A breaking strength that is the size of a flaw

Where a liquid does pull finds water at the top of a tall tree sitting two megapascals below zero absolute, and finds the floor under it in a membrane: a pore tens of nanometres across that lets air through once the tension exceeds what its meniscus can hold. The tree’s limit belongs to its plumbing. It says nothing about how hard water itself can be pulled.

That question has a history of measurements that is unusually uncomfortable. The numbers reported for the tensile strength of water run from a few tenths of a megapascal to a hundred and forty, and the spread is not the ordinary scatter of a hard measurement. Careful methods agree among themselves near one value, a different careful method agrees with the theory near another, and the gap between them is a factor of five.

Stretching water without touching it where it is weakest #

The obvious way to stretch a liquid is to pull on it — a piston withdrawn from a sealed cylinder — and it fails for the reason the tree’s conduits are built to avoid. The tension is applied through a surface: the piston’s face, the cylinder’s wall, the seal between them. Every one of those is a place where gas can hide in a crevice and a cavity can start, and the measurement reports the weakest crevice in the apparatus.

Spinning a tube puts the tension somewhere else. A straight tube full of water, open at both ends and spun about an axis through its middle, holds its water by centrifugal force. In the rotating frame the water is at rest under a body force pointing outward, so its pressure rises outward, and with both menisci at radius $R$ held at atmospheric pressure it is

$$p(r) = p_{\text{atm}} - \tfrac{1}{2}\rho\,\omega^2\left(R^2 - r^2\right).$$

The lowest pressure is on the axis, $p_{\text{atm}} - \tfrac{1}{2}\rho\omega^2R^2$, and it can be made as negative as the shaft will turn. What matters is where that lowest point is: as far from both free surfaces as the tube allows, in the middle of a column of liquid, with the nearest meniscus a full arm’s length away. A gas bubble that forms at a meniscus is flung outward, away from the stretched region, rather than drawn into it. The method stretches the liquid hardest at the one place in the apparatus no gas can reach.

The Z shape of the tube Briggs used in 1950 — the arms bent back at their ends — holds the menisci at a fixed radius as the tube spins up, so that $R$ in the expression is a known length. At a five-centimetre arm, ten thousand revolutions a minute puts −1.3 megapascals on the axis, twenty thousand −5.4, thirty thousand −12.2, and forty-five thousand −27.6. The parabola is drawn above and there is no fitted quantity in it.

What a measured number costs in shaft speed #

The axis pressure falls as the square of the shaft speed and the square of the arm, so the two trade against each other one for one. A five-centimetre arm crosses zero absolute at 2,721 revolutions a minute — an ordinary laboratory centrifuge speed — and reaches −27.7 megapascals at 45,075. Double the arm and half the speed does the same.

The method has no ceiling of its own. What ends a run is the water breaking, and the speed at which it breaks, read back through the parabola, is the measurement. Briggs spun his tubes up until a cavity appeared on the axis, at a range of temperatures, and reported a largest tension of −27.7 megapascals at ten degrees, falling away on either side — a curve that has been reproduced in its shape and roughly in its magnitude by methods that share nothing with it.

Stretching water by cooling it #

The oldest method stretches water without any motion at all, and it has a ceiling that is not in the apparatus but in the water.

Berthelot, in 1850, filled a thick glass tube almost to the top with water at a warm temperature, heated it until the expanding water filled the last of the space, sealed it, and let it cool. The glass holds its volume. The water wets the glass and clings to it. As the water cools it wants to contract, and it cannot: it is held at the density it was sealed at, lower than the density it would take at the cooler temperature, and a liquid held less dense than it wants to be is stretched. To first order in the compressibility κ,

$$p - p_0 = \frac{1}{\kappa(T)}\left(\frac{\rho(T_0)}{\rho(T)} - 1\right),$$

with $T_0$ the sealing temperature. At twenty degrees water’s expansion over its compressibility gives 0.45 megapascals of tension for each degree of cooling, so a modest cooling makes a large tension and the tube eventually breaks with an audible click as a cavity snaps open.

