A calender buys the width

A calender is two heavy rollers and a nip. Cloth goes through it and comes out thinner, denser and much shinier, and the trade’s account of why is that the surface has been smoothed. The surface has not been smoothed. Its crowns are in exactly the same places, at very nearly the same heights, doing very nearly the same thing — and the lustre has gone up by a factor of twenty-four.

The claim #

Calendering multiplies a cloth’s specular area by putting a plane on top of its threads, and the whole of the gain appears in the width factor.

The check is arithmetic rather than rhetorical. Specular area factors exactly into a crown length times a section width, so a change can be attributed: measure both factors before and after, and see where the multiplication went.

The width goes up by 24.4 times. The length goes up by 1.06. The product is 25.9, the repeat itself spreads by 1.14 because a pressed cloth is wider, and the specular area comes out 22.8 times what it was — which is 24.4 × 1.06 ÷ 1.14, exactly, and is asserted as an identity rather than observed to be close.

Why a flat has one normal #

A round thread’s crown is a cylinder, and the normals along a cylinder’s surface sweep the whole half-turn across it. Within a tolerance of two degrees of vertical, only a strip b·sin ε wide qualifies — about six and a half micrometres on an ordinary sheeting yarn, out of a thread 187 micrometres across.

A flattened thread’s section is a racetrack: two quarter-circles of the reduced radius with a flat between them. The flat has exactly one normal, and it is vertical, so the whole of it qualifies at any tolerance whatever.

So the width factor goes from b·sin ε to F + b·sin ε, where F is the flat’s own width. At a moderate pressing the flat is 171 micrometres across against a six-micrometre arc, and the ratio is twenty-six.

That is a change of dimension and not of degree. Before pressing, the reflecting part of the surface is a line with a width set by an angle. After pressing it is a band with a width set by a length. No amount of tolerance-widening turns the first into the second, and no amount of float-lengthening does either.

What was counted, and how #

The pressing is not a parameter of the shine arithmetic. It is the site’s own compression model, run at a stated force per crossing: the state is found by minimising a potential with a bending term, a transverse compression term and the load’s work in it, and what comes out is a pair of aspect ratios, a pair of crimp heights and a new pair of spacings.

The surface is then rebuilt from that state and its specular area recomputed. Nothing is adjusted between the two runs; the same closed form is applied to a different geometry.

At 0.8 newtons per crossing on a two-and-two twill in sheeting, the model gives aspect ratios of 2.38 and 2.59, a flat of 171 micrometres, a thickness fallen from 382 to 220 micrometres — a 42 per cent thinning — and the specular area risen from 0.91 per cent of the plan to 21.5.

Three quantities move and they move for different reasons. The flat widens because the section is being sheared at constant area. The crown line lengthens very slightly because the crimp heights fall and the plateaux take a marginally larger share of the surface. And the repeat spreads because a pressed cloth is wider, which dilutes the same crown line over more plan.

The identity, and why it is the point #

The check that matters is not the size of the gain. It is that the gain lands in the right factor.

A model that computed a lustre gain and reported one number could be wrong in a dozen ways and look right. Factoring it means the arithmetic can be asked where the gain came from, and it has to answer consistently: the specular area’s ratio, times the repeat’s spread, must equal the width’s ratio times the length’s, to the last bit of a double.

It does — 25.92 against 25.92 — and the assertion is written that way rather than against the number, which is the commonest way an assertion goes bad on this site and has been caught here five times before.

The second half of the assertion is an ordering: the width’s gain must exceed the length’s. That is the claim the essay is actually making, it could have come out the other way if the flattening had reduced the crimp enough to lengthen the plateaux substantially, and it does not: 24.4 against 1.06.

The series, and where it saturates #

The gain is not linear in the pressing force, and the shape of the curve says where the finish stops being worth applying.

At 0.05 newtons per crossing — the lightest pressing the model will solve — the specular area has already gone from 0.91 per cent to 3.4, a factor of nearly four, for a flat of only twenty-one micrometres and a thinning of eight per cent. At 0.2 newtons it is 9.1 per cent; at 0.8 it is 21.5; at 1.6 it is 28.9.

The first tenth of the pressing buys half of the total gain. That is the behaviour of a quantity whose driver is a width being created from nothing: the first micrometres of flat are worth as much as an arc’s worth of reflection, and everything after that is incremental.

The thinning does not saturate in the same way. It runs 8, 21, 42 and 51 per cent across the same four states — very nearly linear in the logarithm of the force — so the ratio of lustre gained to thickness spent falls monotonically, and a light calender is by a wide margin the better bargain. That is a design conclusion the arithmetic reaches without being asked, and it matches what finishers do: a lustre calender is a light one, and a heavy nip is used when the object is the thinning rather than the shine.

