The pore that wicks is the pore that leaks

This collection has computed how high a cloth wicks. A curved meniscus in a pore of radius r carries a pressure of 2γcos θ / r across it, that pressure divided by ρg is a height, and the height a fabric lifts water to is set by the finest pore a connected path can be built from — which is inside the yarn, between the fibres, one to two orders finer than the hole between the threads.

The function that computes it refuses a contact angle at or past a right angle, on the grounds that nothing is drawn up. That refusal is right, and it has been silent about something for as long as it has existed.

The pressure is still there. At a contact angle past ninety degrees the meniscus is depressed rather than raised, the same 2γcos θ / r has changed sign, and it is now the thing keeping water out. A cloth’s water resistance and a cloth’s wicking are one expression read at the two ends of an angle.

The claim #

A fabric’s water resistance and its wicking are the same equation with opposite signs, so the geometry cannot be chosen to give both — and they read opposite ends of the pore distribution, so the same cloth is very good at one and very bad at the other by a factor of a thousand.

The first half is an identity: a rise at contact angle θ and a head at 180° − θ are the same number, to the last digit, because the cosine is odd about a right angle. It is asserted over a grid of radii and angles rather than described, because it is the sort of thing that a sign convention can quietly break in one of two functions that are otherwise only ever compared with themselves.

The second half is the consequence, and it is where the arithmetic stops being a restatement.

Why the two questions read different pores #

A rise is set by the finest pore, because the liquid follows the path that lifts highest and a fine pore lifts higher than a coarse one. A muslin’s fine pores lift water 2,975 mm; its coarse ones lift it 56. The cloth wicks to three metres, in principle, and the coarse system contributes nothing to that number except a route to get there.

A leak is set by the coarsest pore that goes all the way through, because water under pressure needs one path and takes the cheapest. The coarse pore gives way at 56 mm of head and the fine ones are never asked.

So the same distribution, and the same expression, and the answers differ by a factor of fifty-three, because one reads the bottom of the distribution and the other reads the top.

A fabric whose pores were all one size would be the only fabric for which one radius answers both questions. No fabric is, and the cloth that comes closest is the one whose holes are all the same — which only a plain weave has, and even then only between its threads, since the fine system inside the yarn is a second population entirely.

The heads themselves, which are small #

At a well-finished 120°, in water:

Two things stand out and both are the point.

The fine column does not move. Every cloth here is the same 0.6-packed cotton, so its fibres are the same distance apart whatever the sett — and the fine pore is a property of the yarn rather than the cloth. Every row is 2,975 mm.

The eight cloths are ordered here by head, which is exactly the reverse of ordering them by pore radius, and it is not the order they come in by anything else — not by weight, not by cover, not by what they pass in air.

The coarse column spans a factor of four and is what the fabric is worth. Eighteen millimetres for a cheesecloth and eighty-one for a sheeting. A hydrostatic head of eighty-one millimetres is what a laboratory calls water repellent and what anybody standing in rain calls wet: a coated fabric is quoted in thousands of millimetres and a membrane in tens of thousands.

So a woven cloth’s water resistance is set by its worst hole and its worst hole is bad. A fabric whose individual threads would hold back three metres of water leaks at fifty-six millimetres, because of the spaces between them.

Where the finish comes in, and why it is not a structure #

Nothing on this site computes a contact angle, and it is worth being exact about why. θ is decided by the chemistry of the fibre’s surface — whether it has been scoured, sized, waxed, or given a fluorochemical or silicone finish — and this collection has no surface chemistry in it at all.

So θ is an argument with a stated default, every result is quoted at the angle it was computed at, and the sensitivity is drawn rather than described.

That division is unusually clean here, and it is the practical content of the whole essay. The geometry decides the magnitude and the finish decides the sign. A close sett and a fine yarn make both the rise and the head large; a water-repellent finish decides which of the two the cloth actually does. And because it is a finish rather than a construction, it wears off, it is defeated by contamination, and a cloth that has lost it reverts to wicking at exactly the rate its geometry always implied.

What was counted, and how #

Young–Laplace across a curved meniscus, Jurin’s law for the height, and the same pressure over ρg for the head. All three are published physics and are used as such.

What is computed here is the geometry they are applied to: the hydraulic radius 2A/P for the hole four threads bound, and the packed-bed radius d(1−φ)/4φ for the space between fibres in a bundle. Neither is a measurement and both are named wherever a number leaves.

The identity is asserted over thirty pairs — five radii by six angles — and it holds to the last digit, as it must. The two refusals are asserted to be complementary rather than overlapping: at exactly a right angle both functions refuse, because there is no pressure either way and nothing to report in either direction.

And the ordering is asserted across the cloth table. Ranking the eight cloths by head must be exactly the reverse of ranking them by pore radius, at every row, because the head is 2γ|cos θ|/ρgr and there is nothing else in it. That check exists because it would catch a sign error, and because nothing else in the table would.

What a coating does to the arithmetic, which is to remove it #

A coated fabric holds back thousands of millimetres and the reason is not that its pores are smaller. It has none. A continuous film has no through-path at all, so there is no radius to substitute and the head is set by whatever pressure will burst or delaminate the film — a strength question rather than a capillary one, and one this collection computes separately.

That is the same structural move as coated being a state rather than a treatment: the coating does not improve a quantity, it removes the mechanism that quantity was a property of.

A microporous membrane is the interesting middle case, and the whole of its design is the distribution argument above. It has pores, so it has a head; its pores are of a single size a fraction of a micrometre across, so the head is enormous; and because the pores are small it passes water vapour by diffusion while stopping liquid water by capillarity. It is, precisely, the fabric with one pore size — the ideal that the two-requirement argument says would be best in both directions at once, built deliberately.

