What a plate takes with it

Take a clean glass plate out of a bowl of water and it comes out wet. The film it carries is about eight microns thick if the plate was drawn out at the speed of an unhurried hand, and the interesting question is what decided that number.

Not the plate: it has no length that enters. Not the bath: its depth does not appear. Not how long the plate was in there. The film’s thickness is set by an argument that takes place entirely in the few tenths of a millimetre where the flat film joins the curved surface of the bath, and it is decided there before the plate has cleared the water.

Two things pulling in opposite directions #

The plate drags liquid upwards by viscosity: no slip means the fluid touching it moves at the plate’s speed, and it takes its neighbours with it. Surface tension pulls the liquid back down, because a thick film on a plate has more area than a thin one and a curved surface has a pressure jump across it.

The ratio of the two is the capillary number, $\mathrm{Ca} = \mu U/\gamma$ — viscosity times speed over surface tension, dimensionless, and small for almost everything. Water at a centimetre a second gives $\mathrm{Ca} = 1.4\times10^{-4}$.

The only length in the problem is the capillary length, $\ell_c = \sqrt{\gamma/\rho g}$, which is 2.7 mm for water and is the size at which surface tension and gravity balance. It is the size of a raindrop on a window, the height water climbs in a wettable tube, and — since it is the only length available — it must be what the film’s thickness is measured against.

So the answer has to have the form $h = \ell_c \times f(\mathrm{Ca})$, and the whole calculation is finding $f$.

The exponent is not fitted #

Here is the argument in the form Landau and Levich gave it in 1942.

Far up the plate the film is flat, moves with the plate, and is not being sheared at all. Down at the bath the surface is a static meniscus: gravity against surface tension, no motion, a shape with a definite curvature $\sqrt2/\ell_c$ where it meets the flat wall.

Between them is a dynamic meniscus in which both matter, and lubrication theory with a capillary pressure applies. Scaling the film by its own thickness $h_0$ and the distance along the plate by $h_0\mathrm{Ca}^{-1/3}$ — which is the only combination that makes the viscous and capillary terms the same size — reduces the whole thing to one equation with no parameters at all:

$$\eta''' = \frac{3(\eta-1)}{\eta^3}.$$

Integrate it downstream from the flat film and its curvature approaches a constant. Set that constant equal to the static meniscus’s, undo the scalings, and the thickness falls out. The two-thirds comes from the scaling and nothing else; it was fixed before the equation was solved.

The number #

The curvature comes out at 1.337577, and matching to a static meniscus of curvature $\sqrt2/\ell_c$ divides it by $\sqrt2$:

$$h_0 = 0.94581\,\ell_c\,\mathrm{Ca}^{2/3}.$$

Nothing in that was quoted. The 1.3376 is a Runge–Kutta march, checked for independence of its own starting condition, and the eigenvalue it starts on — the growing solution near the flat film, with $k^3 = 3$ — is checked too. The other two roots of $k^3 = 3$ are complex with negative real part, which means the approach to the flat film is oscillatory going upstream: there is a ripple ahead of the meniscus, and it is real, and it is why a freshly dip-coated plate has a faint corrugation near the waterline.

Where the work went, which is not where the film is #

This essay sits in a phase about dissipation, and the reason is a fact about the film that its appearance conceals completely.

The flat film costs nothing. It is moving with the plate, uniformly, with no velocity gradient across it at all — a rigid translation, which by the null space of the dissipation function is exactly free. A metre of coated plate carrying eight microns of water is not being sheared anywhere.

Every joule of the extra force needed to withdraw the plate is spent in the dynamic meniscus, in the few tenths of a millimetre where the film is being formed. Integrating the wall stress through that region gives a force per unit width of $\gamma\,\mathrm{Ca}^{2/3}$ times a pure number of about 3.8 — so the withdrawal force has the same two-thirds scaling as the thickness, and it is all incurred in a region a reader would never think to look at.

The same constant in a completely different apparatus #

The strongest evidence that 1.3376 belongs to the join rather than to the plate is that it turns up somewhere with no plate in it.

