The tension that vanishes where the phases meet

The skin that is not a skin established what surface tension is: not a stretched membrane on top of the liquid but an energy per unit area, the extra cost of the molecules at the surface, which have fewer neighbours to attract them than the molecules inside. That account makes the tension sound like a property of the liquid. It is better described as a property of a boundary — between a liquid and its own vapour, or between two liquids that do not mix — and a boundary needs two different things to lie between.

Heat a liquid in a sealed container, so that it stays in equilibrium with its vapour, and the two become steadily less different. The liquid expands and thins; the vapour, compressed by the rising pressure, grows denser. At one temperature and pressure, the critical point, they become the same, and the point at which the two become one followed what the difference in density does as that point is approached. This essay follows the boundary. A surface tension is the cost of the interface between two phases, so when the phases merge the tension must go to zero. How it gets there is a story about what a surface actually is, and its central number turns out to be shared by every critical point of its kind.

Water’s tension, from the tap to the critical point #

Over most of its range water’s surface tension falls almost linearly with temperature, by about a sixth of a millinewton per metre for every degree. At room temperature that is part of why hot water cleans better than cold, and why a warmed spot on a film of water pulls the liquid away from it. It is also why water climbs less high in a warm capillary than a cold one: the rise is proportional to the tension. Extrapolate the straight line and it would cross zero somewhere near 400 °C. The real curve bends down before that and reaches zero exactly at the critical temperature, 373.946 °C, at a pressure of 220 atmospheres.

The curve in the figure is the correlation adopted by the International Association for the Properties of Water and Steam, fitted to the measurements and written in a form that encodes how the tension must end:

$$\gamma = 235.8\ \text{mN/m}\ \times\ \tau^{1.256}\,(1 - 0.625\,\tau), \qquad \tau = 1 - \frac{T}{T_c}.$$

The second factor is a small correction. The first is a power of the distance from the critical temperature, and the power, 1.256, was not chosen to fit water. It was taken from the theory of critical points, which predicts it for every fluid, and then checked against water’s data, which it fits.

The rule that held for a century #

Loránd Eötvös noticed in 1886 that the surface tensions of many liquids, multiplied by their molar volume to the two-thirds power, fall on nearly the same straight line when plotted against temperature, reaching zero near the critical temperature. William Ramsay and John Shields refined it in 1893 by moving the zero six degrees below the critical point. The combination $\gamma V_m^{2/3}$ is the tension’s energy per molecule-sized patch of surface multiplied by Avogadro’s number to the two-thirds, and the rule’s slope, about $2.1 \times 10^{-7}$ joules per kelvin per mole to the two-thirds, works out to about two Boltzmann constants per surface molecule. In other words a molecule moved to the surface of a liquid loses a fixed amount of entropy, about $2k_B$, nearly whatever the liquid is.

The rule is good to a few per cent for simple liquids far from the critical point, and it fails near it, as the figure for water shows: the straight line through the low-temperature data misses zero by twenty-odd degrees. Edward Guggenheim noticed in 1945 that the tension near the critical point of simple fluids fitted a power of $\tau$ better than a straight line, and proposed $11/9$, about 1.22. The modern value is a little larger, and the reason it is not one, nor anything simple, is the subject of the rest of this essay.

Where the two phases come from #

The simplest theory that has a critical point at all is van der Waals’s, the first correction to the gas law that lets molecules occupy room and attract one another, and it is worth seeing its two phases emerge before asking about the boundary between them.

Below the critical temperature a van der Waals isotherm is not monotonic: between the dense liquid and the thin vapour there is a stretch where the pressure would fall as the density rises, which no stable fluid can do. The fluid splits instead into a liquid and a vapour whose pressures are equal and whose chemical potentials are equal — the condition James Clerk Maxwell expressed as equal areas above and below the tie line. At 0.85 of the critical temperature the coexisting densities are 1.81 and 0.32 critical densities; at 0.95 they are 1.46 and 0.58. As the temperature rises the hump and dip flatten and the two densities close in, until at the critical temperature the isotherm has a single flat inflection and the two are one. The part of the curve no fluid follows is the stretch between them, and the interface is how a real fluid crosses it in space rather than in time.

