A dip dyes a depth, not a share

Cut across a denim warp thread and look at the end under a lens, and the colour is not all the way through. There is a blue skin a few fibres thick and a pale core inside it. Every account of why jeans fade the way they do starts from that picture, and the picture is usually offered as a fact about indigo, as though the dye had some particular reluctance to go further in.

It is mostly a fact about a cylinder. A yarn dyed after it is spun is a porous rod that the dye has to reach from outside, and dye moves into a porous rod by diffusion: through the liquor that fills the spaces between the fibres, slowed by the fibres taking it out of the liquor as it passes. Diffusion into a cylinder was solved exactly in the nineteen-fifties, and the solution says three things about any dip, whatever the dye. It says how deep a ring a dip of a given length makes. It says that the depth hardly depends on how thick the yarn is. And it says what more dips do, which is not what they are usually assumed to do.

The yarn is the pore space, not the fibre #

Treat the yarn as the site’s own arithmetic has always treated it: a circle of fibre at a packing factor, so that a count in tex and a fibre density give a diameter. A seventy-four tex cotton yarn packed at 0.6 is 321 micrometres across and holds 435 fibres, each about twelve micrometres thick. Four tenths of the section is not fibre at all. In a dye bath it is liquor, and that liquor-filled network is the road the dye travels.

A cotton’s own water is a twentieth of what a cloth holds counted the same space from the other side, as the channels between fibres in which a wet yarn keeps most of its water. How high a cloth wicks sent liquid along those channels by capillary suction. A dip sends dye across them, from the yarn’s surface toward its axis, and the process that carries it is diffusion rather than flow: the liquor inside a wetted yarn is not moving, and the dye molecules wander through it at random from where they are plentiful to where they are scarce.

Two things slow that wandering below what it would be in open water. The first is the pore network itself: the path round the fibres is longer than the straight line and narrower than the section, and a porous medium’s diffusivity is its liquid’s diffusivity times the pore fraction raised to a power somewhat above one. Archie measured the power for rocks in 1942. Bruggeman’s analysis of packed cylinders gives 1.5, and 1.5 is the value used here, carried with a bracket from one to two.

The second is the fibre. A dye has an affinity for the fibre it is meant to colour, and at equilibrium each unit volume of fibre holds some multiple $K$ of the dye concentration in the liquor beside it. So when a unit of dye arrives at a place in the yarn, most of it is taken out of the liquor by the fibres there, and only what is left is free to diffuse further. The front advances as though the diffusivity had been divided by the yarn’s capacity to soak dye up:

$$D_{\text{eff}} = \frac{D_0\,\varepsilon^{m}}{\varepsilon + (1-\varepsilon)\,K},$$

with $D_0$ the dye’s diffusivity in water, $\varepsilon = 1 - \phi$ the pore fraction and $m$ Archie’s exponent.

Crank’s cylinder #

With the yarn’s surface held at the bath’s concentration from the moment it goes in, the dye inside a cylinder of radius $a$ follows one of the exactly solved problems of diffusion, set out in Crank’s Mathematics of Diffusion. The solution is a series in Bessel functions, and everything in it depends on the time only through one dimensionless group:

$$\tau = \frac{D_{\text{eff}}\,t}{a^2}.$$

The dye arrives as a front. At a dimensionless time of a thousandth it has reached three per cent of the way to the axis; at four thousandths, six per cent; at sixteen thousandths, thirteen. Each fourfold increase in time doubles the depth, which is the signature of diffusion and the reason diffusion is slow at large distances: the front’s depth goes as the square root of the time. The core reaches half the surface’s strength only at $\tau \approx 0.20$.

Since a profile that falls steadily inward divides the section into a dyed ring and an undyed disc at exactly one depth, the depth at which the concentration is half the surface’s is the ring’s edge whatever shape the profile has. That is the depth quoted throughout, and it is what a lens on a cut end would show as the boundary between blue and pale.

Twenty seconds in a denim range #

A rope-dyeing range for denim runs its warp through a sequence of indigo baths, each dip lasting some twenty seconds, and after each one the yarn is squeezed and hung in the air for a minute or two. The airing matters to the arithmetic: indigo goes into the bath in its reduced, soluble form, and in the air it oxidises back to the insoluble pigment wherever it has got to, and stays there. Each dip’s profile is frozen at the end of its twenty seconds.

A small dye molecule diffuses in water at a few times $10^{-10}$ square metres a second, and $4 \times 10^{-10}$ is used here. The affinity is the one input the arithmetic cannot supply: it is chemistry, it depends on the form the dye is in, and for leuco-indigo it is known to depend strongly on the bath’s alkalinity. So every absolute depth below is stated at a named affinity, and each claim that survives any value of it is a claim about a ratio.

