The pore that lifts highest fills slowest

How high water will climb derives Jurin’s law: water in a clean glass tube of radius $r$ rises to $h_J = 2\gamma/\rho g r$, where the pull of the meniscus round the circumference balances the weight of the column. A tube of 0.1 millimetre lifts water fifteen centimetres; one of a micrometre lifts it fifteen metres. The law is exact at equilibrium and silent on the way there.

That silence matters wherever capillarity is used to move liquid rather than to hold it: ink into paper, water into soil and building stone, blood into a diagnostic test strip, fuel along a wick. In each, the useful question is not how high the liquid will end up but how far it will get in the time available. The answer involves the one quantity Jurin’s law leaves out, the viscosity, and a second one that is usually forgotten entirely, the inertia of the column. Between them they give the rise three different laws in succession, and they turn Jurin’s clean inverse proportion into a trade-off.

The equation of a rising column #

Three forces act on the column of liquid in the tube. The meniscus pulls up with a force $2\pi r\gamma$, independent of height. The weight pulls down, growing with the height. And the wall drags on the moving liquid: for flow through a tube, the fourth power in a pipe gives the pressure drop over a length $h$ at mean speed $\dot h$ as $8\eta h\dot h/r^2$. The column’s mass also grows as it rises, so its momentum is $\rho\pi r^2 h\dot h$, and Newton’s law for it reads

$$\rho\frac{d(h\dot h)}{dt} = \frac{2\gamma}{r} - \rho g h - \frac{8\eta h\dot h}{r^2}.$$

At equilibrium the last two terms on the right balance the first with $\dot h = 0$, and that is Jurin’s law. Everything else about the rise is in the other terms, and every figure here integrates that equation directly, starting from the instant the tube touches the liquid.

The narrow tubes do what intuition expects: the water climbs quickly at first, then more and more slowly, approaching Jurin’s height from below. The wide tubes do something Jurin’s law has no room for. The water arrives at its equilibrium height still moving, overshoots it by a third or nearly a half, falls back below it, and oscillates, the swings dying away over a second or more.

Why a wide tube rings #

Near equilibrium the column is a mass on a spring. Displaced by a small amount from $h_J$, it feels a restoring force from the change in its weight, it has the mass of a column $h_J$ tall, and it is damped by the Poiseuille drag. The spring constant is $\rho g$ per unit area, the mass $\rho h_J$, and the damping $8\eta h_J/r^2$. The three ways of coming to rest sorts every such system by one comparison — whether the damping is strong enough to stop the mass before it passes equilibrium — and here the comparison depends steeply on the radius. The drag weakens as $1/r^2$ while the natural frequency $\sqrt{g/h_J}$ grows as $\sqrt r$, so wide tubes are underdamped and narrow ones overdamped, with the boundary where the two match: $r_c = (32\eta^2\gamma/\rho^3g^2)^{1/5}$.

For water that is 0.48 millimetres. The 1 and 2 millimetre tubes are well past it and ring; the 0.2 millimetre tube is well inside it and creeps; the 0.4 millimetre tube, just below it, arrives without overshooting and without much hesitation, which is what critical damping looks like. David Quéré and colleagues observed these oscillations in the 1990s with water and with less viscous liquids, in tubes a millimetre or two across, and found the overshoot and the period close to what this equation gives, with the remaining differences attributed to how the meniscus itself moves and to the flow at the tube’s entrance.

The fifth root makes the critical radius forgiving. Ethanol has a third of water’s surface tension and nearly the same critical radius. The one property that moves it a long way is viscosity, which enters as its two-fifths power: glycerol, fourteen hundred times more viscous than water, needs a tube fifteen times wider before its rise will ring. Most everyday capillaries — the pores of paper, soil, wood and cloth — are far below every one of these radii, which is why capillary rise is almost always seen as a slow creep. The overshoot is real physics that everyday materials are too fine to show.

The ringing column is a resonator like any other, and it can be read the same way. The width that is a lifetime shows that the number of swings a resonator makes before it fades and the sharpness of its response to a periodic push are one number, its quality factor. For the column that factor is the natural frequency $\sqrt{g/h_J}$ divided by the damping rate $8\eta/\rho r^2$, and it grows as $r^{5/2}$: the 2 millimetre tube has a quality factor of about eighteen and rings visibly for more than a second, while a tube a tenth as wide has a quality factor of a few hundredths and cannot ring at all. Shaken gently up and down at its natural frequency — about 6 hertz for a 2 millimetre tube of water — the column would respond like the frequency that gets an answer, with its height swinging in quadrature with the shaking.

