A yarn's voids are not enough

A spun yarn is not solid. It is fibres with air between them, and the fraction of its cross-section that is actually fibre is the packing factor — about 0.6 for an ordinary ring-spun cotton, which is where every diameter on this site comes from.

So when the fibres swell there is an obvious place for the swelling to go. Forty per cent of the section is air; if the fibres take some of it the yarn need not grow at all, and every ratio that a fibre growing across itself disturbs stays where it was.

The obvious thought is wrong, and it is wrong by arithmetic rather than by measurement.

The one line the whole rung is #

Write V₀ for the dry packing factor and A for the ratio by which a fibre’s cross-sectional area grows. If the yarn’s own diameter does not change, the fibres now occupy

$$V_{\text{wet}} = V_0 \cdot A$$

of the section, because the same fibres are in the same circle and each is A times as big. For cotton at a packing of 0.6, with the width-implied area ratio of 1.44, that is 0.864; with the directly measured area ratio of 1.40 it is 0.840.

Those numbers have to fit under whatever the maximum packing is. Requiring it and solving for the yarn’s diameter ratio s,

$$s \;\ge\; \sqrt{\frac{V_0 A}{V_{\max}}}$$

and where V₀·A is already below V_max the inequality says nothing and the floor is one: the voids really could take it.

Everything now turns on V_max, which is the one number here that is a judgement.

Three limits, and only one of them fails to bind #

Three values are worth carrying and all three are.

0.9069 is the closest packing of equal circles in the plane. It is a theorem, it is an absolute upper bound for round fibres, and no spun yarn is anywhere near it.

0.75 is about the densest a heavily compacted assembly of real fibres reaches — a preform under pressure, a wool bale in a press.

0.65 is about the densest a ring-spun yarn reaches at ordinary twist, which is this site’s standing 0.6 pushed as far as it goes.

Against the working limit, 0.864 is impossible and the yarn must grow by at least 7.3 per cent; against the measured area figure, 5.8. Against the spun limit the floor is 15.3 and 13.7. Against the hexagonal bound there is no floor at all: 0.864 is below 0.9069 and the voids could in principle take the whole swelling.

The bound that does not bind is drawn because leaving it out would be choosing. An argument that quoted only the limits that give the answer it wants is an argument with a thumb on it, and the honest form is: the floor exists at every packing a real yarn lives at and vanishes only at a limit for round fibres in perfect order, which a twisted bundle of collapsed ribbons is not.

The threshold, which is the useful form #

Turn the inequality the other way and ask: at what dry packing do the voids stop being enough?

$$V_0 > \frac{V_{\max}}{A}$$

For cotton against the working limit that is 0.521; against the spun limit, 0.451; against the hexagonal bound, 0.630.

So the whole argument applies to any cotton yarn packed above about a half, which is every spun yarn there is. A very open assembly — a card web, a loose roving, a nonwoven batt — is below it, and there the swelling genuinely does go into the air and the assembly does not grow. The question is not whether fibres swelling into voids is possible. It is whether a yarn has enough of them, and a yarn is defined by not having enough.

That is the same shape as several results in this collection — the six-end satin that cannot exist is the cleanest of them: the interesting statement is not a mechanism but the size of a construction relative to a threshold, and the threshold has no free parameter in it.

What the floor is not #

It is a bound, and the distinction between a bound and a prediction is the whole reason this rung is worth writing rather than asserting.

The measured swelling of a real cotton yarn is 15 to 20 per cent on diameter, which is above every bar in the figure. That is where a bound is allowed to be. A model that came out at exactly the measured value would be a fit dressed as a derivation; a model that came out above the measured value would be refuted outright. Coming out below it, by a factor of two or three, is the only outcome that leaves the argument standing and says something.

What it says is that the voids take some of the swelling and not all of it. The yarn grows less than its fibres do — twenty per cent of fibre diameter gives fifteen to twenty of yarn diameter — and the difference is the air being used up. The floor says how much of it is unavailable, and the gap between the floor and the measurement says how much of the rest was actually taken.

The identity that makes the free-swollen state defensible #

Everything downstream of this rung uses a yarn that swells at constant packing, which is to say a yarn whose diameter grows exactly as its fibres’ diameters do.

That is not an extra assumption smuggled in. It is what “the packing factor is unchanged” means, and the arithmetic is one line: if the fibre volume fraction is the same, the yarn’s area has risen in the same proportion as the total fibre area, so the yarn’s diameter has risen in the same proportion as a fibre’s diameter. It is asserted to machine precision across every fibre and three counts, because it is algebra and algebra is entitled to that tolerance.

