Milling holds the cloth a second time

Take a piece of melton — the dense, smooth, weatherproof cloth an overcoat is made of — and cut it with scissors. The edge does not fray. Leave it unhemmed and it stays unhemmed; military uniforms have exploited this for two centuries.

Now do the same to the twill it was woven as, before milling. The edge frays immediately and completely, as any woven cloth does, because a woven cloth’s threads are held only by their crossings and a cut thread has nothing holding its end.

Between those two states, not one thread has moved from above another to below it. The draft is the draft it was woven with. Run this site’s integrity criterion on both and it returns one cloth for both, from the same implementation, correctly.

The criterion is not wrong and it is not enough #

This is the second time this site has met this shape and it is worth putting the two side by side, because the resemblance is exact and the mechanism is different.

The the compound-cloths field found that the integrity criterion cannot separate a V-fastened tuft from a W-fastened one, or a leno gauze from an open plain weave. Both pairs are strongly connected; one member of each pair holds and the other does not. The missing quantity was friction, supplied by the capstan applied to a wrap angle.

Milling is the same gap and a different filler. The criterion consumes contacts between threads and a direction at each, and it decides whether the above-and-below relation is strongly connected. Milling adds no contacts between threads. It adds a different kind of connection entirely — fibre ends that have migrated out of one yarn and into another, mechanically entangled — and there is no place in the criterion’s input for such a thing.

So the criterion answers the question it was asked, exactly and correctly, and the answer is about a fabric whose weave can no longer be seen with a lens.

That is not a defect to repair. A criterion assuming less than any other on this site is why one implementation decides a plain weave, a satin, a double cloth, a braid word, a warp-knit lapping, a pile fabric, a double plush and a leno gauze. Adding fibre entanglement to it would require it to consume a fibre model, and the whole point of it is that it does not.

The second network is one the site already models #

Here is the connection that makes this essay worth writing rather than merely stating.

A nonwoven has no thread crossings at all. It is a mass of fibres laid down at random, and it coheres — when it coheres — because the fibres touch often enough to form a connected network spanning the sheet. That is percolation, this site computes it, and the threshold for randomly oriented sticks is a published constant that the percolation model here carries and checks its own simulation against.

A milled woollen has both. It has a weave, holding it together in the ordinary way, and it has a fibre network of exactly the nonwoven’s kind laid over the top of it — built by the ratchet rather than by a carding machine, but structurally the same thing.

So the two constructions this site has treated as opposite ends of its subject turn out to be the two halves of one fabric. A melton is a woven cloth and a felt at once, and the arithmetic that decides whether its felt half is doing anything is the arithmetic already written for the nonwoven.

Reading the milling on the nonwoven’s scale #

millingProgress accumulates the ratchet’s net displacement per cycle into a migrated-fibre density and expresses it as a fraction of the stick-percolation threshold. That is a deliberate choice of units: it puts a woven cloth’s felting progress on the same axis as a nonwoven’s coherence, so the two can be read against each other.

At the middle of the friction range the model has the fibre network reaching the threshold somewhere around eight hundred cycles of agitation. That number is not a prediction and should not be quoted as one — the constant relating a ratchet displacement to a fibre-network density is fitted to nothing at all, and there is no claim here about how long a milling machine runs.

What the figure does say is the shape: the density accumulates linearly with agitation, the threshold is a fixed line, and there is therefore a fairly definite point at which the second network switches from irrelevant to load-bearing. Milling has a threshold, and a cloth milled halfway is not half-felted in any useful sense; it is a woven cloth with some loose fibre in it.

That matches what the trade does. Milling is run to a specification — a target width and length, checked by measurement — and the finisher’s skill is in stopping at the right place, because the process does not reverse.

What the second network changes #

Four properties, and the first is the one this essay opened with.

Fraying. A cut thread in a woven cloth is held by friction at its crossings alone, and that is why an unfinished edge unravels. A cut thread in a milled cloth is additionally held by every migrated fibre that crosses it, and there are a great many. The edge is stable.

Air permeability. The fibre network fills the interstices the weave leaves, so a milled cloth is far less permeable than its construction suggests. That is the whole point of a loden or a boiled-wool coat: weatherproofing by structure rather than by coating.

Dimensional behaviour. A milled cloth has been deliberately shrunk in area, sometimes by thirty per cent or more, and it is stable afterwards in a way the unmilled cloth was not — because the shrinkage has already been taken and because the fibre network resists the yarns moving relative to each other.

