Two of its three terms cannot be measured independently. A liquid rearranges to relieve surface stress, so its surface energy and its surface tension are one number; a solid cannot, and they are two. Neither $\gamma_{sv}$ nor $\gamma_{sl}$ is directly accessible, so the equation is run backwards from a measured $\theta$, closed with a model. The tensions in this page’s figures come from a geometric-mean closure that satisfies Young by construction and returns $\gamma_{sv} = 68.48$ mN/m for the 20° solid, where a clean high-energy solid is in the hundreds of mJ/m². The drawn arrows are consistent; they are not measurements.

Below about a micrometre the contact line has an energy of its own. Line tension $\tau$ adds a term $\tau/(\gamma_{lv} a)$ to the cosine, with $a$ the drop’s base radius. At the commonly reported $\tau \approx 10^{-11}$ N and $a = 1$ µm that shifts $\cos\theta$ by 0.14, about 9° at a 60° angle; at $a = 1$ mm it shifts it by 0.008°. Reported values of $\tau$ span three orders of magnitude, which is itself a statement about the measurement.

Nothing here moves. A line advancing at speed $U$ has an angle that depends on the capillary number $\mathrm{Ca} = \eta U/\gamma$, which for water at 10 mm/s is $1.4\times10^{-4}$ and negligible; above $\mathrm{Ca}\approx10^{-2}$ the dynamic angle is visibly larger than the static one and a receding line entrains air. The rate at which a wick fills is not on this page either — it is capillary pressure against viscous resistance, giving a distance that grows as the square root of time.

Evaporation is ignored, and it leaves a mark. A volatile drop with a pinned contact line evaporates fastest at its edge, and liquid flows outward to replace what left, carrying everything suspended in it to the rim. That is the coffee ring: a stain darkest where the drop ended, produced by a pinned line and a gradient in evaporation rate alone, and a nuisance in every printed circuit and dried diagnostic assay. In a confined pore the same equilibrium shifts the phase boundary itself, so vapour condenses below its ordinary saturation — which is a boiling point being a pressure read at a curved surface.

The same balance somewhere else entirely #

The three-tension balance is not about tubes and it is not about water, and the range of things it settles is the argument for taking it seriously as an equation rather than as a fact about capillaries.

A water strider stands on a pond because its legs carry hydrophobic microhairs holding an apparent angle near 167°, so the surface deforms under each foot without being pierced — and what decides whether the insect floats is that force against the buoyancy it also has, not against nothing.

Ink on paper is the wicking case, with the width of a printed line set by how far the liquid ran before the vehicle dried. And a crystal pulled from a melt has a meniscus where crystal, melt and gas meet: silicon grows at a fixed 11° angle to the melt surface, and the diameter of a 300 mm boule is held by holding that meniscus steady, in a melt whose own capillary length is 5.3 mm.

The same accounting decides what a wetting film does when it is left on a fibre. A liquid column is unstable to any disturbance longer than its circumference: a pinch shorter than that would add surface and decays, while one about nine radii long grows fastest and sets the spacing of the beads. A thin film coating a wire, a rivulet running down a window and a thread of water leaving a tap all break up under one rule — total interfacial area, priced per unit — which is the rule that fixed the contact angle as well.

Young, who wrote it without a diagram #

Thomas Young published An Essay on the Cohesion of Fluids in the Philosophical Transactions in 1805, and the relation named after him appears in it as a sentence of English prose. There is no diagram, no algebra and no equation anywhere in the paper. Young argued that the cohesion of a liquid to itself and its adhesion to a solid must balance along the edge, and that the angle follows — correctly, and in a form nearly unusable by anyone wanting to compute with it.

Laplace supplied the analysis the following year, in a supplement to the tenth book of the Mécanique Céleste, and it was Laplace’s version that got taught. Gauss then showed in 1830 that both the pressure jump and the contact angle fall out of one variational principle — minimise the interfacial energy plus the gravitational energy — which is the derivation at the top of this page.

The corrections came much later, and from people looking at surfaces rather than at equations: Wenzel in 1936, Cassie and Baxter in 1944, Furmidge in 1962, and Deegan and colleagues on the coffee ring in 1997. Young’s equation was 130 years old before anybody wrote down what roughness does to it.

What the picture cannot show #

The drops are drawn as spherical caps, which is the shape a drop takes when gravity is negligible. For 5 µL of water the Bond number is 0.15, so a real drop that size is measurably flattened: the caps are the small-drop limit, not photographs.

The contact line is drawn as a point where three lines meet. It is really a region a few molecular diameters across, in which the density interpolates between three phases and the continuum stresses used to derive Young’s equation are not defined. The equation survives because that region is small, not because it is absent.

The three arrows at each contact line come from a closure rather than from measurements. Their horizontal components sum to zero by construction, which is what makes the picture consistent — but a picture that satisfies an equation is not evidence for the numbers it satisfies it with, and that is the same warning field lines earn.

Every drop here is drawn at one angle, so hysteresis — the quantity the argument above turns on — is invisible: its whole content is that the same drop would sit differently had it arrived from the other direction. And the tube pictures show only liquids that climb. A depression would have to be drawn upside down, so it appears on the curve of height against angle instead, where the sign is a number rather than a shape.

The ladder from here #

Later rungs on this anchor: the dynamics of wetting, where the angle depends on how fast the line moves. Capillary condensation and the Kelvin equation. Imbibition into a porous network, where the answer is a distribution rather than a height. Electrowetting, where a voltage across the solid–liquid interface changes $\gamma_{sl}$ and steers a drop on demand. And the coffee ring in full.

The neighbouring ladders are the ones this page has been leaning on throughout: the surface as an energy per unit area, the pressure jump across a curved interface, the rise the angle decides the sign of, and a fluid at rest knowing nothing but depth.