A tuft is set so it cannot untwist

A carpet fails in appearance long before it fails in any other way. Walk on it for a year and there is no measurable loss of material, no thinning, no bald patch — and it looks worn out. Everybody who has bought one knows this and the trade has a name for it, appearance retention, which is a polite way of saying that the thing that goes first is not the thing anybody thought they were buying.

This collection has already priced what holds a tuft in and why a crushed pile does not spring back. Neither is what goes wrong. What goes wrong is that the tuft comes untwisted.

The claim #

A cut tuft’s twist is held only by friction along the tuft, so it runs out from the tip over a computable length — and a tuft shorter than that length has no twist left in it at all.

• The run-out length is the same quantity that decides whether a fibre in a yarn slips or breaks: the fibre’s diameter over four times the friction, times a function of the twist angle. For a wool carpet yarn at 30° it is 5.6 mm, its fibre being 22 µm across.
• A cut pile is between five and fifteen millimetres tall, so the run-out length is a large fraction of the tuft — which is why the tips open and the roots do not.
• Heat-setting does not hold the twist in. It removes the torque, which is a different operation with a different failure mode, and the distinction decides what happens when the carpet is cleaned.

The free end, and what it is free of #

Take the pressure argument and ask it about a cut end.

Inside a yarn, a fibre is pressed against its neighbours because every fibre outside it is under tension and curved, pulling inward. That requires tension, and tension requires the fibre to be held at both ends by something further along the yarn.

At a cut, there is nothing further along. The fibres at the tip are under no tension, so they press on nothing, so nothing presses on them. The twist at a cut end is unresisted.

It does not all unwind at once, because a millimetre back from the cut there is a millimetre of yarn to be wound, and that little length is held by friction against whatever is left. The question is how far back the twist survives, and it is exactly the question the fibre-grip argument answers, asked of the whole assembly rather than of one fibre.

The consequence is a gradient rather than a switch. The tip is untwisted, the middle is partly twisted, and the root — which is held by the backing on both sides of the U — is fully twisted. That is exactly what a worn tuft looks like under magnification, and it is what a new one looks like too, only less so.

Why walking on it makes it worse #

The run-out above is what happens with no help. Walking supplies help, and it does so in the way most likely to matter.

A foot compresses the tuft, bends it over, and releases it. Each cycle slides fibres past one another at the tip — which is where there is no pressure holding them — and each slide lets the twist out a little further. The friction that was holding the twist is a static one; once a fibre has started to move, the dynamic coefficient is lower, and the run-out advances.

So appearance loss is a ratchet, in the same sense that felting is a ratchet: each cycle moves the boundary one way and nothing moves it back. There is no restoring force, because a twist that has run out has nothing left to hold it.

The numbers, and the ratio that matters #

The run-out length is the fibre’s diameter over four times the friction and the contact efficiency, times a function of the twist angle — and every one of those is different for a carpet yarn than for a shirting.

A carpet yarn is coarse: 200 to 300 tex is ordinary, against a shirting’s 20. Its fibre is coarse too, at 15 to 20 decitex against cotton’s 1.7 — a factor of ten, and the run-out length is proportional to it. Its twist is high, at 30° to 35° of surface angle, which shortens the run-out. And its pile height is 5 to 15 millimetres.

Putting those together gives a run-out of 5.6 mm for a wool at 30°, against a tuft between five and fifteen millimetres tall — which is the ratio the whole essay turns on: not a small fraction and not, quite, the whole tuft, but half to three quarters of it. That is why a worn carpet looks blurred rather than bald, and why the blur starts at the surface and works down.

And a loop pile has no free end. A tuft that goes down into the backing and comes back up is a continuous length of yarn with tension in it at both ends, so the pressure never falls to zero and the twist is held everywhere. That is a structural prediction and it matches the trade completely: loop-pile carpets keep their definition for far longer than cut-pile ones, and it is not because they are made better.

