How high water will climb

Stand a fine glass tube in water and the water climbs it. It does not stop where the outside level is; it goes past, and stays there. Use a narrower tube and it goes higher, in exact proportion to how much narrower.

The phenomenon has been known for as long as there have been narrow tubes and is routinely explained with a word — suction — that names nothing. What is actually happening is a pressure argument, and every piece of it has already been established on this ladder.

The surface is curved, so the pressure jumps #

Water wets glass, which for now means only that it prefers to be in contact with glass rather than to leave it dry. The consequence is that the water surface inside the tube is not flat: it climbs the wall a little, and the free surface between the walls is concave, curving upward like a shallow bowl.

That curvature has a pressure consequence, and it is the Young–Laplace relation from the previous rung applied in the direction people find least intuitive. For a concave surface the centre of curvature is above the liquid, so the liquid is on the outside of the curve, and the pressure just beneath the meniscus is lower than the air above it by $2\gamma/R_m$, with $R_m$ the meniscus’s radius of curvature.

So immediately under the meniscus the water is at less than atmospheric pressure. But at the level of the bath outside, the water must be at atmospheric — it is connected to the bath, and the bath’s surface is flat and exposed. The only way to reconcile a low pressure at the top with atmospheric pressure at the bottom is a column of water in between, whose weight makes up the difference by the ordinary hydrostatic law.

The column rises until it does. That is the whole mechanism, and nothing in it pulls.

The geometry that fixes the curvature #

The meniscus radius is not free. If the liquid meets the wall at a contact angle $\theta$ measured inside the liquid, and the tube has radius $r$, then the spherical cap that meets the wall at that angle everywhere has radius

$$R_m = \frac{r}{\cos\theta}.$$

Put that into the pressure jump and set it against $\rho g h$:

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

That is Jurin’s law, from 1718. For clean water in clean glass, $\theta$ is near zero, $\gamma = 0.0728$ N/m, and the rise is about 15 millimetres divided by the radius in millimetres. A one-millimetre tube lifts water 15 mm; a ten-micron tube lifts it a metre and a half.

The same result by counting forces #

There is a second derivation, and having both is worth the space because they emphasise different things.

Surface tension is also a force per unit length along a line where the surface ends. The liquid surface meets the glass around a circle of circumference $2\pi r$, and it pulls along the surface, at angle $\theta$ to the wall. The vertical component of that pull is $\gamma \cos\theta$ per unit length, so the total upward force is $2\pi r \gamma \cos\theta$.

The weight of the raised column is $\rho g h \cdot \pi r^2$. Set them equal and $h = 2\gamma\cos\theta/\rho g r$ falls out again.

The force version makes the scaling transparent — a circumference against an area, so one power of $r$ survives — and it is the version that misleads, because it invites the picture of the surface as a rope tied to the glass — the same trap as reading a liquid surface as an elastic film. The pressure version is the one that generalises to shapes with no obvious circumference, and it is the one that says what is true at every point rather than only in total.

Reading the surface tension off the height #

Because every quantity in Jurin’s law except $\gamma$ is easy to measure, the relation is routinely run backwards and used as an instrument. Put a tube of known radius into a liquid, read the height with a cathetometer, and $\gamma$ follows.

It was the standard method for a long time and it is still the most accessible one. Its weakness is the contact angle, which enters as $\cos\theta$ and is difficult to know independently — so the measurement is trustworthy only for liquids that wet the tube completely, where $\theta$ can be taken as zero and the cosine as one. For anything else the method measures the product $\gamma\cos\theta$ and cannot separate the factors.

That is a recurring shape: a clean law with two unknowns in one place gives one equation, and the second has to come from somewhere else. Weighing a plate as it is withdrawn, or fitting the profile of a hanging drop, are the ways round it, and both are the same measurement strategy as weighing a body in and out of water — arrange things so the wanted quantity is the only one left standing.

There is a subtlety in the reading worth naming, because it is a classic source of error. The height to measure is to the bottom of the meniscus, and the liquid above that point has weight too. For a hemispherical meniscus the correction is $r/3$, which at a millimetre radius is a third of a millimetre against a rise of fifteen — two per cent, and larger than the reading precision.

Which way it goes, and why mercury is different #

Everything so far assumed $\cos\theta > 0$. The contact angle is not a property of the liquid: it is a property of three interfaces at once, fixed by Young’s relation balancing the surface energies of solid–vapour, solid–liquid and liquid–vapour where they meet:

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

If the solid gains more from being wet than the liquid loses in making a surface, $\theta$ is small and the liquid spreads. If not, $\theta$ is large. Mercury on glass sits near 140°, whose cosine is $-0.77$, and Jurin’s law then returns a negative $h$: mercury is depressed in a glass capillary, and depressed further the narrower the tube. Both behaviours are the same formula.

