The column that is pulled, not pushed

Water reaches the top of a hundred-metre tree, and every mechanism proposed for getting it there has a number attached that is too small.

Capillary rise is the first candidate, because it is the one that lifts water against gravity with no pump at all. The rung below this one computes it: a curved surface pulls on a circumference while gravity pulls on an area, the two scale differently, and the height is $2\gamma\cos\theta/\rho g r$, with the angle a liquid makes with its solid deciding the cosine.

Why the fine pore is not an option #

The flow through a tube goes as the fourth power of its radius, which is the steepest dependence in the whole of fluid mechanics.

The exponent is steep because narrowing a tube does two things at once. Its flow profile is a parabola pinned to zero at the wall, so making the tube finer removes cross-section and slows what is left — two factors of the radius squared, and a fourth power overall. That is why a fine pore is not an option: halving the radius costs a factor of sixteen in flow, and the rise it buys goes only as the reciprocal of the radius.

So the pore that lifts and the pore that conducts cannot be the same pore, and the account has to be about something other than a meniscus dragging a column up behind it.

There is a tempting way out of that, and it does not work either. Put the fine pores and the coarse ones in series — a long coarse channel with a fine cap at the top — and the fine cap supplies the pull while the coarse channel supplies the flow. That is very nearly the right picture, and the reason it is not a capillary-rise argument is that the water below the cap is then not being held up by its own meniscus: there is no meniscus down there at all. It is being held up by whatever is above it, which is the cap, and the mechanism has quietly become tension rather than capillarity.

Osmosis fails for a different reason. Root pressure is real and measurable, and it is a counting argument: the pressure a solution develops across a membrane depends only on how many particles are dissolved. Xylem sap carries a few tens of moles per cubic metre, which is worth about five metres of water. That is enough to push sap out of a cut vine in spring and nowhere near enough for a redwood.

And a pump is worse. A pump at the top lowers the pressure beneath it, and the atmosphere at the base pushes the column up into the gap. The most the atmosphere can push with is one atmosphere, so the ceiling is 10.3 metres whatever the pump does — a fact a siphon runs into and a suction well runs into and every schoolchild’s straw runs into.

Two ceilings, then, and they are different in kind. The capillary ceiling is a trade against flow: any radius can be chosen, and every choice buys height at the cost of throughput. The barometric ceiling is absolute: 10.3 metres, whatever the tube, because it is set by how hard the atmosphere can push and the atmosphere pushes with one atmosphere.

Neither is what a tall column does, and the tell is that the two ceilings are at completely different heights while the thing being explained sits above both.

What is actually happening #

The something at the top is evaporation. Water leaving a curved meniscus in a fine pore makes that meniscus more curved, and a curved surface has a pressure difference across it — the Laplace pressure, $2\gamma/r$. So the driving pressure is generated at nanometre scale and transmitted through micrometre-scale plumbing, which resolves the contradiction the first two figures set up: the small pore does the pulling and the large pore does the carrying, and they are in different places.

The pressure a curved surface can hold is another inverse law, and it is the same expression as the capillary rise multiplied by $\rho g$ — the generator’s own check confirms the two agree. So the height a column reaches and the pressure a meniscus sustains are one quantity asked in two units, and there is no independent fact in the second.

The arithmetic is worth doing once, because it is the whole mechanism. A pore of radius 20 nanometres in a wall at the top of the column has a Laplace pressure of 2γ/r = 7.3 MPa available if the meniscus curves right into it. Nine atmospheres — 0.9 MPa — is a small fraction of that, so a meniscus in such a pore is nowhere near its limit and the column is held comfortably. The pore has capacity to spare, and the capacity is what makes the arrangement robust rather than marginal.

A tension is not a low pressure. The distinction is worth insisting on because it is where the intuition fails. A gas at low pressure is a gas with few molecules in it, and its pressure has a floor at zero because a gas cannot pull. A liquid can pull: its molecules attract one another — the same attraction that makes a surface cost energy, so a column of it under tension is in a state with a negative absolute pressure, and the tension is a real stress transmitted along the column exactly as it would be along a rope.

What does the pulling is cohesion, and it shows at a free surface. A molecule there has fewer neighbours than one in the bulk, so making surface costs energy — and that cost per unit area is the surface tension. Everything in this essay is that one number: the rise, the pressure, and the tension the column is under are three readings of the same energy.

Why the column does not simply tear #

A liquid under tension is metastable: the state of lowest free energy is liquid plus vapour, and only a barrier prevents it. That barrier is geometry.

A bubble of radius $r$ in a liquid at pressure $P < 0$ costs surface energy $4\pi r^2\gamma$, and the pressure across its wall is the one a small bubble is punished by and gains volume work $\frac{4}{3}\pi r^3|P|$. The first wins at small $r$ and the second at large, so there is a critical radius $r^* = 2\gamma/|P|$ and a barrier $16\pi\gamma^3/3P^2$. Below $r^*$ any bubble that appears shrinks away; above it, one grows without limit and the column snaps.

