A law that is exact as an average

Water climbs a narrow glass tube, and the height it reaches is Jurin’s law from 1718:

$$h = \frac{2\sigma\cos\theta}{\rho g R} = \frac{2\ell^2\cos\theta}{R},$$

with $\ell = \sqrt{\sigma/\rho g}$ the capillary length and $\theta$ the angle the surface meets the glass at. It is always taught as the narrow-tube case, and it is: for a tube a millimetre across it is the height of the meniscus to a fraction of a per cent, and for a tube of a few centimetres, where the water visibly rises only at the edges, it is clearly not the height of anything.

The drop falls at a fold found that Tate’s law for a hanging drop — the same kind of statement, tension round a perimeter against a weight — is wrong by up to a quarter, and located the missing term. This essay finds that Jurin’s law has no missing term at all. It is exact. The thing that fails for a wide tube is not the law but the picture that goes with it.

The balance, cut where the pressure is known #

Take the level of the liquid far outside the tube and draw a horizontal plane there, through the tube. The liquid on that plane, inside the tube and out, is at atmospheric pressure: the two are connected underneath, and the outside surface is flat and open to the air. So the liquid lifted above that plane inside the tube is held up by one thing only, the surface pulling on the wall along the contact line, with a vertical component $\sigma\cos\theta$ per unit length. Whatever shape the surface takes,

$$\rho g\,V_{\text{lifted}} = 2\pi R\,\sigma\cos\theta .$$

There is no pressure term, because the plane was chosen to make it zero. Divide by the area of the tube and the mean height of the lifted liquid is

$$\bar h = \frac{V_{\text{lifted}}}{\pi R^2} = \frac{2\ell^2\cos\theta}{R},$$

which is Jurin’s law, exactly, for a tube of any width.

That was checked, not assumed. The shape of the surface in a tube was found by integrating Young–Laplace outward from the axis — in capillary units the curvature equals the height, so a surface that starts at height $z_0$ on the axis curves until its slope matches the contact angle at the wall, and the tube’s radius is wherever that happens — and then the lifted volume was integrated under it. For twelve tubes, from a twentieth of a capillary length to three, at contact angles of 0, 30 and 60 degrees, the lifted volume is $2\pi R\cos\theta$ to $1.4\times10^{-6}$.

Where the average is not the height #

In a narrow tube all three heights are the same to within a few per cent, and the surface is a hemisphere of the tube’s radius sitting on a column of Jurin’s height. That is the case the law was written for, and the calculation reproduces its standard correction. The hemisphere holds a little less liquid than a cylinder of the same height would, and for a tube of radius $R$ the centre therefore sits at $2\ell^2/R - R/3$ rather than at Jurin’s value — Rayleigh’s correction, from 1916. At a twentieth of a capillary length the integrated centre height agrees with it to $6\times10^{-7}$.

Past a capillary length the three separate, which is the general rule that a group’s value near one is where two descriptions trade places arriving as a Bond number. The centre, in the wide-tube limit, obeys the linearised equation with a modified Bessel function for its solution, and falls roughly as $e^{-R/\ell}$: at twelve capillary lengths — a tube 65 millimetres across for water — the centre rises five hundred-thousandths of a capillary length, about a seventh of a micrometre. The wall height levels off, and the mean, being exact, goes on falling as $1/R$ because the same lifted volume is spread over an area growing as $R^2$.

The exponential is not an accident of this geometry; it is the capillary length doing what a length does. Where the surface is nearly flat its slope is small, the curvature is the Laplacian of the height, and the balance says that Laplacian equals the height itself in capillary units. A disturbance to a flat surface therefore decays over one capillary length in every direction — as $e^{-x/\ell}$ beside a flat wall, and as the modified Bessel function $I_0(r/\ell)$ grows towards the wall of a tube. The centre of a tube is simply the place farthest from any wall, and in a tube twelve capillary lengths in radius it is twelve decay lengths from the only thing lifting it. Nothing about that needs gravity to be weak or the tube to be narrow; it is the same length that makes a drop half a millimetre across a sphere to a per cent, read the other way.

The water is not failing to rise in a wide tube. It is rising exactly as much as Jurin’s law says, and putting nearly all of it in a band one capillary length wide against the glass. A tube much wider than the capillary length is simply a wall that has been bent round into a circle, and the lifted volume is the flat wall’s lifted area, $\ell^2\cos\theta$ per unit length, times the perimeter.

