The film that goes black before it bursts

Blow a soap film across a wire loop, hold it upright, and watch. It runs downwards through a sequence of colours — the same sequence an oil slick shows, growing coarser as the film thins — and then, at the top, a patch appears that is not a colour at all. It reflects nothing. It looks like a hole, and people who have not seen it before assume the film has broken there.

It has not. That patch is the thinnest and toughest part of the film, and the film will burst from it eventually — but not because it is black.

The colours a draining film runs through, and the end of them. The fraction of light a free soap film reflects, against its thickness, at three wavelengths — 450, 550, 620 nm — computed from the sum over multiple reflections rather than from the two-beam approximation. The three curves peak at different thicknesses, which is why a draining film runs through a sequence of colours as it thins. Below about 11 nm every curve is under a tenth of a per cent and the film looks black. At zero thickness the reflectance is exactly zero, checked before the figure is drawn: the two surfaces reflect equally and half a cycle out of step, so a film much thinner than a wavelength cancels itself. That is the whole of why a black film is black. Nothing is absorbing; the film is there and has simply stopped being able to interfere constructively at any visible wavelength — which means the blackness is a measurement, and a film that has gone black is known to be thinner than about a tenth of a wavelength without anything being measured directly. — Fig. 1 The fraction of light a free soap film reflects, against its thickness, at three wavelengths, computed from the sum over multiple reflections. The three peak at different thicknesses, which is why a draining film runs through colours. Below about a tenth of a wavelength every curve is under a tenth of a per cent.

Why thin means black

The reflectance goes to zero as the thickness does, and the zero is exact rather than approximate.

Light reflecting off the front surface of the film comes off a boundary from air to water, which reverses its phase. Light reflecting off the back surface comes off a boundary from water to air, which does not. So the two reflections start half a cycle out of step, and when the film is much thinner than a wavelength there is no extra path to make up the difference. They cancel.

There is a check on that reasoning that costs nothing. If the phase reversal at one surface is what makes a thin film black, then a film with the same medium on both sides of it — an oil layer between two identical glass plates, say — should not go black, because both reflections reverse. It does not: such a film goes bright as it thins, not dark. The sign of the effect is a consequence of the film being free-standing, which is a fact about the arrangement rather than about the liquid.

A film thinner than about a tenth of a wavelength is therefore invisible in reflection, and completely present. That makes blackness a measurement: a film that has gone black is known to be under about fifty nanometres, without anything having been measured directly. It is the cheapest thickness gauge there is, and it is why the two black states in this subject have names.

The same cancellation, arranged deliberately with a quarter-wave layer instead of accidentally with a thin one, is the layer that makes a reflection vanish — the anti-reflection coating on a lens. The soap film gets there by having no thickness at all rather than by having exactly the right one.

Reading a thickness off a colour

Before the black patch, the colours are doing the same job with more resolution, and it is worth knowing how far that can be pushed.

Each wavelength has its own peaks and troughs, so the colour of a film in white light is a specific mixture that changes continuously as it thins. Near a micrometre the orders are packed close together and the colour is a washed-out pinkish white; between about six hundred and two hundred nanometres the sequence is the vivid one — yellow, magenta, blue, then a silvery white — and below about a hundred and fifty nanometres the film goes through a last straw colour and then black.

Matching an observed colour against a computed sequence gives the thickness to a few nanometres, which is why the method is used on everything from oxide layers on silicon to the tear film on an eye. Its weakness is ambiguity: the same colour recurs at every order, so a single reading of a thick film cannot say which order it is in. Watching the film thin resolves it, because the orders come in sequence and can be counted.

The black state is the one unambiguous reading in the set. There is only one way to be black, which is why it is the anchor the rest of the sequence is counted from.

What stops the thinning

Surface tension holds the film taut and does nothing whatever to keep its two surfaces apart. Something else has to, and it is a pressure that exists only at separations of nanometres.