Drive a long bubble along a liquid-filled tube and it leaves a film on the wall behind it. There is no gravity in the problem, no bath, and no capillary length — the geometry supplies its own length, the tube’s radius. Bretherton solved it in 1961 and the answer is

$$\frac{h}{R} = 1.3376\,\mathrm{Ca}^{2/3}.$$

The same constant. What the two problems share is a transition region between a flat film and a curved static surface, governed by the same parameterless equation; what they do not share is anything else at all.

Why there is no length in the answer #

The absence of the plate’s own dimensions from the result is worth dwelling on, because it is the feature that makes the law useful and it is not obvious that it should hold.

A plate a metre wide and a plate a centimetre wide carry the same film. A plate withdrawn from a bucket and a plate withdrawn from a lake carry the same film. A plate that has been in the water for an hour and one dipped for a second carry the same film. None of that is true of, say, the amount of liquid that drips off afterwards, which depends on the plate’s height and on how long it is left.

The reason is that the film is decided in a region whose size is $h_0\mathrm{Ca}^{-1/3}$ — a fraction of a millimetre for water at ordinary speeds — and everything outside that region enters only through one number, the curvature the static surface presents. The bath is summarised by a curvature and nothing else. That is the whole content of a matched asymptotic expansion, and this is one of the cleanest examples of one in the subject: an outer problem that is entirely static, an inner problem with no parameters at all, and a single number passed between them.

It is also the reason the answer is so robust. Deepen the bath, widen the plate, warm the room: none of those changes the curvature at the wall, and none of them changes the coat.

Six things coming out of six liquids #

The law is easy to apply and the range it covers is wider than the arithmetic suggests, because the capillary number is a product of three quantities that each span decades.

Two of those are worth a sentence each. The galvanising line is the case where the law is used in anger: coating weight is a purchased specification, it is controlled by line speed, and the two-thirds exponent means a ten per cent speed change moves the coating by six and a half per cent rather than by ten. And the finger is the case where the law is nearly always misapplied — a finger is not flat, and the curvature of a finger is comparable with the capillary length, so the number above is indicative rather than right.

The energy, done properly #

The claim that the flat film is free deserves the same treatment as everything else here, which is to be computed rather than asserted, and doing so turns up a comparison worth having.

Three energies are in play per unit area of plate coated.

The work done against the extra viscous force is $\gamma\,\mathrm{Ca}^{2/3}$ times about 3.8, which for water at a centimetre a second is 0.7 millijoules per square metre.

The surface energy created is $\gamma$ times the new area — one new interface, so about 72 millijoules per square metre. A hundred times larger.

And the potential energy of the raised film is $ho g h^2/2$, which is a nanojoule per square metre and is negligible against both.

So coating is overwhelmingly a surface-energy transaction with a small viscous overhead, and the overhead is what sets the thickness. That is the resolution of a paradox that would otherwise be awkward: the film’s thickness is decided by a competition between viscosity and surface tension, and the energy is almost entirely surface tension, because the viscous term appears at the two-thirds power of a small number.

Doubling the withdrawal speed raises the viscous overhead by 59 per cent and the surface term not at all. There is no speed at which coating becomes viscously expensive in the sense that a bearing is.

What the picture cannot show #

The plate is perfectly wetted. The derivation assumes the liquid meets the solid at zero contact angle, so that the static meniscus runs smoothly onto the film. A partially wetting liquid has a contact line, the whole matching argument is different, and below a critical speed the plate comes out dry — a transition with no counterpart in the equations above.

The film is thin against the capillary length and the derivation says nothing beyond that. Above $\mathrm{Ca}\approx10^{-2}$ the measured thickness leaves the two-thirds line, gravity begins to drain the film while it is still being formed, and the correct treatment carries a gravity term the scaling above dropped.

The flow is creeping. The film Reynolds number is returned by the solver and checked, and every number quoted here is from a case where it is below one — the same restriction this collection’s other slow-flow essays run under.

The film is uniform and real ones are not. A withdrawn film is unstable to long-wavelength disturbances and to drainage, and what is measured a second later is not what was deposited. The number here is the thickness at the moment of formation.

And there is no evaporation, no solvent and no drying. Every industrial coating is a solution that loses most of its volume after deposition, and the relation between the wet film computed here and the dry one that is sold is a chemistry problem.

What a coater controls, and what it cannot #

The two-thirds exponent has one consequence for process control that is worth stating plainly, because it runs the other way from intuition.

