What water does to a thread

This collection has computed cloth for a long time with no water in it. That is not a small omission in a subject half of whose vocabulary — relaxation, shrinkage, milling, wet-relaxed, washing — names something that happens in a bucket.

It is also not an omission that can be repaired by adding a coefficient somewhere. Water does not scale a fabric. It swells a fibre across and hardly at all along, and a change in one dimension of a thread and not the other is a change in every ratio this collection is built out of.

The one asymmetry everything rests on #

Suppose for a moment that a fibre swelled the way a warm metal bar does, equally in every direction. A cloth of it would come out of water a fifth wider, a fifth longer and a fifth thicker. Every cover factor would be unchanged, because a cover factor is a diameter divided by a spacing and both would have grown by the same factor. Every crimp would be unchanged, because a crimp is an excess length divided by a length. Every jamming sett would be unchanged in units of threads per unit of that cloth. The fabric would be a photograph of itself enlarged, and there would be nothing here to write about.

That is not what happens. A cotton fibre in water is about a fifth wider and about a hundredth longer, so d rises and l does not. The cover factor rises by exactly the swelling ratio. The crimp rises, because a thicker partner is a longer way round. The jamming sett falls. The thickness rises. And every one of those moves in the same direction, because every one of them has the diameter upstairs and a length downstairs.

The mechanism is that short, and this essay is about the number it needs and where that number comes from.

The table, and the two columns that measure the same thing #

The swelling figures are measurements. They are old measurements, made with a microscope on a fibre in a drop of water, and they carry the spread that goes with that.

Cotton widens by twenty per cent, wool by sixteen, silk by eighteen, viscose by thirty-five. Nylon by two and a bit, polyester by essentially nothing. Along their own length, cotton and wool by one and two tenths per cent, silk by one and four tenths, viscose by four and two tenths, flax by one tenth.

The literature also tabulates the area swelling directly, and here the table argues with itself.

A fibre whose section stays similar to itself gains area as the square of its width, so a twenty per cent widening implies a forty-four per cent area gain. The measured figure for cotton is forty to forty-two. For wool the width figure implies thirty-five and the measurement says twenty-five to twenty-six. For silk it implies thirty-nine and the measurement says nineteen. For nylon it implies four and seven tenths and the measurement says two to three.

Four of the ten fibres disagree with themselves, and always in the same direction. The implied figure is never below the measured range and is sometimes far above it.

That is not a transcription error and it is not a licence to pick whichever suits. It says that a swelling fibre does not stay similar to itself: it grows further in whatever direction a microscope was pointed at than the area measurement can account for. A cotton fibre is a collapsed ribbon with a twist in it, and a wool fibre is a scaled cylinder with a cortex of two different cell types. Neither is a circle expanding.

So the area swelling is carried here as an interval — the measured figure at the bottom and the square of the width change at the top — and every result that needs an area is computed at both ends. That is the honest use of two measurements of one quantity that do not agree, and it is asserted rather than described: the implied figure must never fall below the measured range, and the assertion would fail if a later edit reversed the relation.

Why a hundredth is not a rounding error #

The axial column looks like something to ignore. One and two tenths per cent, against twenty. It is the difference between a fibre and the same fibre — surely it can be dropped.

It cannot, and the rung that sets a wet cloth’s construction shows exactly where. When a cloth is wetted at constant thread length there are two competing effects on its size: the thicker partner costs extra crimp and closes the cloth up, and the extra thread length opens it out. The first goes as the square of the thickness over the span, so it is large in a close cloth and small in an open one. The second is flat: a hundredth of extra length is a hundredth wherever it is.

So there is a cover factor below which the second wins, and an open cloth grows in a wash. For a cotton cloth that crossing cover is 0.2299, and a cheesecloth is below it: it comes out of water half a per cent wider than it went in, and the reason is the column of this table nobody looks at.

Suppress the axial swelling in the arithmetic and every cloth shrinks, without exception, because the geometric term is the only one left. Put it back and one of them reverses. That is what a hundredth is worth when it is the only term of its kind.

Regain is a different quantity, and it is worth keeping apart #

Beside the two swelling columns the table carries regain: the mass of water a fibre holds at ordinary room conditions, as a fraction of its dry mass. Cotton 8.5 per cent, wool 16, viscose 13, polyester 0.4.

Regain is not swelling, and the two are not proportional. Wool holds nearly twice as much water as cotton by mass and swells less. Viscose holds half again as much as cotton and swells nearly twice as far. The reason is that regain counts water anywhere in the fibre, including in places that were already there, and swelling counts only the water that had to make room for itself.

