A cut pile sheds in two stages

A blade leaves loose fibre in every tuft counted what a cut pile holds loose. A tuft of staple yarn is a length $L$ of yarn bent into a U — the construction hair, nap and pile are one construction placed at the far end of a continuum — and every fibre end that falls in one of its legs leaves a piece between it and the tip that does not pass under the binding pick. The count came out exact and short: a share $L/(L+\ell)$ of the pieces, a share $L/4\ell$ of the fibre, with $\ell$ the staple length. That is the reservoir a new carpet sheds from.

It ended by saying that the count gives how much is at risk and not how fast it goes, and it named the two lengths that would say. One is the grip length what grips the end of a fibre computed: the length over which a twisted yarn’s pressure holds a fibre end. The other is the tip’s untwisted run-out, which a tuft is set so it cannot untwist found to be the same length, measured from the cut. Put the two together and the reservoir drains in two stages, with the split between them set by one ratio.

At the tip nothing presses on anything #

A twisted yarn holds its fibres by pressure. Each fibre in the outer layers is a helix under tension, and a curved fibre under tension pulls inward on whatever it wraps round, so the fibres inside are pressed together and friction holds them. The pressure is greatest on the axis and falls to nothing at the surface, which is what makes a yarn hairy at its outside.

At a cut end it falls to nothing everywhere. A fibre at the tip of a tuft is under no tension, because there is no yarn beyond the cut to pull on it, so it presses on nothing and nothing presses on it. A millimetre back from the cut there is a millimetre of twisted yarn to hold the twist in, and further back more. The tuft essay found that the twist runs out from the tip over a length $z$ equal to the grip length at the tuft’s twist angle,

$$z = \frac{d_f}{4\,\mu\,c\,Q(\alpha)},$$

with $d_f$ the fibre’s diameter, $\mu$ its friction, $c$ a contact efficiency that is the one measured quantity in the expression, and $Q(\alpha)$ a pure function of the surface twist angle. For a wool carpet yarn at thirty degrees and a contact efficiency of 0.05, $z$ is 5.6 millimetres.

A loose piece inside the run-out is held by nothing #

A loose piece runs from its fibre end, somewhere in the leg, up to the tip. Everything about whether it stays is decided by where that end is.

If the end lies within $z$ of the tip, the whole piece is inside the run-out. No part of it is pressed, so nothing grips it, and the first brush of a foot or a vacuum cleaner’s beater takes it. If the end lies deeper than $z$, part of the piece is in twisted yarn, pressed by the fibres round it, and that part holds the whole piece in place for as long as the twist there survives.

The count established that while the staple is longer than half a tuft, a loose piece’s length is uniform from nothing up to $L/2$. So the share of the loose pieces lying wholly within $z$ of their tip is the run-out’s share of half a tuft, and because each piece’s fibre is in proportion to its length, the share of the loose fibre is that share squared:

$$\text{free at once} = \frac{z}{L/2} \ \text{of the pieces}, \qquad \left(\frac{z}{L/2}\right)^{2} \ \text{of their fibre}.$$

Both are capped at one: a run-out longer than half a tuft frees every loose piece there is. A sampled pile — fibres laid at random along a long yarn, cut into six hundred tufts, and each loose piece checked against the run-out directly — agrees with both formulas to within its own sampling at every construction tried.

A wool carpet sheds a fifth of its reservoir at once #

For the twelve-millimetre wool carpet on a two-millimetre base, a tuft’s yarn is 26 millimetres and half of it is thirteen. The run-out of 5.6 millimetres frees 43 per cent of the loose pieces at once, but only 18 per cent of their fibre, because the free pieces are the short ones. The reservoir is 7.2 per cent of the pile’s fibre, so the first stage sheds 1.3 per cent of it: the fluff a new carpet fills the first few vacuum bags with.

The lengths make a prediction that can be checked without any instrument but a ruler. The first stage sheds only pieces shorter than the run-out, 2.8 millimetres long on average and none longer than 5.6. Everything shed later is longer: pieces between 5.6 millimetres and half a tuft, which were held by the twist and let go when it opened. A new carpet’s first vacuuming should yield a fine, short fluff and its shedding a year later should yield longer fibres, and the boundary between them is the run-out, which the grip arithmetic puts at a figure for each yarn.

