Mercerising is a packing factor

Mercerisation is the most chemically dramatic operation in this field. Cotton is immersed in concentrated sodium hydroxide — around twenty per cent, cold — and the fibre changes structurally: it swells, its lumen closes, its convolutions pull out, and its cross-section goes from the flattened kidney shape of raw cotton towards a round one.

Held under tension while it happens, it keeps its length and gains a lustre that is the whole commercial point. Released, it shrinks by up to a quarter and gains bulk and extensibility instead.

For this site’s purposes, nearly all of that is one number.

The one number #

A yarn is fibres with air between them. This site has computed its diameter from conservation of volume since the counts essay:

d ∝ √(tex / packing)

where the packing factor is the fraction of the yarn’s cross-section that is actually fibre. Ordinary ring-spun cotton runs around 0.6.

Mercerisation does not change the tex. The mass per unit length of the yarn is what it was — no cotton is added and, if the treatment is done properly, very little is removed. What changes is that the same fibre occupies more space, so the packing factor falls, so the diameter rises as one over the square root of it.

Take the packing from 0.6 to 0.5 — a swelling of the fibre volume by twenty per cent — and the diameter rises by √(0.6/0.5) = 1.095, nine and a half per cent.

What follows #

Cover rises. At 60 threads per inch in both directions, the cloth cover goes from 63.4 to 67.8 per cent. The cloth is more opaque and the interstices are smaller, with nothing woven differently.

The jamming sett falls. The closest this cloth could have been set drops from about 88 threads per inch to 80. That matters in a way calendering’s equivalent did not: a cloth woven near its jam and then mercerised can be pushed past it, at which point the fibres have nowhere to go and the swelling is resisted rather than accommodated. Mercerising a very densely set fabric therefore produces a stiff, boardy cloth rather than a lustrous one, and the trade knows to mercerise before the cloth is set too close, or to accept the handle.

Dye uptake rises, which is the second commercial reason for the treatment and the one that is not geometric at all: the swollen, lumen-closed fibre has more accessible surface and takes up dye more deeply, so a mercerised cotton dyes to a stronger shade with the same dye.

And the fibre becomes stronger, by twenty per cent or so, because the cellulose chains reorient. That is a crystallography result and is entirely outside anything this site computes.

What does not follow, and the discipline of saying so #

The lustre is what mercerisation is sold for and this site does not compute it.

The mechanism is optical and it is a shape argument about the fibre, not the yarn. A raw cotton fibre is a flattened, twisted ribbon: light hitting it is scattered in many directions because the surface it hits is curved differently every few micrometres along the fibre. A mercerised fibre is round, straight and untwisted, so a length of it presents a consistent cylindrical surface and reflects specularly along its axis.

That is the same mechanism this site computed for satin’s shine — an uninterrupted length of thread reflecting coherently — at a scale three orders of magnitude smaller, and with a geometry this site’s machinery has no representation of. The yarn model here has no representation of a fibre’s cross-sectional shape at all; only its area.

So the figure says so on its face. The swell generator prints, under the panels, that the lustre the treatment is actually sold for is not computed there and why. That is a deliberate choice and it is this site’s fourth invariant working: a figure that implied the packing factor explained the lustre would be over-claiming a model, which is the risk this site’s own plan named as the one to watch in this whole subject.

Tension is the variable that decides which product #

The same chemistry gives two entirely different fabrics depending on one process choice, and it is worth setting out because it shows where the geometry stops and the mechanics starts.

Under tension — the cloth held to length and width while the caustic acts and is washed out — the fibre cannot shrink. It swells across, straightens, and keeps its length. The result is lustrous, stronger, more dye-receptive, and dimensionally much as it went in. This is what “mercerised cotton” means on a label.

Slack — the cloth free to move — the fibre shrinks along its axis as it swells across. The yarn shortens and thickens, the cloth contracts, and the result is bulkier, softer and more extensible, with much less lustre. This is a different product with a different name, sold for stretch and handle.

The geometric model here describes the first and not the second, because the second changes the yarn’s length as well as its diameter, and the crimp bookkeeping of the shrinkage ladder would have to be run alongside. Slack mercerisation is a shrinkage operation and a swelling operation at once, and this site models each separately and neither together.