The wall on its own #

The flat wall is the wide tube’s limit and has a closed form, which is the second check. Against a single vertical wall the surface is two-dimensional, its only curvature is along the meridian, and balancing the horizontal forces on the lifted liquid gives the height at the wall directly:

$$\tfrac12\rho g h^2 = \sigma(1 - \sin\theta), \qquad h = \sqrt2\,\ell\sqrt{1-\sin\theta}.$$

For water fully wetting glass that is $\sqrt2\,\ell = 3.86$ millimetres, and it does not depend on the size of the wall at all.

Integrating the two-dimensional meniscus from a tiny disturbance on the flat far surface in to the wall recovers those heights to $3\times10^{-10}$, and the whole wetting profile to within $5\times10^{-5}$ capillary lengths of its closed form, $x = \cosh^{-1}(2/z) - \sqrt{4 - z^2} + \text{const}$. The same horizontal balance is where the $\sqrt{2}$ comes from: the lifted liquid’s hydrostatic push on the wall, $\tfrac12\rho gh^2$, is matched by the difference between the surface pulling horizontally far away and the part of its pull the tilted surface at the wall has kept.

The vertical balance holds for the flat wall too, and it gives the lifted area without integrating anything: the wall pulls up with $\sigma\cos\theta$ per unit length and nothing else supports the liquid above the far level, so the cross-section of water a flat wall holds up is exactly $\ell^2\cos\theta$. For water on clean glass that is one square capillary length, 7.4 square millimetres per millimetre of wall — the same for a microscope slide and a dam. The integrated profiles at 0, 30 and 60 degrees enclose $\ell^2\cos\theta$ to $2\times10^{-9}$, which is the flat-wall version of the tube check above.

The tube’s wall does not quite reach that height. Its own curvature — the tube is round, so the surface at the wall is curved around the circumference as well as up the meridian — adds to the pressure deficit it can support, and the wall height sits above $\sqrt{2}$ by about $0.64/R$ in capillary units. At twelve capillary lengths the wall is at 1.467 against 1.414. That approach is slow, which is why a laboratory beaker still has a visibly higher meniscus than a flat plate dipped in the same water.

One identity, cut on two planes #

Now the connection with the hanging drop, which is the reason the two are worth reading together.

A drop hanging from a tube and a column rising in a tube are held by the same thing: the surface pulling round the rim of a circle. Written as a vertical force balance, both say that the weight equals the tension round the perimeter, plus a pressure times an area on whatever horizontal plane the liquid was cut on. The only question is what that pressure is.

For the rising column the natural plane — the level of the liquid outside — is at atmospheric pressure by construction, and the liquid above it is under tension relative to the air, the case where a liquid does pull taken to its gentlest form, so the pressure term vanishes identically and the balance is exact. For the hanging drop the natural plane is the mouth of the tube, and the liquid there is at whatever pressure the drop’s curvature needs, which is not atmospheric. Tate’s law throws that term away and is wrong by it; Jurin’s law never had it.

The difference between an exact law and a rule of thumb here is one choice of plane, and the choice is not available to the drop: there is no level at which a hanging drop’s liquid is at atmospheric pressure except the drop’s own surface. That is the whole asymmetry between two formulas that look, on a page, like the same formula.

What the averaging hides, and when it matters #

The mean height is exact and the local height is not, so what matters is which one a question needs.

Anything that depends on the total is exact at every width. How much liquid a wick or a porous plug lifts against gravity, how much weight a wetted rod gains when dipped, the force a meniscus exerts on a float — these are volumes or forces, they are governed by the perimeter identity, and a tube’s radius enters only through its perimeter. This is why a Wilhelmy plate, which measures the force on a plate dipped in a liquid, measures surface tension without any correction for the plate’s size: it is reading the identity directly.

Anything that depends on the height at one place is not. A tube used to measure surface tension by the height at its centre is accurate only while it is narrow, and the classical method uses capillaries well under a millimetre for that reason. A height read at the wall of a wide vessel measures something closer to the flat-wall value, which depends on the contact angle in a different way — through $\sqrt{1-\sin\theta}$ rather than $\cos\theta$ — and so gives a different answer for a partially wetting liquid.

What a capillary-rise measurement reads #

The distinction has a price in the laboratory, where rise in a tube is a standard way to measure a surface tension, and it can be put in millimetres of water at 20 °C.

In a tube a quarter of a millimetre in radius Jurin’s law gives 59.51 mm and the surface’s lowest point sits at 59.42 — 0.14 per cent low, and Rayleigh’s $R/3$ correction accounts for all of it. At half a millimetre the centre is 0.56 per cent below Jurin’s value and the correction still accounts for it to three parts in a hundred thousand. At a millimetre the centre is 2.2 per cent low, the correction predicts 2.24, and the wall already stands a millimetre above the centre, which is visible by eye. At two millimetres the centre is 8.2 per cent low and the correction, now outside the range where it is a good series, predicts 9.0. At five millimetres the centre is 37 per cent low, the correction predicts 56, and the surface has become a rim five millimetres high around a floor under two.

