Textile Structure
how cloth is put together
Cloth doing a job

A pill is anchored, not made

Every account of pilling starts with how a ball of fibre forms and stops there, which explains why fabrics pill and not why some of them stay pilled. A pill is not a thing that happens; it is a standing population, and the number on a fabric at any moment is a generation rate times a lifetime.
15 min read 6 figures A mechanism, not a materialMeasured, not claimedStructure before fibre

Worth reading first: A hair layer is a balance, not a stock · A yarn's surface is a distribution · Abrasion takes the hairs first.

The standard account of pilling has three steps and stops after the second. Rubbing pulls fibre ends out of the yarn; the loose fibre entangles into a small ball; the ball is held to the cloth by a few fibres that are still anchored in the yarn. That is a correct description of how a pill comes into existence and it explains why fabrics pill.

It does not explain why some fabrics stay pilled and others do not, and staying pilled is the whole of the commercial problem. A garment that develops pills and sheds them is a garment nobody complains about.

The strong fibre is the one that pills. Standing pills per unit area by fibre, relative to wool, at one and the same fuzz supply — every row is the same cloth raised the same amount, so the only thing varying is how long a pill survives once it exists. A pill is not made, it is kept: rubbing generates it and rubbing breaks the anchor fibres that hold it, and an anchor survives in proportion to how much force it takes to break. So polyester carries 18 times wool's standing population from the same generation rate, and the ordering here is exactly the ordering of tenacity and nothing else. Wool sheds its pills because wool anchors break. No two real fabrics have the same fuzz supply, which is why a wool knit still pills more than a cotton shirting in practice — the comparison drawn here isolates the anchor and says nothing about the generation, and reading it as a ranking of fabrics would be wrong.
Fig. 1 Standing pills per unit area by fibre, relative to wool, at one and the same fuzz supply. Every row is the same cloth raised the same amount, so the only thing varying is how long a pill survives once it exists — and the ordering is the ordering of tenacity and nothing else.

The cloth

There are two pilling problems and the trade has one word for them.

A fabric that produces a great deal of fuzz — a soft-spun woollen, a brushed jersey, anything with an open surface — makes pills readily and is a nuisance in the first few wearings.

A fabric that keeps the pills it makes is a different complaint, arrives later, and is the one that generates returns. It is not the same fabrics.

A pilling test grades a specimen after a fixed number of cycles — the same rubbing an abrasion test does for a different reason — and reports one number, which is a snapshot of a population that has been rising and may have started falling. Two fabrics with very different mechanisms can land on the same grade, and two fabrics with the same mechanism can land on different grades depending on where the test stopped.

The claim

The number of pills standing on a fabric is a generation rate times a pill’s lifetime, and the two factors are set by different things. Generation is a fuzz supply and belongs to the finishing and the construction; lifetime is an anchor’s strength and belongs to the fibre. A fabric whose hairs cannot reach one another has a generation rate of zero and cannot pill however strong its fibre is.

The third rate — the one that makes a pill a balance and not merely a product — is the breaking of anchors. A pill does not fall off because it wears out; it falls off because the last fibre holding it breaks, which is a much sharper condition and is where the next rung lives.

Three rates, written as one equation

The shape is the same as the hair layer’s own and it is one line.

Generation. Rubbing frees fibre and entangles it. The supply is the hair population, and the entangling needs neighbours — a hair with nothing within reach cannot become part of a ball.

Stabilisation. An entangled ball is held by some small number of fibres still rooted in the yarn. Call it a handful; the number matters and is the subject of the next essay.

Removal. Rubbing breaks anchors. A pill survives while at least one holds.

So the standing pill count is the generation rate times the mean lifetime, and the mean lifetime is proportional to how much force an anchor takes to break. That is an entirely ordinary balance and its content is in what each factor depends on.

The gate: a cloth that cannot pill

The generation term has a threshold in it and it is this ladder’s own criterion.

Entangling needs a hair to reach another hair, which is the question n_A λ² asks — the hairs per square millimetre times the square of their own length. It is 0.53 for a woven cotton sheeting and 0.003 for a singed one, and above thirty for a raised cloth.