A coating line’s thickness is set by speed, and speed is the variable a line operator has. Because the exponent is less than one, the film is less sensitive to speed than it would be if the relation were proportional — a one per cent speed error gives a two-thirds of one per cent thickness error. That is a helpful direction to be wrong in and it is the reason dip coating survives as a process at all.

The unhelpful direction is viscosity, which enters at the same two-thirds power and is far harder to hold. A bath that warms by ten kelvin over a shift loses a quarter of its viscosity, and the coating thins by seventeen per cent — an enormous drift, invisible to anyone watching the line, and the reason coating baths are temperature-controlled to a fraction of a degree.

And surface tension enters at minus two-thirds through the capillary number and at plus one-half through the capillary length, so it appears overall as $\gamma^{-1/6}$ — a sixth power, which is almost no dependence at all. Contamination that halves a bath’s surface tension changes the film by twelve per cent. Of the three properties, the one a chemist would worry about first matters least.

The ceiling on the one variable a line has #

The two-thirds law says thickness rises with speed, and it says nothing about how fast a line may be run. That limit exists, it is sharp, and it comes from the one part of the problem this derivation assumed away: the contact line.

The derivation above takes the liquid to wet the solid perfectly, so there is no contact line — the static meniscus runs continuously onto the film. A partially wetting liquid has one, where solid, liquid and air meet, and it has to travel along the plate at the withdrawal speed. It manages that by deforming: the apparent contact angle grows with the capillary number, because the liquid wedge near the line has to accommodate a viscous stress that diverges as the wedge thins.

An angle cannot grow past 180°. When it reaches that, the line can no longer keep up, and what happens is that the air wins the race: a film of it is dragged in under the advancing liquid. That is air entrainment, and it is the failure mode every coating process is really limited by. The entrained film breaks into bubbles, and a coating with bubbles in it is scrap.

So the practical maximum speed of a coating line is a wetting instability rather than anything in the film equation, and it is why the two-thirds law is used to choose a speed inside a window whose upper edge the law says nothing about. Higher-viscosity liquids entrain at lower speeds, which sets the awkward trade of the whole business: viscosity is the cheapest way to thicken a coating and the surest way to lower the ceiling on how fast it can be laid.

Two responses to that are worth knowing, because both are counter-intuitive and both are used.

Change the gas. The entrained thing is air, so its properties matter. Running the coating station at reduced pressure, or in a helium atmosphere, delays entrainment substantially and buys real line speed — an intervention with no liquid, no plate and no film in it.

Or stop dipping. Slot-die and curtain coating do not rely on a free contact line advancing against a static bath; a curtain arrives with momentum of its own and helps the line along. Both reach speeds dip coating cannot, and both give up the property that makes dip coating attractive in the first place — that its thickness is set by a law with nothing adjustable in it.

Who found it, and when #

Landau and Levich published in 1942 — five years before the boundary-layer energy equation this collection uses in the third thickness, and by an entirely separate route in the Soviet physics literature, and Derjaguin gave an equivalent treatment the following year, which is why the result carries three names in different places. Bretherton’s tube version came in 1961 and the recognition that the two share a constant came later still.

The surprising connection is with a problem in a different subject that shares the exponent as well as the mathematics. The advance of a wetting front, the spreading of a droplet, the shape a drop takes when it is falling, the rise of a film in a corner and the coating of a plate are all governed by a balance between viscous stress and capillary pressure in a thin wedge, and all of them produce exponents of the form $\mathrm{Ca}^{1/3}$ or $\mathrm{Ca}^{2/3}$ for the same reason: the only way to make a viscous term and a third derivative of a thickness the same size is to scale the length by the cube root of the capillary number. The two-thirds is a consequence of a third derivative, and the third derivative is a consequence of Laplace pressure being a curvature.

Where the ladder goes next #

Below this rung is the capillary length itself, which is the only length in the problem, and the price of a gradient, which is what makes the claim that the flat film is free into a computation rather than an assertion.

Beside it is the drop that is not a tear, which is the same competition between surface tension and something else with the shape as the unknown, and the film that will not be squeezed out, which is a thin film whose thickness is set by a completely different balance and whose behaviour is correspondingly nothing like this one’s.