The reason to state it plainly is that the alternative reading — a yarn that swells by the fibre’s area ratio rather than its diameter ratio — is a factor of 1.44 rather than 1.2 and would make every number in this ladder wrong by a fifth. That confusion is easy and it is exactly the kind that a check on an identity catches and a check on a value does not.

Viscose, where the floor is not marginal #

Cotton’s floor is real and modest. Viscose’s is neither.

Viscose swells thirty-five per cent on diameter, which is an area ratio of 1.82 on the implied figure and 1.50 on the low end of the measured range. At a packing of 0.6 the wet fibre fraction would be 1.09 — greater than one — on the implied figure, which is not merely above a packing limit but impossible outright: there is more swollen fibre than there is yarn.

So a viscose yarn must grow, at every limit including the hexagonal one, and the floor there is 9.8 per cent. Against a working limit it is 20.7. And a viscose yarn does swell enormously in practice, which is why viscose fabrics are notorious for shrinking and for going limp in the wash — and why water tells cotton and viscose apart where nothing in this site’s geometry can — and why a viscose garment is often labelled dry-clean when the same construction in cotton is not.

That is the floor doing what a bound should: on a fibre where it is loose it is quiet, and on a fibre where it binds it binds hard and the fabric behaves accordingly.

What the twist does, and why it is not in the arithmetic #

There is a mechanism this rung deliberately leaves out, and it is worth naming because it points the other way from everything above.

A spun yarn holds together because its fibres run in helices and press inward on one another. That inward pressure is what the twist buys, and it is what makes a bundle of loose staple behave like a cylinder at all. Now swell the fibres inside it. The bundle wants to grow, the helices resist, and a yarn under tension from its own twist is a yarn whose packing can rise rather than whose diameter must.

So a tightly twisted yarn should swell less than a slackly twisted one of the same count, because it has more inward pressure holding the swelling into the voids. That is a real prediction, it is in the right direction, and this collection cannot compute it, because the site’s twist arithmetic gives a helix angle and a strength curve and does not give a radial pressure.

Which means the floor above is a floor for a yarn whose boundary is free, and a real twisted yarn sits somewhere between that floor and its fibres’ own swelling with the twist deciding where. The measured range of 15 to 20 per cent probably has the twist factor inside it, unresolved, and separating the two would take a measurement nobody in this collection has.

That is a limitation with a name and a direction rather than an unknown, which is the most that can be said for it here.

The constant-packing assumption is a pressure balance #

The free-swollen state — a yarn whose diameter grows exactly as its fibres do — is defended above as an identity rather than an assumption, which is right about the algebra and leaves open why a yarn should do it. A pressure balance answers that, and it uses only the two laws already on this page.

A yarn spun to a packing V₀ is held there by the inward pressure its own twist supplies, and by van Wyk’s cube law that pressure is kEV₀³. Swell the fibres and hold the yarn’s diameter: the packing rises to V₀A, and holding it there would need kE(V₀A)³ — a factor of

A³ = 1.44³ = 3.0

more pressure than the twist is providing. It is not providing it, so the yarn grows.

How far it grows is where the requirement falls back to what the twist supplies. With the diameter grown by s, the packing is V₀A/s², and setting kE(V₀A/s²)³ equal to kEV₀³ gives

s² = A, so s = √A = 1.2 —

which is the fibre’s own diameter ratio, and which is constant packing exactly.

So the free-swollen state is not an assumption; it is the equilibrium of a yarn whose twist pressure does not change. The identity the essay asserts to machine precision is the algebra, and this is the reason the algebra describes a real yarn.

And the shortfall is the twist pressure rising #

The balance also accounts for the gap the essay leaves open — that a real yarn swells 15 to 20 per cent where its fibres swell 20.

The twist’s pressure does not stay put. Swelling a fibre inside a helix lengthens its path and tightens it against its neighbours, so the inward pressure rises with the swelling. A rising resistance meets the same falling demand at a smaller s, so the yarn grows less than constant packing.

That is the observed direction and it is the observed size: a shortfall of nought to five points on twenty is what a modest rise in the twist’s pressure buys. The measured range is the balance with a second-order term in it, and the term’s sign is not in doubt.

It also converts the essay’s uncomputable twist prediction into a testable one with a rate attached. A yarn’s swelling should fall as its twist rises, and by the amount that makes van Wyk’s cube law balance the extra helical pressure — so a doubling of the twist factor, which quadruples the radial pressure, should move the swollen packing by the cube root of four and the diameter by its square root:

a fourfold rise in twist pressure buys about a thirteen per cent reduction in the yarn’s swelling — from twenty per cent down to about seventeen.

That is inside the reported range of 15 to 20, which is the point. The spread in the published yarn-swelling figures is the twist factor, unresolved, and a series at three twist levels on one count would separate it in an afternoon.

Which sharpens what the floor is for #

None of this makes the packing floor redundant; it changes what it is a floor on.