So a fabric below the threshold cannot pill at all, whatever its fibre. That is asserted in the model rather than argued: a polyester sheeting at ordinary hairiness is required to return exactly zero standing pills, and a change that let it return a small number would be a change worth noticing.

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.
Fig. 2 The criterion across this site’s constructions, bare. All six are under one, which is why an ordinary shirting does not pill and a brushed one does. And the spread across the whole table is under a factor of two, because a coarser yarn is set more openly — so a weaver cannot construct a fabric out of the pilling regime, only finish it out.

That gate explains two things at once that are usually explained separately. A singed cloth does not pill, and the reason given is that the fibre ends have been removed — which is true and is not sufficient, because the population regenerates and the singed cloth ought to start pilling once it has. The criterion says why it takes a long time: the population has to climb back not merely above zero but above a threshold, and the threshold is a square of a length.

And a knitted fabric pills more than a woven one of the same yarn, which is universal and is usually attributed to the loops being loose. In this language it is the criterion again: a knit presents more yarn surface per unit area and holds it under far less inter-thread pressure, so more ends escape and the criterion clears.

What the balance says about a pilling test

A pilling test is a snapshot of a population that rises and falls, which makes the grade a function of when the shutter opened.

Early in a test the generation term dominates and every fabric with fuzz looks bad. Late in a test the removal term has caught up and the fabrics whose anchors break look better than they did. A test stopped in between ranks fabrics by a mixture whose proportions it does not report.

The standard tests handle this by fixing the cycle count and grading against photographic standards, which is a reasonable engineering answer and which cannot separate a fabric that generates heavily and sheds from one that generates lightly and keeps. Those are the two problems this essay opened with, and they want different remedies.

How much of an abrasion loss is not damage. The share of a reported abrasion mass loss that is hair rather than cloth, for sheeting as woven and raised 32-fold. The first material off a fabric is its hair layer, which is 0.107% of a bare cloth's mass and 3.42% of a napped one's — and which regenerates, so it keeps coming off. A bare cloth is through it by 5344 cycles and the test then reaches the crowns, where the loss means damage. A napped cloth is not through it by 171000, which is more cycles than any standard test runs, so a Martindale on a fleece never measures the fabric at all. Two cloths taken to the same mass loss have therefore not lost the same thing, and the more heavily napped one may not have been damaged. a-cloth-loses-its-strength-before-its-mass made the same point about a different pair of quantities; this is the same failure one layer further out.
Fig. 3 Where the fibres a pill is made of come from. The share of an abrasion loss that is fibre pulled out rather than worn through: a pill is anchored by fibres still rooted in the yarn, so the same rub that supplies the loose fibre supplies the anchor.

The remedy each mechanism wants

The two factors are independently controllable and the interventions do not overlap.

To lower generation: singe, crop, or use a longer staple, a harder twist and a compacted spinning triangle. All of those reduce the hair population, and the criterion’s square makes the return on reducing the length better than the return on reducing the count.

To shorten a pill’s life: use a weaker fibre, or weaken the anchors. That sounds absurd and it is exactly what is done — the anti-pilling polyesters sold for knitwear are lower-tenacity variants, engineered to be weaker so that their pills break off.

A deliberately weakened fibre, sold as an improvement. That is the clearest evidence available that the lifetime term is real and that the trade has been acting on it without a model, and it is the subject of the next rung.

What a pill is not

Two neighbouring processes have to be kept separate, because both involve entangled fibre and only one of them is this.

Felting is not pilling. The ratchet that makes wool felt is a directional mechanism: a wool fibre’s scales let it advance one way and not the other, so agitation drives fibres together irreversibly and the whole cloth shrinks. That is a bulk process with a ratchet in it. A pill is a local ball with no ratchet, and it happens to fibres with no scales at all.

And a nap is not a population of pills. Raising pulls fibre out on purpose and leaves it standing rather than balled. A nap becomes pills when it is rubbed, which is why a raised fabric’s pilling grade is poor and why the two operations sit awkwardly together in the same finishing route.