What it costs #

The lustre is bought and the price is recorded elsewhere in this collection, which is why this essay does not have to pretend the trade is getting something for nothing.

The thickness is spent, permanently. A calender spends the compression for good: the flattening is not elastic, the cloth does not recover its thickness, and everything that depended on that thickness — the warmth, the bulk, the resistance to a bending moment — has gone with it.

The cover goes up, which is the same fact from another direction: a flattened thread is wider, so the cloth’s holes shrink and its air permeability with them.

And the cloth is stiffer in bending and softer in hand, which sounds contradictory and is not: the threads are flatter, so the fabric’s bending rigidity rises as the fourth power of a thickness that has fallen, while the surface a finger meets has changed from a set of ridges to a set of bands.

Why every lustrous cloth is both floated and pressed #

The two levers multiply, so a mill wanting lustre uses both and always has.

The numbers put the choice sharply. Weaving an eight-end satin instead of a plain weave multiplies the specular area by twenty-two, at the cost of shafts, of floats that snag and of a weaker cloth. Calendering multiplies it by twenty-four, at the cost of the thickness and permanently. A calendered satin is worth about five hundred times a plain weave as woven — and that is the fabric that has been sold as satin, sateen and glacé since calenders existed.

The reverse also follows and is less obvious. Calendering a plain weave is nearly useless. Twenty-four times almost nothing is still very little: a pressed plain weave reaches about 1.5 per cent of specular area, which is roughly what an unpressed satin gives for free. The finish cannot supply a length that the draft did not weave in.

What the same states do to the contact #

The pressed states are the same states the contact arithmetic uses, so the two questions can be asked of one cloth at once — and they answer differently, which is worth recording.

A flattened thread’s plateau is a plane rather than a line, and a plane bears its whole area at zero depth exactly as it reflects its whole area at zero tolerance. So calendering changes the bearing curve’s behaviour at the top in the same qualitative way it changes the lobe: from a square root to a step.

A calendered cloth and a cut pile therefore have the same kind of surface at the very top, arrived at by pressing rather than by cutting. That is a genuinely surprising place for the argument to land, and it is exact: both present a finite area of flat at no approach at all, and both have a bounded pressure concentration where a round-crowned cloth’s rises without limit.

The difference is what lies underneath. A pile’s flats are on top of slender columns that buckle; a calendered cloth’s are on top of threads embedded in a fabric. So the two behave alike for the first micrometre and not at all alike thereafter.

Where the model stops #

The racetrack is an idealisation and the flat is not flat. A pressed yarn’s top is fibres, and its local normals scatter at a scale far below the tolerance. So the width factor computed here is an upper bound, and the gap between it and reality is a fibre and finishing property — which is precisely why a filament cloth takes a calender so much better than a spun one.

Nothing here has heat or time in it. A real calender is hot and fast, the fibres are being taken above a transition and set in their new shape, and how much of the flattening survives is a matter of the fibre and the schedule. This collection’s compression model has no rate and no temperature, which is a standing gap recorded again.

The pressing is uniform. A schreiner calender embosses fine lines rather than pressing flat — a relief imposed on the surface rather than one the weave develops for itself — precisely to control the direction the flat faces; nothing here can distinguish the two, because the model has one aspect ratio per system and no orientation.

And the two systems are pressed unequally by the model — 2.38 against 2.59 — which is a real prediction and is not checked against anything, because nobody measures the two aspect ratios of a calendered cloth separately.

What mercerising does instead, and why it is not the same #

The other classical route to lustre on cotton is mercerising, and the surface arithmetic says clearly that it is a different mechanism.

Mercerising is a packing factor: caustic soda swells the fibre, the fibre’s own convolutions are pulled out under tension, and the yarn ends up rounder, smoother and denser. In the terms of this essay it does not create a flat. It works on the fibre-scale roughness that the width factor’s second term contains — the term this arithmetic does not carry — by making the yarn’s own surface less irregular.

So the two finishes act on different parts of the same factor. A calender adds a flat; mercerising removes scatter from the arcs. Both raise the width and neither raises the length, which is consistent with the trade’s practice of using them together and with the fact that neither is ever described as changing the pattern.

It also explains an asymmetry. A calender’s effect is undone by washing and abrasion, because the flat is a pressed shape that relaxes and wears; mercerising’s is not, because it is a change to the fibre. The arithmetic has nothing to say about the durability of either, and the mechanism says which one should be expected to last.

