A wet fibre is stiffer and a wet yarn is not locked

A cloth taken damp off a line handles differently from the same cloth dry, and the usual explanation is the one anybody would reach for: the water is sticking the fibres together. A meniscus between two fibres is a curved surface, the pressure inside it is below atmospheric by two gammas over its radius, and that suction presses the assembly together — so the fibres cannot slide past one another and the yarn bends as a stiffer thing.

That mechanism is real, it can be priced, and it is small.

The arithmetic is available because the same mechanism has already been priced from a different source. A yarn’s stiffness is a bracket, not a number, running from every fibre free to slide to every fibre locked into a solid rod, and what decides where in the bracket a yarn sits is whether the fibres slip when it bends — which is friction times the pressure holding them together, and the pressure comes from the twist through the helix angle.

Water supplies a second pressure, in the same units, and the comparison is one division.

The suction is worth two hundred turns a metre #

A damp yarn’s menisci sit in the yarn’s own pore system, which is computable for every yarn: a packed bed of fibres at a packing factor of 0.6 has a hydraulic radius of 2.3 micrometres for a fourteen-micrometre fibre. The suction in a meniscus of that radius is 2γ over it, which for water is 0.0624 newtons a square millimetre.

The twist’s pressure at the same fibre contacts, for a twenty-tex cotton at eight hundred turns a metre, is 0.943.

So wetting a yarn is worth the first 186 turns a metre and nothing after them. That is a usefully concrete statement of a mechanism that is usually stated qualitatively, and its shape matters more than its size: the suction is a fixed level and the twist’s pressure rises with the square of the helix angle, so the water’s contribution is a share that falls away steeply.

In a very soft yarn — a hundred turns a metre, which is a roving rather than a yarn — the suction is three and a half times the twist’s pressure and is the dominant term. In a crêpe twist at eighteen hundred it is three per cent. The mechanism is real and it belongs to soft yarns.

And it locks nothing #

Being worth a few hundred turns of twist would matter if a few hundred turns were enough to stop the fibres sliding. They are not, and an earlier essay already established by how much.

The crossover is the curvature below which the friction at the contacts can carry the shear a bend demands, and above which the fibres slip. At eight hundred turns it is 0.0089 per millimetre — a bend radius of a hundred and thirteen millimetres. Adding the water’s suction raises it by the same 6.6 per cent, to 0.0095.

A thread in cloth is bent at about twelve per millimetre, which is its own diameter’s curvature and is what its crimp imposes. The margin is a factor of twelve hundred, wet.

So the capillary route does not do what the intuition says it does. It does not lock a wet yarn’s fibres, it does not move the yarn up its bracket, and the change it makes to the pressure is a small correction to a quantity the cloth’s own curvature overwhelms by three orders of magnitude.

What wetting actually does needs no contacts at all #

The other route is geometric and it is the larger of the two by a wide margin.

What water does to a thread established the numbers: a cotton fibre in water is a fifth wider and a hundredth longer, and the anisotropy is the whole reason a wet cloth is a different cloth rather than a bigger one.

A fibre’s own bending rigidity is its modulus times the second moment of its section, and the second moment of a circle goes as the fourth power of the diameter. A yarn’s rigidity at the bottom of its bracket is the fibre count times the fibre’s rigidity, and wetting changes neither the count nor the modulus in this arithmetic — only the diameter.

$$(1 + 0.20)^4 = 2.07$$

A wet cotton yarn’s lower bound is twice its dry one, and a wet viscose’s is more than three times. That is a large effect obtained with no mechanics at all, and it is the answer the intuitive route was reaching for by the wrong road.

It also explains why viscose behaves so differently wet, which is universal experience and is usually put down to the fibre losing strength. Its swelling is nearly twice cotton’s and the effect is a fourth power, so a wet viscose fibre is over three times as stiff in bending before anything else is considered.

Which predicts a cloth that gets stiffer and does not #

There is a difficulty with that conclusion and it has to be faced rather than left implied. A wet cloth does not feel stiffer. It feels limp.

The arithmetic above says a wet cotton yarn’s fibres are twice as stiff in bending, so if a cloth’s bending stiffness were its yarns’ bending stiffness, a wet sheet would be twice as stiff as a dry one and nobody has ever reported that. So one or more of three things must be happening, and the arithmetic can say which are possible and which are not.