The surface is a profile #

Van der Waals also gave, in 1893, a theory of what lies between the coexisting phases, which John Cahn and John Hilliard rediscovered in 1958 and which bears their names as often as his. A boundary between liquid and vapour is not a sharp step. The density changes smoothly from one value to the other over a short distance, and it costs free energy for two reasons: the fluid in the middle is at densities the bulk would never choose, on the forbidden stretch of the isotherm, and the density changes in space, which costs energy in proportion to the square of its gradient because a molecule whose neighbours are at different densities on its two sides is not at its best. A sharp step minimises the first cost and maximises the second; a gentle slope does the reverse. The real profile balances them.

The computed profiles have the form of a smoothed step, and they change shape as the critical point approaches in two ways at once. The height of the step — the difference between liquid and vapour densities — shrinks, and the width over which it happens grows. At 0.70 of the critical temperature the step is twice the critical density high and spread over two molecular lengths, as sharp as a surface can be. At 0.99 it is less than half the critical density high and spread over more than twelve.

The tension is the integral of the excess free energy through the profile, and both changes reduce it: a lower step costs less at each point, and a wider step, though it spreads the cost further, lowers its density more. At the critical point the step has no height and infinite width — which is to say there is no surface at all.

How fast it dies #

On logarithmic axes the approach to zero is a straight line whose slope is the exponent. The computed van der Waals tension falls with slope 1.50: three-halves, the value any theory gives in which each molecule responds to the average of its neighbours rather than to their actual, fluctuating arrangement. Water falls with slope 1.256. The difference does not sound large, but it compounds: a thousandth of the critical temperature away from the critical point, the van der Waals tension has fallen to a thousandth of its value at a tenth away, and water’s only to three thousandths — three times larger than the theory allows.

The mean-field exponent is wrong for the same reason that van der Waals’s order-parameter exponent is wrong. Near a critical point the fluid is not uniform on any scale. The density fluctuates in patches of every size up to a correlation length that grows without limit, and a theory that replaces those patches by their average misses how they soften the free energy. The exponents that survive the fluctuations were found between the 1960s and 1970s, by Benjamin Widom, Leo Kadanoff and Kenneth Wilson, and they are not simple fractions: the correlation length grows as $\tau^{-\nu}$ with $\nu \approx 0.630$.

Widom found in 1965 how the tension’s exponent follows from the correlation length’s. The free energy that holds the fluid together, near the critical point, comes in units of about $k_B T$ per correlated patch. An interface is a layer one correlation length thick, so its free energy per unit area is about $k_B T$ per area of one patch, $\xi^2$:

$$\gamma \sim \frac{k_B T_c}{\xi^2} \propto \tau^{2\nu} = \tau^{1.26}.$$

That is the exponent the water correlation uses. For water a thousandth of the critical temperature from it, where the correlation length is about ten nanometres, $k_B T_c/\xi^2$ is about 0.09 millinewtons per metre and the measured tension about 0.04 — the same size, which is what the argument claims and all it claims.

A surface as thick as the correlation length #

The width of the interface tracks the correlation length, and so it too has an exponent: −0.50 in the van der Waals theory, −0.63 in real fluids. Every tenfold step closer to the critical temperature makes the surface three to four times thicker. A thousandth of the critical temperature away, a real fluid’s interface is tens of molecules thick; a millionth away, thousands — thick enough that it scatters visible light, and a sealed tube of fluid taken through its critical point glows milky white in a band around where the meniscus was. Marian Smoluchowski in 1908 and Albert Einstein in 1910 explained the glow as light scattered by density fluctuations, the same fluctuations that jostle a pollen grain, grown large enough to scatter visible light as strongly as the much smaller fluctuations of the air scatter blue. That is critical opalescence seen from the side of the interface: the same density fluctuations, at the same scale, that the point at which the two become one found filling the bulk.

Charles Cagniard de la Tour sealed liquids into gun barrels and then glass tubes in 1822, heated them, and watched the meniscus fade and vanish. Thomas Andrews mapped the same disappearance for carbon dioxide in 1869 and named the critical point. What both saw going was a surface losing its tension: the meniscus flattens first, because a surface with less tension supports less curvature, and then blurs, because a surface near the critical point is wide.

What a vanishing tension does to everything else #

Almost every process that involves a liquid surface runs through its tension, so its vanishing reaches well beyond the meniscus. A new phase has to climb a barrier to start — a droplet forming in a vapour pays for its surface before its volume pays it back — and the barrier’s height goes as the cube of the tension. Near the critical point that barrier collapses, so a fluid cooled through its coexistence curve close to the critical point separates almost at once, without the long metastable delays that let water vapour supersaturate in a clean cloud chamber or liquid water superheat in a smooth glass. The droplets that form are not sharp-edged drops but diffuse regions, as wide as the interface.

