A coating fills the crowns before it bridges the holes

Coating is described as a layer applied to a fabric, and the picture in the description is of a film lying on a surface. A fabric does not have a surface a film can lie on. It has a height field with a peak-to-valley range of most of a millimetre, and a film poured onto it fills that field from the top down until it has enough material to be continuous.

The claim #

The volume a coating must supply to reach a level is the integral of one minus the bearing area from the top down, and it is dominated by the valleys rather than by the crowns.

Three consequences, and the third is the useful one.

The add-on to make a face flat is larger than the cloth’s own weight. For a sheeting of 135 grams a square metre it is 185, at an ordinary film density, and that is before any thickness of film above the crowns.

The weave decides the first ten grams and nothing after that. At five per cent of the plan covered, a satin needs almost nothing and a plain weave needs fourteen grams; by fifty per cent the three weaves are within eighteen per cent of one another; by the time the crowns are buried the difference is three per cent.

And the film is continuous long after it has covered the crowns. Covering is not bridging. What decides whether a coated cloth holds pressure is whether the film spans the holes, and the holes are at the bottom of the curve.

The construction #

The bearing area at a depth is the fraction of the plan at or above that level. So the fraction not yet reached is one minus it, and the volume of the space between a plane at depth δ and the surface is

$$V(\delta) = \int_0^{\delta} \bigl(1 - A(z)\bigr)\,dz$$

per unit plan area, which is a depth. Multiplying by the film’s density gives an add-on in grams per square metre.

Nothing about the coating enters except its density. The whole of the calculation is a property of the fabric’s surface: a rough cloth needs a heavy film to bury it, a smooth one needs a light one, and the film’s chemistry, viscosity and adhesion are irrelevant to the volume required.

What was counted, and how #

For each weave the surface is built, its bearing curve is taken over a hundred and twenty depths from a ten-thousandth of the cloth’s thickness to the whole of it, and the integral is accumulated by trapezoid.

The results for one sheeting at a film density of 1.2, in grams per square metre needed to reach each stated coverage of the plan:

At five per cent covered — the crowns and nothing else — a plain weave needs 13.9, a two-and-two twill 5.0 and an eight-end satin essentially nothing. At twenty per cent: 46.7, 24.9 and 12.1. At fifty per cent: 102.3, 93.3 and 86.2. At the bottom of the surface: 188, 187 and 184.

The spread closes from infinite to three per cent as the film thickens. That is the shape of the whole result, and it comes directly from the shape of the bearing curves: they differ enormously at the top, converge in the middle, and end at the same place because every one of these cloths has the same cover factor and the same thickness.

Why the valleys dominate #

The integral weights each depth by how much of the plan is not yet covered, and near the top almost none of it is.

At the crowns the bearing area is a per cent or two, so one minus it is nearly one, and every micrometre of descent costs nearly a micrometre of film. That continues for most of the descent. Only at the very bottom, where the bearing area is approaching the cover factor, does the cost per micrometre fall.

So the film’s volume is set by how open the cloth is over most of its thickness, and that is not the same as one minus its cover factor. The cover is a projection — what fraction of the plan the threads occupy seen from above — and it is reached only at the level where both systems are at their full width. Everywhere above that the cloth is more open than its cover says, and the film has to fill the difference.

That is worth stating plainly because it is the opposite of the convenient assumption: for the total add-on, the shape of the surface matters more than the cover factor does, by a factor of about 1.7.

The check that changed the answer #

A volume integral is the kind of thing that can be wrong by a factor and look plausible, so it was checked against an answer that needs no integration at all — and the check reversed what the essay was going to say.

If the surface were a rectangular block — cloth from the top of the crowns to the bottom, with its cover fraction constant throughout — the void volume would be the thickness times one minus the cover, exactly, with no shape in it. For this sheeting that is 382 micrometres times 0.25, which is 94 micrometres of film.

The computed integral is 156. The block estimate is short by two thirds.

So the shape of the surface is worth sixty-seven per cent of the fill, and the reason is visible in the bearing curve: the cover fraction is not reached until the very bottom. Over most of the descent the plan is far more open than the cover factor suggests, because the crowns are thin and the threads only reach their full width at their own mid-height. A block model treats a cloth as though its threads were rectangular, and a fabric’s threads are round.

