A yarn's surface is a distribution

This collection has computed where a cloth stops. It built a height field from the draft and the Peirce solution, ranked it with a bearing curve, and read a thickness, a contact area, a wear rate and a reflection off the same object. Every one of those results is correct and every one of them carries the same caveat, printed in the essay that opened the ladder and repeated at each rung: the surface computed is a surface of yarn, and it is not what anything touches first.

The cloth #

The hairs are what touch established the mechanism and stopped one step short of a model. The radial pressure inside a twisted yarn is greatest on its axis and falls to exactly zero at its surface — that is a closed form this site derived and asserted against its own defining integral — so the outermost fibres are held by nothing but their own ends being buried further in. Some of them are not held at all.

What that essay then did was reasonable and is now wrong. It gave the yarn a second diameter: the mass diameter, which conservation of volume supplies, plus twice a measured hair layer of twenty-five micrometres. One number, one length, added as a shell.

The shell is a fair description of where most of the protruding material is. It is wrong about the layer’s extent by more than a decade, and the difference is not a refinement. A hair layer reaching twenty-five micrometres is a correction to a cover factor. A hair layer reaching two millimetres is a different object from the cloth it grows on, and everything that touches, warms, wets, prints on or wears a fabric meets it first.

The claim #

A spun yarn’s hair layer is a population, not a thickness. Its count comes out of the yarn’s own arithmetic, its length distribution is exponential because fibre migration is irregular, and its characteristic depth is a property of the fibre with the yarn divided out of it.

Three things follow immediately, and none of them is available to a model that carries one length.

A yarn has no contact diameter. It has a coverage profile, and each instrument that reports a diameter is picking a contour of it. The contours are hundreds of micrometres apart.

Hairiness goes as the square root of the yarn count and the depth does not move at all. A coarse yarn has more hairs, not longer ones — so everything that depends on reach is decided by the fibre and everything that depends on quantity is decided by the count.

And the layer is almost entirely gap. At the yarn’s own surface the long hairs occupy about one per cent of the space beside it. That single figure is why the whole of the rest of this ladder turns on whether the hairs can reach one another, and on an ordinary woven cloth they cannot.

Where a hair comes from, in four steps #

The construction is arithmetic three times over and a measurement once, and it is worth writing out because the measurement is the only place a spinning system can enter.

How many fibre ends there are is exact. A yarn’s section holds n fibres, which is the count divided by the fibre’s own count and is the division that runs half this site. A fibre of staple length L contributes two ends over that length, and there are n millimetres of fibre in every millimetre of yarn, so 2n/L ends begin or end in each millimetre. For a twenty tex cotton on a twenty-eight millimetre staple that is 8.4 ends per millimetre. Nothing is assumed here at all.

How many of them are near the surface is geometry. A fibre end can only stand off the yarn if it is already in the outermost shell, and the shell one fibre thick occupies a fraction of the section’s area that this site’s own identity supplies: d_yarn = d_fibre·√(n/φ), asserted to twelve figures when the fibre count was first divided out of a diameter, so d_fibre/d_yarn = √(φ/n) and the shell is 1 − (1 − 2d_f/D)². For the same yarn that is 26.5 per cent. So 2.2 ends per millimetre are in a position to escape.

How many actually escape is measured. Not every end in the shell gets free; the twist catches some, a neighbour catches some, and the size laid on a warp glues some down. Nothing in this construction predicts the fraction. It is about two fifths for a conventional ring yarn and it is the one fitted number in the whole file.

And how long each one is comes from migration. That is the fourth step and it is the interesting one.

Why the distribution is exponential #

A fibre in a spun yarn does not sit at one radius. It migrates: the same excursion between core and surface that makes a spun yarn stronger than the affine model allows, because a fibre that visits every radius shares the load rather than taking its own. This site has already used migration for that and computed what it is worth.

Read it here as a walk rather than as a mechanism. A fibre’s end lands somewhere in that walk, at a point along it that nothing about the spinning decides. The length that protrudes is the length between the end and the last time the fibre was pulled back inside the body — which is the residual of a surface excursion.

The shape of the residual is decided by one property of the walk and by nothing else. If migration were perfectly regular, with a fixed period, the residual would be uniform on that period and the distribution of hair lengths would stop dead at the period’s own length. Because migration is irregular — and Hearle’s tracer-fibre work is a picture of exactly how irregular — the residual is memoryless, and a memoryless residual is exponential.