The barrier has the shape every nucleation problem has: two states separated by a hill, with the hill’s height deciding how long the metastable one survives. A column of liquid under tension is metastable rather than stable — a bubble of vapour would be lower in energy — and what keeps it intact is that a small bubble is higher, so the transition has to start with a fluctuation nothing supplies.

The numbers, and the gap between them. With water’s surface tension the homogeneous nucleation threshold — the tension at which a bubble appears spontaneously in pure bulk water — is around −140 MPa, and inclusions of water in quartz crystals have been taken to about −140 MPa in the laboratory, which is remarkable agreement. A tree operates at −0.5 to −3 MPa, fifty times smaller, and it still cavitates.

The reason is that a real conduit is not pure bulk water. It has walls, and the walls have pores, and a pore that already contains a pocket of gas is a nucleus that has skipped the expensive part of the barrier. Cavitation in a conduit therefore begins when the tension exceeds the Laplace pressure of the largest pore in its wall — which brings the argument back to the same expression a third time, now setting a failure threshold rather than a lifting height.

The mechanism has a name that is a hundred years old and was not believed for most of it. Dixon and Joly proposed the cohesion–tension account in 1894, and the objection to it was exactly the intuition this essay has been dismantling: everybody knew that liquids cannot be pulled, because everybody’s experience of pulling on a liquid was a suction pump, which fails at ten metres. The demonstration that water can sustain large tensions came from spinning it in a Z-shaped tube — a centrifuge produces tension in the middle of the column with no interface anywhere — and reached −27 MPa by the 1950s.

The measurement in quartz inclusions came much later and is the sharper one, because the water in question has never had a free surface at all: it was trapped as the crystal grew, and cooling shrinks it into tension. Those samples reach the calculated homogeneous limit, which is the strongest evidence there is that the barrier calculation above is right.

The pore that decides, and how it is found #

The failure threshold has been attributed to “the largest pore in the wall”, and that pore turns out to be findable — not by looking at it, but by measuring the tension at which the column fails and inverting the Laplace expression.

The route air takes into a conduit is not through the wall in general but through the porous membrane between one conduit and its neighbour. If the neighbour is already empty and full of air at atmospheric pressure, and the conduit in question is at a tension of a megapascal, the pressure difference across that membrane is eleven atmospheres — and air is pulled through the moment the difference exceeds the Laplace pressure of the membrane’s largest pore. One conduit’s failure therefore seeds the next, through a pore whose size is the only thing deciding when.

Inverting that gives an unusual kind of measurement. Apply a known air pressure to one side of a conduit and see when air appears on the other; the threshold pressure divided into $2\gamma$ is the radius of the largest pore in the path. Done this way, the numbers come out in the tens of nanometres — consistent with what electron microscopy of the membranes shows, and arrived at without a microscope.

The population version of the same measurement is the more useful one. Take a sample, subject it to a series of increasing tensions, and measure how much of its conducting capacity survives each. The result is a curve that falls from full capacity to nothing over a few megapascals, and the tension at which half the capacity is lost is a single number characterising the sample.

Those numbers vary enormously and they vary systematically. Species from wet places lose half their capacity at under a megapascal of tension; species from arid places hold on past ten. That is a four-hundredfold range in the largest pore’s Laplace pressure — a tenfold range in pore radius — and it is the clearest evidence that the pore size is a designed quantity rather than an accident of construction.

Measuring a tension without touching it #

The caution that a barometer cannot read a negative pressure leaves the obvious question of how the numbers quoted here were obtained, and the answers are ingenious enough to be worth setting out.

The standard instrument seals a cut sample into a chamber with the cut end protruding, and raises the gas pressure in the chamber until sap just appears at the cut. The reading is the tension the water was under before it was cut: cutting released the tension and pulled the water back into the sample, and the applied pressure is what it takes to push it to where it was. The instrument never touches the liquid and never provides an interface for it to cavitate at, which is exactly the requirement.

A second route measures the vapour rather than the liquid. Water under tension is in equilibrium with a vapour pressure below the saturated one, by an amount fixed by the tension, so measuring the humidity a sealed sample equilibrates with gives the tension. The sensitivity is poor at low tensions — the humidity shift is a fraction of a per cent — and it improves as the tension rises, which is the opposite of most instruments and makes it complementary to the chamber.

And the third route counts failures rather than measuring states. A cavitating conduit snaps, and the snap radiates a burst of ultrasound that a transducer clamped to the outside picks up. Counting bursts against tension gives the vulnerability curve directly, in a sample that is still intact, and it is the only one of the three that watches the event rather than inferring it.

Three techniques, none of which connects a liquid to a gas, all developed because the obvious instrument would destroy the thing it was measuring. That constraint — that the measurement must not provide the nucleus — is what shapes the whole experimental subject.

The engineered version #

The same three numbers are a specification rather than an explanation in one device, and it is worth naming because it makes the argument concrete.

