The hairs decide the sign of the wetting

A drop landing on a fabric does one of two things and there is no middle case. It spreads, and is gone into the cloth in a second; or it sits, beads, and rolls off. Everybody has watched both happen and the difference is usually explained by what the fibre is made of or what has been put on it, which is right and is not the whole of it.

The same finishing operation drives a cloth further into either behaviour, depending only on which it already had.

The cloth #

What water does to a thread established this site’s wetting arithmetic: a fibre has an intrinsic contact angle with water, the cloth’s geometry supplies pores, and how high a cloth wicks follows from Jurin’s law with the pore radius the cloth’s own.

All of that treats the cloth’s surface as the surface of its threads. It is not. A drop landing on a fabric meets the hair layer first, and the hair layer is a set of cylinders with an enormous surface area and almost no volume — which is exactly the geometry that classical wetting theory says does the most to a contact angle.

The claim #

A canopy of protruding fibre multiplies the cosine of the intrinsic contact angle by a roughness ratio well above one. Since a cosine’s sign is decided by whether the angle is above or below ninety degrees, the canopy drives a wetting surface to complete spreading and a non-wetting one to a Cassie state with air trapped underneath. One layer, two opposite outcomes, and the only thing deciding which is the sign of one inequality.

The consequence for a finisher is that raising is not a wetting treatment. It is a multiplier on a decision already made.

Why a hair layer has so much roughness in it #

Wenzel’s relation is one line: a rough surface has more area than its own footprint, so the interfacial energies are all multiplied by the ratio r of true area to projected area, and cosθ_apparent = r·cosθ_intrinsic.

For most rough surfaces r is a modest number — a sandblasted metal might be 1.5. A canopy’s r is large for a purely geometric reason: a cylinder contributes πd of surface for every d of shadow. So the roughness ratio is one plus π times the hairs’ projected coverage, and on a raised cloth that is comfortably above two.

At r = 2, any intrinsic angle below sixty degrees gives an apparent cosine above one, which has no solution — the drop spreads completely. Any angle above a hundred and twenty gives a cosine below minus one, and the surface cannot be wetted at all.

A sixty-degree band of intrinsic angles collapses to the two extremes, and the band widens as the canopy thickens.

The other branch, which is a different state and not a bigger angle #

Above ninety degrees the drop does not follow Wenzel at all, and this is the part that is genuinely different rather than merely larger.

A liquid that does not wet the fibre will not penetrate between the hairs. It bridges from hair to hair and sits on a composite surface of fibre and air, which is the Cassie state, and its apparent angle is set by the solid fraction it actually touches rather than by the roughness:

cosθ = f(cosθ_intrinsic + 1) − 1

with f the solid fraction. A canopy’s f is a few tenths at most and falls with height, so the apparent angle runs toward a hundred and eighty degrees.

That is superhydrophobicity, arrived at with no coating chemistry in it. It is why a napped cloth of a treated fibre repels water so much better than a smooth one of the same fibre, and why the treatments are always applied to fabrics with a surface rather than to films.

The hinge, which the model asserts #

The pivot is at exactly ninety degrees, where the cosine is zero and any multiplier leaves it zero. That is asserted to the last bit of a double rather than checked to a tolerance, because it is the essay’s whole structure: the canopy multiplies, and a multiplier cannot move a zero.

So a fibre at ninety degrees is unaffected by any amount of raising, a fibre at eighty-nine is driven to spreading, and a fibre at ninety-one is driven to beading. There is no continuity in the outcome even though the arithmetic is continuous, because the two branches are two different physical states and the drop has to choose one.

What it means in a finishing route #

The practical reading is short and it inverts the usual order of operations.

A repellent finish should be applied before raising, not after. The chemistry sets the intrinsic angle; the raising amplifies it. Raising first and treating second gives a treated canopy, which is better than nothing, but the treatment has to reach into the canopy to work and a canopy is where a finish is hardest to get even coverage into.

And a cloth intended to absorb should be raised too. A towelling’s job is to take water fast, and raising drives a hydrophilic cotton to complete spreading — which is one of the reasons a raised cotton absorbs faster than an unraised one of the same construction, and is not the one usually given.