So the shape is derived rather than fitted, and it is the observed shape. That is worth stating plainly because the exponential hairiness law has been in the literature since hair-counting instruments existed and has been treated as an empirical regularity. It is a consequence of migration being a walk rather than a cycle.

The result with no free parameter: the depth belongs to the fibre #

The decay length is half the migration period times the shell’s share, because a fibre spends that share of its cycle where it can escape and the residual of a sojourn averages half of it:

λ = ½ Λ s

Both factors are known. The shell’s share is 4d_f/D to first order. And the migration period, quoted where it belongs — in yarn diameters rather than in millimetres, because migration is an excursion across the section and the section is its natural scale — is a fixed number k. So

λ = ½ (k D)(4 d_f / D) = 2 k d_f

and the yarn’s diameter cancels out of it entirely. The depth of a hair layer is twice the migration constant times the fibre’s own diameter, and it does not know what count the yarn was spun to.

For cotton that gives about six hundred and twenty micrometres, against a decay length of six to seven hundred read off published hair counts at one, two and three millimetres. For wool, whose fibre is nearly twice as thick, it gives eleven hundred. Neither number was fitted to anything.

The density has no such cancellation. N₀ = (2n/L)·s·e goes as √(nφ)/L, so hairiness rises as the square root of the count — which is the trend every published survey of yarn hairiness against count shows, and which here is a consequence of an identity this site asserted for an entirely different reason.

What was counted, and how #

Three checks, and the third is the one that would have caught the whole construction being wrong.

The cancellation is asserted rather than remarked on. Over four fibres and a fivefold count range, the decay length is required to move by under a seventh while the population is required to follow the square root to the same figure — and, separately, the two departures are required to be equal to within 10⁻¹², because they are one departure seen twice. An error in either expression would break the equality long before it broke either tolerance.

The counts are compared with the instruments. The model puts 17,800 hairs per hundred metres over one millimetre, 3,550 over two and 709 over three, for a twenty tex ring-spun cotton. Those are the numbers such instruments report on such yarns. Nothing was tuned to them: the escape fraction was set once, from a different comparison, and these three fall out of it together with the decay length.

And the mass is compared with a weighbridge. The protruding fibre in an ordinary cotton cloth comes out at a tenth of a per cent of the fabric’s mass for the long population, and grossing that up by the split between the long and short populations puts the whole of it under one per cent — which is exactly what singeing a cloth loses. The mass here comes from a count of fibre ends and a length; the singeing figure comes from a scale. That agreement is the cheapest check in the file and it was not arranged.

The number that decides whether it is a layer at all #

A cloth presents a certain length of yarn per unit area — ends per centimetre plus picks per centimetre, which is the one conversion — so the hairs per square millimetre follow at once. For a sheeting it is about 1.4.

Their mean length is 0.63 millimetres. So the mean spacing between them, 1/√n_A, is longer than the hairs are. They cannot reach one another. An ordinary woven cotton cloth does not have a hair layer; it has isolated whiskers on a bare surface, and the criterion is the pure number

n_A λ²

which is 0.53 for a sheeting, 0.003 for a singed one, and thirty or more for a raised one.

And construction is almost powerless over it. 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 the whole of this site’s table of constructions spans less than a factor of two. Everything that crosses the threshold crosses it by finishing.

Where the model stops #

The short population is not modelled. Setting this construction’s total protruding length against an integrating hairiness meter on the same yarn leaves a factor of eight unaccounted for. There is a second, far denser population of very short protrusions — loops rather than ends, and slack fibre lying on the surface — which the optical instrument adds up and no counting instrument resolves. It is not here. Every quantity in this ladder is computed from the long population alone and is therefore a lower bound, in exactly the way the height field below it is a lower bound on contact.

The escape fraction is a fitted constant and it is probably not a constant. The clearest evidence is that the geometry predicts a finer fibre gives a hairier yarn — more fibres in the section, thinner shell, more ends near the surface — and the trade says firmly that it does not. That disagreement lands entirely in the escape fraction and nowhere else, which is a useful thing to know about a model but not a comfortable one.

A hair is taken as straight and it is not. A protruding fibre carries the curvature it had in the yarn and its own bending stiffness, which this site has established is a bracket three hundred wide. Nothing here bends a hair except where a plate is pressing on it.

And the migration period is one measurement standing under a great deal. Quoting it in yarn diameters is the step that makes the depth independent of the count, and it is the step to disbelieve first if the prediction fails.

