The skin that is not a skin

A drop of water hanging from a tap bulges, holds, and eventually falls in a shape that looks exactly like a bag of liquid in a stretched skin. Small insects stand on ponds with visible dimples under their feet. A needle laid carefully on water floats, despite being eight times as dense.

All three look like the work of a membrane, and there is no membrane. Water is water all the way to the top.

What a molecule at the surface is missing #

A molecule in the bulk of a liquid is surrounded on all sides by neighbours, each of which attracts it. A molecule at the surface has neighbours below and beside it and none above. It is therefore less tightly bound — it sits at a higher energy than one in the interior, by an amount roughly proportional to the number of neighbours it is short of.

Every molecule at the surface pays that penalty, and the number of them is proportional to the area. So the liquid carries an energy proportional to its surface area:

$$E_{\text{surface}} = \gamma A.$$

The constant $\gamma$ is the surface tension, and this is what it actually is: an energy per unit area. For water at room temperature it is about 0.073 joules per square metre, which is also 0.073 newtons per metre, and the fact that those two are the same units is not a coincidence but the next section.

The magnitude is worth a sanity check. A molecule at the surface is short perhaps a quarter of its neighbours, so the penalty is of order a quarter of the energy needed to remove it from the liquid entirely — a substantial fraction of the latent heat of vaporisation, per molecule. Dividing that by the area a molecule occupies gives tens of millijoules per square metre, which is the observed number. Surface tension is not a separate phenomenon requiring its own theory; it is the cohesion that makes a liquid a liquid, counted at a boundary.

Why an energy per area is a force per length #

Take a rectangular frame with one sliding side, of length $L$, with a liquid film stretched across it. Pull the slider out by $\mathrm{d}x$. The film’s area increases by $L\,\mathrm{d}x$ per surface, so the energy increases by $\gamma L\,\mathrm{d}x$, and the force resisting the pull is

$$F = \frac{\mathrm{d}E}{\mathrm{d}x} = \gamma L.$$

A force proportional to the length of the boundary being extended, with the same constant. Joules per square metre and newtons per metre are dimensionally the same thing, and the two descriptions — an energy that scales with area, a force that acts along a line — are one statement.

The second description is the more familiar and the more misleading. Because the force is proportional to the length of a line, it is easy to picture a stretched sheet pulling on its edges. But a stretched sheet is characterised by a modulus: the further it is stretched, the harder it pulls. A liquid surface is not. Doubling the area of a soap film does not double the tension in it — the tension is the same $\gamma$, because the new surface is made by molecules arriving from the bulk rather than by existing ones being spread thinner. The liquid has an inexhaustible supply of molecules with which to make more surface at a fixed price.

That difference is the reason the word skin is misleading and the reason field lines are worth the same warning: a picture that gets the phenomenon right can smuggle in a mechanism that is wrong, and the mechanism is what gets used to predict the next thing.

Why a free drop is a sphere #

Given a fixed volume of liquid and a cost proportional to area, the shape that minimises the energy is the shape of least area at that volume. That shape is the sphere, and the figure at the top of the page is a demonstration rather than an assertion: four shapes at one volume, with the areas evaluated and the sphere lowest.

The margin is not enormous. A cube has fourteen per cent more surface than the sphere of the same volume, and a cylinder as tall as it is wide has fourteen per cent more too. It takes a genuinely flat shape to get a large penalty — a disc a quarter as tall as its radius carries ninety-one per cent more.

This is why a drop in free fall is round, why a bead of mercury on a table pulls itself into a ball, and why molten metal solidified in a drop tower gives near-perfect spheres. It is also why a drop resting on a table is not round: gravity is competing, and the shape that results is the one that minimises surface energy plus gravitational potential energy together. Which term wins is a question of size, and it has an exact answer.

The length at which surfaces stop mattering #

Compare the two energies for a blob of size $R$. The surface term is $\gamma R^2$; the gravitational term is $\rho g R^4$, since the mass goes as $R^3$ and the height as $R$. Their ratio is $\rho g R^2 / \gamma$, and setting it to one gives a length:

$$\ell_c = \sqrt{\frac{\gamma}{\rho g}},$$

the capillary length, about 2.7 millimetres for water. Below it, surface tension dominates and blobs are round. Above it, gravity dominates and puddles are flat. A puddle of water on a clean surface is a couple of millimetres deep and cannot be made deeper by adding more — the extra spreads sideways — and that depth is $\ell_c$ up to a factor of order one.

The same length settles the insect. A needle or a water strider is held up by surface forces if it is smaller than a few millimetres, and by buoyancy if it is much larger. The reason small things live in a world dominated by surfaces is that this ratio goes as the square of size, and a millimetre is on the wrong side of it.

The general shape of the argument is two effects with different powers of a length crossing at one specific size. Which of them dominates is a question of scale rather than of strength — surface goes as the square of the size and volume as the cube, so their ratio has the dimensions of an inverse length and there is exactly one length at which they balance. That is the same reasoning that decides which term in a potential wins at which distance, and it never depends on either term being large.

