Why felting needs water

A wool fibre is scaled, and the scales point one way. So it slides more easily root-first than tip-first, agitation moves it back and forth, and it nets a displacement in the easy direction. That is a ratchet, it is irreversible in the ordinary thermodynamic sense, and it is why a wool jumper comes out of a hot wash three sizes smaller and never comes back.

This collection computed that several rungs ago and left one thing unexplained: felting is a wet process. Agitating a dry wool fabric does very little. The usual account is that water lubricates the fibres so they can slide.

That account has the sign wrong, and the arithmetic that follows from getting it right has a saturation in it that nobody would guess.

Water does not lubricate; it widens the gap #

Wool’s two static coefficients, scoured, are about 0.22 with the scales and 0.48 against them when dry.

Wet, the with-scale coefficient falls to about 0.15 and the against-scale one rises to about 0.62. Every figure here is a measurement with a spread — the with-scale range is 0.11 to 0.20 wet and 0.19 to 0.26 dry, and the against-scale range is 0.50 to 0.75 wet and 0.40 to 0.55 dry — and what matters is the relation rather than the values.

So water does lubricate in one direction and does the opposite in the other. The mechanism is that a wet scale is swollen and lifted: the scale edges stand further proud of the fibre surface, so they catch harder on a fibre travelling tip-first, while the softened, water-covered surface slides more easily on a fibre travelling root-first.

The directional friction effect — the ratio of the two coefficients — goes from 2.18 dry to 4.13 wet. That is the number this rung is about, and calling it lubrication loses the whole of it.

Rectification is a function of the ratio alone #

The ratchet’s output is a net displacement per cycle, and the useful form of it is dimensionless: what fraction of each cycle’s motion is net rather than cancelled.

Take the distance moved per stroke as inversely proportional to the resisting friction, which is the simplest model with the right property. Then a cycle moves forward by 1/μ_with and back by 1/μ_against, and the rectified fraction is

$$\frac{1/\mu_w - 1/\mu_a}{1/\mu_w + 1/\mu_a} = \frac{r - 1}{r + 1}, \qquad r = \frac{\mu_a}{\mu_w}$$

Both coefficients have gone. The rectification depends on their ratio and on nothing else, which is asserted here rather than derived in prose: two friction pairs with the same ratio and different coefficients must rectify identically, to machine precision, and they do.

That is a real simplification. It means every question about how much of the agitation is being turned into felting is a question about one number, and every question about how fast it happens is a separate question about the coefficients themselves.

And it saturates #

Here is the part that is not obvious. (r − 1)/(r + 1) is a saturating function: it is zero at r = 1, a third at r = 2, three fifths at r = 4, and it approaches one and never reaches it.

So doubling the directional friction effect does not double the rectified fraction. Water takes wool’s ratio from 2.18 to 4.13 — a factor of 1.89 — and the rectified fraction from 0.371 to 0.610, which is a factor of 1.64.

The felting rate rises by less than the ratio does, and the reason is arithmetic rather than chemistry.

The net displacement per cycle rises by more than either — a factor of 2.05 — because that carries the coefficients as well as their ratio, and the wet with-scale coefficient being lower means each stroke is longer. So there are three different multipliers depending on which question is being asked, and quoting one for another is the mistake this rung exists to prevent.

The assertion in this collection’s gate is on the ordering: the rectified fraction must rise by less than the ratio does. That is the statement with no measurement in it, and it would survive any revision of the four coefficients that kept the mechanism intact.

Which of the three multipliers a question wants #

It is worth being explicit about which of the three numbers in that table answers which question, because they are all called “how much water helps” in ordinary speech.

How much does a wash felt a garment? That is the net displacement per cycle times the cycles, so it is the 2.05. A wool jumper in a hot agitated wash felts about twice as fast as one being rubbed dry, per stroke.

What fraction of the machine’s work is going into felting? That is the rectified fraction, so it is the 1.64. Sixty-one per cent of the motion counts wet against thirty-seven dry, and the rest is cancelled by the return stroke.

How far from felt-proof is the fibre? That is the ratio itself, so it is the 1.89. This is the number a treatment has to move, because a treatment acts on the coefficients and not on the machine.

Three questions, three answers, and the arithmetic connecting them is not linear. Quoting the 2.05 to somebody designing a treatment would overstate the job by a fifth; quoting the 1.64 to somebody sizing a milling machine would understate the time by the same.

What a shrink-resist treatment has to do #

Chlorination and the resin treatments that followed do not glue the scales down in any picturesque sense. They degrade or bridge the scales so that the two coefficients converge, and a treatment is therefore describable by one number: how far it closes the gap.

Close it completely and the ratchet has nothing to rectify. That is the claim the whole chemistry is sold on and it is exactly right.

