The surface that pulls toward the stronger side

Pour a glass of wine, swirl it, and a film climbs the glass above the surface, gathers into a ring, and falls back as a row of drops — the same stress that leaves a ring behind when a drop dries, running the other way. The drops are called tears and the effect has been described for two centuries. Nothing about it is capillary rise: the film goes well above where any meniscus stands, it flows continuously rather than sitting still, and the same glass filled with water does nothing at all.

What a gradient in tension actually is #

Surface tension is an energy per unit area, and equivalently a force per unit length along any line drawn in the surface.

Now suppose the tension is not uniform. Take a small square patch of surface. The tension pulls outward on all four edges, and if the pull on the right edge exceeds the pull on the left, the patch feels a net force to the right. That net force per unit area is

$$\tau = \frac{\mathrm{d}\gamma}{\mathrm{d}x},$$

a stress along the surface, in the direction of increasing tension. The surface is pulled toward wherever the tension is strongest. Nothing about this is a pressure: the Laplace pressure jump across a curved interface is a separate effect, perpendicular to the surface, and a flat surface with a gradient in tension has none of it and all of this.

The perpendicular effect is worth setting beside it, because the two are constantly confused. A curved surface has a pressure difference across it of twice the tension over the radius: that acts across the interface and requires curvature. A Marangoni stress acts along the interface and requires a gradient. An ordinary viscous stress acts along it too, and is the thing this one has to beat. A surface can have either, both or neither — a flat surface with a composition gradient has the second and not the first, and a uniform bubble has the first and not the second.

The stress has to go somewhere #

A surface has no mass to speak of, so a stress applied to it cannot accelerate it; it must be transmitted to the liquid underneath, which resists viscously.

Balancing the applied stress against the viscous one in a film of thickness $\delta$ gives $\mathrm{d}\gamma/\mathrm{d}x = \mu U/\delta$, so the surface moves at

$$U = \frac{\delta}{\mu}\frac{\mathrm{d}\gamma}{\mathrm{d}x}.$$

That is why surfaces are so rarely still. Gravity needs a height difference to act; this needs only a difference in composition or temperature along a surface, which is almost impossible to avoid. A cup of coffee cooling by evaporation has a temperature difference across its surface of a fraction of a degree, and the resulting circulation is visible in the motion of the film on top.

The tears, assembled #

The wine glass now falls out in four steps, each of which is one of the pieces above.

Alcohol evaporates faster than water. A thin film clinging to the glass above the bulk therefore loses alcohol preferentially, so its surface tension rises — because ethanol is the component that lowers it. The gradient between the strong-tension film and the weak-tension bulk pulls liquid upward out of the bulk and into the film. The film thickens and climbs until it is heavy enough that gravity wins, gathers into a ring, and breaks into drops that run back down.

Each stage is checkable. The effect vanishes below about six per cent alcohol by volume, which is why beer produces no tears and fortified wine produces many; it vanishes if the glass is covered, which stops the evaporation; and it vanishes if the glass is dirty, because a film cannot climb a surface it does not wet.

The static effect it is most often mistaken for is capillary rise, and the differences are all diagnostic. Capillary rise reaches a fixed height and stops; its height depends on the tube radius; and nothing flows once equilibrium is reached. The tears climb continuously, reach a height that has nothing to do with any radius, and are a steady flow rather than a static balance. The two look similar in a photograph and have almost nothing in common.

Why a trace of something does so much #

The steepness of the tension curve at low concentration is worth explaining rather than merely plotting, because it is what makes the whole subject practical.

A molecule with a water-loving end and a water-hating end lowers its energy enormously by sitting at the surface with one end in each phase. So it accumulates there: the surface concentration exceeds the bulk concentration by a large factor, and Gibbs’s adsorption equation makes the relation exact —

$$\Gamma = -\frac{1}{RT}\frac{\mathrm{d}\gamma}{\mathrm{d}\ln c},$$

with $\Gamma$ the excess amount per unit area. Reading it backwards: a steep fall of tension with the logarithm of concentration is a large surface excess, and the two are the same measurement.

