Indigo is too little to close the pores it dyes through

A denim warp is dipped in indigo six or eight or twelve times, and a fibre that fills dyes deeper with every dip: the dye each dip fixes near the surface uses up some of the fibres’ room there, the next dip’s liquor passes through that skin taking less out of it, and it is taken up further in. For a fibre one dip fills by a fifth, eight dips take the ring from 14.0 per cent of the yarn’s radius to 23.5.

That essay ended on the thing it had left out. Indigo is not fixed as a dissolved dye sitting in sites; it is oxidised in the air between dips to an insoluble pigment, and part of that pigment lodges in the spaces between fibres — the pores the next dip’s liquor has to diffuse through. A skin full of pigment should be a less permeable skin. That pulls the other way: filled sites carry the dye deeper, filled pores hold it back.

Which wins is not a question about mechanism. Both are real and the model below contains both. It is a question about volume: how much of the pores a denim’s indigo is able to fill, given how much indigo a denim carries. The answer is: not much.

The pores the liquor travels through #

The dip is the same radial diffusion as before. Dye is held at the bath’s strength at the yarn’s surface and diffuses inward through the liquor between the fibres at $D_0\varepsilon^m$, Archie’s law with exponent 1.5 on the liquor fraction ε, while the fibres take it up into the room they have left. What is added is one term in two places.

Where the fibres are a share q full, suppose the fixed dye has also taken a share of the pore space, rising with q to a value called the clog at a full fibre. The liquor fraction there becomes

$$\varepsilon_{\text{local}} = \varepsilon\,(1 - \text{clog}\cdot q),$$

and both the diffusivity and the liquor’s share of the capacity follow it:

$$D = D_0\,\varepsilon_{\text{local}}^{\,m}, \qquad C = \varepsilon_{\text{local}} + (1 - \varepsilon)K(1 - q).$$

Between two shells the flux takes the harmonic mean of their diffusivities, so a nearly closed skin throttles everything inside it the way a single tight layer throttles a stack. With the clog at nought every factor is exactly one and the arithmetic is the earlier essay’s, step for step; that is required, and it holds to the last digit.

The clog is the one new quantity, and like the share a dip fills it is not measured here. The calculation is run first across its whole range, from open pores to nine tenths closed, and then bounded from what a denim actually holds.

Filled pores hold the ring back and never stop it #

With the clog at a half, eight dips take the ring to 19.6 per cent of the radius instead of 23.5; with nine tenths of the pores closed at a full fibre, to 15.8. The more the pores fill, the more slowly the ring deepens. None of them stops it. At nine tenths it still creeps outward at every dip, 14.0, 14.2, 14.5, 14.7 and on to 15.8, and the same holds at every share a dip can fill that the calculation was run at.

The reason pore filling never wins outright is in where the two effects act. Saturation acts wherever dye has been fixed, and it lowers the capacity of the skin so that dye passes through it more readily. Clogging acts in the same skin and lowers the diffusivity, so that dye passes through it more slowly. But a closed pore is only closed where the fibres beside it are nearly full, and a nearly full fibre is one that takes almost nothing from the liquor going past. The dye that does get through a clogged skin arrives at empty fibres with nothing taken from it on the way, and it still lays down its load deeper than the dip before.

A clogged skin fills to the same load #

The profiles show the shape of what clogging does.

All three reach exactly the same load at the surface. Below it, the clogged profiles fall away sooner and more steeply. Clogging turns the onion-skin profile of a fast-filling fibre into something nearer a crust: a dense outer layer, and less beyond it. With the pores open the ring reaches a third of the way to the axis; with them nine tenths closed, under a fifth.

That is the qualitative signature the earlier essay predicted: pore filling slows the front and leaves the ring shallower than saturation alone would. It also takes dye. At a fifth a dip, the yarn holds 13 per cent less dye after eight dips with the pores half closed and 25 per cent less at nine tenths, because the empty fibres deeper in see less of each dip.

The surface does not see the pores #

The surface’s load being the same in every profile is not a coincidence of these numbers. It is exact.

The outermost shell of the yarn sits in the bath. Whatever the pores below it do, it meets the bath’s full strength for the whole of every dip, and it fixes what an exposed fibre fixes: a share s of the room it has left. After n dips its saturation is therefore

$$q_{\text{surface}} = 1 - (1 - s)^n$$

at every clog, to the last decimal. A pore can only slow the dye on its way to somewhere; the surface is not on the way to anywhere.

This matters for measurement, because it separates the two effects cleanly. The surface’s darkness depends on the share a dip fills and on nothing else. The ring’s depth depends on the share and on the clog. One cross-section’s surface gives the share; the ring then gives the clog.