And a repellent finish on a woven cloth is neither. It leaves the geometry exactly where it was and flips the sign, so an uncoated water-repellent cotton has a head of tens of millimetres and a coated one has thousands. The difference between the two is three orders of magnitude and no visible change in the fabric.

The figure of merit is the ratio of the smallest pore to the largest #

The two requirements read the two ends of one distribution, so a fabric asked to do both has a single dimensionless number describing how badly it is placed: the smallest pore radius divided by the largest.

For a muslin that is 2.5 micrometres over 134, or about one in fifty-three. For every other cloth in the table it is worse or barely better, because the fine pore never moves and the coarse one is what varies. A microporous membrane’s ratio is very nearly one, which is the whole of why it can hold back a column of water and still pass vapour: it has no coarse tail to leak through.

That is a more useful statement of the design problem than “narrow the distribution”, because it says what narrowing is worth. Halving the coarse pore of a woven cloth doubles the head and does nothing to the rise, so the ratio doubles and the fabric moves half an order of magnitude towards the membrane. Removing the fine system entirely — which is what a filament yarn does, since there are no staple ends and the inter-fibre channels are continuous rather than tortuous — moves nothing, because the fine pore was never the binding one for the leak.

So the only geometric route to a fabric that resists water is to close the coarse system, and there is exactly one way to do that without a film.

Closing the coarse system by jamming, and what does it #

Take the coarse pore to zero and the head does not go to infinity: it jumps to whatever the next-coarsest continuous path holds, which is the fine system, at nearly three metres. That is a woven fabric behaving like a membrane, and it needs only that the threads touch.

A cloth at its jam is exactly that cloth. So the head as a function of sett is not a gentle curve — it rises steeply as the jam is approached and then steps by a factor of fifty, and everything a woven water-resistant fabric does is an attempt to get onto the far side of that step.

The step is also why the trade’s densest cotton fabrics work the way they do. A very closely set long-staple cotton is woven a little under its jam, so it leaks; wet it and the fibres swell, the threads grow into the residual gap, and the coarse system closes. The fabric becomes water resistant by being rained on, which sounds like a marketing claim and is a direct consequence of the arithmetic above: the swelling is a few per cent of a diameter and the gap it has to close is a few per cent of a spacing.

It also says why such a fabric is heavy, stiff and slow to dry, and why it fails the moment it is stretched. Every one of those is the same closeness of sett read for a different purpose, and none of them is a defect that better weaving would remove.

Where the model stops #

A pore is not a circular tube and a cloth is not a bundle of them. The hydraulic radius is a convention for giving a non-circular passage one number, and a real path through a fabric is tortuous, connected, and varies along its length. The scaling survives the transfer and the constant does not.

Breakthrough is not a single-pore event. A fabric under a rising head does not fail at one hole and hold everywhere else: the largest pore gives way, then the next, and the measured quantity — the head at which water appears on the far face — is somewhere in the upper tail of the distribution rather than exactly at its maximum. This arithmetic computes the maximum, which is a lower bound on the measured head.

The contact angle is not one number for a real fibre. Advancing and receding angles differ, sometimes by tens of degrees, and it is the advancing angle that resists a breakthrough while the receding one governs a drop’s departure. Using one angle for both directions is the crudest thing here.

Gravity is in the head and not in the rise’s timing. The equilibrium a rise is quoted at takes a very long time to reach, which this collection has computed separately; the head has no such caveat, because a pressure balance is instantaneous.

And the crimp is in the path. A front travelling along a thread travels further than the cloth is long, which slows a rise and does not affect a head, since a head is a pressure and not a distance.

The generalisation #

When two opposite-sounding requirements turn out to be one expression with a sign in it, no amount of design in the variables the expression shares can satisfy both — and the entire design freedom lives in whatever sets the sign.

That is a strong statement and it is worth being careful about, because it is often false: two requirements usually depend on overlapping but different variables, and the overlap is where a trade-off lives. Here there is no trade-off in the geometry at all. Every geometric change that improves the head improves the rise by the same factor.

The diagnostic is to write both requirements in one expression and look for the shared factor. If it is common to both, the shared variables are not where the design is. Here γ and r are common and θ is not, so the answer is a finish, and the whole of a water-repellent textile industry is downstream of that one fact.

The second lesson is about which statistic a requirement reads. Even with the sign settled, the two requirements read opposite ends of the same distribution — the rise reads the minimum radius and the leak reads the maximum. So narrowing the distribution helps both, and it is the one geometric move that does. A fabric with a single pore size would be the ideal in both directions at once, which is exactly why a membrane is one.

Who found it, and when #

Young and Laplace’s pressure is 1805, Jurin’s height is 1718, and the hydrostatic head test for fabrics — a rising column of water over a clamped specimen — has been standardised since the 1930s.

That the two are the same expression is not a discovery and would not be presented as one; it is in every surface-physics textbook, as the statement that a non-wetting liquid is depressed by the same pressure that raises a wetting one.

What appears to belong to this collection is the pairing with a computed pore distribution: that a woven cloth’s two pore systems put the rise and the head fifty-three times apart, that the head reads the coarse system and the rise the fine one, and that a fabric’s water resistance is therefore a statement about its worst hole while its wicking is a statement about its yarn. Those two properties are quoted on the same data sheets and are usually discussed as though they were traded against each other.

Where the ladder goes next #

If a leak is decided by the largest hole, then anything that makes a hole larger is a water-resistance failure, and one of those is nothing to do with the weave: threads drift. A leno’s hole cannot drift, because its crossing locks the spacing geometrically rather than by friction.

Sideways, the same pore asked about light rather than water gives up the covering rule: one minus the cover is a cloth with no thickness, and a cloth’s openness to the whole sky is a seventh of what it is to a lamp behind it.