A heat pipe is a sealed tube containing a wick and a working fluid. Heat at one end evaporates the fluid, the vapour travels to the cold end and condenses, and the liquid returns through the wick — driven by capillarity, with no pump and no moving parts. Its thermal conductance is enormous, because what carries the heat is a latent heat rather than a temperature gradient in a solid.

Its design constraint is exactly this essay’s arithmetic. The wick’s finest pore sets the maximum pressure the capillary pump can develop; the wick’s overall permeability sets the viscous drop the returning liquid suffers; and the two must balance, with anything left over available to lift against gravity if the pipe is not horizontal. When the pump cannot keep up, the evaporator dries out and the conductance collapses — which is the capillary limit, and it is a cavitation failure with a different name.

So a heat pipe’s wick is specified by a pore size, exactly as a conduit’s vulnerability is set by one, and for the same reason: a small pore pulls hard and conducts badly, a large one the reverse, and the design is the compromise. The device that solved it best is the one with two pore sizes — a fine layer where the pulling is done and a coarse structure where the carrying is done — which is the arrangement this essay described as the resolution to its own opening contradiction.

Where the picture stops #

The whole account rests on continuity. Every step assumes an unbroken thread of liquid from the evaporating surface to the base, because a tension is transmitted along a rope and not across a gap. That is a strong assumption about a hundred-metre-long channel a few tens of microns wide, and it is the assumption that fails first — which is why the failure mode is not the column stretching or the pump running out, but a single break somewhere in the middle.

Nothing here is biology. The account above is hydrostatics, surface tension and nucleation, and it applies to any porous solid with an evaporating surface at the top — a brick wall drawing damp, a wick, a drying ceramic. What a living conduit adds is repair and isolation, and those are the parts this figure has nothing to say about.

Nothing here has any dynamics in it. A conduit that cavitates does so in microseconds, and the bubble that forms does not sit still: it expands, the two liquid columns either side recoil, and the whole event radiates an acoustic click that can be picked up with a transducer on the outside. Counting those clicks is how cavitation is actually measured, and it is a measurement of an event this essay’s figures treat as a threshold.

The tension is not uniform. The hydrostatic line assumes the water is not moving. A transpiring column has a flow through it, and viscous drag adds to the tension — roughly doubling it at the rates a large tree actually moves water — so the operating point is further to the left than the figures draw.

In its ordinary orientation nothing about the hydrostatic line is strange. Pressure rises with depth at $\rho g$ per metre, and the only difference between that and this essay is which end of the column is held: push from below and the pressures are all positive, hang from above and the same gradient runs into tension. The physics does not change at zero; only the sign does.

The critical radius is the number to keep. At −1 MPa it is 2γ/|P| = 146 nanometres, and at −10 MPa it is 14.6. A bubble smaller than that collapses however it got there; a bubble larger than that grows without limit. So the question “will this column survive” is the question “is there a gas pocket anywhere along it bigger than 146 nanometres”, and the answer depends on the largest defect rather than on the average state of anything. That is the signature of every nucleation-limited failure, and it is why such failures are statistical in the size of the sample: a longer column has more places to fail and fails at a lower tension.

And a negative pressure cannot be measured with a barometer. Any instrument that connects the liquid to a gas provides exactly the nucleus the column has been avoiding. The measurements are made instead with a pressure chamber — sealing a cut shoot and raising the external pressure until sap appears at the cut, so that the number read is the tension that was there before the cut — or with a psychrometer reading the vapour pressure the liquid is in equilibrium with.

On a phase diagram the state sits on the wrong side of the vaporisation curve. A liquid held at a pressure below its own vapour pressure ought to boil, and a column under tension is held far below it — so the whole arrangement is thermodynamically forbidden and kinetically fine, which is the usual situation for a metastable state and the reason trees work.

The same three numbers elsewhere #

Strip the biology out and what is left is a recipe that applies to any porous solid with an evaporating face: a pulling pressure set by the finest pore, a conducting resistance set by the widest, and a failure threshold set by the largest defect. The three are governed by the same expression at three different radii, which is why the whole subject can be argued from one figure.

Pressure that only knows depth is the ordinary case, and this is not it. A drying brick does it: evaporation at the surface creates menisci in the finest pores, those pull water from the interior through the coarse ones, and the wall dries from the inside out until the finest pores empty and the front recedes. A wick does it. A heat pipe does it deliberately and at a designed tension, which is why its wick specification is a pore size and not a material.

And a soil does it in reverse. The tension a soil holds water at is set by its finest filled pores, so a clay holds water at tensions a plant cannot overcome while a sand releases it at almost no tension at all — the difference between soil that is wet and useless and soil that is dry and adequate, and it is the same inverse law with the radius changed.

Where this ladder goes next #

Three rungs have now treated a meniscus as a static object with a pressure across it. The next asks what happens when it moves: a meniscus advancing into a dry pore has a different contact angle from one receding out of a wet one, so the pressure it can hold depends on which way it last went, and a porous solid therefore holds a different amount of liquid on wetting than on drying. That hysteresis is the whole reason a soil moisture reading needs to know its own history, and it is measurable in a single capillary.