So the height at the centre is a good measurement of surface tension for bores up to about a millimetre, with Rayleigh’s correction, and a poor one beyond. What stays good at every bore is the lifted volume — which is why the methods that weigh rather than look, a plate or a ring pulled from the surface, work in open vessels where no tube is narrow at all.

Outside a rod dipped into a liquid the same identity holds with the rod’s perimeter: a fibre of radius $b$ holds up exactly $2\pi b\,\ell^2\cos\theta$ of liquid. The difference is that outside a rod there is no far wall, so the lifted liquid spreads out over a region many capillary lengths wide and the height at the rod is small — of order $b$ times a logarithm of $\ell/b$, a classical result that is borrowed here rather than computed. A thin fibre therefore lifts the volume the identity demands and almost no height at all, which is the wide tube turned inside out.

Every check in that table is against something that does not share the integrator’s assumptions: an identity that follows from a force balance, a closed form that follows from a different force balance, or a series that follows from an expansion in the tube’s radius. The lifted-volume check is the one that carries the essay, and it is worth being clear that it could have failed. An integrator with a wrong curvature term, or a surface started at the wrong height on the axis, produces a smooth meniscus with the wrong volume under it — and the volume would then miss $2\pi R\cos\theta$ by the size of the error rather than by a part in a million.

What the picture cannot show #

The surfaces here are static and the contact angle is a single number. A real meniscus in glass climbs to a height that depends on whether the water was rising or falling when it stopped, because the contact line can be pinned anywhere within a range of angles, and the range — the hysteresis — can be tens of degrees on ordinary glass. The first figure shows the advancing and receding cases as the same surface, which they are not.

The first figure also draws a sharp, flat level outside the tube. That level is itself curved near the outside of the tube wall, where the liquid climbs the glass on that side too, and the true reference is the level far from any wall. The drawing puts it at the bottom and does not show the outside meniscus.

Where the model stops #

A uniform, fixed contact angle. Every number assumes the surface meets the glass at one stated angle everywhere round the circumference.

Axisymmetric, and static. The rise in a tube takes time — the column is drawn up against its own viscous resistance, and the approach to the final height can take seconds in a fine capillary — and a tube that is not round, or not vertical, has a surface that is not a figure of revolution. The totals still obey the perimeter identity; the shapes do not follow from the equation used here.

A continuum surface. In pores of a few nanometres the capillary length is irrelevant and the question becomes one of the liquid’s molecular structure; the identity holds as long as surface tension does, and that is a separate question.

The convention the numbers depend on #

Heights are measured from the level of the liquid far outside the tube, not from the bottom of the tube or from the lowest point of the meniscus; lengths are in capillary lengths, 2.73 mm for water at 20 °C. The contact angle is measured through the liquid, so a fully wetting liquid has $\theta = 0$ and rises, and one with $\theta$ above ninety degrees is depressed. Only wetting cases are computed.

Who found it, and when #

James Jurin described the inverse proportionality between rise and bore in 1718, from measurements with glass tubes, and gave the reason in terms of the ring of glass at the top of the column pulling the water up — an attraction round a perimeter, which is the identity above in its original form. Laplace supplied the curvature equation in 1805. Rayleigh gave the correction for the meniscus’s own volume in 1916 and the theory of wide tubes in the same paper. The Wilhelmy plate is from 1863.

Still open: what the rim does when the wall is not smooth #

Everything above puts the contact line at one angle, which is the idealisation that makes the perimeter identity so clean. A real wall has roughness and chemical patches, and the contact line pins on them, stopping at a different angle depending on which way it last moved. The perimeter identity then holds with the actual local angle at every point of the contact line — so the total lift is the perimeter average of $\cos\theta$, not the cosine of an average angle — and the height at the wall depends on the local angle through the flat-wall formula.

The calculation that would follow gives the contact line a spread of pinned angles round the circumference and asks how the lifted volume and the wall height respond: whether a spread of ten or twenty degrees, which is ordinary for glass, moves the total rise measurably, and how much of the scatter in capillary-rise measurements of surface tension it accounts for. Beside it is how a liquid meets a solid when the solid is moving, where the contact line is dragged rather than pinned, and the siphon’s height limit, where a column is held by atmospheric pressure rather than by a rim.