Where a liquid does pull

Nothing sucks makes an argument that is exactly right about the case it is about. The upper surface of a wing is under a lower push, not a pull; the absolute pressure there is 96 kilopascals; and the force does not care where the datum was put. A force forgets the datum, a stress cannot then shows where that last identity stops. Both say, in passing, that a fluid cannot pull, and for air that is a statement with no exceptions: a gas at zero absolute pressure is a vacuum, and there is nothing below it.

A liquid is not a gas. Its molecules hold on to one another, and a liquid held away from anything it could boil on can be stretched to a pressure below zero — not a smaller push but a genuine tension, with the liquid on either side of a surface pulling the surface towards itself. The case that matters is not a laboratory curiosity. It is the water in a tree.

Ten metres, and the reason a pump stops there #

The number every account of suction quotes is ten metres, and it is worth computing before it is questioned. A column of still water standing in a sealed tube over an open reservoir has atmospheric pressure at its foot and loses $\rho g$ per metre of height. The absolute pressure reaches zero at

$$h_{\text{baro}} = \frac{p_{\text{atm}}}{\rho g} = \frac{101\,325}{998.2 \times 9.807} = 10.35 \text{ m},$$

and a suction pump at the top of such a pipe does slightly worse. The water under its piston reaches the vapour pressure, 2.33 kilopascals at twenty degrees, at 10.11 metres, and there it boils. That is the limit the old well-diggers met, and it is the one the siphon’s crown meets when a siphon is primed by sucking.

The reason is worth stating precisely, because the usual statement is subtly wrong. The usual statement is that the atmosphere can only push a column ten metres high and there is no more push to give. The arithmetic does not say that. It says the pressure at 10.35 metres is zero, and at 11 metres it is negative, and the equation for a column carries straight on through zero without noticing.

What stops at ten metres is not the equation but the pump. A pump draws the water up against a chamber holding air and vapour, through a valve with a gap in it, out of water that has been open to the atmosphere for its whole life and carries dissolved gas and a population of microscopic bubbles. It offers the water somewhere to boil, and a liquid with somewhere to boil holds its vapour pressure as a floor exactly as the textbook says. The ten-metre limit is a property of liquids in contact with a gas, which is almost every liquid anybody handles.

The column a tree runs on #

A tree is the case where that contact has been designed out. Water enters the root, rises through a continuous network of dead, water-filled cells — tracheids in a conifer, vessels in a flowering tree — and leaves through the leaves as vapour. The column is unbroken from the soil to the leaf, and it is sealed: the walls between one conduit and the next are crossed through pits covered by a membrane with pores tens of nanometres across, too small for a gas bubble to squeeze through at ordinary pressure differences.

The pressure up such a column falls for two reasons, and they add. Gravity costs $\rho g$ per metre, as it does in the pump. Friction costs the Poiseuille gradient of the conduit, since the flow in a tube a few tens of micrometres across moving a fraction of a millimetre a second is laminar with a Reynolds number of 0.01:

$$\frac{dp}{dz} = -\rho g - \kappa\,\frac{8\mu u}{a^2}.$$

Here $a$ is the conduit radius and $u$ the sap speed, and $\kappa$ is a multiplier for the extra resistance of the pits the water has to cross between conduits. With a conduit twenty micrometres in radius, sap at a quarter of a millimetre a second and the pits doubling the lumen’s resistance — values chosen to be representative of a conifer and stated as chosen, not measured on any tree — the two terms come to 9.79 kilopascals a metre for gravity and 10.02 for friction, which is the rough equality plant physiologists report in transpiring trees.

A column rising a hundred metres at a combined 19.8 kilopascals a metre ends at the top at $p_{\text{base}} - 1.98$ megapascals. The base is not atmospheric either. A moderately moist soil holds its water at about a tenth of a megapascal below atmospheric, so the root starts at 1.3 kilopascals absolute — already below water’s own vapour pressure of 2.33, at the bottom of the tree, before any height has been climbed. The top of the column is at −1.98 megapascals absolute. That is not a small push. It is a pull of about twenty atmospheres, sustained continuously, in every tall tree on the planet.

The convention the numbers are in #

Plant physiology reports the same quantity on a different datum and it is worth naming, because the two readings differ by the whole of what the suction essays were about.

A pressure chamber — the instrument that measures a shoot’s water status by pressurising it until sap appears at the cut — reads the water potential, which for dilute xylem sap is the gauge pressure: the absolute pressure less one atmosphere. A physiology paper saying the xylem at the top of a tree is at “−2 megapascals” means two megapascals below atmospheric, which is −1.9 absolute. The difference is a tenth of a megapascal and does not change the argument, since both are far below zero.

What does change the argument is reading either number as a gauge pressure of the kind a wing’s upper surface carries. A wing’s “−5 kilopascals” is a push of 96 kilopascals. A tree’s “−2 megapascals” is a pull, and no choice of datum makes it anything else — which is precisely the case the argument about datums said a local question about a fluid’s state could not escape. Every number in this essay is absolute, and every one below zero is a tension.