At an affinity of ten, a twenty-second dip dyes a seventy-four tex yarn to 18 micrometres, one and a half fibre diameters: a ring made of the yarn’s outermost fibre or two. At an affinity of five it is 26 micrometres, and at eighty it is 6, half a fibre, so the ring is only the outer part of the surface fibres. Doubling the dip deepens the ring by the square root of two, from 18 to 27 micrometres at forty seconds, and halving it shortens the ring to 13. The observed denim ring, a few fibres at most, corresponds to affinities around five to twenty, and that is as far as the arithmetic goes toward the chemistry.

The same depth in every yarn #

The depth has a property that the pictures of cut denim ends never make explicit. At short times the front’s depth is set by $\sqrt{D_{\text{eff}}\,t}$ and does not contain the radius at all. The radius enters only through the curvature, which slightly concentrates the dye as it converges on the axis, and at these depths that correction is small. At an affinity of ten, the same twenty-second dip dyes a ten tex yarn to 21.7 micrometres, a twenty tex yarn to 19.7, a seventy-four tex yarn to 18.2 and a hundred tex yarn to 18.0. A factor of ten in count moves the ring by a fifth.

So the hero figure is the argument drawn. Both yarns have the same thin ring. The twenty tex yarn is 167 micrometres across and its ring of twenty micrometres is a quarter of its radius and forty-two per cent of its section. The seventy-four tex yarn is 321 micrometres across and its ring of eighteen is a ninth of its radius and twenty-one per cent of its section.

The share falls with the count roughly as one over the diameter, and the diameter goes as the square root of the count. A dip dyes a depth, and a coarse yarn has more yarn behind the same depth. A fifteen tex shirting yarn is nearly half dyed by one dip at an affinity of ten; a denim warp is a fifth dyed; a hundred tex yarn under a fifth.

This is the reason ring dyeing is characteristic of heavy cloths dyed in the yarn, and it needs no special property of the dye. Any short dip of any dye with a real affinity puts a ring in a coarse yarn and something much nearer a full colouring in a fine one. A finer yarn is a worse yarn found the fine yarn paying for its fineness in evenness. In a dip it gets something back: a fine yarn is dyed nearly through by a dip that leaves a coarse one mostly white inside.

Eight dips are one ring eight times #

A denim range does not dip once. It dips six or eight times, and the obvious assumption is that each dip pushes the colour further in, so that a heavily dipped yarn is dyed more nearly through.

The arithmetic says it does not. The airing after each dip fixes the indigo where it lies, and the next dip starts again with a surface held at the bath’s concentration and an interior holding pigment that no longer takes part. If the fixed pigment does not change how the fibre takes up fresh dye, each dip lays down the same profile over the last one.

Eight dips hold eight times the dye of one, all of it in the same ring. The shade deepens and the ring stays where it was. The depth at which the load falls to half its surface value is eighteen micrometres after one dip and eighteen after eight, because a profile multiplied by a constant crosses its own half-maximum at the same place.

That is the rope-dyer’s reason for many short dips rather than one long one. A long dip would deepen the ring as the square root of its length: a dip eight times as long dyes nearly three times as deep. Eight short dips put eight times the dye into the same thin skin. Which of the two a dyer wants depends on what the cloth is for, and a denim maker wants the second, for reasons a later essay on the cut makes exact.

A long bath would dye it through #

The same solution says how long the bath would have to hold the yarn for the dye to reach the middle.

The axis reaches half strength at a dimensionless time of 0.20, and the time scales with the square of the radius, so it goes in proportion to the count. For the seventy-four tex yarn at an affinity of ten it is five and a half minutes: sixteen dips’ worth of immersion in one piece. At an affinity of eighty it is forty minutes. A fine shirting yarn needs about a minute.

That is why a yarn dyed on a package, where it sits in circulating liquor for an hour or more, comes out dyed through, and why a piece-dyed cloth from a jig is coloured to the middle of its yarns. The distinction between a ringed yarn and a level one is not a distinction between dyes. It is the ratio of the time in the liquor to $a^2/D_{\text{eff}}$, and a denim range keeps that ratio small on purpose.

A tighter yarn rings more thinly #

The packing factor appears twice in the arithmetic, pulling opposite ways. A yarn of the same count packed tighter is narrower, $a \propto \phi^{-1/2}$, and a narrower yarn has less section behind the same ring, so the ring is a larger share of the radius. But packing tighter also closes the pores: $\varepsilon$ falls, the pore network’s diffusivity falls with $\varepsilon^m$, and the ring itself gets thinner.

The pores win, under every law tried. From a packing of 0.45 to 0.75, the depth as a share of the radius falls by a factor of 1.5 when Archie’s exponent is one, 1.8 at 1.5 and 2.2 at two. In micrometres, at the middle law, the ring goes from 26.5 to 11.4.

Mercerising is a packing factor found that a treatment which swells cotton in caustic soda changes one number in the yarn’s arithmetic and a long list of consequences follows. Packing is that number here too. Twist is what sets it in a spun yarn, and a hard-twisted warp therefore takes a thinner ring from the same dip than a soft-twisted one. The mechanism is a porous rod’s, and it is the same one that makes a hard-twisted yarn known to wet more slowly.