Both readings are the same statement: the operation is a gain and the sign is set elsewhere.

Why the threshold matters here as much as anywhere #

The whole argument needs a canopy, and an ordinary woven cloth does not have one.

The criterion is n_A λ², and a sheeting sits at about a half — under the line. Its hairs stand alone, its roughness ratio is a few per cent above one, and the amplification is negligible. So an ordinary shirting wets very nearly as its fibre does, which is what makes the fibre and the finish the visible variables and the geometry invisible.

A raised cloth is thirty times over the line and its roughness ratio is above two. The whole of this essay is about raised, brushed and napped fabrics, and it says nothing new about a poplin.

Where this sits against the site’s own wicking arithmetic #

There is a boundary to draw, because this collection already has a great deal of machinery about water in cloth and none of it is about a drop sitting on the outside.

Wicking is what happens after the liquid is in. It is Jurin’s law with the cloth’s own pore radius, it is driven by the same contact angle, and it is slower along a crimped thread than the straight-tube arithmetic says. All of that is interior.

This essay is about the boundary condition on that arithmetic, which is whether the liquid gets in at all. A cloth with excellent wicking and a Cassie surface does not wick, because nothing reaches the pores; a cloth with poor wicking and a spreading surface takes the drop instantly and then holds it near the face.

The two are separately controllable and they are usually specified as one. A moisture-management specification that quotes a wicking height and a spreading time is quoting one interior property and one surface property, and only the second is affected by raising.

What was counted, and how #

Four assertions and one of them is an exact zero.

That a canopy is a great deal more surface than area — the roughness ratio must exceed two, which is a statement about the coverage rather than about wetting.

That a wetting fibre with a canopy on it spreads completely, checked over the intrinsic angles below ninety in the sweep. That a non-wetting one beads harder than the bare fibre would, checked over the angles above.

And that ninety degrees is the hinge, asserted as an exact zero. A model that had introduced an additive term anywhere — a line tension, an offset, a fitted constant — would fail that assertion immediately, and it is the cheapest available check that the construction is multiplicative all the way through.

The one experiment that separates the two branches #

The essay’s central claim is a sorting rather than a shift, and a sorting is falsifiable in a way a shift is not.

Take one construction, raise it in a series of steps, and measure the contact angle of a drop at each step — once with the fabric untreated and once with a repellent finish on it. The two series must diverge, one running to zero and the other to a hundred and eighty, and they must diverge from a common point at the untreated fabric’s own angle.

A model in which raising simply made the surface “more textile-like” would move both series the same way. A model in which the finish and the raising each contributed additively would give two parallel lines. Only a multiplier gives a fan opening from a hinge.

And a third specimen at the hinge should not move at all. A fibre whose intrinsic angle is ninety degrees — which is roughly where a lightly-treated polyester sits — should be indifferent to raising, which is a strange and specific prediction and is the strongest test the essay offers.

How wide the surviving band is, and which fibre sits in it #

The sorting has an exact width and it is worth computing, because it says which fibres are affected and which are not.

Wenzel’s relation has no solution when the apparent cosine leaves the interval from minus one to one, and it does that as soon as |cos θ| exceeds 1/r. So the intrinsic angles that survive as intermediate — neither complete spreading nor a Cassie state — are the ones inside

90° ± arccos(1 ÷ r),

a band centred on the hinge whose half-width shrinks as the roughness ratio grows.

A bare cloth leaves almost the whole range intact — which is why an ordinary shirting wets very nearly as its fibre does, and why the fibre and the finish are the visible variables on such a cloth.

A raised one collapses a sixty-degree band into two points, and heavier raising narrows it further without ever closing it, because arccos(1/r) approaches ninety degrees only as r grows without bound.

Read against the fibres a finisher actually meets, that table sorts them.

Cotton and viscose, at intrinsic angles well under sixty, are outside the band and are driven to complete spreading. A raised cotton takes a drop instantly, which is what a towelling is for.

