A hair layer is a balance, not a stock

Singeing is the cheapest change anybody makes to a fabric’s surface. A cloth passes through a gas flame fast enough that the body of it never gets hot, the fibre ends standing clear burn off, and under one per cent of the mass leaves. Lustre, friction, printability, measured cover and pilling all move at once, and nothing structural has been touched.

That essay priced the operation and asked a question it could not answer: how long does it last?

The cloth #

The question is not idle. A finisher singes a cloth to make it print sharply, and by the time the garment has been worn for a week the hairs are back — or they are not, and the question is which. It is also the question behind a much older observation that nobody has connected to singeing at all: fabrics get fuzzier as they are worn.

Both of those are statements about a population changing over time, and a population that changes over time is a balance rather than a stock. Nothing in this collection had treated it as one.

The claim #

A fabric’s hairiness is a fixed point of two competing rates and not a property inherited from spinning. Rubbing frees fibre ends that spinning left buried, and rubbing breaks off hairs long enough to be caught. The population goes to where the two meet, from either side, with one time constant — and a singeing is undone in a few hundred cycles because a flame changes the stock and leaves both rates alone.

The consequences are three: that singeing before printing works and singeing for durability does not; that a worn fabric’s hairiness has nothing to do with how it was spun; and that the fixed point is reachable from above, which is what a pill actually is.

Two rates, and where they meet #

Write it as a rate equation and the shape falls out immediately.

The supply is the ends spinning did not release. The hair model has 2.2 fibre ends per millimetre lying in the yarn’s outermost shell and releases two fifths of them. The rest are there, in position, held by the twist and by their neighbours. A rub drags at them, and some come free.

The removal is the hairs long enough to be caught. A hair standing clear of the cloth is a hair that another surface can snag, bend and break. The rate is proportional to how many there are.

So dN/dcycle = γ(N_shell − N) − rN, and the fixed point is γN_shell/(γ + r). Everything about the shape follows from that being a linear equation with a stable root:

there is a fixed point, it does not remember the starting state, and the approach is exponential with rate γ + r whichever side it is approached from.

The prediction that is not a fit #

The three properties above are predicted and the position of the fixed point is not.

Where the balance sits relative to the level spinning left needs the ratio γ/(γ + r), and nothing in this collection derives either rate. It is set here to reproduce the one thing everybody has observed — that fabrics get fuzzier as they are worn — and the file says so at the point of setting it.

What survives without the fit is the falsifiable content, and it is unusually clean. Two specimens of one cloth, one singed and one deliberately brushed, rubbed under identical conditions, must converge to the same hairiness and must approach it at the same rate. That is a two-specimen experiment with no calibration in it at all, and it would falsify the model outright if the singed one settled lower.

The model asserts the equality of the two rates to twelve figures, which is trivially true of a linear equation and is worth asserting anyway: it is the statement that the model has one time constant rather than two, and a model with an asymmetric mechanism in it would not.

How long a singeing lasts #

The time constant comes out at a few hundred cycles for the rates chosen, so a singed cloth is half recovered by five hundred rubs and fully recovered by two thousand.

That is a very short time in the life of a garment and a very long one in the life of a printing table. Singeing before printing is permanent for the purpose, because the cloth is printed within hours of being singed and the printed edge is fixed forever at the moment the ink dries. Singeing for the sake of a garment’s appearance in wear is not permanent at all.

The trade knows both halves and states them as separate facts: singeing is a pre-treatment, and fabrics pill after a few wearings. They are the same fact.

A nap decays for the same reason #

The balance runs downward as well, and that is a result about a different finishing operation entirely.

Raising multiplies the population by pulling fibre out of the floats — sixty-fourfold, in the numbers this ladder uses for a flannel. That is far above the fixed point. So rubbing takes it down, at the same rate it takes a singed cloth up, and a raised cloth loses its nap on a timescale of the same few hundred cycles at the places that rub.

Which is exactly what a worn flannel looks like: bright and smooth at the elbows and cuffs, still napped everywhere else. The pattern is a map of where the rubbing was, and its timescale is the balance’s time constant.