That is a genuinely useful correction, and it is the wrong way round from the intuition. The shape matters more for the bulk quantity than the block model allows, and it matters in the direction that costs material. Anybody estimating a coating requirement from a cover factor will under-order by about a half.

Where the sixty-seven per cent comes from #

The block estimate is short by two thirds and the essay attributes it to the threads being round rather than rectangular. That attribution can be made exact, and doing so turns a measured discrepancy into a formula a reader can apply to a cloth this collection has never drawn.

A cylinder occupies π/4 of the rectangle that circumscribes it. So a system of round threads at a plan cover K fills, averaged over its own depth, only (π/4)·K of the volume the block model gives it. The void fraction averaged through the thickness is therefore 1 − (π/4)·K rather than 1 − K, and the ratio between the true fill and the block estimate is

(1 − 0.785 K) ÷ (1 − K).

For the sheeting’s cloth cover of about 0.75 that is 1.64, against the 1.67 the integral measured. The residual is the crimp — the threads are not straight cylinders, so their own axes wander through the thickness — and it is small.

The formula is worth more than the agreement, because it says how the correction moves.

The shape correction is negligible for an open cloth and enormous for a closely covered one, and it grows without bound as the cover approaches one — because the block model’s void goes to zero while the true void does not, the threads never filling their own bounding boxes.

Two consequences follow.

The cheap estimate is right where the coating is expensive and wrong where it is cheap. An open cheesecloth needs a great deal of film and the block model gets it nearly right; a close sateen needs less film and the block model understates it by two thirds. So a coater working from a cover factor will be roughly right on the fabrics that cost most and badly wrong on the ones that cost least — which is the ordering that makes the error survive, because a proportional error on a small number is not noticed.

And it says which fabrics a coating specification should be re-measured on. Anything above a cover of about 0.8 has a shape correction over two, which is past the point where an estimate from a cover factor is worth having at all. Below 0.5 the block model is inside twenty per cent and is fine. That is a usable rule with a threshold in it, and it needs one number off a specification sheet.

The caution is that π/4 is a circular thread’s figure and a pressed cloth’s threads are not circular. A racetrack section fills more of its bounding box, so a calendered cloth’s ratio is nearer one — which means a calendered cloth is cheaper to coat than its cover factor suggests, and by an amount that grows with how hard it was pressed.

Where the surface does matter #

The first few grams, and they matter for a different property.

A coating applied at a low add-on does not fill the fabric; it wets the crowns and stops — which is a wetting question rather than a volume one, and a wetting supplies its own force. What that produces is not a continuous film but a set of coated ribbons on a fabric that is otherwise bare — which is exactly what a light finish is meant to be, and it is why a water-repellent finish weighs a few grams a square metre while a waterproof coating weighs a hundred and fifty.

And the ribbons are where the wear is. The crowns take all the rubbing, so a coating that covers only them is protecting exactly the part that needs protecting, at a twentieth of the weight. A weave with plateaux gets that protection cheaper — a satin’s crowns are covered by essentially nothing where a plain weave’s need fourteen grams — because a ribbon is a more efficient thing to cover than a field of summits.

The practical form: for a barrier, pick the cloth with the smallest holes; for a surface finish, pick the cloth with the most crown line. Those point in opposite directions, since a floated cloth has the most crown line and the largest and most various holes.

Two coatings that weigh the same and are not the same #

The arithmetic separates two fabrics that a specification would call identical.

Take a plain weave and an eight-end satin of one yarn at one sett, and coat both to twenty grams a square metre. The satin is covered over about thirty per cent of its plan; the plain weave over about twelve. The same add-on has produced two quite different coated fabrics, and neither is a barrier — both are surface treatments, one twice as complete as the other.

Now coat both to a hundred and eighty. Both are filled, both weigh the same, and the difference between them has vanished into the third decimal place.