How little of a cloth is standing off it. The long hair population's share of each cloth's areal mass, in per cent. The largest here is voile at 0.145 per cent; grossing the figure up by the measured split between the long and short populations puts the whole protruding mass of an ordinary cotton cloth at under one per cent, which is exactly what a singeing loses. That agreement is the model's cheapest check and it was not arranged: the mass comes from a count of fibre ends and a length, and the singeing figure comes from a weighbridge. It is also the whole argument for why singeing is the cheapest change anybody makes to a surface. Nothing structural is touched, no strength is lost, and the lustre, the friction, the printability and the pilling all move at once.
Fig. 4 How little material a pill is made of. The long population is a tenth of a per cent of a cloth’s areal mass, so a cloth can lose every hair it has and weigh the same — which is why pilling is a surface fault rather than a wear fault.

Why the criterion is a square, which is where the leverage is

The gate is n_A λ² and the exponent on the length is worth dwelling on, because it decides which intervention is worth doing.

The criterion is a count times an area, so it is dimensionless and it has to be: it asks whether the mean spacing between hairs, which goes as 1/√n_A, is shorter than a hair. Halving the population halves the criterion. Halving the length quarters it.

So an operation that shortens the hairs is worth twice one that thins them, per unit of what it removes. And the two operations available do exactly that: compacting a spinning triangle thins the population and singeing truncates it.

That is the arithmetic behind a piece of finishing practice that looks like belt and braces. A mill worried about pilling singes and crops and specifies a compact yarn, and the three are not redundant — the first two act on the length and the third on the count, and the first two are worth more.

It also says what a longer staple is worth, and the answer is less than it looks. A longer staple reduces the ends per millimetre in proportion, so it thins the population and does not shorten it. The length is fifty-six fibre diameters and a staple length does not appear in it at all.

The nap is worth more than the cloth it grows on. Thermal resistance in clo, for sheeting and for the canopy raising puts on it, both faces counted. The cloth itself is 0.037 clo: it is 23% fibre, and fibre conducts about eight times as well as air does. A canopy is two parts in ten thousand fibre, so its conductivity is air's to four figures and every micrometre of it is worth eight micrometres of cloth. At 128× the population the nap is 2.64 mm deep and worth 34 times the fabric — which is the whole reason a flannel is warm and a poplin of the same yarn at the same sett is not. Nothing about the weave enters this comparison except through which cloths can be raised at all, and that is a question about floats that this site answered three phases ago.
Fig. 5 And what the same population is worth when it is wanted. The nap is worth more per gram than the cloth it grows on, so the fibres that pill are the fibres that insulate — one population, two names, and the difference is entirely whether it is anchored.

What was counted, and how

The essay’s assertions are about the gate rather than about the balance, because the balance’s rates are not measured.

That a cloth whose hairs cannot reach one another returns exactly zero standing pills, checked on a polyester sheeting — a fibre chosen because it is the worst-pilling fibre in the table, so the assertion is that the construction gates the fibre and not the other way round.

That the generation term rises with the criterion, and that raising a cloth clears the threshold while singeing takes it two orders of magnitude below.

The rates themselves are proportionalities and are not fitted to a pilling measurement. What comes out of the model is a ranking at fixed generation, and the essay is careful throughout to say that no two real fabrics have the same generation.

Where the model stops

Entangling is a threshold rather than a process. Whether a set of hairs within reach of one another actually forms a ball depends on the fibre’s crimp, its friction and its bending stiffness, and none of the three is in the criterion. Wool entangles far more readily than the criterion knows, and that is a real omission rather than a caveat.

The anchor count is a parameter. A pill is held by “several” fibres and the number is not computed anywhere. It matters a great deal, and the next essay is about how much.

Nothing has a size. A pill in this model is present or absent; a real one grows, and a large pill catches more than a small one, which is a positive feedback the balance has no term for.

And the removal term is linear. A pill that has lost half its anchors is more likely to lose the rest, which would make the lifetime distribution sharper than exponential and would change how a fixed-cycle test ranks fabrics.

Where a pill sits, mechanically

One more thing follows from the contact ladder and it decides which parts of a garment pill.