What holds a film apart, against how thin it is. The disjoining pressure of a free film — what the two surfaces exert on one another per unit area — against the film's thickness, on a logarithmic vertical scale, for several salt concentrations. Three terms are summed: a repulsion between the two charged surfaces that falls off over the Debye length, a van der Waals attraction that goes as the inverse cube of the thickness, and a steric repulsion that only acts when the two surfactant layers touch. The horizontal line is the suction pulling water out, here 60 Pa, and a film rests where the curve crosses it. At 1 mM the crossing is at 75 nm — a common black film, held apart by charge. At 300 mM the electrostatic term has been screened into nothing and the only crossing left is at 4.8 nm, on the wall where the two surfactant layers touch — a Newton black film, which is two monolayers and almost no water between them. — Fig. 2 The disjoining pressure of a free film — what the two surfaces exert on one another per unit area — against thickness, for three salt concentrations. Three terms are summed: an electrostatic repulsion decaying over the Debye length, a van der Waals attraction going as the inverse cube, and a wall where the two surfactant layers touch. The film rests where a curve crosses the suction.

The surfactant molecules that make the film possible sit at both surfaces with their charged heads in the water, so each surface carries a charge and the two repel. That repulsion is screened by the ions in the water and decays over the Debye length, which is the long-range force that does not reach applied to a gap rather than to a plasma.

Against it, the van der Waals interaction across the film is attractive and pulls the surfaces together. And underneath both, at a few nanometres, the two surfactant monolayers meet and stop.

The film rests where the sum of those balances whatever is pulling the water out. In a film held in a frame, that suction is the curvature of the borders at the edges — the Plateau borders — whose concave surfaces have a pressure below atmospheric, exactly the arrangement the small bubble blows up the big one is about, working here to empty the flat film into the curved edges.

The two black films

How thin the salt lets it get. The thickness a film settles at, against the salt in the water, at a suction of 60 Pa. Adding salt shortens the Debye length and therefore the range of the repulsion holding the two surfaces apart, so the resting thickness falls — from 89 nm at the lowest salt to 4.7 nm at the highest, a factor of 19. At the salty end the curve flattens onto the contact thickness: the two surfactant layers are touching, more salt changes nothing, and what is left is a Newton black film — two monolayers and almost no water between them. The upper part of the curve is a common black film, held apart by charge, and the two have names because in a real film the change between them is abrupt: a patch of the thinner state nucleates and spreads across the film in seconds. This model gives a smooth crossover instead, because it contains a contact wall and no short-range hydration repulsion, and the first-order character of the real transition comes from that missing term. — Fig. 3 The thickness a film settles at, against the salt in the water. Adding salt shortens the range of the repulsion, so the film rests thinner — from about ninety nanometres at the lowest salt to under five at the highest, where the curve flattens onto the contact thickness and more salt changes nothing.

The two ends of that curve have names because they behave differently.

A common black film is tens of nanometres thick and held apart by the charge on its surfaces. It is black because it is thin, and it is thin because the repulsion has been screened down to the point where it balances the suction at that separation.

A Newton black film is four or five nanometres — two surfactant monolayers with almost nothing between them. There is no water layer to speak of; the film is two molecules thick and mechanically a single object. It is remarkably durable. The two states are separated by a factor of ten in thickness and by nothing at all in appearance, which is why the boundary between a patch of one and a patch of the other is the only thing that makes the change visible — the boundary refracts, and it can be watched crossing a film, while neither side of it reflects anything. Everything else in this essay is a thickness that can be seen; this is a change of thickness that can only be seen at its edge, and it is the same difficulty the horizon that nothing marks has in a very different subject.

In a real film the change from one to the other is abrupt: a darker patch nucleates somewhere in the black film and spreads across it in seconds, with a visible boundary. The model here gives a smooth crossover instead, and the reason is worth stating rather than hiding — it contains a contact wall and no short-range hydration repulsion, and the first-order character of the real transition comes from that missing term.

What the two states are made of

It is worth being concrete about how much water is in each, because the numbers make the difference obvious.

A common black film at thirty nanometres is about a hundred water molecules thick between its two surfactant layers. That is thin by any ordinary standard and thick enough for the water in the middle to be ordinary water, with the two surfaces influencing it only through the field they set up.

A Newton black film at four and a half nanometres is not that. Two surfactant monolayers are about two nanometres each, so almost the whole thickness is the molecules themselves and what is left between them is a layer of water a few molecules across — bound, oriented, and with properties that are not the bulk liquid’s. The film is a bilayer, and it is the same object a cell membrane is, made of the same kind of molecules for the same reason.

That correspondence is not an analogy. A Newton black film is a surfactant bilayer with air on both sides rather than water, and much of what is known about the forces between bilayers was measured on films of this kind, because they are far easier to make and to look at than a membrane is.