It also puts a bound on something the raising essay treated as permanent. Raising spends the cloth’s strength irreversibly — fibres are pulled out of the yarn and do not go back — and it buys a nap that is not permanent at all. The cost is a stock and the benefit is a balance, and they do not have the same lifetime.

What a pill is, in this language #

The rate equation has a third outcome and it is the one that matters commercially.

A hair that is freed and not broken off can be entangled with its neighbours into a ball, and a ball that stays attached is a pill. In the balance’s terms a pill is a local excursion above the fixed point that has been stabilised — the fibre has come out, it has not gone away, and it is now held by a few anchor fibres rather than by the yarn.

So pilling needs the generation term to be large and the removal term to be small, and the two are separately controllable. Generation is fuzz supply, which is fibre and finishing. Removal is how easily an anchor breaks, which is fibre strength.

That separation is the whole of the next two essays. A pill is anchored, not made computes the balance for pills rather than for hairs, and the strong fibre is the one that pills is what happens when the removal term is small.

What the balance says about a laundering #

There is a third intervention that acts on the population and it acts on both rates at once, which makes it the most interesting of the three and the hardest to predict.

Washing is rubbing with water and detergent in it. It supplies the mechanical action that frees ends, so it raises generation; it swells the fibres, which loosens the twist’s grip and raises generation again; and it removes the freed material into the water, which raises removal. All three effects push the same two terms, and the model has nothing to say about which wins.

What it does say is that the shape is unchanged. A wash cycle is a number of rubs, the population is going to a fixed point at a rate set by the sum of the two terms, and the first few washes therefore move the fabric further than the next few — which is what everybody observes, usually attributed to relaxation shrinkage and to dye loss and only partly caused by either.

The signature that would separate the causes is that a hairiness change goes to a limit and a shrinkage does too, and they have different time constants. Relaxation shrinkage is nearly complete in one or two washes; a hairiness balance takes hundreds of rubs, and a domestic wash is thousands. So a fabric’s hairiness should reach its balance in the first wash, which is a sharp prediction and an easy one to check.

What was counted, and how #

Three assertions, and the second is the one that is really about the model’s shape.

That rubbing takes a cloth to a hairier state than spinning left it in — which is the fitted part, and the file says so.

That a singed cloth recovers at exactly the rate a fuzzy one settles down, asserted as an equality of the two normalised approaches to within 10⁻¹² over eight sampling points. It is a consequence of linearity and it is asserted because a later change that introduced a nonlinear removal term — a plausible refinement, since a long hair is easier to catch than a short one — would break it silently and would change the essay’s central claim.

And that half the recovery happens within five hundred cycles, which is the statement that makes the result matter rather than merely being true.

The ceiling, which needs no fitted rate at all #

The position of the fixed point is fitted and the essay says so. What is not fitted is the highest it could possibly be, and that turns out to be a number with no rate in it whatever — which makes it the one quantitative statement here that a laboratory could test without believing anything about γ or r.

The fixed point is γ·N_shell/(γ + r), and the fraction in front of the reservoir is at most one, reached when nothing is being broken off at all. So

hairiness ≤ N_shell,

and spinning released two fifths of that. A cloth can become at most two and a half times as hairy as it left the spinning frame, by rubbing, ever, whatever the rates are.

That ceiling is worth three remarks.

It is a property of the spinning, not of the wear. The two fifths is the escape fraction — how much of the surface shell the spinning triangle let go — and it is a number this collection computes from the yarn’s own geometry. So the ceiling is quotable per yarn: a compact-spun yarn, which releases less, has a higher ceiling in this ratio and a lower one in absolute terms, and the two must not be confused.

It bounds the fit from the other side too. For the model to reproduce the one thing everybody observes — that fabrics get fuzzier in wear — the fixed point must exceed the spun level, which needs γ/(γ + r) above two fifths, which is r < 1.5 γ. So the fitted rates are not free: the removal rate must be under half again the generation rate, and a fit outside that would predict fabrics that get smoother with use.