Three laws for one rise #

A narrow tube hides a different kind of richness, and it shows on logarithmic axes.

In the first milliseconds the column is too short for the drag to matter and too light for its weight to matter, and the only thing resisting the meniscus’s pull is the inertia of liquid being set in motion. The column rises at a constant speed, $\sqrt{2\gamma/\rho r}$ — 1.21 metres per second in a tenth-of-a-millimetre tube, a speed set by the balance between the surface energy released and the kinetic energy given to the liquid.

As the column lengthens, the drag, which grows with its length, overtakes the inertia. From then on the pull is spent entirely against friction: $8\eta h\dot h/r^2 = 2\gamma/r$, so $h\dot h$ is constant and $h^2$ grows in proportion to time. That is the law Richard Lucas and Edward Washburn found in 1918 and 1921, $h = \sqrt{\gamma r t/2\eta}$, and it is what most descriptions of capillary rise mean. The slope of 0.48 measured off the integration, rather than exactly one half, is the weight already beginning to matter by the middle of the viscous stretch.

Finally the weight catches up with the pull and the column approaches Jurin’s height exponentially. The three regimes are separated by orders of magnitude in time — milliseconds, then seconds — and each is a different balance of two of the three forces. A capillary rise is not one process but three, each with its own law, and Jurin’s is only the last.

The pore that lifts highest fills slowest #

Put Lucas and Washburn’s law next to Jurin’s and the trade-off is immediate. Jurin’s height grows as $1/r$: narrow pores lift higher. Washburn’s rate falls as $r$: narrow pores are slower, because the drag goes as $1/r^2$ and the pull only as $1/r$. A narrow pore will eventually lift water to a great height, but it takes a long time to get there.

For every deadline there is a best pore. To its right, in wider tubes, the liquid has already reached Jurin’s height, and a wider tube only lowers it; to its left, in narrower tubes, the liquid is still climbing, and a narrower tube only slows it. After one second the best radius for water is 170 micrometres and it lifts water 57 millimetres. After a day the best is 3.8 micrometres, and it lifts water two and a half metres. The best radius falls as the cube root of the deadline, because it sits where the Washburn time to reach Jurin’s height, which goes as $1/r^3$, equals the deadline — a sixty-fold longer wait moves the best radius by a factor of 3.9, the cube root of sixty.

The same trade-off decides how porous materials are designed and how natural ones work. Paper for fountain pens has pores large enough to take ink quickly and small enough to hold it; a medical test strip has to deliver a drop of blood across a centimetre in seconds, which fixes its pore size within a narrow range; the stones of a building wick groundwater up to a height set by how fast evaporation from their faces removes it, which sets the balance between the pores that lift high and the pores that feed fast. The column that is pulled, not pushed meets the extreme form of the trade-off: a pore fine enough to lift water a hundred metres up a tree could not carry the flow the tree needs in a year, which is why trees do not use capillary rise to lift water at all.

What the angle and the shape change #

Everything here was drawn for water on clean glass, which wets it completely. On a surface that water wets only partly, the angle a liquid makes enters as a factor $\cos\theta$ on the pull, and it enters everywhere the pull does: Jurin’s height, the inertial speed and the Washburn rate all fall together, and a surface the liquid does not wet at all — an angle beyond ninety degrees — pushes the column down below the outside level rather than drawing it up. The critical radius for overshooting moves only as the fifth root of $\cos\theta$, so the regimes stay where they are while the heights shrink.

The shape of the tube matters more than its material. A tube with corners is not one capillary but several: the corner a liquid never stops climbing shows that liquid runs up a sharp enough corner without limit, in filaments that are thin but have no Jurin height at all, and a square tube therefore fills its corners long before its middle. And a pore too narrow to fill from the bulk liquid in the time available can fill from the vapour instead: the pore that fills from dry air finds nanometre pores condensing liquid from air at half humidity, which is a way round the trade-off that needs no flow along the pore at all.