The wedge, and where it bursts

Fig. 4 The equilibrium thickness of a vertical film against height. The suction is larger higher up by the weight of the water in between, so the film is a wedge — thick at the bottom and thin at the top.

The wedge explains why the black patch always appears at the top and grows downwards, and why a film bursts from its top whatever it is made of and however it is held. The thinnest part fails first, and the thinnest part is decided by gravity.

It does not explain the coloured bands. The whole equilibrium profile is thinner than a hundred nanometres and therefore black; a film showing colours is still far thicker than equilibrium and on its way down. The bands are a drainage problem, and drainage in a soap film is stranger than it looks — most of the thinning happens not by liquid flowing down through the film but by patches of thinner film being pulled up from the borders and swapping places with thicker ones, a process that looks like turbulence and is driven by surface-tension gradients.

So the pretty part of the phenomenon and the calculable part are different parts, and being clear about which figure describes which is most of the honesty available here.

The mechanism behind the swapping is worth naming because it is the same one that drives several other things on this site. A patch of film that is momentarily stretched has a lower surfactant coverage and therefore a higher surface tension, so it pulls on its neighbours — and a gradient of surface tension drags the liquid underneath it along, which is exactly the surface that pulls toward the stronger side. In a draining film that mechanism is not a curiosity; it is the main way the liquid leaves, and it is why the drainage of a real film is far faster than a calculation of viscous flow between two rigid surfaces would predict — the surfaces are not rigid, and the resistance a film offers depends on how mobile its surfactant is, which is the same distinction momentum going sideways draws between a shear carried by a wall and one carried by a free surface.

What the barrier is worth

What it costs to squeeze a film to contact. The work per unit area needed to push a film's two surfaces from far apart down to contact, obtained by integrating the disjoining pressure, for several salt concentrations. 1 mM: 0.79 mJ/m²; 10 mM: 0.64 mJ/m²; 300 mM: -0.03 mJ/m². For comparison, the surface tension of the solution is about 25 mN/m, so a film has two surfaces' worth of energy — of order fifty millijoules per square metre — sitting in it already. The barrier is smaller than that, and it falls as salt is added, which is the same statement the previous figures make in another currency: what stops a film from collapsing to contact is a barrier of the order of a millijoule per square metre, and screening the charges removes it. A film with no barrier left is not less stable in the sense of being closer to bursting — it goes to the contact state and stays there, which is the most stable state of all. — Fig. 5 The work per unit area needed to push a film’s two surfaces from far apart down to contact, obtained by integrating the disjoining pressure. It falls as salt is added, and it is a small fraction of the energy already sitting in the film’s two surfaces.

A film already carries about fifty millijoules per square metre in its two surfaces, and the barrier standing between a thick film and contact is a fraction of one. That ratio is why a film is so easily destroyed by dust, by a dry finger, by anything that locally removes the surfactant: the energy needed to punch through is small compared with what the film is already carrying, and the film’s stability is entirely a matter of the barrier rather than of the surface energy.

The barrier’s size also explains why the recipe matters so much. Adding glycerine to a soap solution is the standard trick for making films that last, and it works by slowing the drainage rather than by raising the barrier — a more viscous film takes longer to reach the thickness at which a chance disturbance can punch through it. Adding salt does the opposite of what one might guess: it lowers the barrier, and the film goes black sooner, which is easy to mistake for the film becoming fragile when it is in fact reaching its most durable state faster. Both of those are readings of the same isotherm, and neither is a statement about surface tension, which barely changes.

It is also why a black film is durable. Once the surfaces are in contact there is no barrier left to cross, because there is nowhere thinner to go. The most stable state and the most fragile-looking one are the same state.

The same balance, on a wetting film

The disjoining pressure is not a curiosity of soap films. It is what decides whether a liquid spreads at all.

A drop of liquid on a solid either beads up or spreads into a film, and the angle a liquid makes with what it sits on settles which by comparing three surface energies. That comparison assumes the film, if it forms, is thick enough for the two interfaces to be independent. When it is not — a film of nanometres, which is what “complete wetting” produces — the two interfaces interact through exactly the disjoining pressure computed here, and the film’s thickness is set by balancing it against the vapour pressure of the surrounding air.