A pill stands proud of the cloth, so it takes the load wherever the garment is pressed — the crossover argument applied to a single object rather than to a population. A pill is a few tenths of a millimetre across and carries the whole of whatever presses on it locally, which means the anchors holding it see a force out of all proportion to the pressure.

That is the mechanism behind the observation that pills appear where garments rub against something: under the arms, at the cuffs, where a bag strap crosses a shoulder, on the seat. Those are the places where the pressure is applied, and the pressure is concentrated onto whatever stands highest.

So the same protrusion that makes a pill visible is what makes it vulnerable, and the balance’s two terms are coupled after all: a taller pill is caught more and broken off sooner. That coupling is not in the model, and its sign is favourable — a real pill population should be more sharply limited in size than a linear removal term predicts.

What a hair layer carries before anything touches the cloth. The pressure a plate feels as it comes down onto sheeting, raised 64-fold, against how far it still is from the cloth's own crowns. Every hair above the plate is bent as a cantilever and carries 3EIδ/ℓ³ until that reaches its own buckling load, after which it lies over and carries no more; integrating over the population gives the curve. At the crowns themselves the layer is carrying 14.74 kPa, so every pressure below that is a pressure at which the cloth has not been touched at all. The standard thickness test presses at 1 kPa and reads the fabric; a light-pressure test reads this instead; and a fabric brushing skin at fifty pascals is entirely inside the hair layer. The model is a bed of independent cantilevers and does not know that a bent hair leans on its neighbour, so wherever a canopy has closed the curve is a lower bound.
Fig. 6 The pressure a raised cloth’s own surface layer carries before the fabric is reached. A pill is the same argument concentrated onto one object: it stands above everything else, so it takes the load, so its anchors are worked hardest of anything on the cloth.

The generalisation

A standing population is a rate times a lifetime, and the two are usually controlled by different people.

The transferable shape is that any complaint about a quantity of something is a complaint about two independent numbers, and asking which one to attack requires knowing both. Attacking generation when the problem is lifetime produces a fabric that makes fewer pills and keeps all of them; attacking lifetime when the problem is generation produces one that sheds a blizzard.

The diagnostic is to watch the population over time rather than at a point. A generation problem shows early and plateaus; a lifetime problem shows late and keeps climbing. A single graded snapshot cannot tell them apart, which is why the standard test cannot.

The trajectory has a shape, and it is already being measured

The measurement proposed below — grade at several cycle counts rather than one — is worth making specific, because the balance has a closed form and the form says exactly what the extra points buy.

With a constant generation rate G and a mean pill lifetime τ, the standing count obeys dN/dt = G − N/τ, so

N(t) = Gτ (1 − e^(−t/τ)).

An exponential approach to a plateau, with two parameters that separate cleanly. The plateau is the product Gτ and the time constant is the lifetime, so the height of the curve gives one combination and its curvature gives the other — and the two are the two commercial complaints.

That also settles what a single-point grade measures, and the answer depends on where the point falls.

Well before the time constant, N ≈ Gt and the grade is a pure measurement of generation — the fuzz supply, which belongs to the finishing.

Well after it, N ≈ Gτ and the grade is the product, in which a fabric that generates twice as much and sheds twice as fast is indistinguishable from one that does neither.

And the standard cycle count decides which, without saying so anywhere.

Which the standard test could already report

The useful part is that the extra points do not need a new method. The Martindale pilling procedure assesses a specimen at a series of intervals — a hundred and twenty-five rubs, five hundred, a thousand, two thousand, five thousand, seven thousand — and reports the grade at whichever interval the specification names.

Six assessments spanning a factor of fifty-six in cycles, made on one specimen, of which one is reported. The trajectory is already being observed and then discarded, and fitting the six points to the expression above returns G and τ with nothing added to the test but the arithmetic.

That is an unusually cheap improvement to a standard: no apparatus, no extra specimen, no extra operator time, and two numbers instead of one.

And it names the dangerous fabric

The fit has a third output which is the diagnostic worth having, and it identifies exactly the fabric that passes a test and fails in service.

