A thin fibre coats by its own radius, and beads by it

What a plate takes with it found the film a plate carries out of a bath by matching two pieces. Far up the plate the film is flat and moves with it; down at the bath the surface is a static meniscus, gravity against surface tension. Between them is a short dynamic meniscus in which viscosity and surface tension compete, and whose curvature at its lower end is a pure number, 1.3376, times $\mathrm{Ca}^{2/3}/h_0$. Setting that equal to the static meniscus’s curvature where it meets the plate, $\sqrt2/\ell_c$, gives Landau and Levich’s film, $h_0 = 0.9458\,\ell_c\,\mathrm{Ca}^{2/3}$. The capillary length $\ell_c = \sqrt{\gamma/\rho g}$ was the only length in the problem, and the essay made a point of it: the film does not depend on the size of the plate.

That is true of a plate because a plate has no size the meniscus can feel. A fibre does. Wire, thread, optical fibre, a hair, a strand of glass being sized: all are drawn through baths, and most are far thinner than the 1.5 to 2.7 millimetres of a capillary length. Roll a plate up into a cylinder much narrower than that and the matching changes, because the static meniscus it has to match is a different shape. This essay finds the new shape, the new film, and then what the film does next — which a plate’s film never has to do, because a film on a fibre is a liquid cylinder, and liquid cylinders break.

The meniscus a fibre pulls up #

A fibre dipped in a bath of a liquid that wets it pulls up a meniscus, as a plate does, but the meniscus is curved two ways. Along the fibre it bends down to the bath, like a plate’s. Around the fibre it is a ring, and a ring of liquid round a thin cylinder is strongly curved.

For the static meniscus, surface tension times the total curvature balances the hydrostatic pressure at each height — the Young–Laplace balance that also shapes a falling drop, with gravity’s pressure in place of the air’s. Taking the capillary length as the unit, with $\psi$ the surface’s angle below the horizontal and $s$ the distance along it outward from the fibre,

$$r' = \cos\psi, \qquad z' = -\sin\psi, \qquad \psi' = -z - \frac{\sin\psi}{r},$$

starting vertical at the fibre, $r = b$, $\psi = \pi/2$, at some height $Z$. There is exactly one $Z$ for which the surface becomes level just as it reaches the bath far away; a higher start turns flat while still above the bath, a lower one reaches the bath still sloping. So $Z$ is found by shooting — integrate, see which way it misses, halve the interval — to fourteen figures.

The first figure shows what changes. On a fibre ten capillary lengths across, the meniscus rises 1.36, close to a plate’s $\sqrt2 = 1.414$, and spreads over a couple of capillary lengths of bath. On a fibre a hundredth of a capillary length across it rises 0.054 — five of the fibre’s own radii — and nearly all of that rise happens within ten radii of the fibre. Near a thin fibre the meniscus is almost a catenoid, the surface of zero mean curvature, whose axial bending is balanced by its bending around the fibre rather than by gravity. Gravity only decides where the catenoid finally meets the level bath, which is why its effect enters as a logarithm.

How high it climbs #

The second figure sweeps the fibre’s radius over six and a half decades. For thin fibres the rise follows the asymptotic law James found in 1974, $Z = b\,(\ln(4\ell_c/b) - \gamma)$, with $\gamma = 0.5772$ Euler’s constant: a few radii, growing only logarithmically as the fibre thins relative to the capillary length. At a thousandth of a capillary length the shooting and James’s formula agree to a part in ten million, and at a tenth they still agree to one in fifteen thousand. For thick fibres the rise approaches $\sqrt2\,\ell_c$; at three hundred capillary lengths it is 0.14 per cent short, which is the fibre’s residual curvature. The two limits are the checks that the shooting has the right equation and the right sign on the ring term, and they are the two ends of the capillary length’s reach: below it the object’s own size takes over, above it the bath’s.

What the meniscus asks of the film #

The film on the fibre is a thin liquid cylinder, and its surface is curved round the fibre with curvature $1/b$, so its pressure is higher than the air’s by $\gamma/b$. The static meniscus at the fibre, at height $Z$, is at a pressure $\rho g Z$ below the air’s. The dynamic meniscus between them has to take the liquid from one pressure to the other, and it can do that only by bending along the fibre, with an axial curvature

$$\kappa = \frac1b + \frac{Z}{\ell_c^2}.$$

On a plate the first term vanishes and $Z = \sqrt2\,\ell_c$, which gives back $\sqrt2/\ell_c$. On a thin fibre the first term dominates. The dynamic meniscus itself does not care what it is matched to — it is the same short lubrication region as on the plate, with the same universal equation and the same 1.3376 — so the film is

$$h_0 = \frac{1.3376\,\mathrm{Ca}^{2/3}}{\kappa} \;\longrightarrow\; 1.3376\,b\,\mathrm{Ca}^{2/3}\quad(b \ll \ell_c),$$

which is the law Quéré measured on fibres in 1999. It is also Bretherton’s law for a long bubble in a tube, which the plate’s essay met sharing its constant, where the tube’s radius plays the same part, and for the same reason: in each case the static shape the film must join is curved on the scale of the object, not of gravity.