Why the water does not boil #

A liquid at −2 megapascals is two megapascals below its vapour pressure, and by the rule the pump obeyed it should boil at once. It does not, and the reason is the whole of the physics.

Boiling is not something a liquid does at a pressure. It is something a cavity does. A pocket of vapour of radius $r$ inside a liquid is held shut by its own surface tension, which pushes inward with the Laplace pressure $2\sigma/r$. It grows only if the liquid around it is at less than $p_v - 2\sigma/r$, and a liquid with no cavity bigger than $r$ in it can sit, perfectly quietly, anywhere above that.

For a cavity a millimetre across the correction is 150 pascals and the vapour pressure is, for every practical purpose, the floor. For one a nanometre across it is a hundred and forty-five megapascals. So the vapour pressure is the floor of a liquid that has a large cavity in it, or a free surface, or a gas-filled crevice in a wall — and it is not the floor of a liquid that has none. The same relation that fixes the shape of a drop fixes how far a clean liquid can be stretched, and it does so with the same σ, 72.7 millinewtons a metre for water at twenty degrees.

A tree’s conduits are that clean liquid. They grow full of water, they are never exposed to air, and their walls are wetted cellulose. There is nothing inside them for a cavity to start on, and the water at −2 megapascals is metastable in exactly the sense a supercooled liquid is: below the state it would rather be in, and staying where it is because the route to that state has a barrier on it.

The floor is a pore #

A metastable column has a floor, and the floor is not in the liquid. It is in the membranes between the conduits.

When a conduit does fail — from frost, from a wound, from an insect — it fills with gas at about atmospheric pressure, and its neighbours are still carrying water under tension. Between them sits a pit membrane. A gas–water interface in one of its pores has the same Laplace pressure a cavity has, $2\sigma/r$ for a perfectly wetting pore of radius $r$, and it holds as long as the pressure difference across it is smaller than that. When the water side falls further, the meniscus is pulled through the pore, a bubble enters the working conduit, and the tension in that conduit snaps it into a gas-filled void. That is air seeding, and it is the standard account of how a working conduit is lost.

The numbers are worth reading as instructions. A fifty-metre column at the reference sap speed needs its pit pores no wider than 133 nanometres. A hundred-metre column needs them under 70. A column 140 metres tall needs them under 51. The limit on how hard a plant can pull its water is a length on a membrane, and that length is tens of nanometres — which is the scale pit-membrane pores are actually measured at.

For the reference membrane, with pores fifty nanometres in radius, the floor is −2.81 megapascals absolute. The hundred-metre column’s top at −1.98 sits 0.83 megapascals above it. That margin is what the plant has in hand, and it is small: less than half of the tension already being carried.

Drinking faster, not growing taller #

The friction term has a consequence that the gravity term does not. Gravity is fixed by the height and cannot be changed by anything the tree does. Friction is proportional to how fast the sap is moving, which is set by how fast the leaves are losing water, which follows the sun.

With nothing flowing, the top of the hundred-metre column is at −0.98 megapascals, which is gravity alone. At a millimetre a second friction has added four megapascals more and the top is at −4.98. The line between is straight, because Poiseuille friction is linear in the speed, and it crosses each pore’s floor at a definite sap speed: 0.457 millimetres a second for a fifty-nanometre pore, and 0.094 for a hundred-nanometre one.

That turns the failure of a conduit from a question about a tree’s size into a question about its afternoon. A tree does not lose conduits because it grew; it loses them because the air dried and the leaves drew water faster than the column could supply without its top dropping through the floor. The physiological reading is that the leaf’s pores, the stomata, close as the xylem pressure falls, and that closing them — which also stops the leaf taking in carbon dioxide — is the control loop that keeps the column above the line drawn here. The figure is that control problem’s plant: a straight line in sap speed, with a hard floor on it set by a membrane.

How tall the arithmetic allows #

Put the two effects together and ask the question the other way round: for a given pore, how tall can a column be before its top reaches the floor?

The answer is a ratio of the floor to the gradient, and it halves by day. A still column with fifty-nanometre pores could stand 287 metres before its top reached −2.81 megapascals; a column carrying sap at the reference speed has twice the gradient and reaches it at 142. The tallest tree measured is about 116 metres, and to stand that tall while transpiring at this speed it needs no pit pore wider than 61 nanometres — at night 118 would do.

This is the surprising connection, and it is exact within its model. The height of the tallest trees on Earth is bounded, in part, by the curvature a meniscus can hold in a pore — the same Young–Laplace relation that decides how large a raindrop can be before it flattens and how high water climbs a capillary. The branch of the subject that prices the radius a vessel should have and the branch that prices a drop’s shape meet at the top of a redwood, and the quantity they meet in is σ.

