A fibre that fills dyes deeper with every dip

A warp yarn for denim is dipped in indigo, aired, and dipped again, six or eight or twelve times, and each dip darkens it. A dip dyes a depth, not a share worked out what one dip does: the dye diffuses in through the liquor between the fibres, the fibres take it out of the liquor as it passes, and in twenty seconds a ring forms whose depth is the same number of micrometres in a fine yarn and a coarse one. It then found that eight dips deepen the shade and not the ring, because every dip meets the same fibres and lays down the same profile on top of the last.

That essay named the assumption that made the second result true: each dip’s fixed dye was taken to leave the fibre’s affinity unchanged. It said, in its own list of what was assumed, that if the fixed dye used up the fibre’s room, the ring would deepen with the number of dips.

It does, and the amount it deepens is a measurement of the one quantity dyeing arithmetic usually leaves out.

A fibre has room for so much dye #

The single dip is Crank’s cylinder: dye held at the bath’s strength at the yarn’s surface, diffusing through the pores at $D_0\varepsilon^m$, and slowed by the fibres, which take up $K$ times the liquor’s concentration. The front moves as though its diffusivity were divided by the capacity $\varepsilon + (1 - \varepsilon)K$ — the liquor’s share of the section plus the fibre’s share times its appetite.

A fibre’s appetite is not bottomless. It has sites a dye can sit in, and once a dip has fixed dye in some of them the next dip finds fewer. Written with $q$ as the share of the fibre’s room already filled, the next dip meets an affinity

$$K_{\text{eff}} = K(1 - q),$$

and the share one dip at the bath’s full strength fills in an empty fibre is $s = K/S$, where $S$ is the fibre’s whole capacity. That share is the new quantity. When $s$ is nought the fibre has room to spare and every dip is the first; when it is a fifth, five dips at full strength would fill the surface if nothing else changed.

Dye taken up less travels further #

Near the surface, where the earlier dips fixed their dye, the fibres are partly full, and the capacity there is smaller:

$$\frac{\partial}{\partial t}\Big[\big(\varepsilon + (1 - \varepsilon)K(1 - q)\big)c\Big] = \frac{1}{r}\frac{\partial}{\partial r}\left(r\,D_0\varepsilon^m\frac{\partial c}{\partial r}\right).$$

The liquor’s dye passes through a partly full skin taking less out of it on the way, so it arrives deeper before the empty fibres inside start taking it. A saturated skin is a skin the next dip crosses at the liquor’s own speed. The front of each dip therefore starts further in than the last one did, and the ring — the depth at which the fixed dye falls to half its value at the surface — moves inward.

After each dip the liquor is squeezed out and the yarn aired, which fixes what the fibres hold and adds it to $q$ in proportion to the room that was left: $q \leftarrow q + s(1 - q)c$. That is the whole of the model, and with $s = 0$ it is Crank’s cylinder repeated.

Room to spare gives one dip’s ring #

The first thing required of the calculation is that it gives back the earlier essay’s answer when nothing fills. It does: with $s = 0$, the profile after one dip matches Crank’s series to one part in ten thousand, the dye taken up matches his uptake to four figures, and eight dips leave the ring at one dip’s 14.0 per cent of the radius every time. The shade is eight times one dip’s, and the ring has not moved.

A fibre that fills lets the ring grow #

With the fibre filling, the ring deepens at every dip, and the rate depends on $s$ alone. At five per cent a dip, eight dips take the ring from 14.0 to 16.1 per cent of the radius; at ten, to 18.5; at twenty, to 23.5; at forty, to 33.7. The ring after eight dips is 1.15, 1.32, 1.68 and 2.41 times one dip’s.

The profile shows how. The surface’s load rises towards the ceiling the fibre’s room sets — five single-dip loads when a dip fills a fifth — and slows as it nears it, while the foot of the profile walks inward dip by dip.

At high enough saturation the profile begins to look like an onion: a plateau near the surface where the fibre is nearly full, and a front beyond it where the latest dips are laying down their dye. Each dip builds a layer under the last, and a yarn dipped often enough into a fibre that fills would be dyed in shells.