This is why the figure takes the contact angle as an input rather than assuming zero. A generator that hard-wired perfect wetting would be unable to draw half the phenomenon.

The tree, which does not work this way #

The most persistent misuse of this page’s law is that it explains how water reaches the top of a tall tree. It cannot, and the arithmetic is short enough to settle it.

A hundred-metre rise needs, from Jurin’s law with perfect wetting, a radius of about 0.15 microns. The xylem vessels that actually carry water in trees are between about 10 and 300 microns across — a factor of a hundred or more too wide. Capillarity in vessels of that size lifts water a few centimetres.

What actually happens is that evaporation from the leaves puts the water column under tension: the water is pulled from above rather than pushed from below, and the pressure in the xylem is genuinely negative — measured at tens of atmospheres below zero. That is possible because water resists being pulled apart, and it is metastable rather than stable, which is why an air bubble entering a vessel is catastrophic for it — the same metastability that lets a clean liquid be heated past its boiling point without boiling.

Capillarity’s real role is at the very top: the pores in the leaf’s cell walls are tens of nanometres across, and those can support the enormous curvature that sustains the tension. So the law is not irrelevant — it is doing its work at the last micron rather than over the whole hundred metres.

What it does run #

The places where capillary rise is the dominant transport mechanism all have one thing in common: pores in the micron range and distances in the centimetre range.

Soil. Water rises from a water table through the pore network by exactly this law, and the height reached depends on grain size — centimetres in coarse sand, metres in clay. Whether a field drains or waterlogs is largely this arithmetic.

Paper, cloth, wicks. A wick is a bundle of capillaries, and a paper towel is a disordered one. The same law governs how far a stain spreads, and the rate is a separate question answered by viscosity rather than by this page.

Chromatography. A solvent climbing a paper or a thin layer, carrying a mixture at different rates, is a capillary rise used as an instrument.

Heat pipes. A sealed tube with a wick returns condensed working fluid to the hot end by capillarity, which is why a heat pipe works in any orientation and needs no pump.

Where it is a nuisance. Damp climbing a wall from the ground is capillary rise in the mortar’s pores, and a damp-proof course is a deliberate break in the pore network — a physical interruption of this equation rather than a coating.

What decides the height is the ordinary weight of the fluid underneath. Capillarity supplies a pressure deficit at the top of the column and nothing else; the column then settles to whatever height makes the hydrostatic pressure of the liquid below balance it. That is the same hydrostatic balance an atmosphere settles into, and it is worth separating the two contributions because only one of them is about surfaces at all.

The length that decides whether it matters #

Jurin’s law grows without bound as $r \to 0$, which is a sign that something else must eventually intervene. What it competes with is the same comparison that appeared on the surface-tension ladder: the capillary length

$$\ell_c = \sqrt{\frac{\gamma}{\rho g}} \approx 2.7\ \text{mm for water}.$$

For tubes much narrower than $\ell_c$, surface tension dominates, the meniscus is a spherical cap, and Jurin’s law is accurate. For tubes comparable with $\ell_c$ or wider, gravity flattens the meniscus, the cap approximation fails, and the rise is small anyway. That is why capillary rise is invisible in a drinking glass and decisive in a soil: the glass is on the wrong side of a length that belongs to the liquid rather than to the container.

The figure refuses to draw tubes wide enough for the rise to be smaller than the meniscus is deep, which is the same boundary expressed as something the picture cannot honestly show.

Many tubes, and the shape of a wet powder #

The most common form of this phenomenon is not a tube at all. It is a heap of grains with liquid in the gaps between them, and the same arithmetic applies to each gap.

Two touching spheres with a little liquid between them hold a ring-shaped meniscus whose curvature is concave, so the liquid in it is below atmospheric pressure — which means the air outside is pressing the two grains together. The force is roughly $2\pi r_g \gamma$ with $r_g$ the grain radius, and it is why damp sand holds a vertical face and dry sand does not.

The strength of a sandcastle is therefore not adhesion in any chemical sense. It is atmospheric pressure, admitted between the grains through a curved surface, and it disappears when the sand dries and again when it is saturated — because a fully flooded pore has no curved surfaces left. The strongest damp sand is at a few per cent water by volume, which is a real optimum rather than a rule of thumb.

The same mechanism sets how much water a soil retains against gravity after rain, which agronomists call field capacity: the water held in pores whose capillary rise exceeds the depth to the water table stays; the rest drains. A clay’s fine pores hold much more than a sand’s coarse ones, and the difference is one power of the pore radius.

A real pore network makes all of this statistical rather than exact. Its throat sizes have a broad distribution, so every result on this page holds pore by pore while the aggregate is an average over that distribution — which is why soil physics quotes curves rather than single heights, and why a wet powder has a range of behaviours rather than a height.

