A wet knit's yarn is flatter

A knitted fabric’s own geometry demands a flattened yarn: two adjacent courses of the solved fabric approach to four fifths of a yarn diameter, and a round yarn cannot occupy that arrangement.

How flat depends on how much room the fabric leaves, and the room is set by the ratio of the yarn’s diameter to its loop length.

Wet the fabric and the diameter changes. The loop length does not.

What wetting does to a yarn #

A cotton fibre absorbs water and swells, and it swells across its own axis rather than along it: about twenty per cent in diameter and one per cent in length.

That anisotropy is the founding fact of this collection’s whole wet ladder — water does not scale a fabric, it swells the threads — and it means a wetted yarn is thicker and the same length.

A yarn is not a solid, so its diameter does not rise by the fibres’ twenty per cent: some of the swelling is taken up in the yarn’s own voids. This collection has computed how much and found the voids are not enough: a yarn must grow, and the working figure for a cotton yarn’s wet-to-dry diameter ratio is about 1.10, with a range from 1.05 to 1.20.

So a wet knitted fabric has a yarn a tenth thicker in the same loop length.

What the fabric does about it #

The fabric’s loop length is fixed — it is a length of yarn and nothing about wetting changes it — so a thicker yarn in the same loop is a tighter fabric in the model’s only dimensionless sense.

The tightness factor is the yarn’s diameter over the loop length, up to a constant. Raise the diameter by a tenth and the tightness factor rises by a tenth.

And the flattening a fabric demands falls with the tightness factor.

So a wet fabric demands a flatter yarn. For a twenty tex cotton at a three and a half millimetre loop, the demanded flattening goes from 0.780 dry to 0.761 wet at the working swelling — and to 0.744 at the top of the range.

Why the computation is a substitution #

The arithmetic is worth setting out because it is simpler than it sounds.

A yarn’s diameter goes as the square root of its count at a fixed density and packing. So a yarn whose diameter has risen by a factor s has the diameter of a dry yarn of s² times the count.

A ten per cent diameter rise is therefore the same geometry as a twenty tex yarn becoming a 24.2 tex one, and every figure and number on this rung is computed that way: the collection’s own machinery run at a substituted count, with the substitution stated.

That is exact for the geometry and it is not exact for everything else — a swollen yarn has more mass in it than a heavier dry yarn does not, and its stiffness is different — but the flattening is a purely geometric quantity and the substitution is right for it.

What a section should show #

The prediction is a measurement anybody with a microtome could make.

Section a knitted fabric dry, and again wet. The yarn should be flatter wet by about two per cent of its own diameter at the working swelling, and by up to five per cent at the top of the range.

Two per cent is a small difference and it is not below what a careful section measures. What makes it worth attempting is that the direction is not obvious: a swollen yarn is a bigger yarn, and the naive expectation is that a bigger yarn in the same fabric is rounder rather than flatter, because there is more of it.

The geometry says the opposite. The fabric’s clearance is set by its spacings and the spacings are set by the loop length, so a bigger yarn in the same clearance has to give more.

The complication the fabric supplies #

The prediction as stated holds the fabric’s spacings fixed, and a wet fabric’s spacings are not fixed.

A knitted fabric wetted goes from its dry-relaxed state to its wet-relaxed one, which is a shrinkage of several per cent in both plan directions. That brings the courses closer together, which makes the overlap worse still.

So there are two effects and they point the same way: the yarn is thicker and the courses are closer, and both reduce the room.

The collection’s own relaxation constants supply the second, and computing the two together gives the wet-relaxed flattening rather than a dry fabric with a swollen yarn in it.

That is the number a section would actually measure, and it is smaller than either effect alone suggests: about four per cent flatter than the dry-relaxed state.

Which the other ladder already half knew #

The result overlaps with something this collection found on a different ladder and did not connect.

The demanded flattening was computed across three relaxation states and found to move by two per cent between dry-relaxed and fully relaxed — with the fully relaxed state the flatter one, because relaxation brings the courses closer.

That was computed at a fixed dry diameter, so it is the second of the two effects with the first left out. Putting the swelling in doubles it.

So a fabric’s demanded flattening moves by about four per cent across its own wetting-and-relaxing cycle, and the collection had half the answer for two rungs without noticing the other half was missing.

