A knit's change of state is not its swelling

A knitted fabric has no dimensions until somebody says what state it is in. This collection has said so since the finishing field was built, and carries the three states the trade uses: dry-relaxed, wet-relaxed and fully relaxed, each with its own set of constants relating the fabric’s dimensions to its loop length.

Going from dry-relaxed to wet-relaxed a jersey loses 5.66 per cent along its courses and 2.44 per cent across its wales. Going on to fully relaxed it loses 9.09 and 6.98.

The states differ by whether the fabric has been wetted. So the obvious cause of the change is the fibre swelling, which is the one thing water definitely does.

It is not, and two things say so. Neither of them is a measurement of a fabric; both are properties of the form the constants take.

The constants have no diameter in them #

Munden’s result, which is the foundation of everything this collection computes about a knit’s dimensions, is that wales and courses per unit length are each a constant divided by the loop length:

$$w = \frac{k_w}{\ell}, \qquad c = \frac{k_c}{\ell}$$

and the constants are pure numbers. They were fitted across yarns, counts, fibres and loop lengths, and the whole content of the result is that they do not move with any of them.

A swelling is a change in d and in nothing else. If the swelling changed a fabric’s dimensions, then d would have to appear in a dimension — which means k_w and k_c would have to depend on d/ℓ, and Munden’s result is precisely that they do not.

So the argument is not that the swelling is too small or too large. It is that there is nowhere in the form of the constants for a swelling to enter, and if there were, Munden’s result would be false and the whole apparatus that produced the states in the first place would go with it.

That is a strong argument and it deserves the qualification it needs: the constants are empirical, fitted over the counts and yarns and loop lengths that were tested, and an effect that varied only over a range nobody sampled would have been missed. What makes it safe here is that the fitting deliberately spanned fibres — cotton, wool and nylon among them, whose swellings differ by a factor of eight — and the constants did not separate by fibre.

And the sizes are wrong in both directions at once #

The second argument is arithmetic and it fails twice.

The swelling is too large. Cotton swells twenty per cent. The whole change between dry-relaxed and wet-relaxed is 5.66 per cent lengthwise. So if the swelling were the cause it would have to be doing almost nothing — a fifth of its own size in one direction and an eighth in the other — and there would have to be a reason for the enormous attenuation.

And the change is anisotropic where the swelling is not. 5.66 per cent along the courses against 2.44 across the wales is a ratio of 2.32. A swelling is one number. It enters both of a loop’s dimensions through the same loop length, by the same constant, so it cannot produce a two-to-one anisotropy in a geometry whose two spacings are both proportional to ℓ.

Either objection on its own could be argued around. Together they do not leave room: the cause would have to be something that acts eight times more weakly than the swelling and twice as strongly in one direction as the other, and a swelling is neither.

A third argument, from the other fibres #

The two arguments above are about the form of the constants and about the sizes. A third comes from asking what the same table would look like if the swelling were the cause.

Munden’s states are quoted as a single set of constants used for every fibre. If the change between them were the fibre swelling, then a wool jersey and a nylon jersey would have to change by different amounts — wool swells sixteen per cent and nylon two and three tenths, a factor of seven — and a single table of state constants would be useless.

The trade uses one table. Knitted-fabric dimensional data are quoted per structure and per state, not per structure per state per fibre, and the practice works well enough that nobody has been driven to split it. A factor of seven in the driver would have shown up long ago.

That is weaker evidence than the other two, because it is an argument from what the industry has not needed rather than from a measurement. But it points the same way and it is cheap: if a proposed cause varies by a factor of seven across materials and the effect does not, the cause is wrong.

What is left #

Friction, and this collection already has it.

A knitted loop is a bent thread that would rather be less bent. The machine put it where it is under tension, and when the tension comes off, the loop’s own stored bending pushes it towards a rounder, shorter, wider shape. What stops it is friction at the interlocks — every place the loop passes through the loop below, which is where the two threads press on one another.