Water’s density has a maximum, and it is at 3.98 degrees. Above it, cooling makes the water want to be denser and the sealed tube’s tension grows. Below it, cooling makes the water want to be less dense, which is the reverse of what stretched it — and the tension relaxes. So a sealed tube can be stretched only while it is cooling towards the density maximum and not past it, whatever temperature it was sealed at. The same anomaly that lets a lake freeze from the top down puts a ceiling on the oldest method of stretching water.

The ceiling, a few degrees above freezing #

The deepest tension a sealed tube reaches is not quite at the density maximum, and the reason is worth having because the obvious answer is wrong by a measurable amount.

If the compressibility were a constant, the tension would follow the density ratio alone, and its deepest point would sit exactly at 3.98 degrees for every filling temperature. It is not constant: cold water is eleven per cent more compressible at freezing than at twenty degrees, and a more compressible liquid takes less pressure to hold at a given stretch. So as the tube cools towards the maximum, the stretch is still growing but each unit of it is worth a little less pressure, and the deepest point moves up.

Filled at twenty degrees the tube reaches −3.48 megapascals at 4.7 degrees. Filled at thirty it reaches −8.69 at 5.7. Filled at forty it reaches −15.72 at 6.9. The harder the tube was stretched the more the compressibility’s variation weighs, and the further the deepest point moves from 3.98. The pale curve with the compressibility frozen lands on 3.98 exactly, which is the check that the shift comes from κ and from nothing else.

The calculation stops at a forty-degree filling for a stated reason. Beyond it the tension asked for reaches tens of megapascals, water under that much tension is measurably more compressible than its atmospheric value, and the first-order form above stops being a description of anything. A real sealed tube also has glass that stretches a little and a wall where the liquid can let go — which is why Berthelot tubes have tended to report tensions of a few megapascals to a few tens, well short of the spinning tube, rather than a clean ceiling.

Every measured strength names a flaw #

Two methods, one spinning and one sealed, and a set of numbers that do not agree with the theory. The relation that makes sense of them is the one the tree’s floor was built on.

A cavity of radius $r$ in a liquid is held shut by surface tension pushing inward with $2\sigma/r$, and it grows without limit once the liquid around it falls below

$$p_{\text{grow}} = p_v - \frac{2\sigma}{r}.$$

A liquid whose largest cavity has radius $r$ therefore breaks at about that pressure, and read the other way round, a measured breaking pressure is a measurement of the largest cavity the sample held.

A bubble a micrometre across — the size of the gas nuclei that ordinary tap water is full of — lets go at 0.29 megapascals of tension, which is why water that has been standing open cavitates at barely below zero. Briggs’ −27.7 megapascals reads as a cavity 5.25 nanometres in radius. The quartz inclusions’ −140 reads as one of 1.04. And the critical cavity of the theory of a perfectly pure liquid, computed below, is 0.98 nanometres — a sphere a few molecules across.

The connection this makes is to the strength of glass. Griffith showed in 1921 that a brittle solid breaks not at the stress its atomic bonds could stand but at the stress its worst microscopic crack concentrates, so that a freshly drawn glass fibre is enormously stronger than a pane of the same glass, and the strength of a sample is a statement about the size of its worst flaw. A liquid’s tensile strength is the same argument with a cavity in place of a crack. A spinning tube and a sealed tube measure the same thing a tensile test on glass measures: not the material, but the largest defect the particular piece happened to contain.

The number the pure liquid gives #

A liquid with no defect at all still breaks, because its own thermal motion makes cavities. Classical nucleation theory prices them. A critical cavity costs a free energy

$$W = \frac{16\pi\sigma^3}{3\,(p_v - p)^2},$$

and cavities appear at a rate per unit volume and time of $J_0\,e^{-W/kT}$, with the kinetic prefactor $J_0 = n_l\sqrt{2\sigma/\pi m}$ — about $4\times10^{40}$ per cubic metre per second for water. The liquid breaks, in a volume $V$ watched for a time $\tau$, when one such event is expected:

$$(p_v - p)^2 = \frac{16\pi\sigma^3}{3\,kT\,\ln(J_0 V \tau)}.$$

For a cubic millimetre watched for a second at twenty degrees, $\ln(J_0V\tau) = 72.8$ and the pressure comes out at −148 megapascals.