What heat-setting is for #

Every cut-pile carpet yarn is heat-set, and the operation is usually described as “setting the twist”. That description invites a wrong mechanism.

It does not add friction, and it does not glue the fibres. What it does is remove the torque: the yarn is twisted, heated above the temperature at which the fibre’s internal stresses relax, and cooled. The fibre’s memory of its untwisted shape is erased, so the helix becomes the shape the fibre wants rather than the shape it is being held in.

The distinction matters because the two mechanisms fail differently.

If the twist were held by friction, wetting the yarn would release it — water swells fibres, lowers friction and lets fibres slide, which is precisely why a cloth relaxes when it is washed. A carpet would come untwisted the first time it was cleaned.

Because the twist is held by the absence of torque, wetting does very little. There is nothing to release; the fibres are already where they would rather be.

That is a testable difference and it is the reason the operation is worth its cost. An unset carpet loses its tuft definition in the first wet clean; a set one does not.

Two further things follow.

Heat-setting is available only to fibres that can be set, which means thermoplastics and wool — wool by a quite different chemistry that this collection does not have. A cotton or a viscose carpet yarn cannot be set, which is one of several reasons those fibres are not used for cut pile.

And the setting fixes the twist at the level it was set at, so the twist has to be right before the heat is applied. A yarn set at the wrong twist cannot be corrected, which makes the twist decision irreversible in a way that it is not for a weaving yarn.

What the twist is doing in the first place #

It is worth asking why a carpet tuft is twisted hard at all, since the twist is causing all the trouble.

To keep the tuft together as a unit. An untwisted bundle of fibres standing on end is not a tuft; it is a smear. The visual definition of a cut pile — the reason it reads as a surface of discrete points rather than as a mat — is entirely the twist.

To resist crushing. A hard-twisted tuft is a stiffer column, and a crushed pile does not recover through its fibres, so whatever recovery there is comes from the tuft’s own stiffness as a body.

And to hold the tuft in the backing, since the capstan grip on a tuft depends on the tuft’s diameter and coherence at the point where it wraps the ground threads.

The three requirements point the same way — more twist — and the appearance failure points the other. Heat-setting is what makes the conflict go away, and without it a carpet designer would have to trade tuft definition against how long the definition lasted.

The frieze, and the other way round #

There is a construction that takes the argument and inverts it, and it is worth describing because it shows that the mechanism is understood in the trade even where it is not written down.

A frieze carpet is made from yarn twisted far harder than an ordinary cut pile — well past 40° — and heat-set at that twist. The tufts do not stand straight; they curl, because a very hard-twisted yarn heat-set in the twisted state has a helical shape it wants to keep and a short tuft cannot straighten it.

The surface is deliberately irregular, and the reason is precisely the failure this essay is about. A surface that already looks irregular cannot come to look irregular, so a frieze has no appearance to lose. The tufts open exactly as any other cut tufts do, and nothing shows.

That is a design response of the same kind as the crepe weave’s: a crepe’s whole point is a surface with no repeat in it, which is a surface in which no defect can be seen as a departure. Both are cases of arranging that the failure mode has no signal.

The cost is the same in both cases too. A frieze is made of yarn at an angle where the obliquity has taken half the strength and where the migration bracket is at its widest, so a frieze tuft is the weakest and least predictable tuft in the catalogue. It survives because a tuft is not asked to carry a load along its length.

The fibre chosen for resilience is the worst for twist retention #

Two decisions are made about a carpet fibre and they pull opposite ways through the same number.

A carpet wants a coarse fibre. Resilience against crushing is a bending stiffness, and a fibre’s bending stiffness goes as the fourth power of its diameter while the number of fibres in a tuft of fixed weight goes as the inverse square — so a tuft’s resistance to being flattened rises as the square of the fibre’s diameter. That is why carpet fibre is 15 to 20 decitex where apparel fibre is 1.7, and it is the single largest difference between the two trades’ raw material.

And the run-out length is proportional to the fibre’s diameter. So the same choice that makes the tuft springy makes its twist run out further from the tip.