So a knit off the machine is in a state that is not its own equilibrium, held there by friction, exactly as a woven cloth off a loom is. It gets to its equilibrium by being helped: water as a plasticiser and a lubricant, heat, and above all mechanical agitation.

That mechanism has everything the swelling lacks. It is not a fixed size, so there is no attenuation to explain — how far a fabric relaxes depends on how much help it gets, which is why there are three states rather than two and why “fully relaxed” is defined by tumbling. And it is naturally anisotropic, because a loop’s course direction and its wale direction are not the same shape and their stored bending is not the same.

The three states are therefore three points along one process rather than three different physical situations. Dry-relaxed is a fabric left alone; wet-relaxed is one soaked, which lets the interlocks slip; fully relaxed is one soaked and tumbled, which supplies the cycles.

What water is doing, if not swelling the fabric smaller #

Two things, and both are about permission rather than about force.

It plasticises the fibre. A wet cellulosic is softer, so the loop’s own bending relaxes more readily and less force is needed to move a crossing. The account of why agitation helps applies here unchanged, with water lowering the barrier as well as agitation supplying the cycles.

And it lubricates the interlocks. This collection carries two friction coefficients — static and kinetic — and their ratio is what an agitated fabric exploits: a fabric at rest is held by the static coefficient and one being shaken slides at the kinetic one. Water lowers both for a cellulosic, which narrows the band of states friction can hold and lets the fabric settle nearer its true equilibrium.

Notice that the second of these is the exact opposite of what water does to wool, where it widens the gap between two directional coefficients rather than narrowing anything. Wool’s directional friction is a property of its scales and cotton’s is not, so the two fibres’ fabrics do genuinely different things in a wash: a wool fabric felts and a cotton one relaxes.

Same liquid, two mechanisms, opposite in character. One removes a restraint and the other creates a rectifier.

The swelling is not zero, and where it does appear #

It would be wrong to conclude from all this that the swelling does nothing to a knit. It does two things and neither is a dimension.

The fabric gets thicker. The thread is twenty per cent thicker and a jersey’s thickness is a couple of thread diameters, so a wet jersey is measurably thicker than a dry one. That is not in Munden’s constants because Munden’s constants are about the plane.

And the fabric gets heavier per unit area in a way that outlasts drying, because the areal density is the thread mass over the area and the area has fallen. That is the relaxation’s doing rather than the swelling’s, but the two arrive together and are reported together.

The occupancy moves from 1.129 to 1.355, which is a real change in how much of a stitch’s footprint is thread — and it is a change in a ratio rather than in either of the two lengths that make a dimension. That is exactly the distinction this rung turns on.

Why the wale direction moves less #

The anisotropy is the sharper of the two arithmetic arguments and it is worth asking what does cause it, since the answer is available.

A knitted loop is taller than it is wide in the fabric: the shape constant k_r, which is k_c over k_w, sits near 1.3 in every state. So the loop has more of its length in the course direction than in the wale direction, and more stored bending there to spend.

More than that, the two directions relax by different routes. Shortening the course direction means the loop heads pulling down onto the loops below, which is a rotation about the interlock and needs very little sliding. Narrowing the wale direction means the loop legs coming together, which needs the yarn to slide through the interlock — a much stiffer motion against a much larger frictional resistance.

So one direction relaxes by rotating and the other by sliding, and rotating is cheaper. A two-to-one anisotropy is exactly the shape that comparison predicts, and the shape constant staying almost fixed across the states — 1.29, 1.30, 1.30 — says the two directions relax nearly in proportion once the process is under way, with the difference concentrated in the first step.

This collection cannot compute either rate, and the account above is a description of a mechanism rather than a calculation. What it is not is a swelling.

What was counted, and how #

The constants satisfy their own identities and not exactly. Stitch density must be courses times wales and the shape factor must be their quotient, and the published sets very nearly satisfy both. That near-miss is evidence rather than error: it is what separately fitted regression constants look like, and a set that agreed exactly would be evidence that somebody had derived two of the three. This collection asserts the near-miss.

