The drop that takes a decade

Momentum going sideways defined viscosity as the rate at which momentum diffuses across a flowing fluid, and every essay on still and slowly moving liquids since has used the definition. The liquid that remembers then took a material that is a solid or a liquid according to how long it is watched — pitch, which shatters under a hammer and flows through a funnel — and made the general argument that the classification belongs to the observation, not the substance. It cited the funnel without opening it.

The funnel is real, it is in a glass case in a corridor of the University of Queensland, and it is the longest-running laboratory experiment in the world. Thomas Parnell, the university’s first professor of physics, heated a sample of pitch in 1927, poured it into a glass funnel with its stem sealed, and let it settle for three years. In 1930 he cut the stem, and the pitch began to flow. It has produced nine drops since. Each one is a measurement, and read together they say more than that pitch is viscous: they say how viscous, how strongly that depends on temperature, and what happened to the building.

A viscometer read once a decade #

The drops fell in December 1938, February 1947, April 1954, May 1962, August 1970, April 1979, July 1988, November 2000 and April 2014. No one has ever seen one fall in Brisbane. The drop forms over years, hangs from the stem by a neck that thickens and then thins, and finally separates quickly; the drop of 1988 was missed by a few minutes, and the camera installed to catch the drop of 2000 failed on the day it fell.

Each interval is a measurement of the pitch’s viscosity, for a reason that does not depend on the funnel’s exact shape. In a flow this slow, inertia plays no part at all: the pitch’s weight is balanced at every instant by its viscous resistance, so the flow is what is called Stokes flow, and in Stokes flow every speed in the problem is proportional to $\rho g L^2/\eta$, the density times gravity times the square of a length, divided by the viscosity. Double the viscosity and every speed halves, every time doubles, and the shape of the flow is unchanged. The interval between drops is therefore proportional to the viscosity, with a constant set once and for all by the funnel’s geometry and the drop’s size.

R. Edgeworth and B. J. Dalton, colleagues of John Mainstone, who looked after the experiment for over fifty years, worked the constant out in 1984, treating the flow through the funnel’s stem as the flow through a pipe that the fourth power in a pipe derived, and put the pitch’s viscosity at about 2.3 × 10⁸ pascal-seconds. That is two hundred and thirty billion times the viscosity of water. A second, unrelated pitch-drop experiment, set up at Trinity College Dublin in 1944, had a drop filmed falling in 2013, the first ever caught on camera.

The drops are also unlike the drops from a tap in what sets their size. A drop of water hangs from a tap until its weight overcomes the surface tension holding it, which is why water drops are all about the same few millimetres across. Surface tension in pitch is of the same order as in other organic liquids, but a drop of pitch is centimetres across before it falls, and at that size its weight exceeds what surface tension could hold by a factor of a hundred or more. What holds the pitch drop up for years is not its surface but its viscosity: it hangs from a neck that is flowing, and it falls when the neck has thinned enough. The water drop is a balance of forces; the pitch drop is a race between two slow flows.

Where pitch sits among things that flow #

Viscosity spans a wider range than almost any other property of matter, and pitch sits near the middle of it. The second figure puts it in company. Water and olive oil pour; honey and peanut butter ooze; molten glass at the temperature a glassblower works it is a few times stiffer than peanut butter. Pitch is seven decades beyond honey. Glacier ice, which flows down valleys at tens of metres a year, is another five decades beyond pitch, and the solid rock of the Earth’s mantle, which flows in the slow convection that drives the plates, another eight beyond that.

The same scaling that makes the drop a viscometer turns the scale into times. A layer a centimetre thick sags by its own thickness under its own weight in about $\eta/\rho g L$: a tenth of a second for honey, three weeks for pitch, four thousand years for ice. At each step along the axis, a material is a liquid to anyone who waits that long and a solid to anyone who does not. Nothing in the scale marks where solids begin.

A degree a year #

The seven intervals before 1988 scatter between 7.2 and 9.3 years, and the two after it took 12.4 and 13.4. The cause is not in the pitch. In the late 1980s the building was air-conditioned, and pitch’s viscosity is extremely sensitive to temperature.