The volume and the time sit inside a logarithm, and that is the property that makes the theory’s number worth quoting. Across twenty-one decades of volume times time — a femtolitre watched for a millisecond at one end, a cubic metre watched for a quarter of an hour at the other — the breaking pressure moves only from −175 to −126 megapascals. No apparatus that could be built changes the theory’s answer by more than a third. Temperature moves it more: −162 megapascals at freezing, −148 at twenty degrees, −121 at sixty.

A gap nobody has closed #

The measurements fall into two groups and the gap between them is the subject’s open problem.

Water sealed as microscopic droplets inside quartz crystals — inclusions a few tens of micrometres across, formed and then stretched by cooling in exactly Berthelot’s manner — has been reported holding around −140 megapascals, which is where the theory puts a pure liquid. Nearly every other careful method — spinning tubes, focused acoustic pulses, sealed glass, flows through constrictions — breaks water in the region of −25 to −30 megapascals, with a temperature dependence of roughly the same shape across methods.

Two readings of that gap are argued, and neither is settled. One is that every method but the inclusions is heterogeneous: that some impurity, dissolved species or surface is present in all of them at the 5-nanometre scale the Laplace relation reads −27 megapascals as, and that a quartz inclusion is simply the cleanest sample anybody has made. The other is that the lower figure is a property of water itself, connected to the same anomalies that give it a density maximum, and that the inclusions are the exception rather than the rule. The arithmetic here computes both numbers and the size of the flaw each implies. It cannot say which of them is the liquid’s.

What it can say is that the gap is not about the sample’s size or the length of the experiment, because the logarithm has already shown that twenty-one decades of both move the theory by a third. The five-fold disagreement is about what is in the water, or about whether the theory’s picture of a cavity is right at a nanometre — and that is a much more interesting place for a discrepancy to be.

What these calculations do not settle #

The cavity holds only vapour. A real nucleus contains dissolved gas as well, which pushes outward and lowers the tension it can survive. Blake’s treatment includes it and shifts every threshold in the nucleus figure towards zero; it does not change the scaling or the reading of a measured strength as a size.

Surface tension is the flat-surface value. At a nanometre a surface’s tension is itself in doubt, since the interface is only a few molecules thick and its curvature is comparable with its thickness. The correction usually argued for lowers the tension of a very small cavity, and it makes the theory’s number less negative — which narrows the gap, and does not close it.

The prefactor is an estimate. The kinetic prefactor $J_0$ is uncertain by orders of magnitude. It sits inside the same logarithm as the volume and the time, so an error of a thousand in it moves the breaking pressure by a few megapascals, which is why the estimate survives its own crudeness.

The spinning tube is rigid and isothermal, the sealed tube’s glass is rigid and its compressibility is first order, and neither calculation contains the wall at which a real Berthelot tube tends to let go. The measured values drawn are borrowed and are drawn as borrowed, and Briggs’ was taken at ten degrees rather than the twenty the computed values use.

Each result is computed a second way that could disagree with the first. The spinning tube’s axis pressure is also found by integrating $\rho\omega^2r$ inward from the meniscus, and the two agree to a part in a million with the lowest point of the tube on the axis. The sealed tube’s curve, rerun with the compressibility frozen, puts its deepest point on the density maximum to within a hundredth of a degree — which is how the shift above is known to come from κ and from nothing else — and the shift grows as the filling warms. And the nucleation threshold, recomputed at the two ends of eighteen decades of volume and time, moves by under forty per cent, which is the logarithm’s claim measured rather than quoted.

Pressing the flaws out #

If a breaking strength is the size of the worst cavity, then removing the cavities should raise it — and that prediction has been tested, in a form that owes nothing to a spinning tube.