It is also a ceiling that is not the only one. The measurements on the tallest redwoods argue that growth at the crown — a leaf expanding its cells against a water potential that low — is limited before the column is, and the hydraulic line drawn here and the physiological one sit within a few tens of metres of each other. Which binds first is a biological question the arithmetic cannot settle. What it does settle is that the ceiling is a pore size and a sap speed, and that ten metres is not in it anywhere.

Where a tree sits among the tensions #

A tree’s two megapascals is a large number against ten metres of water. Against what water can be made to hold, it is small.

The top of the column is at −1.98 megapascals and its pore floor at −2.81. Water in careful breaking experiments lets go at around −26, a spinning tube at −27.7, and water sealed into microscopic inclusions in quartz at around −140, which is close to what the theory of a perfectly pure liquid predicts. A tree works about a hundredth of the way down that scale, and the reason is not that water is weak. It is that a tree’s floor is set by the weakest pore in any membrane between a working conduit and a failed one, and a membrane full of pores is a much weaker thing than water.

That is also the answer to why a pump cannot do what a tree does. Neither is limited by the water. A pump is limited by the gas it lets the water touch, at a floor of +2.3 kilopascals; a tree by the pores it lets the gas reach, at a floor of −2.8 megapascals. Both floors belong to the plumbing, and the tree’s plumbing is better by three decades.

What the straight line leaves out #

The column is uniform. Real xylem tapers, branches, and changes its conduit size between the roots, the trunk and the twigs, and the friction gradient at the top of a tree is usually steeper than at the bottom. A tapering column’s profile is curved rather than straight and its top is lower than the uniform column’s for the same average; the essay’s conclusions depend on the order of magnitude of the gradient, which the tapering does not change.

The parameters are prescribed. The conduit radius, the sap speed, the pit resistance, the soil’s water potential and the pore radius are representative values, stated as such, and no tree was measured for this calculation. The equality of the gravity and friction terms is a consequence of the values chosen, which were chosen to reproduce that equality as it is reported rather than to discover it.

Perfect wetting. The air-seeding threshold $2\sigma/r$ assumes a contact angle of zero. A pore with a finite contact angle holds $2\sigma\cos\theta/r$, which is less, and real pit membranes are not circular holes of one size — the floor is set by the largest pore in the whole area of membrane a conduit has, which is a statistical quantity rather than a geometric one.

Nothing living. The column has no osmotic term, no living cells around it, no storage in the tissue that buffers a daily cycle, and no mechanism for refilling a conduit that has failed, which trees do have and which is argued over. The profile is steady; a real tree’s is a cycle, with the morning’s tension rising from the night’s.

And the account is itself argued. The cohesion–tension account is the standard one and the calculation here is written inside it. It has had challengers, largely over whether the indirect pressure-chamber readings and direct probes inserted into single conduits report the same tension. The arithmetic shows that the standard account is self-consistent at the pressures reported; it does not show that the pressures reported are the pressures present.

The two checks the calculation is held to are the ones that could fail it. A still column with a wet foot must cross zero absolute at exactly $p_{\text{atm}}/\rho g$, which it does to a part in a billion; and the straight-line profile must agree with a numerical integration of its own gradient, and the height limit with a bisection on the profile, each to a part in a million — because a closed form for a straight line is exactly what a sign error looks like when it is right at both ends.

A mechanism named before it was measured #

The cohesion account of the ascent of sap is Dixon and Joly’s, from 1894, with Askenasy arriving at the same argument independently in the same year. It was a remarkable piece of reasoning for its date: it required water to hold tensions of tens of atmospheres at a time when the few measurements of a liquid’s tensile strength were of a few atmospheres, made by stretching water in glass, and it required nobody to have seen the tension directly.

Nobody did for seventy years. The pressure chamber that made the measurement routine is Scholander’s, from 1965, and it reads the tension indirectly by applying an equal and opposite pressure until sap returns to the cut end of a shoot. The air-seeding account of how conduits fail, and the identification of the pit-membrane pore as the thing that sets the floor, followed in the 1980s.

The order is the interesting part. The argument that a liquid could pull came from a tree before it came from a laboratory, and it was accepted on the strength of the tree: a fifty-metre plant is a demonstration that no pump, however good, could perform, and the only account that explained it required the conclusion that every essay about suction says in passing is impossible.

Still open: what breaks water when nothing is in it #

The tree’s floor is a membrane, and a membrane is not water. What sets the floor of a liquid with no pore anywhere near it — no membrane, no crevice, no gas — is a different question with a stranger answer, and the measurements of it disagree with one another by a factor of five while each is careful.

That is a breaking strength that is the size of a flaw: water stretched in a spinning tube, where the tension is greatest at the one place no free surface can reach, and in a sealed tube, where a property of water’s own density puts a ceiling on the method. What every such measurement turns out to report is not quite a property of the liquid.

Beside it is the other thing that ends a column, which is gas already in the water. A siphon that its heights say can never break is broken by the air dissolved in it, which comes out of solution at the crown because the crown is where the pressure is lowest — and a cavitation threshold quoted as a pressure is the long-hold limit of the same competition between a liquid’s tension and what is in it to give way.