The sensitivity has the form the exponential that decides everything found in every thermally activated rate. For the molecules of a bitumen to slide past one another they must find, among their tangled neighbours, a configuration that lets them move, and the chance of finding one at any moment falls as $e^{-E/RT}$, with an apparent activation energy E that for bitumens near room temperature is around two hundred kilojoules per mole. At room temperature that is a steep dependence. Each degree cooler multiplies the viscosity, and so the interval between drops, by 1.32. The third figure draws the consequence. A room at 22 °C gives the early intervals; the lengthening to almost thirteen years needs the average temperature to have fallen by only one and a half degrees, well within what a building’s air-conditioning does to a corridor in subtropical Brisbane.

The same steepness accounts for the scatter before 1988. Brisbane’s summers are hot and its winters mild, and a drop hanging by a neck that the flow is slowly stretching does not fall at a uniformly chosen moment: it falls when the pitch in the neck is softest, so a warm spell can bring a drop that has hung for years. The scatter of a year either way corresponds to differences of under a degree in the average temperature from one interval to the next. The experiment is a viscometer and, through its viscosity, a thermometer that integrates over a decade.

What pitch is, and why roads soften #

Pitch is not a single substance. Parnell’s sample was a coal-tar pitch, the residue left when coal tar is distilled, and the bitumen used on roads is its petroleum cousin, the residue of crude oil. Both are mixtures of many thousands of large molecules, most of them built from flat rings of carbon stacked and tangled together, with no regular arrangement to crystallise into. They never freeze. As they cool they simply grow stiffer, continuously, through the whole range from a pourable liquid at a couple of hundred degrees to a brittle glass well below zero, and room temperature happens to fall in the stretch where the change is steepest.

That steepness is a nuisance on roads, and it is measured there in a way that is a crude cousin of the funnel. The penetration test, used to grade road bitumens for a century, presses a standard needle into a sample at 25 °C for five seconds and records how far it goes, in tenths of a millimetre; a softer bitumen, with a lower viscosity, lets the needle in further. Engineers choose a harder grade for hot climates and a softer one for cold, because the same thirty-odd per cent per degree that lengthened the Queensland intervals makes a road surface flow under heavy lorries on a hot afternoon. Ruts form in summer and cracks in winter, from the same temperature dependence read at its two ends.

Measuring without waiting #

Nobody who needs a bitumen’s viscosity waits eight years for it. Rheometers measure at higher temperatures, where the material flows in minutes, or apply small oscillating loads at many frequencies and read the response, and the results at different temperatures are slid along the time axis onto one another, because a change of temperature acts on a material like this mainly as a change of clock. The same exponential that let the Queensland funnel slow down from eight years to thirteen does the sliding.

Run the funnel’s clock the other way and the numbers become vivid, if the single activation energy is trusted far beyond the range it was taken from. At 30 °C the funnel would drip about once a year. At 40 °C, a hot Brisbane summer’s day in an unshaded building, it would drip every few weeks. At 60 °C, the temperature a dark road surface reaches in strong sun, every few hours. Parnell’s experiment is a demonstration of slowness only because it is kept at room temperature, and its slowness at room temperature is an accident of where bitumen’s softening happens to lie.

Solid for a hammer, liquid for a decade #

The liquid that remembers argued that whether a material behaves as a solid or a liquid depends on the ratio of its relaxation time to the time it is watched for, the Deborah number. The drop record turns that argument into numbers for pitch. Its glassy stiffness — the modulus it shows to a sudden load, before any flow — is about a gigapascal, like a hard plastic, and the time it takes to relax a stress by flowing is the viscosity divided by that modulus: a quarter of a second.

The fourth figure follows how far the pitch gives under a load held for a given time. For anything faster than a quarter of a second it gives a fixed, small amount, springing back when released, and a hammer blow lasting a millisecond meets a material two hundred times stiffer than its flow allows: it shatters, like glass. For anything slower the elastic give is swamped by flow. Held for an hour the pitch has flowed sixteen thousand times further than it sprang, and over the eight years a drop takes, the elastic part of the motion is one part in a billion. The funnel experiment and the hammer are measuring the same material on two sides of a time it carries in itself.