The second stage is the twist opening #

The remaining four fifths of the reservoir by mass is held by twisted yarn, and it stays until the twist that holds it goes. The tuft essay found what makes it go. Each time a foot compresses a tuft and bends it over, fibres at the tip slide past one another where nothing presses them, the twist lets out a little further, and nothing puts it back. The tip’s twist angle falls, and the grip length at a lower angle is longer.

As set, at thirty degrees, 43 per cent of the pieces are free. With the tip’s twist opened to twenty-five degrees the run-out is 7.6 millimetres and 58 per cent of the pieces, a third of the mass, are free. At twenty degrees the run-out is 11.2 millimetres and 86 per cent of the pieces and three quarters of the mass have gone. Below about nineteen degrees the run-out has passed half a tuft and the reservoir is empty. The grip length rises steeply at low angles, which is why a tuft’s shedding accelerates once its tip begins to open.

The faint curves are the bracket the measured input puts on the mass share. At a contact efficiency of 0.02, the run-out is two and a half times as long and most of the reservoir is free the day the carpet is laid; at 0.2 it is a quarter as long and almost none is. The contact efficiency is fitted, not derived, and the shares move a long way across its plausible range. What does not move is the shape: a first stage, a second stage paced by the tip’s twist, and a boundary between them in the shed fibre’s length.

The pile height decides the split #

The formula has the run-out over half a tuft in it, and half a tuft is roughly the pile height. So the same yarn at the same twist sheds in very different proportions depending on how tall the pile is cut.

Up to about four and a half millimetres the two curves coincide. Half a tuft is inside the run-out, every loose piece is free at once, and the whole reservoir leaves in the first stage. Above that height they part. The reservoir keeps growing with the height, since the tuft’s yarn is longer and has more fibre ends in it, while the first stage shrinks, since the run-out is a fixed length and becomes a smaller share of a taller tuft. At six millimetres 2.5 per cent of the fibre goes at once of 3.9 in all. At twelve, 1.3 of 7.2. At twenty-five, 0.66 of 14.4.

The constructions line up in the order the height puts them. A cotton velvet, two and a half millimetres tall on a soft twist, sheds everything it will ever shed at once, 5.4 per cent of its fibre, and then nothing, because nothing in it is loose and held. A wool plush at six millimetres sheds two thirds of its reservoir at once. A wool carpet sheds a fifth. A nylon staple carpet, on a staple twice as long, has a smaller reservoir and sheds a tenth of it at once. A shag sheds almost nothing at first and has the largest reservoir of all, fourteen per cent of its fibre, held in the lower parts of its long tufts and released only as walking opens the tips.

Why a new carpet stops shedding, and an old one starts again #

The two stages explain an observation the trade knows and does not usually explain. A new cut-pile carpet sheds heavily for a few weeks, then settles; and a carpet that has been walked on for some years begins to shed again as it wears. It is usually put down to “loose fibres from manufacture” in the first case and to “wear” in the second, which are two names for two processes.

The arithmetic says they are the same reservoir, split by the run-out. The first shedding is the pieces inside the run-out, and it stops when they have gone. The second is the pieces below it, released as the tips’ twist opens under foot, and it continues as long as the tip angle keeps falling. For a medium-height carpet the second reservoir is four times the first by mass, so the later shedding is not a nuisance tail on the early one but most of the fibre that will ever leave.

That also puts heat-setting in its place. Setting removes the torque that would otherwise untwist a tuft’s tip the first time it was wetted, so it holds the tip angle against washing and cleaning. It holds the second stage back; it cannot touch the first, which is decided by the run-out at the set angle and has left before the carpet is a month old. A cotton or viscose pile, which cannot be set, runs through its second stage at the first wet clean.

Shearing a pile starts a fresh first stage #

A pile is sheared in finishing to level its tips, and a worn carpet is sometimes sheared again to tidy it. Both are a second blade, and the arithmetic says what a second blade does to the drain.

It makes a new tip, and a new tip has a new run-out. The twist a millimetre below the old tip was held; cut there, and it becomes a free end with nothing beyond it, so the twist runs out from the new cut over the same length $z$. Every loose piece whose fibre end lies within $z$ of the new tip is free at once. Some of them were loose already and held by the twist that the cut has just removed; others were anchored pieces, passing under the binding, which the cut has shortened without freeing. So shearing a pile releases a fresh batch of the second stage’s pieces into a new first stage, and a sheared carpet should shed again for a while however long ago it last stopped.