Why the loop length really is fixed #

The whole argument rests on the loop length not changing, and it is worth defending because it is not obvious.

A loop length is a length of yarn, measured along the yarn. Wetting a cotton raises the fibre’s length by about one per cent, so a yarn’s length rises by about that — and a loop’s length with it.

One per cent against a diameter change of ten is a tenth of the effect, in the opposite direction, so the net is a nine per cent rise in the ratio rather than ten.

That is the correction and it is small. What matters more is that nothing else changes the loop length: the fabric’s own relaxation moves its spacings and does not move the amount of yarn in a stitch, because the yarn cannot slide from one stitch to the next without friction letting it.

This collection established that separately, and it is the reason a knitted fabric’s dimensions are all functions of one length: a loop has no closure condition fixing its shape, so the loop length is the only thing that survives every state change.

So the fixed quantity is genuinely fixed, and it is what makes the substitution above the right one.

What was counted, and how #

The swelling figures are the collection’s own wet-ladder table: a cotton fibre’s diameter swelling of twenty per cent, and a yarn’s wet-to-dry diameter ratio of 1.10 with a range of 1.05 to 1.20.

The geometry is computed by the substitution above — the same machinery at a count scaled by the square of the diameter ratio — and the substitution is exact for a diameter.

The relaxation states are the collection’s own Munden constants, unchanged.

The two effects are computed separately and together, and the together is the one quoted, because that is what a section of a wet fabric would see.

What it says about a fabric that is dried under restraint #

The two effects can be separated by a process, and the separation is something a finisher already does.

A fabric dried free takes its wet-relaxed dimensions and keeps them, so both effects operate: the yarn was thick while the spacings settled, and the spacings settled to what a thick yarn wanted.

A fabric dried under restraint — on a stenter, held to width — is prevented from taking its wet-relaxed spacings. Its yarn was thick and its spacings were held, so only the first effect operated while it was wet and neither operates once it is dry.

So the two processes should leave sections that differ, and the difference is the clearance the fabric settled at. That is a comparison between two finishing routes on one fabric, which is exactly the kind of thing a finishing trial already produces.

Nobody sections them, because a section is not what a finishing trial measures. What it measures is width, weight and shrinkage — and the section would say why those came out where they did.

Where the model stops #

The yarn’s swelling ratio has a range of its own, from 1.05 to 1.20, which is a factor of four in the effect. So the prediction is between one and five per cent, and only its direction is sharp.

The substitution is exact for the diameter and nothing else. A swollen yarn’s bending rigidity, its packing factor and its fibre volume fraction are all different from a dry yarn of the substituted count, and the loop is solved with the substituted count’s properties throughout.

That is a real approximation. The loop’s shape depends on the diameter and on the bending rigidity, and the substitution gets the first right and the second wrong. How much that matters is not computed.

And the fabric may not be at equilibrium. A wet fabric is relaxing while it is wet, and how far it has got depends on how long it has been wet and whether it has been agitated.

Why the flatter section is not the same as a wetter one #

A confusion worth heading off, because the two things a wet section shows are different.

A section of a wet fabric shows a yarn that is larger — its area has grown by the square of the swelling, forty per cent for a cotton at the working figure — and that is the obvious and expected change.

What this rung is about is the shape at that larger area: how flat the larger ellipse is. The area is set by the swelling and the flattening by the clearance, and the two are independent.

So a measurement has to report both, and reporting only the minor axis would confuse a yarn that has grown with a yarn that has been squashed. The right quantity is the aspect ratio, which is dimensionless and is what the geometry predicts.

That is a small methodological point and it is the kind that decides whether a measurement means anything. This collection has made the same point about a fabric’s dimensions, where a width without a state is not a measurement, and it applies with equal force to a section without an aspect ratio.

The other quantity wetting moves #

For completeness: the swelling changes the pressed fraction as well as the flattening, and the second is the more visible of the two in a profile.

A thicker yarn is inside a diameter of its neighbour over more of its length, so the fraction of the yarn that is being pressed rises. For a ten per cent diameter rise it goes from about a fifth to about a quarter.

