A cloth shrinks most the first time

A laundering test runs a fabric through five domestic cycles and measures it after each. The numbers for an unfinished cotton are recognisable to anybody who has washed a shirt: about four per cent the first time, about one the second, a half, a third, a fifth. The total is six or seven per cent and most of it happens once.

Everyone who has quoted those numbers has treated the shape as obvious — things settle down, the cloth is approaching its final size, later washes have less to do. It is not obvious. It is a strong constraint on what is happening inside the cloth, and this rung is about reading it.

The claim #

Progressive shrinkage is not a rate and a second wash is a measurement.

Nothing about a wash is slow. A domestic cycle agitates a fabric tens of thousands of times in half an hour, in warm water, with a surfactant that lowers friction. If every crossing in a cloth had the same frictional barrier, then that barrier is either above the driving force — in which case nothing ever happens, in any wash — or below it, in which case everything that is going to happen happens in the first few minutes of the first cycle and the second cycle measures zero.

The second cycle does not measure zero. It measures a quarter of the first. So the barriers are spread, and how far they are spread is a number a laundering test reports without anybody having noticed.

What a wash actually does, in this collection’s terms #

The machinery is already here and is worth restating in one paragraph.

A cloth off the loom is not at the least-energy state of its own locus. It has been held stretched in the warp by the beam tension and driven up in the weft by the reed, and it comes off carrying an excess — a distance along its locus from where it would sit if nothing were holding it. What holds it there is friction at the crossings, and this collection computes the interval that friction can hold as the resting band: a cloth may come to rest anywhere inside it, and its width is friction over stiffness.

Agitation lowers the effective friction. It does not add energy in any useful sense — the energy to move is already stored in the bent threads — it removes the barrier, which is why a cloth in a washing machine relaxes and a cloth soaking in a bucket does not.

So a wash narrows the band, and every crossing whose band is now narrower than the excess the cloth is carrying slips to the edge of its own. The cloth’s new excess is the average over its crossings.

That recursion is x → E[min(x, a·w)], where w is a crossing’s own band edge and a is the agitation factor. It is monotone, it converges to zero, and — this is the point — it takes many steps precisely because w is spread. Give every crossing the same w and it takes one.

The refutation, which is what makes it an account #

An explanation is worth having only if it could have failed, so the machinery is fed the case it must refuse.

Run the same recursion with the friction range collapsed to a point — every crossing identical — and the series has exactly one non-zero term. The first wash takes 0.442 per cent and the second takes zero, to machine precision, and so does every wash after it.

That is asserted rather than described, and it is the load-bearing check in this ladder. A laundering test that reports a second figure has measured a spread, and the spread is a property of the cloth rather than an uncertainty in anybody’s measurement of a friction coefficient.

How wide the spread has to be #

Here the account meets a difficulty, and the difficulty is the result.

Yarn-on-yarn friction for cotton is reported between 0.2 and 0.4. Take that seriously as a range across the crossings of one cloth — each crossing with its own coefficient, its own band, its own answer to whether this wash moves it — and the band edges span a factor of 1.34. Run the recursion on that and the first wash takes three quarters of a per cent, the second takes eight hundredths, and the tail is over by the third cycle.

The reported test says the second wash is 1.1 per cent, which is fourteen times larger.

So invert the question. Take the band edges to be spread log-uniformly over a factor S, centred on the band this collection computes at the middle of the reported friction range, and fit S against the second wash. It comes out at 37.2.

That is a wide spread and it is not a friction table’s spread. It is a statement about a fabric: what governs whether one crossing moves is that crossing’s own normal force, its own contact geometry, its own local moisture and whatever the neighbouring crossings are doing, and the variation in those across a fabric is not the same quantity as the variation between laboratory measurements of μ on a yarn.

A washing test is one of the very few instruments that sees it.

Two fitted, three predicted #

The fit uses two numbers and there are five data. The excess the cloth came off the loom with cannot be computed here — it depends on the beam tension, the take-up and everything the finishing did — so it is fitted, against the first wash, and comes out at 7.13 per cent. The spread is fitted against the second.

Washes three, four and five have nothing left to adjust. They come out at 0.506, 0.284 and 0.179 per cent against reported 0.5, 0.3 and 0.2 — within one per cent, five per cent and eleven per cent respectively, with the error growing steadily in the direction of the model under-predicting the tail.