The numbers are severe. A monolayer of a typical surfactant is about $3\times10^{-6}$ moles per square metre. Spread over a square metre of water one millimetre deep, that is a bulk concentration of three millimoles per cubic metre — a few parts per million. A concentration far below anything that changes the bulk liquid measurably changes its surface completely, which is why washing-up liquid works at the dilution it does, why a trace of oil kills the ripples on a pond, and why surface-tension measurements are the most contamination-sensitive routine measurements in physical chemistry.

It is also why the effect is so hard to switch off. Removing the last part per million of surface-active contamination from a water surface takes deliberate work, and an ordinary water surface in an ordinary room is not clean in the sense this subject requires.

A liquid that pulls itself apart #

Because the stress drags liquid away from wherever the tension is lowest, a Marangoni flow tends to thin the region that started weakest — which makes the gradient steeper and the flow faster.

A step in velocity spreads into the fluid by viscous diffusion, and the depth reached grows as the square root of the time. That is what carries the surface’s motion downward into a film — so a thin film responds to a surface stress almost immediately and a deep pool hardly at all, which is why the effect belongs to films. Below about a millimetre the whole depth is entrained within a fraction of a second; above a centimetre the surface slides over fluid that never learns about it.

Drop a little detergent into a clean water surface dusted with pepper and the pepper is driven violently outward, because the detergent has lowered the tension where it landed and the surrounding stronger surface pulls the whole film away from it. The same mechanism drives a camphor boat and the paper boat with soap behind it: the vessel is not pushed by anything expanding, it is pulled forward by the intact surface ahead of it.

Whether a film survives that thinning is a competition. A soap film has a restoring mechanism the pure-water film lacks: stretching a region of soap film spreads its surfactant thinner, which raises the local tension and pulls liquid back in. That is the Gibbs–Marangoni elasticity, and it is the whole reason soap makes bubbles and water does not.

Left alone, a liquid column pulls itself apart: surface energy makes long-wavelength disturbances grow, and the thread breaks into drops. A surfactant does not remove that instability — nothing does — but it slows it, and the mechanism is the one this essay is about. Pinching a region locally concentrates the surfactant there and raises the tension where the thread is thinnest, which pulls liquid back into the neck. The instability is opposed by the very gradient it creates.

The ring a drying drop leaves, and the flow that fights it #

A drop of coffee drying on a table leaves a dark ring rather than a uniform stain, and the explanation involves two competing surface flows.

The first is not Marangoni at all. The drop’s edge is pinned by roughness, so it cannot retreat; evaporation is fastest at the edge, because the vapour there can escape into a larger solid angle; and liquid must flow outward from the middle to replace what the edge loses. That outward flow carries every suspended particle to the rim, where it is deposited. The ring is a consequence of pinning and geometry, and it happens with no tension gradient anywhere.

The second is Marangoni, and it opposes the first. Evaporation cools the surface, most at the edge where it is fastest, and the cooler edge has the higher tension — so the surface is pulled outward at the top and returns along the bottom, a recirculation that carries particles back toward the centre. In water the effect is weak, because water’s surface is so easily contaminated that the tension gradient is smothered by whatever surfactant is present. In a pure organic solvent it is strong, and the deposit is uniform.

Adding a surfactant deliberately, or mixing two solvents of different volatility, changes which flow wins, and that is how the ring is defeated in printing and in the deposition of thin films from solution. The physics is a competition between a geometric flow and a thermal one across a surface a hundred micrometres wide, and the visible outcome is whether a printed line has ragged edges.

What destroys a gradient #

The flow exists only while the gradient does, and two processes work to remove it.

What destroys a gradient is diffusion, and its rate sets how long any of this lasts. A composition gradient along a surface is erased at a rate set by the diffusivity over the square of the distance, so a millimetre-scale gradient in a small molecule survives seconds and a centimetre-scale one survives minutes. Thermal gradients go faster still, since heat diffuses about a hundred times more quickly than dissolved matter in water — which is why thermal Marangoni flows need a maintained heat source and compositional ones do not.