The window that did not flow #

Pitch is often introduced with a story about glass: that old church windows are thicker at the bottom because glass is a very slow liquid and has flowed downwards over the centuries. Momentum going sideways dismissed it with a viscosity for glass at room temperature. That number is an extrapolation from temperatures hundreds of degrees higher, and estimates of it differ by tens of decades depending on the law used to extrapolate, which leaves the story’s defenders a foothold. The same scaling that reads the pitch’s viscosity from its drops removes the foothold without extrapolating at all.

Ask instead what viscosity a pane would need for the story to be true. A pane three millimetres thick and half a metre tall, standing under its own weight for eight hundred years, would have to have sagged enough to thicken its bottom edge by half a millimetre — a sixth of its thickness, enough to see. The weight per unit area at the bottom of the pane, the time and the strain together fix the viscosity: no more than about 10^15.3 pascal-seconds. The last figure draws soda-lime glass’s viscosity through the three temperatures at which glass-makers measure it, and that value is reached at about 490 °C. Below the strain point, near 516 °C, glass is so stiff that the stresses in it can no longer relax in hours, and every law fitted above it predicts viscosities that grow without limit as the temperature falls further. The window would have had to stand at the temperature of an annealing oven for the whole of its life.

The panes that are thicker at one edge were made that way. Medieval window glass was blown and spun into discs or cylinders that were never of uniform thickness, and glaziers commonly set the thicker edge at the bottom, where it sat more securely in the lead. Panes fitted thick edge up exist too, and they have not flowed upwards.

How far the scaling reaches #

The arguments rest on two assumptions about the pitch and one about the glass. The pitch’s flow is treated as Newtonian — stress proportional to rate — which bitumens are at these very slow rates but not at fast ones, where they shear-thin. Its viscosity is taken to follow a single activation energy over the few degrees that matter; over wider ranges bitumens are better described by laws whose effective activation energy changes with temperature, and the value assumed here, two hundred kilojoules per mole, is typical rather than measured on this sample. The building’s actual temperature record was not kept with the precision the argument would need, so the fall of one and a half degrees is what the change requires, not a measurement of it. And the glass calculation uses a fitted law for a typical modern soda-lime composition. Medieval glass was richer in potash and lime and of different viscosity, but the conclusion needs the viscosity at room temperature to exceed 10^15 pascal-seconds by any margin at all, and no composition of glass has been proposed for which it does not.

What the curves cannot show #

The curves show averages. They cannot show the drop itself — the slow growth of a bulb of pitch on the stem’s tip, the thinning of its neck as the bulb’s weight stretches it, and the last instant in which the neck, by then a thread, lets go. That final stage is where the drop’s timing is decided, and it depends on the flow in a thinning thread, where the stress grows as the thread narrows and the pitch’s shear-thinning and elasticity matter more than in the steady flow above. They cannot show the funnel’s geometry either, which the calculation needs and which no one can measure without disturbing the experiment. The published figure for the viscosity assumed a funnel stem and a drop of particular sizes, and a different, equally plausible assumption would shift the answer by a factor of order one. What the record gives without any geometry at all is the ratio of one interval to another, and that is what the temperature argument uses.

Still open: how the thread lets go #

The tenth drop is expected in the 2020s or 2030s. The question the experiment poses now is not about pitch’s viscosity, which is well enough known, but about its last few seconds: how a viscous, slightly elastic thread under its own weight thins to nothing, whether it breaks by a viscous thinning that accelerates in a predictable way or by some faster process once the thread is thin enough for its elasticity or surface tension to matter, and how long the final separation takes. The Dublin footage of 2013 shows the final separation happening quickly after years of slow growth. A model that predicts the date of the next Queensland drop from the shape of the hanging bulb, rather than from the average of the previous intervals, has not been made.

The habit worth carrying away is that a long enough record turns a curiosity into an instrument. The interval between drops from a funnel of pitch is proportional to its viscosity — 8.3 years apart before 1988, a viscosity of 2.3 × 10⁸ Pa·s — and with an activation energy of 200 kJ/mol each degree cooler lengthens it by 32 per cent, so the slowdown to 12.9 years after air-conditioning needs only 1.6 °C; the same scaling says a window pane could have thickened visibly in 800 years only if its viscosity were below about 10^15.3 Pa·s, which window glass reaches at about 490 °C. The pitch has been a liquid for nearly a century, and the window has never been one.