The fibre’s modulus falls when it is wet. That would cancel the geometric factor directly, since the rigidity is the modulus times the second moment, and it is a measured quantity for every fibre — wool’s falls a great deal, cotton’s rather little. The factor here is exactly 2.07 at constant modulus, so the modulus would have to fall to a half to cancel it, and for cotton it does not.

The cloth’s stiffness is not its yarns’ stiffness. A fabric’s rigidity is dominated by what happens at the crossings — whether the threads can rotate and slide past one another — rather than by the threads’ own bending, on any cloth that is not jammed solid. Water lubricates those crossings, and a lubricated crossing is a cloth that bends more easily however stiff its threads are.

And the cloth is swollen as well as the fibre. A wetted cloth is set closer than it was woven, its cover rises by an eighth, and its crimp redistributes — so its geometry has changed at the same time as its threads, in a direction the bending arithmetic has not been run for.

The arithmetic is decisive about the first and silent about the other two. A cotton’s modulus does not fall by half in water, so the geometric factor is not cancelled at the fibre; whatever makes a wet cloth limp is happening at a scale above the fibre, and the two candidates above are both at the crossings. That is a real conclusion about where to look, obtained by pricing the fibre-scale terms well enough to rule them out.

The two routes differ in saturation, and only one has a peak #

There is one prediction the two mechanisms make differently, and it is the cleanest way to tell them apart by experiment.

The geometric route saturates. A fibre swells as it takes water and stops when it is saturated, so the factor rises and then flattens. There is no maximum: a fully wetted fibre is as stiff as it will get and stays there.

The capillary route has a maximum and it is at neither end. At no water there is no meniscus and no suction. At full saturation there is no meniscus either — a yarn whose pores are entirely full has no air–water interface inside it, so nothing is curved and nothing pulls. The suction exists only in between, and it is largest when the remaining menisci sit in the finest crevices.

So a measurement of bending stiffness against moisture content separates them by shape rather than by size. A monotone rise to a plateau is the geometry; a rise and a fall is the capillarity; and a rise, a fall, and a plateau above the dry value is both, with the peak’s height giving the capillary term and the plateau’s giving the geometric one.

That experiment is the one this account most wants and cannot do. It needs a cantilever or a heart-loop test on one cloth at a series of regains, which is an ordinary laboratory operation and not an arithmetic.

The two routes are not alternatives and they do not add #

It would be convenient if the two mechanisms were two contributions to one total, and they are not — they act on different quantities and only one of them has a number in the same units as the other.

The geometric route changes the rigidity itself. It multiplies the fibre’s own second moment, so it multiplies the free bound of the bracket and the coherent bound alike, and it would apply to a yarn with no twist, no friction and no contacts whatever.

The capillary route changes where in the bracket the yarn sits. It adds a pressure, the pressure buys friction, and friction decides whether the fibres act separately or together. It has no effect at all on a yarn whose fibres are already locked, and no effect on one whose fibres could never lock.

So the right arithmetic is a product rather than a sum: the rigidity is the geometric factor times whatever position in the bracket the friction supports. And this yarn’s position does not move, because the cloth’s own curvature is twelve hundred times the crossover. So for a woven cloth the product collapses to the geometric factor alone, and the capillary route — the one everybody reaches for — contributes nothing at all.

There is one construction where it would not collapse, and naming it is the useful half. A yarn whose fibres are nearly locked already — a very high twist, a heavily sized warp, a bonded nonwoven — sits near the crossover, and a six per cent change in pressure there is a six per cent change in a quantity that matters. Nothing in a woven cloth is near it.

What was computed, and how #

The geometric route is the fourth power of one plus each fibre’s own measured transverse swelling, applied to the free bound of the collection’s stiffness bracket — the fibre count times the fibre’s rigidity — with the modulus held at its dry value and carried as a parameter so that a reader with a wet modulus can read off the consequence. The capillary route is 2γcosθ over the hydraulic radius of the yarn’s own packed bed, expressed as a pressure and set beside the pressure already computed from a twist’s helix angle and the fibre’s residual axial stress. The equivalent twist is found by bisection on that pressure. The crossover curvature is new here, computed at both pressures.