So a Marangoni flow is a steady state rather than a transient only when something maintains the gradient — evaporation in the wine glass, heating from below in a pan, a continuously fed surfactant in an industrial process. The dimensionless number comparing the driving to the smoothing is the Marangoni number, and it plays the part the Rayleigh number plays for buoyant convection: below a threshold nothing happens, and above it a pattern appears.

That pattern is worth naming because it was mistaken for something else for fifty years. Bénard’s 1900 experiment on a thin layer of spermaceti heated from below produced the famous hexagonal cells, and they were taken as the type case of buoyancy-driven convection. They are not: in a layer that thin, with a free upper surface, the driving is the temperature dependence of surface tension, and Pearson showed in 1958 that surface tension alone accounts for them. The buoyant version exists and needs a thicker layer.

The sign, and the forty parts per million that reverse it #

The most consequential instance of all this is in a welding pool, and it turns on the sign of one derivative.

An arc heats the centre of a molten pool most, so the centre is hottest. For a clean metal, surface tension falls with temperature, so the surface is weakest at the centre and strongest at the rim — and the surface is pulled outward. The flow carries hot metal outward, spreading the pool wide and shallow.

Add a few dozen parts per million of sulphur or oxygen and the sign of $\mathrm{d}\gamma/\mathrm{d}T$ reverses. These are surface-active in molten steel, they desorb as the temperature rises, and above a threshold the tension increases with temperature. Now the surface is strongest at the hot centre, the flow runs inward, and it carries hot metal down into the middle of the pool. The weld becomes narrow and deep, and its penetration can double.

This was for years an unexplained variability: two casts of nominally identical steel, welded with identical settings, giving welds of different depth. The cause is a trace element at a concentration nobody was specifying, acting on a derivative rather than on a value. Modern specifications for steels intended for automated welding state a sulphur range rather than a maximum, because too little is as bad as too much.

It is a good example of what makes the Marangoni effect awkward as engineering: the driving quantity is not a property of the material but of how a property varies, and a derivative is far more sensitive to trace composition than the property itself.

The teaspoon that measured a molecule #

The remark above that a trace of oil kills the ripples on a pond is the oldest observation in this subject and it turned into the first measurement of the size of a molecule.

Benjamin Franklin, in 1774, reported to the Royal Society that he had poured “not more than a tea spoonful” of oil onto the pond at Clapham Common and watched it spread and calm a surface of “perhaps half an acre”. He described the spreading carefully — it ran out as fast as he could pour, and stopped — and drew no quantitative conclusion.

The conclusion is one division. Half an acre is about two thousand square metres, and a teaspoon is five cubic centimetres. Five millionths of a cubic metre spread over two thousand square metres is a film about two and a half nanometres thick.

That number is a statement about matter. The oil stopped spreading at a definite area, which means it reached a definite thickness rather than thinning without limit, which means it is made of something with a size. Rayleigh did the measurement deliberately in 1890 with olive oil and obtained about 1.6 nanometres, and identified it as an upper bound on the length of a molecule — one of the first such numbers anybody had.

The person who turned that into an instrument was working at a kitchen sink. Agnes Pockels, with no laboratory and no formal training, built a trough out of a tin tray in which a barrier could sweep the surface, and measured the surface tension as a function of the area available to the film, using a small button on a thread as a balance. She had been doing it since 1882. In 1891 she wrote to Rayleigh describing the results; he had the letter translated and published it in Nature, with a note saying that her work anticipated much of his own.

What her curves show is a surface tension that stays at water’s value while the film is sparse, falls sharply as the barrier compresses the molecules into contact, and then resists further compression — a clear signature of a monolayer being squeezed. That trough, with a mechanical barrier and a tension measurement, is the direct ancestor of the Langmuir trough, and it is still how a monolayer’s area-per-molecule is measured.