What happens when the competing term is removed #

The capillary length is a ratio of surface tension to weight, so it says what happens in orbit at once: with $g$ effectively zero, $\ell_c$ is unbounded and there is no size at which gravity takes over. Capillary forces organise the liquid in a whole tank.

That is not a curiosity for anybody designing a spacecraft. A rocket engine must be fed liquid rather than vapour, and in free fall the propellant is wherever it happens to be — possibly nowhere near the outlet. The old answer was to fire small thrusters before every burn to settle the tank, which costs fuel and complexity. The modern one is a propellant management device: an arrangement of vanes and sponges inside the tank that holds liquid over the outlet by surface tension alone, permanently, with no moving parts.

The geometry those devices exploit is a result about corners rather than tubes. A liquid meeting an interior corner of half-angle $\alpha$ will wick along it without limit, drawn by the ever-tightening curvature the corner forces, provided the contact angle satisfies $\theta + \alpha < 90°$. A tube of the same material and the same wetting would rise a definite height and stop; a sharp enough corner has no such stopping point, because there is no radius setting a scale.

The most charming application is a coffee cup. A vessel flown on the space station has a sharp interior corner running up one side, and the coffee climbs it to the drinker’s lip and stays there, so it can be sipped rather than sucked through a bag. It is a container designed entirely around the condition in the previous paragraph.

The pressure that destroys a machine #

Everything above treats the solid as rigid. If it is not, the same negative pressure under a meniscus is a force that bends things, and there is a length at which it wins.

Compare the stiffness of a slender object with the capillary pull on it and a characteristic length falls out, built from the bending stiffness and the surface tension. Shorter than that, the object holds its shape; longer, a meniscus between it and its neighbour bends it into contact. That is why a wet paintbrush comes to a point and a dry one does not, and why hair clumps when it is washed.

It is also a manufacturing disaster. A micro-machined device — a cantilever, a comb of fingers a few microns thick and hundreds of microns long — is finished by etching away the layer beneath it and rinsing. As the rinse liquid evaporates, menisci form between the released structure and the substrate beneath it, pull the two together, and they stay together: the surfaces bond on contact and no force available will separate them. The failure has a name, release stiction, and for a while it was the dominant yield problem in the field.

The fix is the one this page’s mechanism suggests. Every term in the argument requires a curved liquid–vapour interface, so the answer is to arrange that no such interface ever exists. The rinse liquid is exchanged for liquid carbon dioxide, and the vessel is then taken around the critical point — above the critical temperature and pressure the liquid and its vapour cease to be distinguishable, so the fluid can be vented as a gas without any surface ever having passed through the device. Supercritical drying is a standard step now, and it is a piece of process engineering whose entire justification is the absence of a meniscus.

Where the model stops #

The contact angle is a single number. Real surfaces give a range: a liquid advancing over dry solid meets it at a larger angle than one receding over wet solid. That hysteresis is why a raindrop can sit still on a sloping windscreen — the front is advancing and the back receding, and the difference in angles supplies a retaining force that Young’s relation alone does not contain.

The surface is smooth and uniform. Roughness amplifies whatever the flat surface does — a rough hydrophilic surface is more hydrophilic, a rough hydrophobic one more hydrophobic, which is the basis of engineered water-repellent surfaces and is a shape effect rather than a material one in the same sense the metacentre is.

Only equilibrium is described. Nothing here says how fast the liquid rises. That is a balance between the same capillary pressure and viscous resistance, giving a distance growing as the square root of time — a result of the same shape as everything else diffusive, and one this page cannot reach.

The tube is a circular cylinder. Real pores are irregular, connected and of varying width, so a soil’s behaviour is governed by a distribution of radii and by which throats are narrowest.

Evaporation is ignored. In a wick that is being used, the top is evaporating, and the steady state is a flow rather than a height — governed by the resistance a fluid offers to being sheared rather than by anything on this page.

The history #

Jurin published the inverse-radius law in 1718, and it was empirical: he measured. The mechanism came from Young and Laplace independently around 1805, once surface tension was understood as an energy and the pressure jump across a curved surface had been written down. Young’s relation for the contact angle dates from the same moment.

That order — the law first, the mechanism a century later — is common in this subject, and it is worth noticing that the empirical law was correct and complete. What the mechanism added was not accuracy but reach: it explains mercury as well as water, settles what happens in a non-circular pore, and connects a rise in a tube to a bubble’s pressure and a drop’s shape as three consequences of one statement.

The ladder from here #

Later rungs on this anchor: the contact angle properly derived, and the hysteresis real surfaces show. Wetting and superhydrophobicity, where roughness is used deliberately. The dynamics of imbibition, and the square-root-of-time law. Capillary condensation, where a vapour condenses in a small pore below its ordinary saturation. And capillary adhesion — why damp sand holds a shape and dry sand does not, which is many small menisci each doing the arithmetic on this page.