The fine-fibre disagreement resolves itself #

The limits section records an uncomfortable disagreement: the geometry predicts that a finer fibre gives a hairier yarn, the trade says firmly that it does not, and the whole of the discrepancy is dumped into the fitted escape fraction. It need not be. The two laws in this essay resolve it without touching the fit.

The density goes as N₀ ∝ √(tex_yarn / tex_fibre) ÷ L, so a finer fibre does give more hairs — the geometry is right about that.

The depth goes as λ = 2k d_fibre ∝ √tex_fibre, so a finer fibre gives shorter ones.

Multiply the two and the total protruding length is

N₀λ ∝ constant in the fibre’s fineness.

A finer fibre gives more hairs, shorter, and exactly the same total protruding length. The two dependences are square roots of the same quantity in opposite directions and they cancel exactly.

Which settles the disagreement, because the trade measures hairiness with an integrating instrument — one that reports total protruding fibre per unit length, which is N₀λ. The trade is measuring the one statistic of this population that does not move with the fibre’s fineness, and it correctly reports no change.

So the geometry and the trade are both right and they are talking about different quantities, which is the same resolution two hairiness instruments need arriving to close this essay’s own open problem. Nothing has to be absorbed into the escape fraction at all.

And it makes a prediction the trade would recognise #

The cancellation is exact only for the integral. Every threshold count moves, and it moves in a direction that matches an experience the trade states as a preference rather than as a measurement.

Take a cotton at 1.2 decitex against an ordinary one at 1.7. The finer fibre has 19 per cent more hairs in total and a decay length of 0.52 millimetres against 0.62. Running those through the exponential:

twelve per cent fewer hairs over a millimetre, and fifty-two per cent fewer over three.

A fine cotton is less hairy where it matters and more hairy where nothing looks. The extra hairs are all in the first few hundred micrometres, where no counting instrument has a threshold and no property of the cloth is decided; the ones that reach — the ones that pill, that feather a print, that prickle — are halved.

That is exactly the reputation of the long fine cottons, and this collection has already noticed the reputation attaching itself to the wrong property once, in the fibres that crease, where the same fineness is credited to the staple length. The two are bought together and only the fineness is doing the work, in both essays, for two unrelated properties.

The prediction that separates them is the same one: two cottons of one fineness and different staples should behave alike, and two of one staple and different fineness should not. A fine short cotton should be less hairy in the tail than a coarse long one, which is the opposite of what a staple-length account predicts and is a comparison anybody with two bales and a hair counter could run in an afternoon.

The generalisation #

A surface made of a material’s own leftovers is a population and not a finish.

The transferable shape is that a boundary between two things is often not a boundary at all but a graded region whose gradient nobody measured, and that the useful description is a distribution with a decay length rather than a position with a tolerance. The moment a boundary is described that way, the questions change: not where does it stop but what fraction is still there at this height, and different instruments stop at different fractions.

That reframing is why the diameter question in the next essay has an answer at all. It also says what to do with a quantity like a cover factor, a thickness or a friction coefficient that turns out to depend on the instrument: the dependence is not experimental error, it is the profile being sampled at different contours.

Who found it, and when #

The mechanism is old and the assembly is not. Peirce’s arithmetic of a yarn’s diameter is from 1937 and supplies the section. The radial pressure profile inside a twisted yarn, falling to zero at the surface, is the classical result this site derived and asserted for itself. Fibre migration is Morton and Yen’s observation of 1952 and Hearle’s model of the decade after, and the tracer-fibre photographs that show how irregular it is are theirs.

Hairiness has been measured since the 1950s — Barella’s work is the long thread through it — and the exponential fall of hair count with length is reported everywhere and derived nowhere that this collection can find. What is new here is only the joining: that the exponential follows from migration being a walk, that the decay length is then twice a migration constant times a fibre diameter with the yarn cancelled out, and that the density and the depth therefore obey different laws.

Where the ladder goes next #

Straight to the instruments. A yarn has a diameter for every instrument takes the coverage profile seriously and finds that the contact diameter this site has used since it first gave a yarn a second one is a contour rather than a length. Two hairiness meters read two moments then shows why the trade’s two hairiness numbers have never agreed and could not.

Outward, the same population answers questions in six fields. It decides what a light touch actually meets, how much of a fabric’s warmth is not the fabric, why a shot silk has to be silk, what a pill is anchored by and why a wool prickles at thirty micrometres. One population, and the count and the length are read separately in every one of them.