Regain matters here for one reason and it is a bookkeeping one: every mass on this site is a conditioned mass. A yarn count is a mass per unit length measured at standard conditions, which for cotton means the yarn is 8.5 per cent water by weight before anybody starts. So the tex number a cloth is specified in already has water in it, and the diameter this collection computes from a count is a conditioned diameter and not a dry one. Saturating the fibre takes it further, and the swelling figures above are quoted from the dry state, so a cloth going from the shop to the wash moves less far than the table’s twenty per cent suggests.

That correction is not applied here, and it is the largest single thing this rung leaves undone. It is a shift of the origin rather than a change of mechanism, and every result below scales with it.

The correction the origin needs, and how large it is #

The section above names the conditioned origin as the largest thing this rung leaves undone. It can be estimated two ways, the two agree, and the answer is a third of the quantity the whole ladder runs on.

By volume. A conditioned cotton holds 8.5 per cent of its dry mass as water. Adding that water’s own volume to the dry fibre’s — 0.658 cubic centimetres per gram at cellulose’s density, plus 0.085 — gives a volume ratio of 1.129, so a linear ratio of 1.063.

By regain. Cotton’s saturation regain is about 25 per cent, so a conditioned fibre is a third of the way to saturation; a third of a twenty per cent swelling is 6.8 per cent.

Two routes, six and a bit per cent either way. A conditioned cotton fibre is already six or seven per cent swollen before anything is wetted, and the swelling remaining between the shop and the wash is

1.20 ÷ 1.065 = 12.7 per cent, not twenty.

Which scales the whole ladder by a third #

Every result downstream carries the driving quantity linearly or through a ratio, so every one of them moves.

The cover multiplier falls from 1.2 to 1.127, so a wet cloth’s cover rises by an eighth rather than a fifth.

The threshold at which a wetted cloth’s holes close completely is one over that multiplier, so it moves from a dry cover of 0.833 to 0.887 — a harder specification for a shower-proof cotton than the uncorrected figure suggests, and one that fits the reputation of such cloths as very difficult to weave.

And the crossing cover at which a cloth stops shrinking and starts growing is a ratio of the two swellings, so it moves less: both the transverse and the axial figures are measured from dry and both are correspondingly reduced, and the ratio between them survives better than either.

That last is worth noticing because it is the pattern this ladder keeps producing. The results that are ratios are robust to the origin and the results that are levels are not, and the essay’s own list — critical swelling, cover, pressure — is mostly levels.

And it is worst for the fibres that hold the most water #

The correction is not the same for every fibre, and it is largest exactly where the swelling figures are largest.

Wool conditions at 16 per cent regain against a saturation regain near 33, so a conditioned wool fibre is half way to saturation already. Its tabulated sixteen per cent transverse swelling is, from the conditioned state, nearer eight.

Polyester conditions at four tenths of one per cent, so its origin correction is nothing at all — and its swelling is nothing at all either, so the correction changes nothing.

Viscose conditions at 13 per cent against a saturation regain around 30, so it is 43 per cent of the way, and its thirty-five per cent becomes about twenty.

So the ordering of the fibres by practical swelling is not the ordering of the table. The hygroscopic fibres arrive at the wash already part-swollen, and the more hygroscopic they are the more of their tabulated swelling has already happened.

Cotton and viscose keep their gap — twenty against twelve rather than thirty-five against twenty — so the pair the geometry cannot distinguish is still distinguished, by a smaller factor. Wool moves closest to cotton, from sixteen against twenty to eight against thirteen, which brings two fibres the table separates comfortably into much the same place.

None of that changes a mechanism. It changes which fibres a mechanism separates, and it does so by moving an origin nobody had written down.

What the fibre supplies, and what the structure does #

There is a tension in this ladder that is worth naming rather than letting a reader find.

This collection’s standing position is structure before fibre: what a cloth does is decided mostly by how it is put together and only secondarily by what it is made of, and the site’s own worked examples are that two cloths of the same yarn in different weaves set and firm quite differently while two cloths of different fibres in the same weave behave almost alike. Water is the first subject here that is unambiguously a property of the fibre.

The position survives, and in an interesting form. The fibre supplies exactly one number — how much its diameter grows — and everything after that is geometry. The swelling a cloth can accommodate, the shrinkage it undergoes, the cover it reaches, the pressure it builds: none of those is looked up. They are computed from the same Peirce solution and the same closure condition the rest of this collection uses, with one input changed.