Where the surprise is #

The surprising connection is between ring dyeing and a result that belongs to wicking. The quantity that decides how deep a dip reaches is a diffusivity divided by a square length, and the quantity that decides how fast a cloth wicks is a permeability, which is also set by the pore size. In both, the yarn’s own geometry supplies the length and its packing supplies the pore. A denim warp and a sportswear yarn are the same porous object put to opposite uses. One is meant to keep a liquid near its surface for twenty seconds, the other to pull a liquid through its whole length in minutes — the race a wick reaches its ceiling in the time its cloth takes to dry timed — and both are governed by one fraction, $\varepsilon$, in the same way.

The second connection is with floats and abrasion, which explained that a denim’s face is three quarters warp and fades where its floats are worn. What it could not say is why the fading is abrupt, white arriving at the creases rather than a gradual paling. The ring is the answer, and the answer is quantitative: a denim warp’s colour lives in its outer fifth, and a denim yarn is coarse because a coarse yarn is the one a dip rings.

The model and its inputs #

The diffusion is Crank’s cylinder: a circular rod whose surface is held at the bath’s strength from the moment it goes in, with a constant diffusivity and linear, instantaneous sorption. The series is summed to sixty terms of the zeros of $J_0$; below a dimensionless time of 0.012, where the series converges slowly, Crank’s short-time expansion is used instead. The two are required to join to a part in a thousand at the switch, and the uptake is required to equal the profile integrated over the section at four times.

The yarn’s radius and porosity come from the count and a packing factor of 0.6, the value every thread diameter in these essays has been computed at. The dye’s diffusivity in water, $4 \times 10^{-10}$ square metres a second, is quoted as an order of magnitude for a small dye molecule, not measured. The tortuosity is Archie’s law, exponent 1.5, bracketed from one to two. The affinity is assumed, at three values that span a factor of sixteen; the ring’s depth goes as its inverse square root.

The claims that do not depend on any of these are that a dip’s ring has the same depth in micrometres at every count to within the curvature’s correction, that more dips of the same length multiply the dye without moving the ring, and that the time to dye through goes as the count. Each of those is required of the arithmetic, and the depth’s independence of count is required to hold to within ten per cent from twenty to a hundred and fifty tex.

What the pictures cannot show #

A real yarn’s fibres are not at fixed radii. They migrate between the surface and the core as they run along the yarn, which is what a straight fibre cannot share the load found makes a twisted yarn strong. Dyed after spinning, a yarn is coloured by position and not by fibre: pull one fibre out of a denim warp and it is blue along the stretches where it was near the surface and pale where it dived into the core. The sections drawn here put fibres on a lattice and shade each by its radius, which is right for the colour’s geometry and silent about any single fibre’s history.

A real bath is not a surface held at one concentration. The squeeze rolls, the dye’s depletion near the yarn in a poorly stirred bath and the fixed pigment already on the fibres all change the boundary condition. The account assumes they do not, and the one that matters most is the last: if pigment already fixed in the ring reduces the fibre’s affinity for the next dip’s dye, successive dips reach slightly further in, and the ring deepens a little with the number of dips rather than not at all.

And the colour is not computed. How blue a yarn looks depends on how much pigment is in the layers an eye’s light reaches, which is an optical question about scattering in a fibre assembly. The arithmetic here says where the dye is. It does not say what shade that makes.

Who worked out which part #

The diffusion equation for a cylinder is Fourier’s and its series solution goes back to the nineteenth century; the form used here, with its short-time expansion, is from John Crank’s The Mathematics of Diffusion (1956, second edition 1975), which remains the standard source. The porosity-to-diffusivity law is G. E. Archie’s, from electrical measurements on sandstones published in 1942, and the exponent of 1.5 is D. A. G. Bruggeman’s 1935 result for a medium of packed particles.

The practice of rope dyeing and the dependence of indigo’s ring on the bath’s alkalinity are the dyeing trade’s; the relation between pH, the ionic form of leuco-indigo and the depth of penetration was worked out on production ranges in the late 1980s and 1990s, notably by J. Nolan Etters.

What is put together here is the yarn’s own count arithmetic as the cylinder’s radius and pore fraction, and the three consequences that follow without any chemistry: the depth that does not know the count, the dips that deepen the shade and not the ring, and the packing that thins it.

Still open: whether fixed pigment slows the next dip #

Every result about multiple dips rests on each dip seeing the same fibre. Indigo fixed in the outer fibres occupies some of the sites a fresh dye molecule would be taken up by, and a fibre that is part-saturated has a lower effective affinity for the next dip, which would let the dye front run slightly further in each time.

If that is so, the ring deepens with the number of dips, and by an amount that measures the saturation. The measurement is direct: cut ends of one warp taken after one, two, four and eight dips, with the pale core’s diameter read off each under a microscope. A core that does not shrink confirms the fixed-profile account; a core that shrinks as the square root of the dip count would say that the affinity falls as the fibres fill. Either answer is also a number for the affinity itself, which is the one input this account has had to assume.