The two effects are not of the same strength, which is what makes the trade decidable rather than deadlocked: resilience goes as the diameter squared and the run-out goes as the diameter to the first power. Coarsening the fibre is therefore worth doing, and it is worth doing up to the point where the run-out reaches the pile height — beyond which the tuft has no twist anywhere and there is nothing left to protect.

That gives a design inequality with nothing fitted in it but the contact efficiency:

pile height ≥ 2 × run-out length.

The factor of two is the judgement — it says that at least half the tuft should still be twisted when the top half has opened — and everything else is computed. For the wool at 30° on this page the run-out is 5.6 mm, so the rule asks for a pile of at least 11 mm, which is the deep end of the ordinary domestic range and is where wool cut piles in fact sit.

Read the other way it is a constraint on the fibre. A short pile — a 5 mm contract carpet, chosen because a short pile crushes less in the first place — needs a run-out under 2.5 mm, which at 30° of twist needs a fibre under about 10 decitex. That is a finer fibre than the resilience argument wants, and it is why short-pile contract constructions are made from finer filament than deep domestic ones despite facing much heavier traffic. The usual explanation is soil-hiding, which is true and is not the whole of it.

And it explains the one construction that escapes both arguments. A loop pile has no free end, so its run-out is irrelevant and the fibre may be chosen for resilience alone — which is exactly what contract loop constructions do, at 17 to 20 decitex, coarser than any cut pile in the catalogue. The fibre coarseness of a carpet is therefore readable from its construction, and the reading is not about wear at all.

What was counted, and how #

The run-out length is the same function as the fibre-grip critical length, called at the tuft’s own twist angle rather than re-derived, so a change to one is a change to both. Writing a second body for the same closed form is how two arguments about one mechanism come to disagree.

The pressure profile is a closed form checked against its own integral, and the zero at the surface — which is the whole of the free-end argument — is exact rather than asymptotic.

And the contact efficiency is the same fitted number as everywhere else in this ladder, named in every caption that depends on it. What survives it is the ordering: a coarser fibre runs out further, a lower twist runs out further, and a lower friction runs out further, all of them by the same group.

One consequence of that thickness is worth keeping. A carpet yarn’s diameter is 0.64 mm — four times a shirting warp’s — so the tuft is a body whose bending stiffness goes as the fourth power of it, and a tuft is therefore some two hundred times stiffer than a shirting thread of the same fibre. That is why a tuft stands up at all, and it is the quantity the stiffness bracket is least able to pin down, since the bracket widens in proportion to the fibre count and a carpet yarn has thousands.

Where the model stops #

The run-out is computed as though the tuft were a yarn with one free end, and a cut tuft is a yarn with a free end and a very short length. Whether the expression, which was derived for an end embedded in a long yarn, applies to a tuft only a few run-out lengths tall is not established here — and the interesting regime is exactly the one where it does not obviously apply.

Nothing here computes what an opened tuft looks like. The step from “the twist has run out over three millimetres” to “the carpet looks worn” is a perceptual one and this collection has no instrument for it.

Heat-setting is described and not modelled. The temperatures, the times and the chemistry are outside this collection entirely; what is offered is the distinction between removing a torque and adding a friction, and the different failure each implies.

And the walking argument is a mechanism rather than a rate. It says why appearance loss ratchets and not how fast, which would need the number of cycles, the load, and how much twist each cycle releases — none of which is here.

Where the ladder goes next #

Sideways, into the same free-end argument at a smaller scale. Every fibre end at a yarn’s surface is a place where the pressure is zero, which is why a yarn is hairy and why the cheapest way to change a surface is to burn them off. A tuft is that argument with one very large end instead of many small ones.

And back into the pile ladder, where the quantity this essay is about — the tuft as a coherent body rather than as an anchored one — is the missing term in the crushing argument. A crushed pile is not held down by its fibres; what it is held up by is its own twist, and that is a stiffness this collection brackets rather than computes.