The capillary length — the size below which surface tension beats gravity, a few millimetres for water at room temperature — shrinks as well, because the tension falls faster than the density difference that gravity acts on. A meniscus near the critical point is flat on every scale anyone can see. Bubbles and drops, whose roundness is a contest between tension and gravity, flatten into layers. And the coexistence pressure, which fixes where the two phases meet at all, ends at the same point the tension does, because both are properties of the two-phase region and the region ends there.

One number for many fluids #

The exponent 1.26 belongs to water only because water belongs to a class. Argon, carbon dioxide, xenon and sulphur hexafluoride approach their critical points with the same exponent, within the error of the measurements; so does the interfacial tension between two liquids that stop being immiscible at a critical mixing temperature, such as methanol and cyclohexane near 45 °C; and so does the free energy of the wall between up and down domains of a magnet in which the spins can point only along one axis. All of them are described, near their critical points, by the same scaling, because what matters there is only the dimension of space and the number of components of the quantity that changes — one, the density, the concentration or the magnetisation.

That is universality, and the surface tension is one of its cleanest demonstrations because it is measured directly: the height a liquid rises in a capillary, or the shape of a hanging drop, gives $\gamma$ without any model. Eötvös’s rule, which was a regularity among liquids far from their critical points, and Widom’s relation, which is a law for all of them near it, describe the same quantity in two regimes, and the crossover between them is visible in the first figure as the bend.

Where the model stops #

The van der Waals curves in this essay are computed exactly from a theory that is known to be wrong near the critical point. It is drawn because it is the simplest model with two phases, a boundary and a critical point, and because the size of its error is informative: its exponents are the mean-field values, and comparing them with water’s shows how much the fluctuations matter. Far from the critical point it gives tensions and widths of the right size and shape; near it, it gives the wrong powers.

The water curve is a correlation, not a theory. It is fitted to measurements that stop a little short of the critical point, where gravity itself interferes: near the critical point the fluid is so compressible that its own weight stratifies it, the density varying from the top to the bottom of a sealed cell by more than the difference between the phases. The closest approaches to critical points have been made in orbit, in experiments on the space shuttle and the International Space Station, to remove that effect.

The domain of the argument is a pure fluid in equilibrium with its vapour, or two liquids near their critical mixing point. Inside it the tension vanishes at the critical point with a universal exponent. Outside it — a liquid against a solid, or against a gas that is not its own vapour, such as water against air at high temperature — the tension does not vanish, because the two sides never become the same.

What the pictures cannot show #

The profile figure draws smooth curves. A real interface near the critical point is not a smooth profile but a rough, fluctuating surface: thermal ripples, called capillary waves, displace it up and down by amounts that grow as the tension falls, and an instantaneous snapshot would show a ragged boundary whose average is the smooth curve drawn. The width of a real interface includes those ripples, and separating the intrinsic width from the capillary-wave broadening is a problem the square-gradient theory does not pose.

And none of the figures can show what the vanishing of a tension looks like: a meniscus flattening, a drop losing its roundness, two phases in a tube losing the line between them until the tube is uniformly glowing. The numbers are the content; the disappearance is the experience.

Still open: how a surface ends #

The exponent of the tension is settled, and so is its relation to the correlation length. What is not settled in full is the structure of the interface itself near the critical point — how the intrinsic density profile and the capillary-wave roughness combine, whether the width measured by X-ray reflectivity and by ellipsometry is the same quantity, and how the interface of a real fluid crosses over from the molecular sharpness of the low-temperature surface to the broad, fluctuating zone near the critical point. Simulations of large systems and measurements at synchrotrons have narrowed the questions; the answer depends on which width is meant, which is itself a sign that “the surface” stops being one thing.

A surface tension is the price of a boundary, and a boundary needs two sides. Water’s surface tension falls from 72.7 mN/m at room temperature to exactly zero at 373.946 °C because the liquid and vapour it separates become one fluid there; the interface spreads from a molecule’s width to the correlation length as it goes, and the tension dies as kT per correlated patch, τ^2ν = τ^1.26 — a number every fluid shares and that the simplest theory, at 1.50, misses.