The floor says the voids cannot take the swelling at any pressure. The balance says what pressure would be needed and where the yarn actually settles. The first is a statement about geometry and the second about a competition between two elastic effects.

So the two results sit on top of one another rather than in place of one another: the floor rules out one answer and the balance picks the answer that is left, which is the ordinary relationship between a bound and a mechanism and is the one this collection has been looking for on this quantity since it started.

The measurement this bound is standing in for #

It is worth being clear about why a bound is being computed at all rather than a number being looked up, because “the swelling of a cotton yarn” sounds like something that should simply be in a table.

The trouble is in the measuring. A dry yarn’s diameter is already a difficult quantity — a yarn is fuzzy, compressible and not round, so the figure depends on how hard the instrument presses and every published diameter is really a diameter at a stated pressure. A wet yarn is worse in every one of those respects: it is softer, so it flattens more under the same pressure; it is hairier, because water lifts the surface fibres; and it changes while it is being measured.

So the reported spread of 15 to 20 per cent is not sloppiness. It is what a hard measurement of a badly defined quantity looks like, and it is the reason a constraint that needs only a dry packing factor and a fibre swelling is worth having.

This is a pattern rather than an accident. Several quantities in this collection are like it — the yarn’s transverse stiffness, the contact force at a crossing, the bending rigidity of a spun yarn — and in every case what made progress possible was finding something easier to measure that the hard quantity had to be consistent with. A packing factor is an easy measurement: weigh a length of yarn, measure its outside, divide.

What was counted, and how #

Three assertions, and one of them is on the shape of the answer rather than its value.

The floor inverts correctly. A yarn swollen by exactly the floor must have its wet fibre fraction land exactly on the maximum packing, by construction. That is checked as an identity, and it is the check that would catch a square root taken the wrong way round — an error that would give a floor of 0.93 instead of 1.073 and would look entirely plausible.

The free-swollen yarn keeps its packing. Asserted across eight fibres and three counts, to machine precision.

And every guard refuses. A packing factor above one, a packing limit that does not exist, a fibre with no water data: each is fed to the machinery and must complain. An assertion that has quietly stopped rejecting anything is worse than no assertion, because the build stays green and nobody looks again.

The counting here is thin by this collection’s standards, and deliberately: there is nothing to enumerate. What there is instead is a bound with two inputs, both of which are stated, one of which is a measurement with a spread, and the result reported over that spread rather than at a point.

Where the model stops #

The fibres are treated as round. They are not: a cotton fibre is a collapsed ribbon that rounds out as it swells, which is exactly the reason the two area columns disagree in the rung below. A bundle of ribbons packs differently from a bundle of cylinders and there is no clean maximum for it, which is why the working limit is quoted as a judgement and the hexagonal one as what it is.

The twist is not in the arithmetic. A real yarn’s fibres run helically and its packing varies with radius — denser at the core, looser at the surface — so a single packing factor is an average. A swelling yarn also untwists slightly, which changes the packing again. None of that is here.

And the yarn’s own boundary is treated as free. In a fabric it is not: the yarn is held by the threads it crosses, which is what the swelling a cloth cannot take is about, and a yarn that cannot grow freely is a yarn under pressure.

The generalisation #

A bound is worth writing down when its inputs are more certain than the thing it bounds.

The packing factor of a spun yarn is known to a few per cent. The area swelling of a fibre is known to a few per cent. The swelling of a yarn is a much harder measurement — it needs a wet yarn measured without squashing it — and is reported over a much wider spread. So a bound on the third from the first two is not a poor substitute for the measurement; it is a constraint the measurement has to satisfy, and it was available without going near a microscope.

The wider habit is to ask, of any quantity that is hard to measure, whether something easy to measure constrains it. Very often something does, and very often nobody has written the inequality down because a bound feels like a weaker result than a number. It is a different kind of result, not a weaker one: a number can be wrong and a bound can only be vacuous.

Who found it, and when #

Fibre swelling and yarn swelling have both been measured since the 1930s and the discrepancy between them — a yarn swelling less than its fibres — has been remarked on for as long. The usual explanation given is exactly the one this rung starts from: some of the swelling goes into the interstices.

The packing bound does not appear in that literature in this form as far as this collection can tell, probably because the packing factor and the swelling were measured by different people for different reasons and nobody put the two beside each other and divided.

What is this collection’s is the inequality, the threshold packing that follows from it, and the observation that the measured yarn swelling sits above the bound rather than at it.

Where the ladder goes next #

The yarn now has a diameter that grows and a length that does not. That is exactly the input the cloth geometry needs, and the next question is what a cloth does when its threads thicken and it has nowhere to put them — which is where the geometry stops having an answer at all.