A treated fibre, at a hundred and ten or more, is outside the other side and is driven to a Cassie state. That is the superhydrophobic route, and it needs the raising as much as it needs the chemistry.

And untreated polyester, at around eighty degrees, sits inside the band at every raising a finisher would apply. It is the one common fibre the sorting does not sort: raised, it moves from eighty to about seventy, which is a real change and not a collapse. So a polyester fleece is neither absorbent nor repellent, is described by everybody who handles one as “not very good at either”, and is the case the essay’s own hinge argument predicts — a fibre near ninety degrees is barely moved by any amount of canopy.

That is a testable statement about the commonest napped fabric there is, and it is the reason such fabrics are always sold with a finish on them. The construction cannot decide the question for polyester and the chemistry must.

Where the model stops #

Which state a real drop reaches is not predicted. Near the hinge both Wenzel and Cassie states are available, and which one a drop finds depends on how it arrived — dropped from a height, condensed, or placed. The model computes both branches and does not choose between them, and the metastability there is genuine physics rather than a modelling gap.

The solid fraction is a single number and it falls with height. A drop bridging a canopy sits at whatever height its own weight and surface tension put it, and the fraction it touches there is not the fraction at the base.

Nothing here is chemistry. The intrinsic contact angle is an input and every result is a transformation of it. What a fibre’s angle is, what a finish does to it and how it changes when the fibre swells are all outside this collection’s subject, and swelling in particular is one of the more violent things water does to a fibre.

And a hair is rigid. A real hair is bent by a drop’s own surface tension — that is the mechanism behind elastocapillary clumping, and a clumped canopy has a completely different solid fraction from a standing one.

Why the effect is stronger on a cloth than on any other rough surface #

It is worth asking why a fabric is such an extreme case, because the answer says something about the geometry rather than about textiles.

A roughness ratio is a surface divided by a footprint, and for most rough solids the two are made of the same material and the ratio is bounded by how steep the roughness can be. A machined groove, an etched pit or a sandblasted pit all have their surface attached to the substrate at every point, so doubling the area means doubling the depth of something.

A canopy is not attached at every point. Its surface hangs in the air, held at one end, so each hair contributes its whole circumference for the price of its shadow — a factor of π before anything else — and it does so at whatever height it happens to reach. That is a much cheaper way to buy area than roughening a solid, and it is the reason the most water-repellent surfaces in nature and in engineering are hairy rather than merely rough.

The same geometry explains why the two branches of the essay are so unequal in their robustness. The Wenzel branch needs the liquid to follow the surface into the canopy, which needs the canopy to be wetted all the way down; the Cassie branch needs only that the liquid bridge the tops, which needs nothing of the interior. So a repellent canopy is easy to make and a fully-wetting one is not, and a hydrophilic raised cloth that has been allowed to dry with its hairs clumped will not spread as the arithmetic says it should.

The generalisation #

A multiplier on a quantity whose sign matters is not a gradual effect; it is a sorter.

The transferable form is that whenever a treatment scales something that can be positive or negative, the treatment does not shift the population along a scale — it drives it to the ends, and the direction each member goes was decided before the treatment was applied. That is an unusual shape for an intervention and it is easy to mistake for two different interventions working on two different materials.

The diagnostic is to look for the hinge. If there is a value of the input that the treatment leaves alone, the treatment is multiplicative and the outcomes are being sorted rather than shifted.

Who found it, and when #

Wenzel’s relation is from 1936 and Cassie and Baxter’s from 1944, and the observation that roughness amplifies both wetting and non-wetting is theirs. Its application to textiles is standard: every account of superhydrophobic fabrics rests on it.

What is added here is the roughness ratio computed from a hair population rather than measured, the threshold that says which fabrics are in the regime at all, and the reading of raising as an amplifier with the sign set upstream — which is a statement about a finishing sequence and does not seem to be written down.

Where the ladder goes next #

The same canopy holding a drop out is the canopy holding air in, which is most of a fabric’s warmth; and the same canopy taking ink off a printed edge is why a print is only as sharp as the hairs are long, where the liquid wets the fibre and travels rather than sitting on it.

Both of those are the wetting branch of this essay with something else riding on it.