That drift is worth naming rather than smoothing over. A log-uniform distribution has a hard upper edge, and a real distribution of barriers presumably does not; the crossings that survive four washes are the extreme tail, and the model’s tail is cut off where the real one continues. Fitting a distribution with an unbounded tail would improve washes four and five and would also be a third parameter, which is a bad trade against five data points.

Why this changes what a shrinkage figure means #

A residual shrinkage on a label is a number with a test attached: so many cycles, so many per cent. Read through this account, that number is not a property of the cloth alone. It is a property of the cloth and of how many washes the test ran for, and the two cannot be separated without knowing the spread.

Two cloths can have the same five-cycle figure and different tails — and neither figure separates this mechanism from the hygral one that shares its tape measure. The one with the wider spread has more left, and it will go on giving it up for as long as it is washed. That is the mechanism behind the familiar complaint that a garment “kept shrinking” after it was supposed to have stopped: the test stopped, the distribution did not.

It also says what a pre-shrinking process has to do. Compressive shrinkage takes the cloth mechanically to where the washes would have taken it, and the question of how much to apply is exactly the question of how far into the tail to go.

What was counted, and how #

The band edges are computed by the frictional ladder at each of a grid of friction coefficients, and the recursion is run on the resulting list rather than on a formula, so that the distribution’s shape is an input that can be changed rather than an assumption baked into an integral.

The recursion is asserted to be monotone at every step — a wash never makes a cloth longer — and each wash is asserted to take less than the one before it. That second assertion is a claim about the mechanism rather than about the data: a recursion of this form cannot produce a growing series, and if it ever did, the arithmetic would be wrong rather than the cloth surprising.

The fit for the spread is a bisection, and it is bracketed and checked: the reported second wash must lie between what the narrowest and the widest spreads can produce, or the model cannot represent the data at all and says so rather than converging on an endpoint.

The comparison with the computed spread is an assertion too — the fitted spread must be several times the computed one — because that gap is the finding and an arithmetic slip that closed it would otherwise pass unnoticed.

Why the first wash is not special #

One consequence of the account deserves separating out, because it contradicts the way the first cycle is usually described.

The first wash is not doing anything different from the fifth. It is not “releasing the loom strains” while later washes do something subtler; it is the same operation, applied to a cloth that happens to be carrying more excess. What makes it large is that at the start the excess is bigger than nearly every crossing’s band, so nearly every crossing moves. By the fifth wash the excess is smaller than all but the widest bands, and only the extreme tail is still free to slip.

So the sequence is a filter passing progressively narrower and narrower windows over one fixed distribution, and the cloth’s remaining excess is doing the filtering. That is why the ratios between successive washes are not constant and yet look constant: they are the ratios of successive integrals over a distribution’s tail, and for a log-uniform distribution over a wide range they change slowly.

It also explains a practical observation that has no obvious cause otherwise. Washing a cloth harder shortens the series without changing its total. More agitation is a smaller a, which scales every band down at once, which lets more crossings past on each cycle. The cloth arrives at the same place in fewer washes. That is precisely what a commercial pre-shrinking range is doing when it works the cloth mechanically, and it is why a domestic test and an industrial one report different numbers of cycles for the same fabric.

Where the model stops #

The distribution’s shape is chosen and not derived. Log-uniform is the simplest two-parameter family that is scale-free over a wide range, and nothing here argues that a fabric’s barriers are log-uniform. A different shape would give a different number for the spread and the same qualitative conclusion, which is that it must be much wider than a friction table.

The agitation factor is a constant. Whether a wash lowers the effective friction by a factor of 0.6 or 0.5 or 0.8 changes the fitted excess and does not change the shape, because it enters as a scale on every band alike. It would matter if different crossings were agitated differently, which they certainly are.

Nothing here is time-dependent, on purpose. A wash is treated as an event with a number attached rather than as a duration. That is defensible for agitation and is not defensible for the fibre’s own swelling, which happens in minutes and which this collection prices in its own ladder — a wet cloth is a different cloth before any crossing has moved.

And the excess is fitted rather than computed. It is the one quantity here that a mill actually controls, and this collection cannot predict it because it would need the beam tension, the take-up motion and the finishing route. Seven per cent is a plausible answer and it is a fitted plausible answer.

The number of washes is the logarithm of the spread #

The spread is fitted above against the second wash, which needs a model and two parameters. There is a much more direct reading of the same data, and it comes out of the recursion’s own asymptotics.