The generalisation #

A finish that changes the dimension of a feature beats a finish that changes its size.

The transferable statement is that a surface property which depends on a measure — a length of contact, a width of reflection, an area of adhesion — can be raised either by making the feature bigger or by raising the dimension of the set that participates. The second is worth far more, because it turns a small quantity governed by a tolerance into a finite quantity governed by a length.

Pressing a cylinder into a flat does exactly that: the participating set goes from a strip of angular width to a band of physical width, and the ratio between them is the reciprocal of a small angle. Whenever a process converts a curve into a plane, expect a factor of one over the tolerance rather than a factor of a few.

One consequence of that is checkable and nobody appears to have checked it. If a calendered cloth’s bearing curve has a step in it, then its compression curve should lose the two-thirds exponent this collection predicts for a floated cloth and become much steeper at the very light end — because the area is already there and the pressure rises immediately. A light calender should therefore change the shape of a compression curve and not merely its position, and by an amount that is largest at the smallest loads.

The exponent the step predicts, and how large the step is #

The prediction that a calender should change the shape of a compression curve can be made specific, and specific enough that the experiment has a number to look for.

A floated cloth’s contact area opens as the square root of the approach, which gives a pressure proportional to δ^(3/2) and an approach proportional to P^(2/3). Put a flat on top of every crown and the area at zero approach is no longer zero: it is the flat’s own area, so the pressure becomes proportional to δ times a constant, and the approach becomes proportional to

P to the power one.

Two thirds to one, at the light end, from a finish. That is a much larger change than anything else this ladder attributes to a weave, and it is measurable with the same specification the compression essay sets out — a decade of pressure below a kilopascal, thicknesses to five per cent.

And the step is not a small one. The flat is 171 micrometres across, and the crown line of this pressed twill works out at about 1.2 millimetres per square millimetre, so the flat area is

about a fifth of the plan, in contact at no approach at all.

Set that beside an unpressed twill, which reaches 4.4 per cent of its plan at the five kilopascals a hand applies. A calendered cloth at zero load is already touching five times what an unpressed one touches under a finger.

Which is why a glazed cotton feels cool #

That factor of five is the answer to a question about handle that is usually put down to the finish’s smoothness.

Heat leaves a hand into a fabric through the real area of contact, so a fabric with five times the contact takes heat five times as fast at the same touch — and a surface that takes heat quickly is a surface that feels cool. A chintz, a glazed cotton, a calendered percale: all of them are described as cool and crisp to the hand, and the coolness has always been attributed to the smoothness rather than to the area.

It is the area, and the area is computable. Nothing about the fibre changed, nothing about the weave changed, and the thermal contact went up fivefold because a cylinder was pressed into a plane.

The same number says why the effect is so much stronger than the mass of the finish suggests. A calender adds nothing and removes nothing; it moves 42 per cent of a cloth’s thickness and, at the surface, converts a strip governed by a tolerance into a band governed by a length. Every contact property follows the band.

And a prediction for two calendered cloths #

One consequence runs further and is worth stating as a test rather than a result, because half of it is outside this collection’s reach.

Two fabrics touch on about the square of what one fabric touches a plate on. A fifth squared against four and a half hundredths squared is a factor of twenty-two, so calendering should raise fabric-on-fabric contact by twenty-two times while raising fabric-on-plate contact by five.

Whether that shows up as twenty-two times the friction is a different question, because friction is a contact area times an interfacial shear strength, and a pressed, glazed surface plausibly has a lower shear strength than a fibrous one. The geometry says the area rises steeply and the chemistry may take some of it back, and the two can be separated by measuring the same pair of cloths against a plate and against each other.

If the plate measurement rises fivefold and the pair measurement rises twenty-two, the shear strength is unchanged and the whole effect is geometric. If the pair rises much less, the glaze is doing something the area cannot account for.

Who found it, and when #

Calendering is medieval and its effect is not in doubt. Kemp’s racetrack section, 1958, is what makes the flat computable; the compression model that produces the aspect ratio at a stated load is this collection’s own.

What the surface adds is the attribution. The trade’s language — smoothing, polishing, closing the surface — describes an appearance and does not say which geometric quantity changed. The factorisation says: not the length, the width, and by the ratio of a flat to an arc.

Where the ladder goes next #

To the last rung of this ladder, where the two systems are separated rather than the two factors: turn the cloth and the shine changes hands, because a warp crown’s normals lie across the warp and can only mirror light arriving from that direction.

Sideways, the same flattening that buys the width is what erases the crowns for every other purpose: it is the top of the compression curve, where the geometric regime ends and the cloth stops being a surface.