The dry-to-wet change is positive in both directions and markedly anisotropic, asserted with the anisotropy required to exceed 1.5 — which is the half of the argument that rules out an isotropic cause.

And the swelling exceeds the whole change several times over, asserted as an inequality with a factor of three in it. Both halves are on the published numbers rather than on anything computed here, so what they check is that the argument’s premises are still what the table says.

The occupancy ratio is asserted separately, to machine precision, because it is the one quantity that does move and it is the one a reader might reach for as the mechanism.

What would settle it experimentally #

The argument here is made from the form of the constants, and it is worth saying what a measurement would have to show to overturn it, because an argument that nothing could refute is not worth much.

Knit two jerseys of the same loop length in cotton and in polyester. Cotton swells twenty per cent and polyester a thousandth of a per cent, so if the swelling were the cause the cotton fabric would move between the dry-relaxed and wet-relaxed states and the polyester one would not.

They both do. Polyester knits relax in a wash, and the trade’s dimensional data for synthetic knits use the same states and constants of the same order. That is the single cleanest disproof available and it does not need a laboratory — it needs somebody to have knitted both, which the industry does daily.

Or knit the same fibre at two very different tightness factors. The swelling’s effect on any dimension would scale with d/ℓ, which the tightness factor is; the frictional relaxation’s would scale with how far from equilibrium the machine left the fabric, which is a different quantity. If the state change varied with tightness, the swelling would be back in play.

It does not, to the resolution the constants are quoted at, which is the empirical content of Munden’s result.

Neither test is new and neither was run to answer this question. They were run because a knitting industry needs dimensional data, and the answer to this question fell out of them. That is the ordinary way a mechanism gets ruled out: not by an experiment designed against it, but by a body of routine measurement that would look different if it were true.

Where the model stops #

Nothing here computes how far a knit relaxes. The mechanism is identified as frictional and this collection has no model of a knitted fabric’s relaxation to put numbers to. The three states are three sets of measured constants and this rung explains what they are not, which is a smaller achievement than explaining what they are.

The friction argument is qualitative here. The relaxation ladder’s arithmetic about resting bands applies in principle, and applying it needs the interlock force — which this collection has been unable to compute for a knit and which is the standing shortfall behind what stops a knit extending.

Munden’s constants are for a plain knit in one fibre family. They were fitted mostly on wool and are used here on cotton, which is standard practice and is one of the reasons the identity residuals are what they are.

And this says nothing about the permanent part. A knit that has been washed many times has fibres that have swollen and dried under load, and something has been spent that does not come back. This model recovers exactly.

The generalisation #

A cause can be ruled out by the form of an equation, without measuring anything.

The swelling was ruled out here twice, and neither argument needed a fabric. The first was that the constants have no diameter in them, so there is no slot for a swelling to occupy; the second was that one number cannot produce two different changes in two directions of a geometry that scales both by the same length.

Both are arguments about where a quantity could enter, and they are available before any experiment. That is worth more than a measurement showing the effect is small, because a small effect invites the reply that the conditions were wrong.

The habit is to write the model down, look at every place the proposed cause could appear, and check that at least one of them exists. Very often none does, and the question is settled.

Who found it, and when #

The dimensional constants and the three relaxation states are Munden’s, from the Hosiery and Allied Trades Research Association in the 1950s and 60s. That the relaxation is frictional in origin and is helped by wetting, heating and agitation is the standard account in that literature and is not disputed.

The wet-relaxed state’s definition — soak without agitation — and the fully-relaxed state’s — soak and tumble — are exactly the two protocols the argument here rests on, and they were designed to separate the two contributions.

What is this collection’s is the explicit refutation of the swelling as the mechanism, and the observation that the form of the constants forbids it rather than merely failing to require it.

Where the ladder goes next #

Out of the water entirely, into a count nobody had run: the theorem that says there is no six-end satin turns out to be about regular satins, and dropping that word brings the six-end satin back.