The third figure is the answer across the whole range. Below the capillary length the coat is set by the fibre, not the bath. The two laws cross at $b = 0.71\,\ell_c$, where the true coat is 0.61 of either, the transition being broad because the rise changes only logarithmically. At a tenth of a capillary length the coat is 0.137 of what the plate’s law gives; at a hundredth, 0.014. The capillary length has not disappeared from the problem — it is still in the logarithm of the rise, and it still sets how thin a fibre must be before this applies — but it has stopped setting the film.

Three fibres in one oil #

The fourth figure puts numbers on it for a laboratory case: a silicone oil of 20 mPa·s, surface tension 20 mN/m and density 950 kg/m³, whose capillary length is 1.47 mm and whose capillary number happens to equal the drawing speed in metres per second. Drawn at a centimetre a second, a fibre of 25 µm radius carries 1.5 µm of oil, one of 125 µm carries 7.6 µm and one of 500 µm carries 25 µm. The plate’s law would give each of them the same 64 µm. For the thinnest, that is wrong by a factor of forty.

All three rise as the two-thirds power of the speed, since the exponent comes from the dynamic meniscus and that has not changed. What changed is the prefactor, and with it the scaling in the one variable a process engineer controls besides speed: the coat is proportional to the fibre’s radius. Double the diameter of a wire and it carries twice the varnish in thickness and four times in volume per unit length, at the same speed.

Why the ring makes the film thinner #

It is worth seeing the mechanism rather than only the formula, because it runs against the plate’s intuition. On a plate the flat film and the air are at the same pressure, and the only thing sucking liquid back towards the bath is the static meniscus’s modest curvature, $\sqrt2/\ell_c$. On a fibre the film’s own surface is curved round the fibre, so the film is at a higher pressure than the air, by $\gamma/b$, and the meniscus at its foot is at a lower one. The film is being squeezed back into the bath by its own ring curvature, harder the thinner the fibre, and the viscous drag of the moving fibre has to hold up a film against that. The balance settles at a thinner film.

The capillary number measures the same competition it did on the plate, and for a drop held in a shear flow: a viscous stress against the stress a curved surface can supply. What the fibre changes is which curvature supplies it. That is also why the result is so different from the film that will not be squeezed out between two solid surfaces, which is thinned by a load rather than by its own curvature and has no two-thirds law in it at all.

The film is a liquid cylinder #

A plate’s film, once it has left the meniscus, is finished. It is flat, it moves with the plate, and nothing drives it to change. A fibre’s film is a cylinder of liquid round a solid core, and a liquid cylinder has more surface than a string of beads of the same volume. Any ripple along it longer than its circumference lowers the area, and grows — the Rayleigh–Plateau instability that breaks a jet into drops.

For a film thin against the fibre, lubrication theory gives the film’s evolution. Its pressure is $\gamma(1/b - h/b^2 - h_{xx})$, higher where the film is thinner and more sharply curved, and liquid flows down that pressure gradient through a film whose mobility is $h^3/3\mu$. Measured in the fibre’s radius along the fibre, the drawn film’s thickness across it, and the time $3\mu b^4/\gamma h_0^3$, the equation has no parameters left at all:

$$H_t + \bigl[H^3\,(H + H_{XX})_X\bigr]_X = 0.$$

A ripple of wavenumber $K$ in units of $1/b$ grows at $K^2(1 - K^2)$: every ripple longer than the fibre’s circumference grows, and the fastest has $K = 1/\sqrt2$ — a wavelength of $2\pi\sqrt2 \approx 8.9$ radii — growing at a quarter.

The fifth figure checks the march that the rest of the essay leans on. The film equation is marched with a conservative difference scheme — the flux of liquid between neighbouring cells is computed once and subtracted from one and added to the other, so no liquid can appear or vanish — from a ripple of a millionth, and the measured growth rate matches $K^2(1 - K^2)$ to a fraction of a per cent at every wavenumber tried: 0.1343 against 0.1344 at $K = 0.4$, 0.2500 at the fastest, 0.1543 against 0.1539 at 0.9, where the grid is coarsest against the wavelength.

From ripple to bead #

The sixth figure follows one fastest wavelength of film from a ripple a thousandth of its thickness until its crest is twice the drawn film. For most of that time nothing can be seen: at sixteen units the ripple is still under a tenth of the film. Then, in the last few units, the liquid gathers into a single collar per wavelength and the film between thins, the collar sitting on a film that drains into it. The crest reaches twice the drawn film at 30.0 units.

How long it takes depends on how big the ripple was to start with, but only as a logarithm, which is what exponential growth at a rate of a quarter gives. From ripples of a tenth, a hundredth, a thousandth and a ten-thousandth of the film, the times to bead are 11.5, 20.8, 30.0 and 39.2 — four times the natural logarithm of the inverse amplitude, plus 2.3 for the non-linear finish. A ten-fold change in how smooth the fibre, the bath and the drawing are changes the answer by about a third.