But the gap a treatment has to close is the wet one. A treatment works in a wash, and in a wash the gap is 0.15 to 0.62 rather than 0.22 to 0.48. So a treatment that would have flattened a dry fibre’s ratchet entirely leaves a wet one running.

Ask how far a treatment must close the wet gap to bring a wet fibre’s rectification down to what an untreated fibre has when dry — which is a fair statement of “as felt-resistant in a wash as it already is in a drawer” — and the answer is 62 per cent.

That is a demanding target and it is why shrink-resist treatments are aggressive: chlorination is a real chemical attack on the fibre surface, and it costs handle, lustre and some strength. A treatment that closed a third of the gap would be gentle and would achieve very little.

Why agitation and water are not interchangeable #

There is a temptation, given that both help, to treat water and agitation as two ways of doing the same thing. They are not, and the ratchet says so cleanly.

Agitation supplies the cycles. The net displacement is per cycle times the number of cycles, so agitation is a linear multiplier on the total and does nothing at all to the rectified fraction. A fabric agitated twice as long felts twice as much, in this model.

Water supplies the ratio. It changes what fraction of each cycle counts, and it does so with a saturating dependence.

So a long cold quiet soak does almost nothing — the ratio is favourable and there are no cycles — and a vigorous dry tumble does little either, because there are plenty of cycles and the ratio is poor. Felting needs both, and the trade’s recipe of hot water plus mechanical action plus alkali is exactly a recipe for maximising the two factors independently.

Which also says why a wool garment survives dry cleaning. A dry-cleaning solvent does not swell the scales, so the ratio stays near its dry value, and the mechanical action is gentle. Two factors both minimised.

Where the swelling itself comes in #

There is a second thing water does to a wool fibre and it is easy to conflate with the first.

The fibre swells sixteen per cent transversely, so the fibres in a yarn are pressed harder against one another and the normal force at every fibre-to-fibre contact rises. Friction is a coefficient times a normal force, so both coefficients act on a larger force wet and the absolute resistance to sliding goes up.

That does not change the ratio, so it does not change the rectified fraction at all. What it changes is the force needed to make the fibres move, which is a statement about how much agitation is enough rather than about how much of it counts.

So the swelling and the scale-lifting are two effects of one wetting acting on two different terms, and this collection can compute the second’s consequence exactly and the first’s not at all — because the normal force between fibres inside a compacted yarn is precisely the quantity the compaction arithmetic can only bracket.

What was counted, and how #

The rectification is a function of the ratio alone, asserted twice: against the closed form (r − 1)/(r + 1) at both states to machine precision, and by requiring two pairs with the same ratio and half the coefficients to rectify identically. The second is the one that would catch a change making the model depend on the coefficients separately.

Water widens the ratio rather than lowering both coefficients, asserted as an inequality on the table. A future edit that made wet wool simply slippery would fail it.

And the rectified fraction rises by less than the ratio does. The saturation, asserted as an ordering rather than a size.

The closure needed is found by search rather than solved, sweeping the wet gap in two thousand steps until the rectification drops to the dry untreated value. That is a search on a monotone function and the result is bracketed to a twentieth of a per cent.

How a treatment closes the gap matters as much as how far #

The sixty-two per cent is computed for a treatment that closes the gap from above — degrading the scales so that the against-scale coefficient falls towards the with-scale one. That is the picture chlorination suggests and it is not the only way to close a gap.

Suppose instead a treatment converges the two coefficients on their mean, raising the easy direction as it lowers the hard one — which is what a polymer laid over the whole surface does, since it covers the scale faces as well as their edges. Then the wet pair moves from 0.15 and 0.62 towards 0.385 from both sides, and the ratio required — 2.18, which is simply the dry ratio the target is defined by — is reached at

39 per cent closure, against 62.

Two-fifths of the job instead of two-thirds, for the same result, and the whole of the difference is which end of the gap the chemistry works on.

That is worth having because the two routes cost quite different things. Closing from above means attacking the scales, which is a chemical attack on the fibre surface and is paid for in handle, lustre and strength. Closing from both sides means covering the fibre, which is paid for in add-on, in hand and in a slightly different surface — and needs a third less of whatever it is doing.

It also agrees with what the ratchet’s own sensitivities say from the other direction. There the net displacement responds to the with-scale coefficient about five times as strongly as to the against-scale one, because a reciprocal’s derivative goes as one over the square and the small coefficient is the small one. Here the ratio needs less closure when the small coefficient is moved as well. Two different quantities, two different derivations, one instruction: work on the easy direction.

Where on the saturation curve an intervention lands #

The saturating form also says how much a further improvement is worth, and it says it as a derivative rather than as a fraction.