What was counted, and how #

mercerise() computes the two diameters from diameterFromTex at the two packing factors, and the covers and jamming setts from those. The only assertion is that swelling at constant mass gives a thicker yarn, which is nearly trivial and is exactly the kind of check that catches a reciprocal written the wrong way up.

The care in this function is entirely about units, and it is worth recording because the mistake was made.

diameterFromTex returns millimetres. A sett is quoted per inch. A cover factor computed from a millimetre diameter and a per-inch sett is twenty-five times too large — and a cover factor of 15 is not obviously wrong, because cover factors above one are meaningful on this site: they mean the specification asks for more thread than there is room for. So the wrong number landed in a range where the site’s own vocabulary had a reading for it, and the first version of this function reported a cover of exactly 1.000 before and after, saturated, with the change invisible.

The fix is one line — convert the sett rather than the diameter — and the lesson is the general one: a unit error that produces a value the domain already has a meaning for is the hardest kind to see.

The two products the same bath makes #

It is worth closing on the practical form of the tension distinction, because it is the clearest case on this site of one operation with two outcomes.

Mercerised cotton, held to length, is what a sewing thread, a poplin shirting and a lustrous knitted yarn are made from: stronger, deeper in shade, and shiny. The dimensional change is small because the fabric was not allowed one.

Slack-mercerised cotton is what a stretch denim’s cotton component and various bulked yarns are made from: shorter, thicker, more extensible, and matt. The dimensional change is the product.

Same chemistry, same bath, same concentration. The difference is entirely whether the cloth was held, which is a mechanical decision taken by whoever set the machine — and it is the same variable, with the same power, that this field’s shrinkage ladder is about.

Where the model stops #

The packing factor is an input. How much a fibre swells in caustic of a given concentration at a given temperature is chemistry, and nothing here derives it. The figures use a fall from 0.6 to 0.5 because that is the middle of the range reported for a mercerised cotton, and the drag range covers the rest.

Fibre swelling and yarn swelling are not the same thing. A yarn’s diameter rises less than its fibres’ do, because some of the swelling is accommodated by the air between them — a yarn at a packing factor of 0.6 has forty per cent of its section available to swell into. Treating the swelling as a pure packing-factor change assumes none of that, which overstates the diameter change. A model that let the fibres swell into the interstices first would give a smaller and more defensible number, and this site does not have one.

The cross-section change is ignored entirely, and it is the change the treatment is famous for.

And nothing is time-dependent. Mercerisation is a reaction with a rate, and an incompletely mercerised cotton is a real and common product.

The shape this field keeps taking #

This is the fourth operation in this field to reduce to a parameter this site already had, and it is worth naming the pattern before the last ladder.

Relaxation is the crimp. Calendering is the racetrack’s flattening. Raising is a question about the float map. Mercerisation is the packing factor. In every case the finishing operation is not a new phenomenon but a move along an axis the site’s existing model already had a coordinate for.

That is not a coincidence and it is not quite a triumph either. It is a consequence of the models being geometric: a geometric model of cloth has parameters for every dimension of the thing, so anything that changes the cloth physically must show up as one of them moving. The models earn their keep by saying which parameter and by how much.

What they cannot do is anything about the parameters they do not carry — the fibre’s cross-sectional shape, its crystallinity, its surface chemistry — and every one of this field’s honest admissions is of that form. The boundary of what this site can compute about finishing is exactly the boundary of what geometry can say about a fibre, and the lustre of a mercerised cotton is on the far side of it.

The order of operations decides the product #

Finishing operations do not commute, and mercerisation is the clearest case on this site, so it is worth a section.

Mercerise then dye and the fibre’s improved dye uptake is available for the dyeing: deeper shades from less dye, and better fastness. This is the usual order.

Dye then mercerise and the caustic strips or shifts the colour, because most dyes are not alkali-fast at twenty per cent sodium hydroxide. The operation is possible and is done for particular effects, and it is not the same fabric.

Mercerise before or after singeing changes what is singed. Singeing burns off protruding surface fibre, and a mercerised yarn has fewer protruding fibres because the swelling has drawn them in — so singeing after mercerising removes less and leaves a different surface.

And mercerising after a resin finish is impossible, since the cross-links prevent the swelling the whole operation consists of.