Two drops become one, and the difference goes somewhere #

Merging two drops conserves volume and reduces area, because a sphere of twice the volume has $2^{2/3}$ times the surface of one, not twice. So the pair loses a fifth of its surface, and with it a fifth of its surface energy.

Where it goes is worth following, because energy bookkeeping is where a picture like this can be checked. It goes into motion: the merged drop is left oscillating violently, ringing between prolate and oblate shapes, and viscosity damps that ringing into heat over some tens of milliseconds. A high-speed film of two coalescing drops shows exactly this, and the frequency of the ringing is set by surface tension against inertia in the same way a spring’s is set by stiffness against mass.

For millimetre drops the released energy is a fraction of a microjoule and warms the drop by a negligible amount. For micron drops it is a much larger fraction of the drop’s total energy, and coalescence in a fine mist is correspondingly violent. This is the reason foams and emulsions need something added to them: left alone, every merge is downhill, so a foam’s existence is always temporary and its lifetime is a question of how slowly it goes down the hill rather than whether it does.

Coalescence is a slope with no minimum on it. Every merge lowers the total surface and therefore the energy, so there is no resting place until everything has merged into one body — which is why a foam or an emulsion is never at equilibrium. It is somewhere on that slope, held up by whatever slows it: a surfactant, a viscosity, a solid particle wedged at the interface. Stability in these systems is always kinetic and never thermodynamic.

Soap films, which solve the problem by settling into it #

The clearest demonstration that a liquid surface minimises its area is a soap film on a wire frame, because there is no volume constraint at all — the film is a sheet with two surfaces and negligible thickness, so its energy is simply $2\gamma$ times whatever area it adopts.

Dip a frame of any shape into soap solution and the film that forms is the minimal surface spanning that frame. This is a genuine variational problem, hard to solve on paper for anything but simple boundaries, and the film solves it by relaxing into it in a fraction of a second. Two parallel rings give a catenoid, which is the surface of revolution of a hanging-chain curve; a cubical frame gives a configuration of thirteen faces meeting at angles that are not obvious in advance.

Those angles are the part worth knowing, because they are forced rather than chosen. Where three films meet they meet at exactly 120°, and where four edges meet they do so at the tetrahedral angle of about 109.47°. Both follow from the tensions balancing at the junction, in the same way three forces on a body must close a triangle: three equal forces in a plane can only balance at 120°, and there is no arrangement of more than three films along one edge that balances at all. These are Plateau’s laws, stated in the 1870s from observation and proved a century later, and they are why foam looks the way it does at every scale from a bath to a metal casting.

Underneath the energy argument is a counting one, and here they point the same way. A surface is a place where molecules have fewer neighbours and therefore fewer arrangements available at the same energy — so minimising surface minimises a cost that is entropic as well as energetic. That the two agree is why surface tension falls with temperature rather than rising: the entropic part is the part temperature multiplies.

What changes the number #

Surface tension is a property of the interface, not of the liquid alone, and three things move it.

Temperature. $\gamma$ falls as temperature rises, roughly linearly, and reaches zero at the critical point — which it must, because above the critical point there is no distinction between liquid and vapour and therefore no interface to cost anything. Water goes from 0.076 N/m at 0 °C to 0.059 at 100 °C. A gradient in temperature is therefore a gradient in surface tension, and a liquid surface pulls toward the colder region: that is thermocapillary flow, and it is why a temperature difference can drive convection with no buoyancy involved at all — an effect that becomes the only one available in orbit, where there is no pressure gradient to make buoyancy out of.

What is dissolved in it. Molecules that prefer the surface to the bulk accumulate there and lower the cost of surface. These are surfactants — soap is one — and they cut water’s surface tension by a factor of two or more. The same gradient effect applies: a difference in surfactant concentration produces a flow along the surface from low tension to high, which is the Marangoni effect and is what drives the tears of wine on a glass. A drop of detergent touched to a water surface dusted with pepper scatters it violently for the same reason, and the speed of that scattering is a direct measurement of how large the tension difference is.

What is on the other side. The number quoted for water is really water against air. Water against oil is different, water against glass is different again, and it is the three-way balance between them that decides whether a liquid climbs a tube or is pushed down it — which is the contact angle, and the next-but-one rung on this anchor. It is another instance of a quantity that belongs to a boundary rather than to a bulk, which is what charge sitting entirely on a conductor’s outside is too.

The measurement, and what it is a measurement of #

Surface tension is measured, not calculated, and the methods are worth a paragraph because each of them is one of the arguments above run backwards.

The oldest is the capillary rise itself: put a tube of known radius in the liquid and read the height, which gives $\gamma$ directly and is the subject of a later rung. It is the same style of measurement as weighing a body in and out of water — arrange things so the wanted quantity is the only one left. The Wilhelmy plate hangs a thin plate so its lower edge just touches the surface and weighs the extra pull, which is the force-per-length picture used as an instrument. The du Noüy ring does the same with a ring, and needs a correction because the surface it drags up is not a simple cylinder.