That has a consequence for friction rather than for shape. A fabric’s friction lives where its yarn is pressed, so a wet fabric has a larger fraction of its yarn engaged — which is one mechanism by which a wet knit is stiffer and less free to move than a dry one.

Whether it is the dominant mechanism is not settled here. Water also lubricates, swells the fibres against one another inside the yarn, and softens the cellulose, and each of those pushes differently.

The generalisation #

The rung is a small instance of something worth noticing about how this collection’s ladders interact.

The flattening result came from a contact ladder that knew nothing about water. The swelling figures came from a wet ladder that knew nothing about contact. Neither would have produced this, and putting them together took one substitution.

The cheapest results in a collection this size are the ones that need no new machinery, and they are found by asking of each new quantity what every existing ladder would do with it.

That is worth doing deliberately rather than opportunistically. This work produced two genuinely new quantities — a flattening the geometry demands, and a torsional rigidity — and the number of existing ladders that would say something new if given either is larger than the number that have been asked.

The fibre that shows it best #

The effect goes with the fibre’s swelling, and the collection’s own table has a factor of seventeen across it.

Viscose swells thirty-five per cent in diameter — the most of anything here — so a viscose knit’s demanded flattening should move about twice as far as a cotton’s on wetting.

Cotton at twenty per cent is the middle case and the one computed above.

Wool at sixteen per cent is a little less.

And polyester at under two per cent is effectively unaffected: a wetted polyester knit’s yarn is the same thickness it was, and its demanded flattening does not move at all.

That gives the sharpest available test. Section a viscose knit and a polyester knit, wet and dry. The first should show a measurable change and the second none, and the two are being measured by the same person with the same instrument on the same afternoon.

A comparison between two fibres removes every systematic error a single-fibre before-and-after comparison carries, which is why it is the measurement worth proposing rather than the cotton one.

What a dryer does that a wash does not #

One more separation, because it decides which state the fabric is in when it is looked at.

Wetting swells the yarn. That is immediate and it reverses on drying.

Relaxing moves the spacings. That is slower, it needs the fabric to be free to move, and it does not reverse on drying — a fabric that has relaxed stays relaxed.

So a fabric that has been wetted and dried without being allowed to move has had the first and not the second, and a fabric that has been washed and tumbled has had both.

That means the flattening a section shows depends on the fabric’s whole history rather than on whether it is currently wet, and the two effects have to be separated by the treatment rather than by the moment of measurement.

It also means the demanded flattening rises again on drying — the yarn shrinks back and the spacings do not — so a washed and dried fabric should be flatter than a virgin dry one and rounder than a wet one.

Three states, three different flattenings, all of them a consequence of one length staying fixed while a diameter and two spacings move.

Who found it, and when #

That fibres swell across and not along is old and is the founding fact of every account of wet fabric behaviour.

That a yarn’s voids cannot absorb its fibres’ swelling is this collection’s own, from its wet ladder.

The flattening a fabric’s geometry demands is this collection’s, from its contact ladder.

What is new here is only the composition, which is a substitution and a paragraph — and which produces a measurement nobody has made because nobody had a reason to expect a difference.

What this is worth against the swelling’s own bracket #

A last piece of accounting, because the prediction’s precision is set by the worst-known input rather than by the geometry.

The geometry is exact: given a diameter ratio, the demanded flattening follows from a solve with no fitted constants.

The diameter ratio is not. The collection’s own table gives 1.10 with a range of 1.05 to 1.20, which is a factor of four in the effect — one per cent at the bottom and five at the top.

So the honest statement is: the flattening falls, by somewhere between one and five per cent of a diameter, and the fibre’s swelling decides where in that range.

That is a weaker claim than the dry result, where the geometry was the only input and the answer was sharp. It is also the ordinary condition of anything in this collection that involves water, and the reason is always the same: a fibre’s swelling is a measurement with a wide spread, and everything computed from it inherits the spread.

What survives it is the direction and the fibre ordering, and both are testable without any absolute number at all.

Where the ladder goes next #

The same substitution says something sharper about a limit rather than about a section. A knitted loop’s tightest bend is already at twice the yarn’s own radius, which is the hard limit below which a rod occupies its own space — and a thicker yarn in the same loop is bent harder relative to itself.

What wetting does to the bending limit is the same tenth of a diameter read against a constraint rather than against a clearance.