Take the band edges log-uniform over a factor S and write u for the logarithm of the excess measured from the narrowest band. Working through E[min(x, b)] for an excess well inside the distribution, the recursion becomes, to leading order,

uₙ₊₁ − 1 = (uₙ − 1) × (1 − 1/ln S),

so u approaches one geometrically with a decay constant of ln S washes.

That is the whole shape of a laundering series in one line. The number of cycles over which a cloth keeps shrinking is the natural logarithm of its barrier spread, and nothing else enters — not the excess, not the agitation, not the friction coefficient’s own value.

At the essay’s fitted spread of 37.2, ln S is 3.6 washes. The standard test runs five, which is one and a half characteristic times — enough to see most of the series and not enough to finish it, which is exactly the behaviour the five figures show, with their ratios still climbing at the fifth cycle.

Which makes the test a spread meter without any fitting #

Inverting is the useful direction, and it needs no model of the distribution’s shape beyond its being broad.

S = e^N, where N is the number of cycles over which the shrinkage decays by a factor of e.

So a cloth whose shrinkage is essentially over by the third wash has a spread of about twenty. One still giving up a tenth of a per cent at ten washes has a spread of twenty thousand. The instrument is the number of cycles, and it is already being recorded — it is simply being read as a test protocol rather than as an output.

That is a stronger version of the essay’s own claim. The essay says a laundering test measures a spread and demonstrates it by fitting; this says the measurement is a count, available by eye off the series, and it converts an eleven-per-cent disagreement at the fifth cycle into an unimportant detail because the count is robust to the distribution’s exact shape.

And it says what a test length is worth #

The same expression prices the standard’s own choice of five cycles, which is otherwise a convention.

Since the log-distance to the end falls by a factor of e every ln S washes, capturing ninety-five per cent of it takes about three characteristic times — which for a spread of thirty-seven is eleven cycles, and for a spread of a thousand is twenty.

Five cycles is a compromise chosen against a cloth with a moderate spread, and it under-reports a cloth with a wide one in exactly the way the wide-spread cloth is the one a customer will complain about. A test that ran to the point where successive cycles were equal — which is what the count above is — would be self-terminating, would report both the total and the spread, and would take longer on the fabrics that deserve it.

That is a rare thing in this collection: a change to a standard that costs nothing to justify and would produce a number nobody currently has. Run the test until two consecutive cycles agree, and report how many it took. The first figure is the shrinkage and the second is the spread, and the second is what says whether the first is the whole of it.

The generalisation #

A relaxation that takes many steps when nothing in it is slow is a measurement of heterogeneity. The number of steps is set by the spread of the barriers and not by any rate, and the ratio between successive steps is the most accessible measurement of that spread there is.

The shape recurs wherever a population of pinned elements is released by a common driving force: a stack of grains settling under repeated taps, a magnet’s Barkhausen jumps under a rising field, a network of contacts in a granular pile, a set of dislocations under a load. In every case, the tell-tale of a distribution is that repeating the same disturbance produces a smaller response rather than none, and the tell-tale of a single barrier is that the second attempt does nothing at all.

The lesson for reading a specification is narrower and sharper. A number reported after N repetitions of a test is a number about N, and quoting it without N invites the reader to treat it as an asymptote. The five-cycle shrinkage of a cotton cloth is not what it eventually shrinks; it is 96 per cent of it, and the remaining four per cent arrives at the rate of a fifth of a point a wash for a very long time.

Who found it, and when #

Progressive shrinkage and its shape are as old as laundering tests, and the standard practice of specifying a fixed number of cycles is an acknowledgement of it. Textile finishing has always understood relaxation shrinkage as the release of strains put in at the loom, and the effect of agitation is why every test standard specifies the machine.

That relaxation is hysteretic — that a cloth rests in a band whose width is friction over stiffness — is this collection’s own, from the ladder that gave the locus a force. What is added here is reading the number of washes as data rather than as a test protocol, and the finding that the barrier spread a real series implies is more than an order of magnitude wider than the friction coefficient’s own range.

Where the ladder goes next #

The same distribution of barriers governs a cloth held under load rather than washed, where it decides how much of a tension survives the night — and there the barriers set a floor rather than a schedule.

Sideways, the excess this rung fits is put in at the loom, and the construction it is put into is not the construction the finished cloth has. And a knitted fabric goes through the same shape for the same reason, with three named states standing in for a series: a knit relaxes for as long as it is allowed to.