That is the mechanism behind an adsorbed film on any surface exposed to a vapour: a few molecular layers of water on almost every solid in a humid room, thickening as the humidity rises and diverging as it approaches saturation. The thickness follows the same isotherm, read with the chemical potential of the vapour supplying the suction instead of a curved border.

The same isotherm decides a third thing: whether a bubble in a liquid survives long enough to matter, since the film between two approaching bubbles is the same object and coalescence is that film failing. One function of one variable, asked three different questions, and the fact that it answers both is the reason it has a name of its own rather than being folded into surface tension.

Where the model stops

The disjoining pressure is DLVO plus a wall. Real films have short-range hydration and steric forces from the surfactant heads and tails, and those are what make the transition between the two black films first order. The model gets both thicknesses and the salt dependence and misses the discontinuity.

Everything is in equilibrium. The interesting behaviour of a real film — the drainage, the marginal regeneration, the bursting — is dynamic, and the equilibrium profile is only where the film is heading.

The surfaces are treated as uniformly charged planes. They are a monolayer of molecules with heads of finite size, and at four nanometres of separation the smeared-out picture is being asked to describe a gap two molecules wide.

And the film is assumed pure. A real soap film contains micelles, and at high surfactant concentration the disjoining pressure acquires oscillations — the film thins in discrete steps as layers of micelles are squeezed out one at a time, which is a genuinely different phenomenon and is visible as stepwise darkening.

Why a film bursts at all

Nothing above says how a film ends, and the answer is not that it gets too thin.

A film at its equilibrium thickness is stable against uniform thinning: push the two surfaces closer and the disjoining pressure pushes back. What it is not stable against is a hole. Once a hole of some critical size opens, the surface tension pulling on its rim exceeds what closes it, and the hole runs outwards at a speed set by the tension and the mass it has to accelerate — tens of metres per second for a soap film, which is why bursting looks instantaneous.

The critical size is set by comparing the energy released by removing two surfaces against the energy of the rim, and for an ordinary film it is of order a micrometre. Anything that opens a hole that big does it: a speck of dust, a dry fibre, an evaporating patch, or a local loss of surfactant.

So a film’s lifetime is a nucleation problem rather than a thinning problem, and it is why the answer to “how long does a soap film last” depends far more on the cleanliness of the air than on anything in the figures here. A film in a sealed jar can last for days at a thickness that would fail in seconds in a room.

What the pictures cannot show

None of the figures shows a film. They show a reflectance, a pressure and a thickness, and a soap film’s most useful property is that all three are visible at once: the colour is the thickness, so a photograph of a film is a contour map of it, and no plot conveys that the whole measurement is available by looking.

Nor do the figures show the borders. Everything here treats the flat film and takes the suction as given, and the suction comes from the curved Plateau borders at the edges — which are also where the liquid goes, and whose own geometry changes as they fill. A complete account solves the film and the borders together.

The measurement this is all built on

The disjoining pressure is not inferred from soap films; it is measured directly, and the instrument is worth a sentence because it is what turns the curves here into data.

A thin-film pressure balance holds a small film in a porous ring, and the pressure in the liquid feeding the ring is set from outside. That fixes the suction; the film’s thickness is then read from its reflectance, exactly as the first figure describes. Sweeping the applied pressure and recording the thickness traces out the isotherm point by point, over four or five decades of pressure.

That is an unusually direct measurement of an interaction between surfaces, and the same isotherm measured on the same surfactant in a surface-force apparatus — two crossed mica cylinders, with the separation read interferometrically — agrees with it. Two instruments with nothing in common but the quantity they measure is the right kind of evidence, and it is why the DLVO terms are trusted where the model works and why their failure at short range is known to be a real failure rather than an experimental artefact.

Where the ladder goes next

The surface-tension ladder began with the skin that is not a skin, where the tension is an energy per area rather than a membrane, and went through the pressure inside a bubble, the thread that cannot stay a thread, the Marangoni stress, the ring a drying drop leaves, and the angles a film has no choice about. This rung takes the film down to where the two surfaces can feel each other, which is a regime none of the earlier ones needed.

The rung after it is the foam: many films meeting at borders, draining into one another, coarsening as gas moves from small bubbles to large. The habit worth carrying is the one this rung is built on: when a system gets thin enough, look for a force that only exists at that thickness — because the quantity that governed everything at larger scales will have stopped being the one that decides.

The film that goes black before it bursts