Textile Structure
how cloth is put together
Concept

Escape fraction — where it appears

The fraction of the fibre ends lying in a yarn's surface shell that actually get free rather than being caught by the twist, a neighbour or a size film. It is the one measured number in this site's hair model, and it is where a spinning system enters.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

Whether a cloth's hairs can reach one another. n_A λ² for each construction in this site's table — the hairs per square millimetre times the square of their own length, which is the pure number that asks whether a hair can touch its neighbour. It is a count times an area, so it has to be a pure number. Every one of them is under one, which means no ordinary woven cotton cloth has a hair layer at all: it has isolated whiskers on a bare surface. The dashed line is the threshold. The spread across the whole table is only 1.9-fold, because the density goes as the sett times the root of the count and those move in opposite directions as a cloth is made finer — so construction is almost powerless here, and everything that crosses this threshold does so by finishing rather than by weaving.

A knit gives up its fibres more easily

Knitwear pills and shirting does not, and the fibres are often the same fibres. The difference is a count of yarn per unit area and a pressure between threads, and both of them push a knit over a threshold that a woven cloth of the same yarn cannot reach.

knits · Knit A 20 tex cotton yarn and the fibre standing off it. 6 mm of a 20 tex ring-spun cotton yarn with the hair population this site computes from the yarn's own count and staple — 0.89 hairs per millimetre, every one of them drawn. The two axes are at different scales and have to be — the yarn is 167 µm across and its hairs reach past a millimetre, so a picture at one scale is either a bare line or a black rectangle. Along the yarn is 99 pixels to the millimetre and off it is 74, a 1-fold exaggeration of the vertical. Lengths are drawn from the exponential the model predicts, mean 621 µm; the rules mark one, two and three millimetres with the count a hair-counting instrument reports at each, and the hairs crossing each rule in the drawing are the ones those counts are about. At the yarn's own surface the long hairs cover 1.1% of the space beside it, which is why the picture is mostly gap. Nothing here is the short population, which carries most of the protruding length and none of the reach; and a hair reaching past the room the canvas has is drawn to the edge of it, so the very longest few are shortened in the drawing and not in the arithmetic.

A yarn's surface is a distribution

This collection has computed where a cloth stops, and every one of those numbers is a statement about yarn. What a finger, a plate, a droplet or a ray of light actually meets first is a population of fibre ends standing off the yarn — and it is a population, with a count and a length, rather than a layer with a thickness.

cloth · Hair layer One hairiness reading does not fix the other. Every yarn on this curve has exactly the same total protruding fibre length — the quantity an integrating hairiness meter reports — and they differ in how that length is distributed. The count of hairs at least three millimetres long runs from 170 to 4354 per hundred metres, a factor of 26, across decay lengths real yarns actually have. The two instruments read two functionals of one population: the first moment N₀λ and the tail N₀e^(−3/λ). A correlation between them can exist only if λ is fixed across the yarns being compared, and λ is a fibre property, so it is not. That is the whole of why the trade's two hairiness numbers have never agreed, and it is arithmetic rather than instrumentation. What the figure cannot show is which of the two predicts anything: the tail does, because pilling, prickle and a printed edge all need reach.

Two hairiness meters read two moments

The trade has two instruments for yarn hairiness and thirty years of failing to predict either from the other. They are not measuring the same thing badly. One reports the first moment of a distribution and the other reports a tail probability, and two functionals of one curve are related only through a parameter neither of them reports.

cloth · Hair layer A hair layer is a balance, so singeing does not stay done. The hair population of a 20 tex cotton yarn under rubbing, started from a singed cloth and from an unusually fuzzy one. Abrasion does two opposite things: it frees ends that spinning left buried, from a supply of 2.23 per millimetre in the surface shell, and it removes hairs that are long enough to be caught. Where the two meet is a fixed point at 1.43 per millimetre, and the cloth goes there from either side with the same time constant — 248 cycles to halve the distance, whichever direction it is travelling. A singeing is therefore undone in a few hundred rubs, because the flame changed the stock and not the balance. What is predicted here is that a fixed point exists, that it does not remember the starting state, and that one rate serves both directions; where it sits relative to the spun level needs two rates the model does not supply, and it is set to reproduce the one thing everyone has noticed, which is that fabrics get fuzzier as they are worn.

A hair layer is a balance, not a stock

Singeing takes under one per cent of a cloth's mass and changes its lustre, its friction, its printability and its pilling. It also does not stay done, because rubbing frees fibre ends as fast as it breaks them off, and a flame changes the stock while leaving the balance exactly where it was.

finishing · Hairiness The depth belongs to the fibre and the density to the yarn. Over a 5-fold range of cotton yarn counts, the hair layer's decay length moves by 6.2% and its population moves by 2.37-fold against a square root of 2.24. Both follow from one cancellation. The shell's share of the section is 4d_f/D, the migration period is a fixed number of yarn diameters, and λ = ½(kD)(4d_f/D) — the yarn's diameter divides out and leaves λ = 2k·d_f, a length belonging to the fibre alone. The density has no such cancellation and goes as √(nφ)/L. The two small departures visible here are not two facts: they are the shell's second-order term, and they are the same number to the last bit of a double. The consequence for a spinner is that a coarse yarn is hairier and its hairs are no longer, so everything that depends on reach — pilling, prickle, a printed edge — is decided by the fibre and not by the count.

Hairiness goes as the root of the count

A coarse yarn is hairier than a fine one and everybody knows it. What nobody has said is that its hairs are no longer — the count and the length obey different laws, one rises as a square root and the other does not move at all, and the identity behind both was asserted on this site for an entirely unrelated reason.

setting · Hair layer Compacting a spinning triangle moves one instrument and not the other. What happens to each hairiness reading when a 20 tex cotton yarn is spun compact instead of ring, as a percentage of the ring value. The total falls by 8% and the long hairs by 65%, a ratio of 8.3. The asymmetry is a prediction rather than a fit. Compaction removes ends that were unbound over a long stretch of the spinning triangle, which is the long population and nothing else; the short population is untouched, and it carries about 88% of the length the integrating instrument is adding up. So the instrument that sees everything barely moves and the one that sees only the tail collapses. The model under-states the fall in the total, because compaction certainly does something to the short population too and nothing here models it — the direction of that error is stated and it is the conservative one.

The spinning triangle decides the hair

A ring frame converges a flat ribbon of fibres to a round yarn, and for the length of that convergence the fibres at the ribbon's edges are held by nothing. Everything a spinner can do about hairiness is done in that triangle, and the two hairiness instruments respond to it by wildly different amounts.

setting · Twist

Named alongside it

The objects these essays reach for when they reach for this one.

Hair layerProtrusion lengthHairinessMigration periodStaple lengthSurface shellFibre countFibre finenessFibre migrationPillingTwo populationsAbrasion

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