If the fitted time constant exceeds the test’s length, the fabric has not been tested. Its trajectory is still on the rising part at seven thousand rubs, its grade is a reading of Gt rather than of Gτ, and the plateau it will reach in wear is higher — possibly much higher — than anything the laboratory saw.

Those are the low-generation, long-lifetime fabrics: little fuzz, strong anchors, few pills forming and none coming off. They grade well because the count is low when the shutter opens and they go on accumulating for the life of the garment.

The opposite fabric — a blizzard early and a clear surface later — grades badly and behaves well, because its time constant is short and its plateau is reached inside the first interval.

So the single-point grade ranks the two exactly the wrong way round, and the trajectory ranks them correctly. That is the sharpest form of this essay’s complaint about the test, and it comes with the remedy attached: fit the six points that are already there, and reject any fabric whose time constant is longer than the test.

The one number that would make the model predictive

Everything above is a ranking and none of it is a rate, and there is a single measurement that would change that.

A pilling test already produces the raw material: a specimen graded at several cycle counts rather than at one gives the population’s whole trajectory, and a trajectory that rises and rolls over determines both rates. The rise’s slope is the generation term and the plateau’s height is the ratio of the two.

Two numbers out of a test that currently reports one, from the same apparatus and the same specimen, at the cost of stopping four times instead of once. And the two numbers are the ones the two commercial complaints are about.

It would also settle the anchor count, which the next essay shows is the parameter everything in a blend turns on. A fabric’s plateau height against its fibre’s tenacity, across a few fibres, is a measurement of how many anchors a pill has — because the lifetime is proportional to the anchor count times the tenacity and the tenacity is known.

Who found it, and when

The three-step account of pilling — fuzz, entangle, anchor — is standard and dates from the 1950s, when the problem arrived with the synthetic fibres. That strong fibres pill worse is equally standard and is the reason low-pilling polyester exists.

What is added here is the balance: that the standing count is a product of two independently controlled factors, that the generation term has a geometric threshold in it which a construction cannot cross, and that the two commercial complaints usually called pilling are complaints about different factors.

Where the ladder goes next

To the lifetime term, which is where the surprising arithmetic is. The strong fibre is the one that pills computes what happens when a pill survives while any of its anchors holds — and finds that a fifth of a strong fibre in a blend buys most of the pure strong fibre’s pill life, which is a maximum over a small sample rather than an average and which no mixing rule produces.

Where this goes next

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. A fifth of a strong fibre buys most of its pilling. How long a pill survives on a wool fabric as nylon is blended into it, relative to the pure wool. A pill is held by several anchor fibres and survives while any of them holds, so its life is set by the strongest anchor it happens to have — and the chance that a pill with 8 anchors has at least one strong one is 1 − (1 − x)^8, which is already 83% at a fifth. The blend therefore gets 83% of the pure strong fibre's pill life while keeping the whole of the weak fibre's fuzz supply, which is the worst of both. Nine tenths of the way arrives by 30%. No average of the two fibres' properties produces this curve: it is a maximum over a small sample, and a maximum is not an average. It is also why a fifteen-per-cent polyamide in a wool knit is notorious, and the arithmetic says the reputation is deserved. The strong fibre is the one that pills A pill survives while any one of its anchors holds, which is a maximum over a small sample rather than an average — and a maximum behaves nothing like an average. A fifth of a strong fibre in a blend buys four fifths of the pure strong fibre's pill life while leaving the whole of the weak fibre's fuzz supply in place. The ratchet a wool fibre is. A fibre with its scales, and the two strokes of one cycle of agitation. The push is the same in both directions; the distance is not, because the scales resist tip-first motion more than root-first. Every cycle therefore nets a displacement in one direction, and no amount of further agitation undoes it. The ratchet that makes wool felt A wool fibre is covered in scales pointing one way, so it slides more easily root-first than tip-first. Agitate it and the motion is symmetric while the result is not — every cycle nets a displacement in one direction, and no amount of further agitation undoes it.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

AbrasionCanopy criterionFeltingHair balanceHair layerPill anchorPillingRaisingSingeingTenacity