Six things are checked. A wet cotton fibre is about twice as stiff in bending, between 1.9 and 2.2, which is the fourth power checked rather than restated. The capillary suction is under a fifth of an ordinary twist’s pressure. It matters less the harder a yarn is twisted, at every step of a sweep from a hundred turns to eighteen hundred. And in a very soft yarn it exceeds the twist’s own pressure, which is the other end of the same claim and would fail if the suction had been computed in the wrong units. The curvature a thread carries in cloth is more than a hundred times the crossover even wetted, which is the statement that nothing is locked. And every swelling fibre gains at least half again in its own rigidity.

The swellings, the fibre densities and the contact angle are quoted; the pore radius, the helix angle and the pressures are computed from the yarn’s own construction.

What it says about the stiffness bracket #

The stiffness bracket has been this collection’s standing admission of ignorance: a yarn’s bending rigidity is known to a factor of several hundred, the lower bound is quoted wherever a number is unavoidable, and every result computed from it is stated as a bound.

This account adds one thing to that position and it is worth stating separately. Water moves the bracket and not the yarn inside it, so every earlier result quoted as a lower bound stays a lower bound when the cloth is wetted — multiplied by two for cotton, by 1.8 for wool, by 3.3 for viscose, and by nothing that depends on the unknown.

That is an unusually clean statement to be able to make about a quantity nobody can pin down. The uncertainty is multiplicative and the wetting is multiplicative, so they commute: whatever fraction of the bracket a dry yarn occupies, a wet one occupies the same fraction of a bracket whose ends have both moved by the fourth power of the swelling.

So the loop that presses with tens of millinewtons, the bouclé loop’s peak force and every other force this collection quotes at the bracket’s lower end can be read wet by one multiplication, and the multiplication is a measured swelling raised to the fourth. That is a better position than the dry one, because the dry quantity has one unknown in it and the wet one has the same unknown times a known.

Where the model stops #

The wet modulus is a parameter and it is the thing most likely to matter. Every geometric factor here is at constant modulus, and the modulus of a wet fibre is measured rather than derived. Wool’s falls substantially, cotton’s changes little, viscose’s falls a great deal — so the three factors here should be read as geometry times a material number that is not in hand.

The suction is at one pore radius. A yarn’s pores are a distribution and the menisci sit in whichever are draining, so the suction varies through the drying and the single figure here is the value at the mean hydraulic radius. Its variation is the whole of the saturation dependence the last section wants and is not computed.

The contact angle is nought. Every suction is proportional to its cosine, so a fibre that does not wet has no suction at all and a treated one has a negative one — which would push the fibres apart rather than together, and which is the same sign question the hairs decide for a drop.

And the cloth is not computed at all. Everything here is a yarn. What a cloth does when its threads stiffen, swell and lubricate at once is a question about crossings, and this collection’s crossing arithmetic has not been run wet.

Still open: what water does at a crossing #

The essay ends by naming where the answer must be, and the apparatus to go there already exists.

A cloth’s bending is dominated by its crossings: two threads pressed together at an angle, held by friction, which must slide or rotate for the cloth to bend. Every crossing is a force computes the normal load at one; a thread is gripped where it turns computes the friction that load supplies.

Water changes both and in opposite directions. The swelling raises the normal load, because a swollen thread presses harder on the thread it crosses — a wetting supplies the force the criterion needs computes exactly that and gets a figure within seven per cent of the measured route. And water lowers the friction coefficient at a fibre surface, which is why a wet cloth’s threads slip.

So the wet crossing is a product of a rising load and a falling coefficient, both of which are available, and the sign of the product decides whether a wet cloth is stiffer or limper at the scale that matters. That is one multiplication away and it has not been done here, because the friction coefficient of wet cotton on wet cotton is a measurement rather than a geometry, and it is not in hand.

Who worked it out #

Capillary cohesion between fibres is standard colloid physics and its role in damp fibre assemblies is long established; the swelling figures are the standard tabulated ones. Expressing the suction as an equivalent twist, setting the two routes at one yarn, and finding the geometric route the larger by a factor of several were done here.