Each dip adds less, and puts it deeper #

A fibre that fills takes less from each dip than from the one before, because less of its room is left where the dye arrives first.

The eighth dip fixes 86 per cent of what the first did when a dip fills five per cent, 74 at ten, 55 at twenty and a third at forty. So the dyer’s familiar diminishing return — each dip darkening the yarn less than the last — and the ring’s growth are one effect seen twice. The shade slows exactly because the ring deepens: the dye that no longer fits near the surface is not refused, it is carried further in, where it adds depth and adds less to what the eye sees on the yarn’s face.

Two cross-sections measure the share #

The share $s$ is a chemical quantity: it depends on the dye, the fibre, the bath’s strength and the dip’s time, and nothing in this account measures it. But the model turns it into something a microscope can see.

Cut a yarn after its first dip and after its eighth, and measure the two rings. Their ratio is a curve of $s$ alone, rising from one at room to spare through 1.32 at ten per cent to 2.41 at forty. A ratio of 1.2 says a dip fills about seven per cent; a ratio of two says about thirty. The same two sections give the surface’s darkness, which the model says should have climbed to a stated share of its ceiling, and the two readings of $s$ have to agree — the check that the saturation, and not something else, is what moved the ring.

The same dye in more dips sits a little deeper #

The share a dip fills is set by the bath as well as the fibre. In the linear regime the fibre takes up $K$ times the liquor’s strength, so a bath twice as strong fills twice the share per dip, and a dyer choosing between a strong bath and a weak one is choosing $s$.

That makes a comparison the model can answer directly: the same amount of dye, laid down in few strong dips or many weak ones. For a fifth of the fibre’s whole room fixed across the section, sixteen dips at five per cent leave the ring at 18.7 per cent of the radius; eight at ten per cent, 18.5; four at twenty, 18.0; two at forty, 16.9. And one dip strong enough to do it alone would leave the ring at one dip’s 14.0, since a single dip’s depth does not depend on the bath’s strength at all.

So how the dye is split hardly matters once it is split at all — sixteen dips against four move the ring by under a percentage point of the radius — and what matters is how much dye the fibre has been asked to hold. The dyer’s habit of many dips from a weak bath is not, on this account, what makes the ring deep; the total is. What the many dips buy is evenness, which is outside this model entirely.

Where in the fibre the room is #

The capacity $S$ is a number here, and it stands for something with a location. A cotton fibre swells in water, and what water does to a thread is to thicken it by more than its own voids can take up: the fibre’s inside opens, and that opened inside is where a dye’s sites are. So the room a dip fills is room the bath itself has made, in the swollen state, and it closes again as the yarn is squeezed and aired.

That is the same distinction a cotton’s own water drew for moisture: the fibre’s own capacity is small against what the spaces between fibres hold. For dye it runs the other way. The liquor between the fibres carries the dye but keeps none of it after squeezing, and everything fixed is inside the fibres, so their capacity is the whole of the shade and the share one dip fills of it is the whole of this essay.

Extra dips delay the fade only if the fibre fills #

A ring-dyed yarn whitens all at once: an abrader cutting it flat shows nothing until the cut passes the ring, then white in a square-root rush. The ring is a wear indicator, and the depth at which it sits decides how much of the yarn must go before the first white shows.

With room to spare, eight dips make a darker yarn that shows white exactly as soon as a one-dip yarn: 3.1 per cent of the yarn worn away. With a fibre that fills, the extra dips buy wear: 6.6 per cent at twenty per cent a dip, 11.1 at forty, twice and three and a half times as much. So the trade’s belief that a deeply dyed denim fades later is true only if its fibres fill. A dyer who adds dips to a fibre with room to spare is buying shade and nothing else; one whose fibres fill is buying shade and a later fade together, and cannot have one without the other.

That carries straight into the cloth. A float fades late and hard, and a crease across a twill fades in dashes that join into one line along the twill at a stated wear; every one of those wears is measured from the ring’s depth, and every one moves in proportion to it.

A depth in micrometres, still #

The earlier essay’s headline was that a dip dyes a depth in micrometres, the same in a fine yarn and a coarse one, because the front’s position depends on the time and the diffusivity and not on the yarn’s size. For the 50 tex yarn the first dip’s ring is 18.5 micrometres; after eight dips at twenty per cent it is about 31.