How much pigment a denim holds #

The clog has a natural ceiling, because pigment is matter and occupies its volume.

A yarn that ends its dips holding a stated share of its own weight in indigo has, averaged across its section, a share q̄ of its fibres’ room filled. A full fibre would therefore hold that weight divided by q̄. That much pigment, at its own density, in a yarn of cotton packed at 0.6, occupies a definite share of the yarn’s volume; set against the 0.4 of the yarn that is pore, it is the clog the pigment would make if every grain sat in the pores and none inside the fibres:

$$\text{clog} = \frac{\varphi}{1 - \varphi}\cdot\frac{\text{indigo on the yarn}}{\bar q}\cdot\frac{\rho_{\text{cotton}}}{\rho_{\text{indigo}}}.$$

q̄ itself moves a little with the clog, so the two are solved together. Indigo is taken at 1.2 grams per cubic centimetre, at the light end of what is quoted for it, against cotton’s 1.52; a lighter pigment takes more room, so this makes the bound larger. And two shades are taken: a light one carrying 1 per cent of the yarn’s weight in indigo and a dark one carrying 3, near the heavy end of what a denim warp is dyed to.

A dark denim’s indigo, all of it in the pores, fills at most a fifth of them at the yarn’s surface — 19.8 per cent when a dip fills 3 per cent of an empty fibre, 14.4 when it fills a fifth, 9 when it fills three fifths. A light shade fills between 3 and 6 per cent. The surface is where the pigment is densest, so deeper in the share is smaller still.

The bound falls as the share rises, which looks backwards and is not. A fibre that fills slowly has a great deal of room, so the same weight of indigo is a small share of its capacity, and its “full fibre” would hold a great deal of pigment; the clog at a full fibre is then large, but the fibres are nowhere near full and the pores they actually fill are the same few per cent. What matters to the dye is the product, the share of the pores filled where the dye has been fixed, and that is what the figure draws.

Saturation wins, by four to fifteen times #

Put the bound into the dips and compare.

At a fifth a dip, saturation deepens the ring by 9.5 points of the radius and the dark shade’s pigment, all of it in the pores, takes back 1.3 — 13.8 per cent of the deepening. In micrometres the open ring goes from 18.5 to 31.1 and the clogged one to 29.3: pore filling moves the ring by under two micrometres. A light shade takes back 4 per cent.

Across every share from 3 to 60 per cent a dip, the dark bound pulls the ring back by between 1.0 and 1.9 points of the radius. As a share of the deepening that is large only where the deepening is small: at 3 per cent a dip saturation deepens the ring by about one point, and the bound takes back most of it. From a tenth a dip upward, where saturation moves the ring by more than four points, it outweighs the clogging by between four and fifteen to one, and that is with every grain of the pigment put where it does the most harm.

So the earlier essay’s answer stands, with a correction of about a seventh. A fibre that fills dyes deeper with every dip; the pigment narrows the road but does not close it.

Two sections still tell them apart #

The earlier essay proposed a measurement: cut the yarn after its first dip and after its eighth, and read the share a dip fills from how much the ring has grown. Clogging makes the ring grow less, so a ring read as though the pores were open reads the share short.

A true share of a fifth with the pores half closed grows the ring 1.40 times, and read off the open-pore curve that says 12 per cent; at nine tenths closed it says 4. At the dark shade’s bound the misreading is small — a fifth reads as 17.5 per cent — and that is the most the pigment can do.

But the two-section measurement already had a second reading in it, the surface’s darkness, which the earlier essay offered as a check that the two readings agree. With the pores in the model the check becomes a measurement. The surface gives the share exactly. If the ring then reads less, the gap is the clog, and the curves above convert one into the other. Two cross-sections, after one dip and after eight, measure both unknowns the model has.

What the pores do to the fade #

A ring-dyed yarn whitens all at once, at the wear that cuts through the ring, and the earlier essay found that eight dips into a fibre that fills a fifth delay that from 3.1 per cent of the yarn worn away to 6.6. At the dark shade’s bound the ring is 22.2 per cent of the radius and the first white comes at 6.1 per cent. Clogging gives back about a sixth of the extra wear that saturation buys — half a per cent of the yarn out of three and a half — and the conclusion that deep dyeing delays the fade only if the fibre fills is unchanged.

The same holds for every wear-measured result that hangs on the ring — a float fades late and hard, a crease across a twill fades in dashes and along it as one line. Each is measured from the ring’s depth and moves with it, so each gives back about the same share of what saturation added.

Where the pigment actually sits #

The bound puts every grain in the pores. Real indigo does not do that. Much of it is fixed inside the fibres, in the swollen cellulose that what water does to a thread opens and that closes again on drying — which is where a dye’s sites are, and where a solid narrows nothing the liquor uses between fibres. With half the pigment inside the fibres the clog at a fifth a dip falls to 0.085 and the ring is pulled back by 6.7 per cent of its deepening instead of 13.8.