So the honest statement is not that structure comes before fibre, and not that it does not. It is that a fibre property enters through a single scalar, and the structure decides what that scalar does. Two cloths of the same cotton at different setts do completely different things in water. That is the site’s thesis with a fibre constant in it, and it is stronger for having one.

Two fibres this collection cannot tell apart #

The clearest evidence for that reading is a pair of fibres.

Cotton and viscose are both cellulose. The density table gives both 1.52 grams per cubic centimetre; the mechanics table gives both a modulus of 8 gigapascals and a fineness of 1.7 decitex. So a 20 tex yarn of either has the same diameter to the last figure — 0.1671 millimetres — and therefore the same crimp at any construction, the same cover, the same jamming sett, the same bending bracket and the same shear locking angle.

Every geometric result on this site is the same number for the two fibres. Water separates them twice over: viscose swells 1.75 times as far across and three and a half times as far along, and it loses half its strength wet where cotton gains a tenth.

That is not a small addition to a table. It is the first thing this collection carries that distinguishes two materials its geometry treats as one, and it does so through the one number the geometry was missing.

What was counted, and how #

Nothing in this rung is a new measurement. What is done here is arithmetic on measurements, and there are four assertions on it.

The three material tables name the same fibres. The density table, the mechanics table and the swelling table are three files, and a fibre that exists in two of them and not the third is a fibre whose results come back undefined three functions later. The check is one string comparison and it runs at import.

Every working value sits inside its own reported range. A swelling of twenty per cent quoted against a range of fourteen to twenty-three is a choice inside a measurement; a swelling of twenty-five quoted against the same range is a typing error, and this is what catches it.

The implied area swelling never falls below the measured one. The half of the two-column comparison that holds, asserted as the bound it is: a fibre whose measured area grew faster than the square of its measured width would be one that had lengthened as well, and none of them has.

And swelling is anisotropic in every fibre that swells at all. The ratio of the transverse figure to the axial one is required to exceed 1.2 for every fibre in the table, which nylon only just satisfies and flax exceeds by a factor of a hundred and sixty. It is asserted as a property of the whole table rather than as a value for cotton, because a later change that made the two dimensions comparable would make every result in this ladder uninteresting and this is what would say so.

Where the model stops #

Nothing here is time-dependent. A fibre takes minutes to reach equilibrium with liquid water and hours to reach it with humid air, and every number in this ladder is the equilibrium. A cloth taken out of a wash and hung up is somewhere between two states this collection can compute and at neither of them.

There is no chemistry. Swelling is an input here, not a prediction. Why cellulose swells and polyester does not is a question about hydrogen bonding between chains in an amorphous region, and it is not in this collection. What is in this collection is what a swelling of a stated size does to a fabric of a stated construction.

The fibre is treated as growing uniformly along its length. A real fibre has thicker and thinner places, and its swelling is not uniform either. This matters more here than it might, because the whole ladder turns on a comparison between a diameter and a spacing and a diameter that varies is an average being used as a value.

And there is no plasticity anywhere. Everything computed from these numbers is recoverable: dry the cloth and it comes back. Real repeated wetting and drying does not fully recover, which is why a garment shrinks most in its first wash and less in each one after. That behaviour needs a model this collection does not have.

The generalisation #

A material constant that enters a geometric model at exactly one place is worth more than one that enters at several.

The swelling enters here as a multiplier on a diameter, and nowhere else. That is why the ladder above can compute a critical swelling, a shrinkage, a crossing cover and a pressure without ever asking what the fibre is again — and why a result that came out independent of the yarn count could be trusted rather than checked for luck.

The converse is the warning. A constant that entered at three places would make every result a function of three uncertainties that move together, and no amount of care in reporting would separate them. This ladder is tractable because water was allowed in through one door.

Who found it, and when #

The swelling of fibres in water was measured systematically from the 1930s onwards and the figures used here are the standard ones tabulated in Morton and Hearle’s Physical Properties of Textile Fibres, which is where a textile physicist still goes for them. The anisotropy was known immediately and was never controversial: it follows from the fibre’s own structure, in which long molecules lie roughly along the axis and water gets between them rather than into them.

The wet-to-dry strength ratios are equally old and equally standard, and the fact that cotton is stronger wet was reported as a curiosity long before anyone explained it.

What is this collection’s is nothing about the constants and everything about what follows from them, which is the subject of the rungs above.

Where the ladder goes next #

Straight up. The next rung asks what a yarn does when its fibres swell, and finds that a yarn’s voids are not enough to absorb them — a bound with no measurement in it, and where the arithmetic starts.