And it is the quantity a pilling test is really bounded by. A pill is fibre that has come out and stayed, so the material available to be pilled is the same reservoir. Two and a half times the spun hairiness is the whole stock a cloth can ever put on its surface without losing fibre from the yarn’s interior — and past that, further rubbing has to reach below the surface shell, which is a different mechanism with a different rate and is where the linear equation stops being defensible.

The measurement that would test it is unusually blunt. Rub a cloth to its balance and measure its hairiness against the same cloth’s as-spun value. The model says the ratio cannot exceed 2.5 and, on the fitted rates, sits near 1.5. A measured ratio above 2.5 would falsify the escape fraction rather than the balance — which is a useful thing for a prediction to do, because it would point at the part of the chain that is a geometric computation rather than at the part that is a fit.

Where the model stops #

Both rates are invented. Their ratio is fitted to a qualitative observation and their sum sets the time constant, which is therefore a number with no measurement behind it. The few hundred cycles should be read as an order of magnitude.

The removal term is linear in the population and should not be. A long hair is far more likely to be caught than a short one, so removal ought to weight the tail, which would make the balance’s shape different from the population’s — a worn cloth would have proportionally fewer long hairs than a new one at the same total. That is a real and testable difference and the model cannot produce it.

Nothing is entangled. A freed fibre in this model either stands as a hair or is broken off. In reality a large fraction of them are entangled with their neighbours instead, which is a third fate the equation has no term for, and it is the fate that produces pills.

And the shell is a fixed reservoir. Rubbing hard enough and long enough would exhaust it, and the model would then predict a decline that nothing in it can reach. That limit is far outside the range where the linear equation is defensible.

The one thing the balance cannot restore #

There is an asymmetry the rate equation does not contain and it is the reason the whole argument is not a counsel of despair for the finisher.

A hair that has been burnt is gone and a hair that has been laid down is not. Singeing removes material; calendering, and the size laid on a warp, merely hold the hairs against the cloth. So the two treatments look identical the day they are applied and diverge completely afterwards: the sized warp’s hairs are released by desizing and the singed cloth’s are not, and what comes back on a singed cloth has to be pulled out of the yarn body first.

That is the difference between changing the stock and masking it, and it means the balance the singed cloth returns to is reached from a genuinely lower starting point rather than from a temporary one. The recovery is real and the material recovered is different material.

It also decides which treatment is worth what. A mask is worth exactly as long as it stays on, which for size is until the desizing bath. A stock change is worth one relaxation time, which is a few hundred rubs. And a rate change — a compact spinning frame, a shorter staple removed by combing — is worth the life of the fabric, because it moves the fixed point itself.

The generalisation #

A quantity that is maintained rather than possessed does not remember what it was given.

The transferable form is that an intervention which changes a stock buys a benefit lasting one relaxation time, while an intervention which changes a rate buys one that lasts. Singeing is the first kind; compact spinning, which changes how many ends can escape in the first place, is closer to the second — although it too changes only what is available to be released rather than the release rate itself.

The diagnostic question is the one to carry: is this a stock or a balance? For anything that is worn, rubbed, washed or weathered, the answer is nearly always the second, and nearly every finishing specification is written as though it were the first.

Who found it, and when #

That fabrics become hairier and then pill in early wear is universal knowledge and is the basis of every pilling test, which is why such tests run to a fixed number of cycles and report the worst state rather than the final one — the standard tests were built around the fact that the population rises and then falls back.

Singeing’s impermanence is likewise known to finishers and stated as practice. What is added here is the rate equation, the observation that it has one time constant in both directions, and the consequence that a singeing for print and a singeing for wear are different operations with different lifetimes.

Where the ladder goes next #

To what the balance does to a measurement. Abrasion takes the hairs first shows that a mass loss reported at a fixed cycle count is a mixture of hair and cloth in proportions that depend on the finishing, and that a napped fabric’s test never reaches the fabric at all.

And to the excursion that does not come back: a pill is anchored, not made, which is this essay’s equation with a stabilising term in it.