And tear strength falls. This one runs the other way and is worth stating because it is the cost. Tear strength in a woven cloth depends on yarns being able to slide and group up ahead of the tear, sharing the load between several threads at once. A fibre network that locks the yarns in place prevents exactly that. A milled cloth tears more readily than the fabric it was made from, and the cloth resists everything else better.

What was counted, and how #

The accumulation is the ratchet’s per-cycle displacement multiplied by the cycle count and by a fitted constant, read against web.js’s STICK_PERCOLATION, which is 5.63726 and is a published value that this site checks its own simulated web against rather than assumes.

Two assertions run. An unmilled cloth has no migrated fibre network at all — the zero-cycle row is exactly zero, which would fail if the accumulation had picked up an offset. And milling accumulates one, so the last row exceeds the first, which would fail if the ratchet’s sign had inverted.

The threshold comparison is the honest part and the constant is the dishonest part, and the figure’s note says so: the model is a mechanism, not a process. What is defensible is that the density grows with agitation, that there is a threshold, and that the threshold is the nonwoven’s. What is not defensible is any particular cycle count, and none is quoted anywhere except on a figure that labels it as a model.

The other end of the sweep is where the second hold stops being an addition to the cloth and starts being most of it.

The operation has no undo, and that decides how it is run #

Every other operation in this field can be adjusted after the fact. A cloth calendered too little goes through again; a cloth stretched too far on the stenter relaxes back; a cloth pre-shrunk too little is run again.

Milling cannot be reversed at all, and a cloth milled past its target is scrap. So the operation is run short and checked, repeatedly, with the piece measured between passes — which makes it slow and skilled in a way the rest of a finishing route is not.

That is the practical consequence of the irreversibility argument the previous rung makes, and it is worth seeing that a statement about a mechanism’s thermodynamics turns directly into a statement about how a machine is operated and how much the operator is paid.

Where the model stops #

The fibre network’s density is not derived from anything. The constant relating displacement to density carries the fibre length, the fibre diameter, the yarn’s surface fibre population and the geometry of migration, all rolled into one fitted number.

Percolation of sticks is a two-dimensional model and a milled cloth is three-dimensional, with the fibre network threading through the fabric’s thickness as well as across it. The threshold in three dimensions is different and this site does not carry it.

Entanglement is not contact. The percolation model asks whether fibres touch. What holds a felt together is that fibres are wound around one another and cannot be pulled straight without friction, which is a stronger condition than touching and closer to the capstan’s business than to percolation’s.

And nothing here is about the yarn. Milling also consolidates each yarn internally, and a milled cloth’s threads are individually different objects from the ones that went in.

Milling raises the density twice, and the second way is free #

The accumulation above counts migrated fibre and divides by an area. Both terms move, and only one of them is in the model.

A milled cloth shrinks — a third of its area in a heavy milling — so every fibre already in the network becomes denser simply because the sheet it is in has got smaller. A thirty per cent area loss multiplies the areal density of whatever fibre is present by 1/0.7, which is 1.43, with no migration at all.

So milling drives the network up two ways at once, and the second is a pure consequence of the first. That has a consequence for the shape of the process which the linear accumulation cannot show:

the density grows faster than linearly, because shrinkage feeds back into it. More entanglement draws the cloth together, the smaller cloth has a denser network, a denser network entangles faster. It is autocatalytic in the mild sense that its rate depends on its own output.

Which explains the operating practice better than the threshold does. A process that accelerates as it proceeds is one that cannot be run to a time; it has to be run short and measured, repeatedly, with the operator stopping on a dimension rather than on a clock — which is exactly what a milling machine’s operator does and why the job is skilled. A linear process with a threshold would be run to a cycle count and checked once.

And the three-dimensional threshold is not the one #

The limits section records that stick percolation is two-dimensional and that the three-dimensional threshold is different. It is different in a direction that matters, and working out how far settles what the model is really about.

Slender rods percolate in three dimensions at a volume fraction of order their own diameter over their length. A wool fibre thirty millimetres long and twenty micrometres thick has an aspect ratio of fifteen hundred, so its critical volume fraction is of order a thousandth.

A woollen cloth’s fibre volume fraction is twenty or thirty per cent.

So the cloth is some hundreds of times above the three-dimensional contact-percolation threshold before it is milled at all. Every fibre in it is already part of a connected touching network, and has been since it left the loom.

That is not a correction to the arithmetic; it is a correction to what the arithmetic is a picture of. Contact percolation cannot be the switch that milling throws, because it was thrown before the process started. The two-dimensional reading is a useful proxy precisely because it is not a contact model — it counts migrated fibre only, and migrated fibre is the part of the population that has crossed from one yarn into another.