The general point is one this field’s last essay collects: a fabric is not described by the operations it has had, but by the operations it has had in order. A specification listing a finishing route without its sequence is under-specified, and two mills following the same list in different orders will not produce the same cloth.

Why the geometry understates the change #

A closing observation about the model’s honesty, since this essay has been careful about what it does not compute.

Treating the swelling as a pure fall in packing factor assumes the fibres swell and the yarn swells with them, in proportion. A real yarn has forty per cent of its cross-section as air, and a swelling fibre can expand into that air rather than pushing the yarn’s boundary outward — so the yarn’s diameter rises less than the fibres’ volume does, and by an amount that depends on how tightly the yarn was spun.

That means the diameter ratio here, at 1.095 for a twenty per cent volume increase, is an upper bound. A loosely spun yarn with room to absorb the swelling internally would show less; a tightly spun one with no room would show all of it.

A better model would let the fibres fill the interstices first and only then push the boundary, and it would need a packing model rather than a packing number. This site does not have one, and the honest consequence is that every cover and jamming figure in this essay is quoted at the top of its plausible range rather than in the middle.

One consequence of the square-root relation deserves stating on its own, because it runs against intuition. A large change in the fibre is a small change in the fabric. Swelling the fibre’s volume by a fifth — a dramatic chemical event, visible under a microscope as the lumen closing and the convolutions pulling out — moves the yarn’s diameter by under ten per cent, and moves the cloth’s cover by four points. The chemistry is spectacular and the geometry is modest, which is exactly why the treatment is valued for the properties it changes that are not geometric at all.

The bracket the swelling actually sits in #

The essay’s own closing caution — that the fibres can swell into the air between them, so the diameter ratio of 1.095 is an upper bound — is worth turning into the collection’s usual shape. Both ends of the bracket are computable, the lower one is exact, and the interval turns out to be the whole of the effect.

The upper bound is the packing-factor reading. Every bit of swelling pushes the yarn’s boundary outward, the packing stays where it was in the sense that the fibres keep their arrangement, and the diameter goes as the square root of the volume: 1.095 for a twenty per cent fibre swelling.

The lower bound is exactly one. If the fibres swell entirely into the interstices, the yarn’s boundary does not move at all. That is available here because there is room: a yarn at a packing factor of 0.6 taking twenty per cent more fibre volume rises to 0.72, which is still well under a close packing. The yarn never runs out of room, so nothing forces the boundary outward and the diameter need not change by anything.

So the honest statement is that mercerisation changes a yarn’s diameter by between nought and nine and a half per cent, and every cover, jam and thickness figure in this essay is at one end of that.

That is a wider admission than the essay’s caveat and it is the right one, because it names what decides the answer. What sets the position in the bracket is how free the fibres are to move, which is the twist: a hard-twisted yarn has its fibres pressed against one another with nowhere to go and swells outward; a soft-spun one has room and absorbs the swelling internally.

Which is exactly the shape this collection keeps meeting. The bending bracket is two limiting arrangements of the same fibres with twist deciding the position, and the strength bracket is another. A third one, in a finishing operation, decided by the same variable, and the collection’s standing habit applies: compute both ends, refuse the interpolation, and say what would decide it.

The practical form is a testable ordering rather than a value. A hard-twisted mercerised yarn should thicken measurably and a soft one should barely thicken at all, both at the same chemistry, and the two ought to differ in the finished cloth’s cover by several points. A sewing thread — hard-twisted, mercerised as a matter of course — is the case at the outward end, and a mercerised knitting yarn is at the inward one.

Who found it, and when #

John Mercer patented the treatment in 1850 and it was a commercial failure: his process gave the strength and the dye uptake and not much lustre, because he worked slack. Horace Lowe’s 1890 patent added tension, the lustre appeared, and the process became universal within a decade.

Forty years between the chemistry and the product, with the difference being whether the cloth was held. That is a good illustration of the point this essay’s tension section makes: the same reaction produces two fabrics and the mechanics decides which.

Mercer’s own name is on the process he did not make work commercially, which is one of the more graceful outcomes in industrial history.

Where the ladder goes next #

The surface ladder ends here. What remains is the field’s third mechanism — the one that takes fibre out of the cloth’s structure and stands it up on the surface — and it begins with a question that is decidable exactly from the draft: which cloths can be raised at all.