How much fibre comes out smooth #

The fibre is moving. A bead forms at a place on the fibre a time $t_b$ after that place left the meniscus, by which time it is $U t_b$ further on. The length of fibre that leaves the bath still coated smoothly is therefore

$$L = U\,t_b = U\,\frac{3\mu b^4}{\gamma h_0^3}\,\hat t_b,$$

where $\hat t_b$ is the scaled time the march gives. On a thin fibre, putting in Quéré’s film, the viscosity and surface tension cancel into the capillary number and what is left is short:

$$L \approx 1.25\,\hat t_b\,\frac{b}{\mathrm{Ca}} \approx 38\,\frac{b}{\mathrm{Ca}}$$

for a ripple of a thousandth. It is the same construction a jet’s break-up length is made of, a growth time multiplied by a speed, with one twist that reverses the obvious expectation.

Drawn faster, a fibre beads sooner along its length. A faster draw makes a thicker film, as the two-thirds power of the speed; a thicker film beads faster, as the cube of its thickness, so the time to bead falls as the square of the speed; and carrying that time at the higher speed leaves a length falling as one over the speed. The seventh figure shows it for the three fibres in the silicone oil. At a centimetre a second the 25 µm fibre comes out smooth for 9.4 cm, the 125 µm one for half a metre and the 500 µm one for 3.4 m. At ten centimetres a second each is a tenth of that.

For a process that has to cure or dry a coat before it beads, that is the number that places the oven. And it says that the obvious response to a beaded coat — draw faster, so the coat reaches the oven sooner — makes it worse: the coat reaches the oven sooner in time, but not in length, and length is what the oven is placed by.

What was checked #

The eighth figure is the ledger. The meniscus reaches a plate’s $\sqrt2$ to 0.14 per cent at three hundred capillary lengths, and James’s thin-fibre rise to parts in a million at a thousandth and a ten-thousandth. The coat is Quéré’s law to five figures on a thin fibre and Landau–Levich’s to a part in a thousand on a thick one — the same 1.3376, taken from the plate essay’s own Runge–Kutta march, matched to two different static shapes. The marched film grows at $K^2(1 - K^2)$ at three wavenumbers and keeps its volume to a part in $10^{14}$ all the way to a bead.

What the model leaves out #

A film thin against the fibre. Both halves assume it: the dynamic meniscus’s matching, and the film’s pressure written as $1/b - h/b^2$. At a capillary number of a hundredth the film is 6 per cent of the radius and the assumption is sound; at a tenth it is 28 per cent and it is not. Quéré’s own data leave the $\mathrm{Ca}^{2/3}$ line there, thickening, as the plate’s do.

Gravity in the film. Once on the fibre the film is taken to feel only surface tension. Gravity drains it downwards at a rate that competes with the capillary flow once the fibre’s radius is a sizeable fraction of the capillary length, which puts the 500 µm fibre, at a third of the capillary length, at the edge of the model.

Inertia. At drawing speeds of metres a second the liquid’s inertia thickens the film beyond the two-thirds law, and the dynamic meniscus is no longer a lubrication flow. No slip is assumed at the fibre throughout.

A clean surface. Surfactants rigidify a film’s surface and thicken it, and a surface-tension gradient along the fibre drives flow of its own. The price that gradient exacts is computed elsewhere; here the surface is taken as clean.

One wavelength. The march follows one fastest wavelength with its neighbours identical. On a real fibre the ripples start from noise with a spread of wavelengths, and neighbouring collars later merge; the time to the first bead is set by the fastest, which is what the march gives.

Who found it, and when #

Landau and Levich derived the plate’s film in 1942 and Bretherton the bubble’s in 1961. The meniscus on a thin fibre is James’s, from 1974. White and Tallmadge treated wire coating in the 1960s, and Quéré, measuring films on fibres drawn from silicone oils in 1999, established the $1.34\,b\,\mathrm{Ca}^{2/3}$ law and its thickening at larger capillary numbers. The instability of a film on a fibre goes back to Goren in 1962, and the thin-film equation used for it here is the standard lubrication reduction. What is added is the whole curve between the two laws from one shooting calculation, and the smooth length that falls out of putting the two halves of the problem together.

Still open: when the film beads before it forms #

The smooth length $38\,b/\mathrm{Ca}$ assumes the film exists before it beads. It is a long length at small capillary numbers — thousands of radii — but it falls as the capillary number rises while the film thickens, and at some drawing speed the growth time of the Rayleigh–Plateau instability becomes comparable to the time liquid spends passing through the dynamic meniscus itself. Past that point the coat never exists as a uniform film at all: the fibre comes out of the bath already carrying a train of drops.

The next calculation drops the thin-film assumption from both halves — the film’s full pressure $\gamma/(b+h)$ and the growth rate of a thick annulus — and marches the meniscus and the film together, to find the capillary number at which beading moves into the meniscus, and how that number depends on the fibre’s radius against the capillary length.