The most accurate modern method uses the shape of a hanging drop. A pendant drop’s profile is the solution of the Young–Laplace equation with gravity in it, and the solution family is one-parameter: fit a photograph of the drop to that family and the parameter returned is $\gamma/\rho g$. Nothing touches the liquid, which matters for anything that would be contaminated by a plate — and contamination is the great hazard here, because a monolayer of the wrong molecule can halve the number being measured.

That sensitivity is itself informative. A quantity that a single molecular layer can change by fifty per cent is a quantity that lives entirely in a single molecular layer, which is exactly what the argument at the top of this page claims it is.

The surface tension of a nucleus #

The argument on this page used nothing about water. It used only that a constituent at the boundary has fewer neighbours than one inside, and that is true of anything held together by short-range attraction — including a nucleus.

The nuclear force is short-ranged, so a nucleon at the surface of a nucleus is short of neighbours in exactly the way a water molecule is, and the nucleus’s binding energy carries a term proportional to its surface area. That term is in every semi-empirical mass formula, at about eighteen million electronvolts times the two-thirds power of the mass number, and dividing by the area gives a nuclear surface tension of around $10^{17}$ newtons per metre — eighteen orders of magnitude above water’s.

It does what surface tension does. It makes small nuclei less tightly bound per nucleon than large ones, which is why the binding-energy curve rises steeply at the light end and therefore why fusion releases energy at all. And it makes a nucleus resist being deformed, because any departure from a sphere increases the area.

Fission is the competition between that resistance and the electrostatic repulsion, which falls when the charge is spread out. Deform a nucleus slightly and the surface term rises while the Coulomb term falls; the two scale differently with the deformation, so whether the total goes up or down depends on a single dimensionless ratio, essentially $Z^2/A$. Above about fifty, the surface cannot hold the charge and the nucleus comes apart spontaneously at any provocation. Uranium sits at thirty-six, which leaves a barrier — low enough that a neutron’s arrival can supply the energy to cross it, and high enough that the nucleus survives for thousands of millions of years otherwise.

So the shape of a raindrop and the possibility of a reactor are two consequences of the same counting argument about neighbours, with the numbers differing by eighteen decades.

Making a solid by removing its surface #

The essay’s last caveat says a solid cannot rearrange to minimise its surface. Given enough temperature it can, slowly, and an entire manufacturing method is built on making it do so.

A metal or ceramic powder has an enormous surface area — a kilogram of micron-sized particles has thousands of square metres — and solid surface energies are of order a joule per square metre, a thousand times water’s. That is a large amount of stored energy sitting in a bag of powder, and it is all downhill.

Press the powder into a shape and heat it to somewhere below its melting point. Atoms become mobile enough to diffuse, and every diffusive step that removes surface lowers the energy: necks grow between touching particles, the necks thicken, the pores between them shrink and close, and the loose powder becomes a dense solid without ever having been liquid. That is sintering, and it is how essentially every ceramic component and a great many metal ones are made.

The driving force is precisely this page’s $\gamma$ times the area removed, and the rate is set by how fast atoms can diffuse. Nothing pushes the particles together except their own preference for having less surface.

The same term is why very small particles behave oddly. A nanoparticle has so much of its material at its surface that the surface energy is a substantial fraction of the whole, and the consequence is measurable: gold that melts at 1,064 degrees in bulk melts several hundred degrees lower at a few nanometres across, because melting removes a solid surface and replaces it with a cheaper liquid one.

Where the model stops #

The interface is not infinitely thin. It is a few molecular diameters across, over which the density falls from liquid to vapour smoothly. Treating it as a mathematical surface with an energy per area is an excellent approximation for drops millions of molecules across and a poor one for a nanoscale droplet, where the surface tension itself becomes size-dependent.

Curvature is ignored here and is not ignorable. A curved surface has a pressure difference across it, and that is the whole of the next rung on this anchor — including the result that makes bubbles behave in a way nobody expects, which is that the smaller of two connected bubbles empties into the larger.

Static only. A surface being created rapidly has a higher effective tension than an equilibrated one, because surfactants take time to reach it. Dynamic surface tension is a real and separately-measured quantity, and it is what matters in inkjet printing and spraying.

Gravity was set aside and then only partly restored. The capillary length says which term dominates; it does not give the shape when the two are comparable, which requires solving the Young–Laplace equation with a gravitational term and produces the pendant-drop profiles used to measure $\gamma$ in the first place.

Solids are excluded. A solid surface has a surface energy too, but it cannot rearrange to minimise it, so the two quantities that were identical for a liquid — energy per area and force per length — come apart for a solid. Confusing them is a standard error in the wetting literature.

The ladder from here #

Later rungs on this anchor: the pressure jump across a curved interface, and what it does to bubbles. Capillary rise and the contact angle. The instability that makes a thread of liquid break into drops. Minimal surfaces and soap films on wire frames, which solve a variational problem by settling into it. Surfactants and micelles, and how detergency actually works. Marangoni flows. And the surface as a place where chemistry happens differently, because a molecule with a missing neighbour is a molecule with something available.