The rectified fraction is (r − 1)/(r + 1), so its slope is 2/(r + 1)². At wool’s dry ratio of 2.18 that is 0.198 per unit of ratio; at its wet ratio of 4.13 it is 0.076.

So a unit of directional friction effect bought at the wet end is worth less than two fifths of what the same unit is worth at the dry end. An intervention is being applied where the curve is flattest, and that is not a choice — a wash is where felting happens, so the wet ratio is the only place a treatment operates.

Read the other way it is a small consolation. A treatment that gets a wet fibre only part of the way is working on the steepening part of the curve as it goes, so the last stages of a closure are worth more than the first. A half-treated wool is worse than half-protected, and a nearly complete treatment is nearly completely effective. That is an unusual shape for a chemical treatment and it is the arithmetic’s rather than the chemistry’s.

Which prices the dry case #

One more number falls out and it is worth stating because it is often asserted without one. Dry felting is not impossible; it is slower, by the net-displacement ratio of 2.05.

A dry process needs about twice the strokes of a wet one for the same migration — not a thousand times, and not never. That is consistent with what happens to a wool garment worn without washing over years: it does felt, at the collar and the cuffs where it is rubbed, slowly, and nobody attributes it to water because there is none.

The direction the fibre travels, and why it does not average out #

A reasonable objection to the whole ratchet account is that a fabric contains fibres pointing every way, so whatever migration happens should cancel.

It does not, and the reason is that the ratchet is not about the fabric’s directions but about each fibre’s own. Every wool fibre has a root end and a tip end, and every one of them migrates root-first relative to itself, whatever direction that happens to point in the cloth. So the fibres do not all move the same way in space; they all move the same way along themselves, which is what makes the ratchet a fabric-scale effect rather than a fibre-scale curiosity.

The consequence is not a bulk translation of the fabric but a progressive entanglement: fibres burrowing past their neighbours, in every direction at once, until the assembly is knotted. That is why felting shrinks a fabric in both directions and why the shrinkage is not anisotropic in the way a woven cloth’s relaxation is.

It is also why felting cannot be undone by stretching. A relaxation shrinkage is a fabric that has moved to a state it can be pulled back out of; a felted fabric has fibres that have physically moved past one another and are now held by the tangle. Pulling on it breaks fibres before it untangles them.

The ratchet explains the migration and the tangle explains the permanence, and this collection has a model for exactly one of the two.

Where the model stops #

The friction figures are static coefficients and a felting fibre is sliding. This collection carries a static-to-kinetic ratio of about three quarters for fibre on fibre and does not apply it here, because the directional pair was measured statically and applying one ratio to both would leave the quotient unchanged anyway — which is the one thing that saves it.

The displacement per stroke is taken as inversely proportional to the friction. That is a shape rather than a prediction of millimetres, and it is chosen because it is the simplest thing with the right property. Any other decreasing function gives a different rectified fraction and the same ordering, so the results here are safe and the numbers are not.

Nothing here has entanglement in it. Felting is not only fibres migrating; it is fibres migrating and then locking, and a felted fabric is held by a three-dimensional tangle this collection has no model for. The ratchet explains the migration and stops there.

And there is no alkali. Real felting is done at pH 8 to 10, which swells the fibre further and changes the scale geometry again. The direction is known and the size is not.

The generalisation #

When an effect depends on two quantities only through their ratio, find out early — because it changes what a measurement is for.

The rectified fraction here has two coefficients in it and one degree of freedom. Once that is known, a laborious measurement of both coefficients is worth exactly as much as a much easier measurement of their ratio, and any experiment that varies both while holding the ratio is measuring nothing.

The saturation is the second half of the same point. A quantity that depends on a ratio through (r − 1)/(r + 1) has diminishing returns built in, so an intervention that doubles r is worth much less at r = 4 than at r = 1.5. Knowing the functional form says where on that curve an intervention would land before anybody builds it.

Who found it, and when #

The directional friction effect in wool was identified in the 1950s, and Makinson’s work through the 1960s and 70s established both the wet and dry coefficients and the ratchet mechanism. That water increases the effect rather than lubricating is in that literature and is not a new observation.

Chlorination as a shrink-resist treatment is much older, from the nineteenth century, and was in use long before anybody knew what it did.

What is this collection’s is the closed form for the rectified fraction, the observation that it depends on the two coefficients only through their ratio, and the saturation that follows — together with the 62 per cent closure a treatment needs, which falls out of putting the wet pair and the dry target in the same arithmetic.

Where the ladder goes next #

Into the knits, where a fabric’s change between two relaxation states looks exactly like a swelling and turns out not to be one — and the form of the constants is what says so.