That is the reason the trade shears before it sells: how a tuft is held is untouched by a shear, which leaves the binding as it was, and the fresh first stage is taken off in the finishing works rather than on the customer’s floor. It is also a reason corduroy, whose tufts are cut from floats only a few millimetres long, barely sheds after its cutting: half its tuft is already inside the run-out, and there is no second stage for a shear to open.

A pill needs both stages #

A pill is anchored, not made: a tangle of loose fibre held to the surface by a few fibres still attached to the cloth. On a cut pile the tips do the anchoring, which are held by the binding, and the tangle is made of loose pieces that have worked up to the surface.

The first stage’s pieces leave too easily to make pills. They are short, gripped by nothing, and taken by the first vacuuming before they can tangle. The pieces that make pills are the second stage’s: long enough to wrap round a tip, held by the twist until a foot works them partly free, and then dragged across the surface still attached at their lower ends. So a carpet should pill once its second stage is under way, and the pills should be made of pieces longer than the run-out. Both are predictions a cut-pile carpet and a length measurement of its pills could test.

What the model assumes #

The grip is the site’s own arithmetic: the grip length of what grips the end of a fibre, read at the tuft’s twist angle as the length over which the tip’s twist runs out, with a contact efficiency of 0.05 unless stated and bracketed from 0.02 to 0.2. The fibres are wool at twenty-two micrometres, nylon and cotton at their own diameters and frictions.

The pieces are the count’s: fibre ends uniformly scattered, so that while the staple is longer than half a tuft a loose piece’s length is uniform up to $L/2$. The drain is counted only for such staples, which is every construction here; a staple shorter than half a tuft leaves whole fibres lying in a leg, and the arithmetic refuses it rather than extending itself.

“Free” is a threshold. A piece wholly inside the run-out is taken to be held by nothing, and a piece reaching below it to be held completely. The pressure does not in fact switch at $z$; it rises from nothing at the tip over roughly that length. So the boundary between the two stages is softer than drawn, and the first stage is a slight underestimate for pieces just longer than $z$ and a slight overestimate for pieces just shorter.

The shares are required to match a direct count of sampled pieces against the run-out, by count and by mass, at four constructions; a run-out past half a tuft is required to free the whole reservoir; and a twist opened from thirty-five degrees to twenty is required to free more at every construction tried.

What the figures cannot show #

How fast the second stage runs. The drain is set against the tip’s twist angle, and what lowers that angle is walking: the number of treads, the weight of each, whether the tuft is bent over or crushed straight down. Why agitation helps a cloth relax and the tuft essay both describe the mechanism, a ratchet in which each cycle lets a little twist out; neither gives a rate, and without one the second stage is a function of an angle rather than of a year.

And the fibre broken by wear. Walking also breaks fibres, and a broken fibre inside the tuft becomes two pieces, at least one of them loose. The count and the drain describe the reservoir a blade leaves. A worn carpet adds to it, and the pieces it adds are shorter than the second stage’s, so the neat split in the shed fibre’s length would blur as a carpet ages.

Who worked out which part #

Carpet shedding is common knowledge, and so is the distinction between early shedding and wear. The grip length is the fibre-grip arithmetic of the yarn essays, and the run-out is that length applied at a cut end by the tuft essay. The count of loose pieces is the blade essay’s.

What is put together here is the drain: the run-out divided by half a tuft as the share of the reservoir that goes at once, its square as the share by mass, and the consequence that the pile height, not the fibre or the finishing, decides how a cut pile’s shedding is split between its first weeks and its later years.

Still open: a rate for the tip’s twist #

Every second-stage number is a function of the twist angle left at the tip, and nothing in the arithmetic says how fast walking lowers it. The mechanism is a ratchet: a tread bends the tuft, fibres at the tip slide where nothing holds them, and some of the twist that was held by friction is let out.

The rate at which that ratchet advances is measurable directly: tufts pulled from a carpet after a counted number of treads on a walking tester, their tip angles read under a microscope, and the run-out computed from each. That would give the second stage its clock, and with it the shed fibre per year that a construction should be expected to lose. The same measurement would also say whether heat-setting slows the ratchet or only protects the angle against water, which is the question on which the choice between a set and an unset pile yarn turns.