Which sharpens the essay’s own caveat rather than undermining it. The limits section says entanglement is not contact, and the three-dimensional number turns that from a caution into a demonstration: if contact were sufficient, an unmilled woollen would already be a felt. What milling builds is not a network of fibres that touch; it is a network of fibres that are wound around one another, and the threshold for that is a much later and much sparser condition with no published constant behind it.

So the honest reading of the figure is narrower than the axis suggests, and the narrowing is worth stating plainly. The percolation threshold is standing in as a unit of density — a scale to measure migrated fibre against, borrowed because this collection already had one — and not as the condition being crossed. The condition being crossed is an entanglement threshold nobody here can compute, and the reason the borrowed unit is useful is that any entanglement threshold must be some multiple of it.

The general shape, which recurs #

There is a lesson here that outlives wool, and it is about what a criterion is for.

This site’s integrity check is its sharpest instrument precisely because it assumes so little. It consumes contacts and directions, it is exact, it is decidable in linear time, and it has decided every construction the site has drawn across the whole collection. The price of assuming little is being blind to everything that is not a contact between threads — friction was the first thing it could not see, and a second, independent connection system is the second.

The right response in both cases was the same and it is worth naming as a method: do not extend the criterion. Add a second quantity, compute it separately, and print the two side by side. The figures in this field print the criterion’s verdict and the percolation reading beside each other rather than blending them into one score, exactly as the compound-cloths field printed the criterion’s verdict beside the capstan’s number.

A blended score would be more convenient and would be a number with no model behind it. Two numbers with two models is the honest presentation, and it has the practical advantage that when the two disagree the disagreement is visible rather than averaged away.

What the finisher is actually controlling #

Milling is run to a target and the target is dimensional, which is worth setting beside the percolation reading because they are different descriptions of the same process.

A milling machine works the cloth wet, warm and under compression, and the operator measures the piece periodically — width and length — and stops when it reaches specification. A heavily milled cloth may have lost a third of its area; a lightly milled one a few per cent.

So the process variable is area shrinkage, and it is a proxy for the thing that is actually wanted, which is the density of the fibre network. The proxy works because the two go together: fibres migrating and entangling draw the structure together, so area loss and network density rise in step.

That relationship is not modelled here and would be the natural next piece of machinery. What this site can say is that they are two readings of one process, and that the finisher’s stopping rule is a dimensional measurement standing in for a structural one.

The cost of the proxy is that two cloths milled to the same width can have different amounts of entanglement, depending on how they got there — a cloth milled gently for a long time and one milled hard for a short time are not the same fabric.

Why a woven cloth is milled rather than a felt made #

Since the fibre network is what gives a melton its useful properties, it is fair to ask why the weave is there at all — a felt made directly from fibre needs no loom.

True felts exist and are made, and their limitations say what the weave is for. A felt has no yarns, so it has no yarn strength: its tensile strength is the strength of the entanglement, which is far lower than the strength of continuous twisted threads. It tears readily, it cannot be tailored into shapes that carry load at a seam, and it stretches in every direction.

A milled woven cloth has both. The weave supplies the tensile strength, the dimensional framework and the ability to hold a seam; the felt supplies the surface, the wind resistance and the stable cut edge. Neither structure alone gives an overcoat.

That is the same argument as the pile fabric’s third thread system and the double cloth’s second layer: when one structure is asked for two things it cannot do at once, the answer is two structures occupying the same space, each doing what it is good at.

Who found it, and when #

Fulling is one of the oldest textile operations there is — the fuller is a Roman trade, the fulling mill is medieval, and the mechanism of it was empirical for the whole of that history. What the twentieth century added was the recognition that felting is directional friction rather than hooking, which is the previous rung’s subject.

The percolation reading is not a standard way to look at milling and is offered here as an analogy with arithmetic attached rather than as an established result. Its justification is that the two mechanisms genuinely are the same mechanism — a random fibre network becoming connected — and that this site had already built the machinery for one of them.

That is the sort of connection a collection built on shared generators is supposed to produce, and it is worth saying that it was not planned. web.js was written for the nonwoven essay earlier here, and its threshold turned out to be the natural unit for a question about wool.

Where the ladder goes next #

If felting is a ratchet, then stopping it is a matter of removing the asymmetry — and the last rung of this ladder asks what a shrink-resist treatment must actually achieve, which is a narrower and more precise requirement than the chemistry is usually described by.