Half the energy, whatever the route #

The last figure asks where the energy goes, and the answer is the same for every tube.

As the liquid wets the tube’s wall it releases surface energy in proportion to the height it has climbed: $2\pi r\gamma h$. Some of that goes into lifting the column, some into its motion, and the rest is lost to viscous friction and to the swirl where liquid enters the tube. In the ringing 1 millimetre tube the kinetic energy sloshes back and forth with the oscillation, and the potential energy overshoots and falls back. When everything has settled, the column holds 0.499 of the surface energy that was released and the rest has become heat. In the narrow tube, which rises without ringing and far more gently, the split at the end is exactly the same: half and half.

Half the surface energy is always lost, however slowly the rise happens. The reason needs no dynamics. Jurin’s height is the height at which the column’s weight equals the pull, $\rho g h_J \cdot \pi r^2 = 2\pi r\gamma$. The surface energy released is the pull times the height, and the potential energy of a uniform column is its weight times half its height. So at equilibrium the stored energy is exactly half the released energy, and the other half must have gone somewhere else — into friction, whichever way the column got there. Slowing the rise down only spreads the loss over a longer time.

The same half turns up in a circuit. A capacitor charged from a battery through a resistor stores exactly half the energy the battery delivers, whatever the resistance, for the same reason: the battery pushes with a constant voltage while the capacitor’s voltage rises from zero to that value, so the capacitor holds the average of the two and the resistor takes the difference. The meniscus is the battery, pushing with a constant force; the column’s weight is the capacitor, pushing back in proportion to how far it has been filled; and viscosity is the resistor. Any process that works against a constant force into a store that pushes back linearly loses half.

Where the rising column stops #

A perfectly wetting, steady contact angle. The equation takes the meniscus to be a hemisphere from the first instant. A real meniscus moving fast has a dynamic contact angle larger than its static one, which weakens the pull early in the rise; in the inertial regime this matters, and it is one of the main reasons measured early speeds fall below $\sqrt{2\gamma/\rho r}$.

Fully developed flow. The Poiseuille drag assumes the parabolic profile of long, steady flow. Near the entrance and in the first instants the profile is still developing, the drag is different, and the kinetic energy of liquid entering from the reservoir has to be counted; the model includes that last term in its energy account and nothing more.

A single straight tube. Real porous materials are networks of pores of many sizes, connected in series and in parallel, and they fill by a front that is not the height in any one pore. The square-root law survives on average in many of them, with an effective radius that is a measurement rather than a pore size; the overshoot does not survive, because a network of fine pores is overdamped.

Constant properties and no evaporation. A liquid that evaporates from the wetted wall, or a tube open to dry air, reaches a height set by the balance between supply and loss, below Jurin’s; the figures assume a tube in saturated air.

The flow that reverses first at the wall #

Every figure is a height or an energy against time; none shows the flow. The column in a wide tube during an overshoot is not a rigid plug: near the wall it is slowed by friction, at the centre it runs faster, and during the fall-back after the first overshoot the flow reverses first at the wall. The Poiseuille law averages that profile into a single drag. Nor do the figures show the meniscus, whose shape changes as it moves and whose own inertia and dissipation — at the contact line, where the liquid is sheared hardest — are neglected here and are not neglected by nature.

Still open: what happens at the very start #

The first instant of capillary rise is not well described by any simple equation. At contact the column has zero length, the equation above predicts a finite speed from the first moment, and the real liquid has to form a meniscus, accelerate from rest, and satisfy a contact angle that is changing as fast as the liquid moves. High-speed measurements show that the early speed depends on the tube’s surface treatment and on how the tube is brought to the liquid, and models that include the dynamics of the contact line and the developing flow describe some experiments and not others. How much of the earliest rise is inertia and how much is the contact line is still being argued, and for the fastest applications — microfluidic devices that fill in milliseconds — it is the part that matters.

The habit worth carrying away is to ask of any equilibrium law how the system gets there and what it costs. An equilibrium says where a process will stop, not how long it takes or what it loses on the way, and the property that improves the destination is often the one that slows the journey. A finer pore lifts water higher, and reaches its height later; a gentler rise loses the same half of its energy as a violent one. The best pore is the one sized for the time available.