So an add-on specification is meaningful only at the heavy end. At the light end it does not say what has been coated, because the same weight of film covers different fractions of different weaves — and the fraction covered, not the weight applied, is what decides whether the finish does what it is for.

Where the model stops #

The film is taken as a liquid that levels perfectly. A real coating is applied by a knife, a roller or a transfer paper, it has a viscosity and a surface tension, and it does not fill a fabric’s interstices completely — it bridges some of them, trapping air, which is why a coated cloth’s measured add-on and its computed fill do not agree.

The fabric is not penetrated. A coating that wets the yarn goes into it, filling the spaces between fibres as well as between threads, and a yarn is about forty per cent air. That is a large additional volume, it is the difference between a coating and an impregnation, and nothing here distinguishes them.

And the cloth is not compressed by the application. A knife over roller presses the fabric hard, so the surface being filled is a pressed surface with a smaller relief and a smaller void volume than the relaxed one computed here. That would reduce the required add-on by tens of per cent — and it would also reduce it more for the weaves that press most.

No bridging, no meniscus, no cure shrinkage. Every one of those changes the geometry at the hole, which is where the failure is, and none of them is in a volume integral.

What it says about a laminate #

The volume argument also explains why a laminate is a different object from a coating, and why the trade keeps them apart.

A laminate is a film made elsewhere and bonded to the fabric. It does not fill anything: it sits on the crowns, bonded wherever the adhesive reaches, and the interstices beneath it stay full of air. So a laminate reaches a continuous barrier at a fraction of the weight, because it never pays the fill.

The price is where the bond is. A laminate is held at the crowns and nowhere else, which is a per cent or two of the plan, and everything about its durability follows: it fails by delamination at the crowns, it fails first where the crowns are fewest, and it fails under exactly the flexing that moves the crowns relative to one another.

That is a testable ordering. A laminate should adhere best to the weave with the most crown line, since the bond area at a given pressure is the bearing area, and worst to a plain weave. It is also why laminating substrates are usually knitted or lightly napped — a raised surface presents fibre ends over the whole plan, which is a great deal more bond area than any woven crown pattern can offer.

The generalisation #

Filling a rough surface costs the integral of its empty space, which is a bulk property, and covering it costs the integral of its top, which is a surface property.

The transferable statement is that “coating” names two operations that are separated by two orders of magnitude in material. Covering the highest few per cent of a surface takes almost nothing and changes everything about how the surface behaves — its friction, its wear, its wetting. Filling the surface takes as much material as the substrate weighs and changes what it is.

The corollary is a way of reading a specification. An add-on in grams per square metre says which of the two operations has been performed, without anything else being known: a few grams is a surface treatment, a hundred is a filled composite, and there is very little in between that is a sensible thing to make.

The one number worth carrying #

If a single figure is to be taken from this, it is that a fabric’s void volume is about 1.7 times its thickness times one minus its cover, and that is the material a coating has to displace.

For the eight cloths in this collection’s table the block estimate alone runs from 94 micrometres of film on a close sheeting to 273 on an open cheesecloth — a factor of three, driven by the thickness and the cover together — and the true fill is about 1.7 times each of those.

The corollary is that an open cloth is expensive to coat and a close one is cheap, in film rather than in fabric, and the film is usually the costlier material. That is a real constraint on how coated fabrics are built and it is arithmetic rather than practice: no coating chemistry can avoid filling the space that is there.

Who found it, and when #

The bearing-curve integral is Abbott and Firestone’s construction read as a volume, which is standard in surface metrology, where the same integral is called the material ratio and is used to specify how much of a bearing surface must be worn away before it seats.

The textile application appears not to exist, which is a little surprising: coated fabrics are specified by add-on, add-on is a volume, and the volume available is a computable property of the construction. What the trade does instead is measure the pick-up empirically for each cloth and each coating, which works and produces no transferable number.

Where the ladder goes next #

To the last case where a surface meets something that is not a plate: friction is two surfaces rather than one, where a fabric meets a fabric and the two bearing curves have to be combined rather than read.

Sideways, the coating result closes a loop with the wear result: the crowns are where the rubbing lands and where a light finish sits, so what wears and what is protected are the same few per cent of the cloth.