That makes the conclusion stronger rather than weaker. The only arrangement in which the pores matter at all is the one the pigment is least likely to take. And it puts this effect in the same family as a cotton’s own water: the fibre’s own space and the space between fibres are two different reservoirs, and what lodges in one does little to the other. A yarn is two fifths pore — a space large enough to carry the liquor, though still too small to take a fibre’s swelling — and a few per cent of indigo is a small part of it.

It also says why pore filling matters in other finishing and not here. A coating fills the crowns before it bridges the holes because a coating is laid on at many times a dye’s weight. Indigo is a few per cent, and in its pores it is a film on the walls, not a plug.

The model named #

Each dip is twenty seconds of radial diffusion into a 50 tex cotton yarn packed at 0.6, its surface held at the bath’s strength, through pores whose liquor fraction is $\varepsilon(1 - \text{clog}\cdot q)$ with $D_0 = 4\times10^{-10}$ square metres a second and Archie’s exponent 1.5, harmonic-mean diffusivities between shells, and sorption linear and instant into the room left, $K(1 - q)$ at $K = 10$. Between dips the liquor is removed and $s(1 - q)c$ is added to the saturation. The radius is cut into 160 shells and stepped explicitly within the stability limit set by the fastest shell and the emptiest. The bound sets the clog from indigo at 1.2 grams per cubic centimetre, cotton at 1.52, and 1 or 3 per cent of indigo on the yarn after eight dips, iterated with the mean saturation it produces.

What was counted #

Eight dips at clogs of 0, 0.2, 0.5 and 0.9 and shares of a tenth, a fifth and two fifths; the bound at seven shares from 3 to 60 per cent for each shade; the ring-growth curves for four clogs across seven shares, and the reading of a share from them. Required of it: the open model reproduced exactly at a clog of nought; the surface’s saturation equal to $1 - (1 - s)^n$ at every clog, every share and every dip; the ring shallower at every larger clog and deeper than one dip’s at every clog up to nine tenths; the bound reproducing its own mean saturation, falling when pigment is put inside the fibres, and taking back less than half the deepening at a fifth a dip; and the surface reading the share exactly while the ring alone reads it short.

What the model assumes and cannot show #

The clog grows in proportion to the fibres’ fill. Pigment might instead collect where the liquor slows, at fibre contacts and in the narrowest pores, which would close the paths that matter with less pigment than an even film needs. The bound is a bound on volume, not on placement, and a pigment that plugged the narrowest throats first could do more than this calculation allows — though not more than the volume it has.

Indigo is fixed between dips, all at once. The leuco dye is oxidised by the air as the yarn hangs, and in a thick yarn the core may oxidise later than the surface; a dip that arrives before the last one’s pigment has formed meets open pores. That would weaken the clogging further.

Archie’s law is the pore model. Its exponent is taken at 1.5, and its validity at a porosity reduced by a fifth is assumed. A steeper law makes clogging count for more: across the range of one to two that the first dip essay allowed, the dark shade’s bound at a fifth a dip takes back 0.9 to 1.7 points of the radius, 7 to 23 per cent of the deepening. Even at the steepest, saturation wins by more than four to one.

The share and the shade are assumed, as before. Nothing here measures either; the two cross-sections are how to.

Who worked out which part #

Archie’s law is from his work on porous rock in the 1940s, and diffusion with a filling capacity and a narrowing medium is the standard account of pore blocking in packed beds and ion-exchange columns. That indigo is fixed as a pigment and builds up in a ring is the dyer’s working knowledge. Putting the pigment into the pores of the earlier essay’s dip model, bounding the clog from the pigment’s own volume, finding that a dark denim’s indigo takes back about a seventh of the ring’s deepening, and turning the surface’s darkness from a check into a measurement of the share were done here.

Still open: whether a slow fibre lets the front run ahead #

Everything here, and in the two essays before it, takes the fibres’ uptake to be instant: the liquor arriving at a fibre gives up its dye at once, in the proportion the fibre’s room allows. A cotton fibre is itself a porous solid, and dye entering it diffuses in over a time that may not be small against a twenty-second dip. A fibre that takes up slowly lets the liquor run further ahead of the fixed dye, and that would deepen the ring by a mechanism that has nothing to do with either the room or the pores. Whether it does — whether a dip is short enough that the fibre’s own uptake rate, rather than its capacity, sets where the dye ends — is a two-scale diffusion, the yarn’s pores and each fibre’s interior, and the dip model would need a fibre inside every shell to answer it.