Why a film of oil calms a wave #

The calming itself needs an explanation, because the obvious ones are wrong. The film is one molecule thick, so it cannot be damping anything by its own viscosity, and it is far too thin to weigh the water down or to shield it from the wind.

What it does is change a boundary condition, and the change is exactly the Marangoni effect.

A surface wave’s water moves in small orbits, and at the surface those orbits alternately stretch and compress the surface itself. A clean water surface does not care: it can be stretched and compressed freely, because the tension is the same however much area there is. A surface carrying a monolayer cares a great deal. Compressing it crowds the surfactant, which lowers the local tension, and the resulting gradient pulls back against the compression. The film resists being stretched and squeezed, which is precisely the Gibbs elasticity named earlier.

The consequence is that the surface becomes nearly inextensible. Instead of a free surface that can slide, there is effectively a rigid lid, and the water just beneath it must shear against a stationary boundary. That creates a viscous boundary layer where there was none, and a boundary layer dissipates. The wave’s energy is drained into heat in a layer a fraction of a millimetre thick, at a rate that can be tens of times what a clean surface offers.

The mechanism has a frequency preference, and it is the one that matches what is seen. The Marangoni restoring force takes a certain time to act — the surfactant has to redistribute — so the damping is strongest when that time matches the wave’s period. For ordinary surfactants that puts the peak on ripples of centimetre wavelength, and it explains the observation exactly: a slick flattens the small chop that roughens a surface and does nothing whatever to the long swell rolling underneath it.

That selectivity has become an instrument. Radar looking down at the sea gets its return almost entirely from centimetre-scale waves, because those are the ones whose spacing matches the radar wavelength and scatter coherently back. A surfactant film damps exactly those waves, so a slick appears on a radar image as a dark patch — and satellite radar is now the standard way of finding oil spills, and of mapping the natural films that plankton produce.

Where the model stops #

The surface is treated as a mathematical plane with a tension. A real interface with surfactant on it has its own rheology — a surface viscosity, an elasticity, sometimes a yield stress — and none of that appears in a single number $\gamma$.

The concentration at the surface is treated as given. In reality it is set by an exchange with the bulk: molecules adsorb onto the surface and desorb from it at rates that depend on concentration, and if that exchange is fast compared with the flow, the gradient is replenished from below and the analysis changes completely.

The film is thin and flat. The balance $\tau = \mu U/\delta$ assumes a linear velocity profile across the film, which requires the film to be thin compared with the distance over which the tension varies. It fails at the edge of a spreading drop, where the film thickness goes to zero and the stress would give an infinite speed — a singularity the same shape as the one at a moving contact line.

What the pictures cannot show #

The gradient figure plots surface tension against composition and against temperature on one axis read two ways, which is a convenience for comparison and not a physical statement: there is no quantity of which mole fraction and hundredths of the temperature range are both values.

Nothing here shows the surfactant molecules. The whole subject rests on a monolayer one molecule thick whose concentration is the actual variable, and every figure works with the tension instead — which is the observable, and is one inference removed from the cause.

And no figure shows the flow. The speeds computed here are surface speeds under a stated gradient; what a real tear or a real Bénard cell does is a two-dimensional circulation with a return flow underneath, and drawing one would require the full velocity field rather than a balance at a point.

Where the ladder goes next #

Surface tension has appeared here three times already — as a skin that is not a skin, as the pressure inside a bubble, and as the instability that breaks a thread — and in all three it was a constant. Making it a field with a gradient is the fourth rung, and the ones after it are the surface rheology proper, the coupling between Marangoni and buoyant convection in a layer thick enough to have both, and the drying-drop problem, where evaporation, capillary flow and Marangoni flow together decide whether a stain has a ring around it.

The habit worth carrying away is what happens to a constant when it is allowed to vary. Surface tension entered this collection as a number with units of energy per area, and every argument built on it treated the number as fixed. Letting a constant become a field usually adds a term proportional to its gradient, and that term is frequently a force in a direction the original argument had no way of producing — along a surface, here, rather than across it.