The deepening should be a depth too, since each dip’s front starts from where the saturated skin ends and the skin is set by the same diffusion. If it is, the consequence falls on the share: a fine yarn’s ring is already a larger share of its section, and a ring that deepens by much the same number of micrometres a dip reaches a fine yarn’s core in fewer dips. A fibre that fills would then dye fine yarns through before coarse ones, which a fibre with room to spare never does. The calculation here is for one count, and the claim across counts is a prediction from its form rather than a result.

The model named #

Each dip is twenty seconds of diffusion into a cylinder whose surface is held at the bath’s strength, through pores at $D_0\varepsilon^m$ with $D_0 = 4\times10^{-10}$ square metres a second and Archie’s exponent 1.5, and its sorption is linear and instant into the room the fibre has left, $K(1 - q)$ at an affinity $K = 10$. Between dips the liquor is removed and what the fibres hold is fixed, adding $s(1 - q)c$ to the saturation. The yarn is 50 tex cotton at a packing of 0.6, the light denim of the worn-face essays. The radius is cut into 160 shells and stepped explicitly within the stability limit of the emptiest one, and the ring’s edge is interpolated between shells.

What was counted #

Eight dips at five shares, and eight at nine shares for the read-off curve; for each dip, the ring’s depth, the surface’s saturation, the dye taken up and the wear to first white. For the comparison of strong baths with weak ones, sixteen dips at each of four shares, read at the dip where the dye fixed across the section first reached a fifth of the fibre’s whole room: the sixteenth at five per cent, the eighth at ten, the fourth at twenty and the second at forty, which fix between 19.5 and 20.8 per cent of the room — close enough to compare, not identical. Required of it: the one-dip profile against Crank’s series and uptake, the unchanged ring when nothing fills, and at twenty per cent a ring deeper and a dip weaker at every step.

What the model assumes and cannot show #

The share is assumed. Nothing here measures how much of a cotton fibre’s room one indigo dip fills, and the answer is the whole of the result: at one per cent the ring barely moves, at forty it more than doubles. The two-section measurement is the way to get it.

Sorption is linear into the room left. A real fibre’s uptake is closer to a Langmuir curve, and its sites are not all alike; a fibre whose easy sites fill first would deepen faster early and slower late. And indigo is not a dye that sits in sites and stays: it goes in as the soluble leuco form, is oxidised to insoluble pigment by the air, and much of it ends up as particles in the fibre’s pores rather than on its molecules. A pigment that fills pores narrows them, which slows the next dip’s diffusion as well as lowering its uptake — an effect in the opposite direction on the ring, not in the model at all.

The profile is drawn, not seen. Every section here is shaded from the computed profile; how a microscope shows a real one — ragged, fibre by fibre, darker on fibres that face the surface — is not modelled, and a ring’s edge measured from a real section has a scatter of its own.

Who worked out which part #

Crank’s cylinder is from his Mathematics of Diffusion; diffusion with a filling capacity is how chromatography and ion exchange have been modelled since the middle of the last century, and the shrinking-core pattern it produces is a textbook result. Ring dyeing and its dependence on the number of dips are the dyer’s working knowledge.

What is done here is the connection: that the unchanged ring of the earlier essay is the limit of room to spare; that the ring’s growth over eight dips is a curve of one unmeasured share and so measures it; and that the delayed fade which deep dyeing is supposed to buy is bought only if that share is not small.

Still open: what the pigment does to the pores #

Everything here lets the fixed dye change how much the fibre takes up and nothing else. But oxidised indigo is a solid, and it sits partly in the spaces the next dip’s liquor has to travel through, so a heavily dyed skin should also be a less permeable one. That pulls the other way: filled sites carry the dye deeper and filled pores hold it back.

The two effects have different signatures. Saturation deepens the ring and flattens the surface’s growth together; pore filling slows the front and would leave the ring shallower than saturation alone predicts, with the surface still filling. Whether a real indigo range’s rings deepen with the dips or stay put is, on this account, a measurement of which of the two wins.