Ordinary water is full of gas nuclei: bubbles too small to rise, kept from dissolving by sitting in crevices on dust particles and container walls, where the gas–water surface can curve the way that holds the gas in rather than squeezing it out. The crevice account of cavitation nuclei dates from the 1940s, from work on why animals form bubbles when the pressure on them falls, and it makes a specific prediction. Squeeze the water hard enough for long enough and the gas in every crevice is forced into solution, its meniscus retreating into the crack until there is nothing left to grow.

The prediction holds. Water held at hundreds of atmospheres before a test and then released breaks at a far larger tension than the same water taken straight from the tap — a treatment that changes nothing about the liquid and a great deal about its flaws. That is the Laplace reading of a breaking strength made into an experiment: the tension the sample survives rose because its largest cavity was made smaller, and it rose in the direction $2\sigma/r$ says a smaller cavity is worth.

It also explains why the strength of water depends so heavily on its history. Water boiled and cooled, water filtered through a membrane, water left to stand while its larger bubbles rise out, and water drawn from a pipe a moment ago are chemically one liquid and mechanically four, because the population of cavities in them differs and the population is what a tensile test reads. The crown of a siphon holds its column on the same account, and a siphon that its heights say cannot break is at the mercy of the same population: the tension it can bear belongs to the liquid in that tube, not to the liquid in a handbook.

What squeezing cannot remove is the thermal cavity. The theory’s critical cavity, 0.98 nanometres in radius, is not a flaw waiting in the water to be pressed out. It is made and unmade continually by the liquid’s own thermal motion, and no treatment reaches it. So there is a floor under the flaw reading, and the floor is the theory’s number: a sample can be made cleaner until its strength approaches −148 megapascals and no further. The quartz inclusions sit near that floor. Everything else sits well above it, which is the gap again, restated as a question about how clean a sample can be made.

And the reading has a limit of its own at the bottom. A surface tension is a statement about a surface many molecules thick, and the continuum picture of a fluid is precisely the approximation that stops being safe at a nanometre. A cavity five nanometres in radius is some thirty molecular diameters across, and the Laplace relation is a fair description of it. One of a nanometre is about six, and the relation that reads the inclusions’ −140 megapascals as a cavity of that size is being used at the edge of where a cavity with a sharp surface means anything. The balance that fixes the largest raindrop at millimetres is being asked about a sphere of a few hundred molecules — and that a relation built for a surface nothing sucks on still gives the right order there is itself one of the stranger facts in the subject.

A century and a half of stretched water #

Berthelot’s sealed tube is from 1850, and he reported tensions of about fifty atmospheres, which is five megapascals — larger than any tension then imaginable and small against what the theory would allow. Osborne Reynolds whirled a tube in the 1880s and reported a few atmospheres. Briggs’ spinning Z-tube is of 1950 and gave the temperature curve with its maximum near ten degrees. The quartz inclusions came in 1991, and the review literature since has been largely about the disagreement between them and everything else.

Dixon and Joly needed tensions of tens of atmospheres in 1894 to explain how sap reaches the top of a tree, at a time when the measurements available were Berthelot’s. They were accused of asking more of water than water could give. The tree turned out to need two megapascals; the laboratories found twenty-seven; the inclusions found a hundred and forty; and the question of which number is water’s own is older than the airplane and still open.

Still open: whether water’s own number is the lower one #

The measurement that would settle the gap is one that stretches water to −140 megapascals by a method that is not an inclusion — or one that finds, in an inclusion, the flaw the other methods share. Neither has been made convincingly.

The nearest open question brings time into it. A breaking pressure is a threshold, and a threshold that is also a duration finds that a cavitation pressure quoted as a single number is the long-hold limit of a curve, with a short pulse needing forty times as much. The acoustic methods that give the −26 megapascal group apply their tension for a fraction of a microsecond, and how that interacts with a nucleation rate is the next calculation.

Beside it is the case where the flaw is not an accident but a supply. Water that has stood open to the air is saturated with dissolved gas, and wherever its pressure falls — a propeller’s suction side, a siphon’s crown, the collapse of a bubble that hammers — that gas comes out and makes its own cavities, at a pressure barely below the ambient rather than megapascals below zero.