A meniscus closes the gap only at two-thirds cover

A ridge in the matrix is not a ridge in the cloth broke a prediction and named the one thing that could mend it. A 2/2 twill’s crowns join into a diagonal across its matrix and a 2/2 hopsack’s do not, so a film on the twill ought to be whole from the first gram. On the cloth’s height field that fails. Each step of the ridge crosses the gap between two ends, and a film of a given depth meets nothing across that gap until it reaches the pick lying below. Both weaves join at the same depth, well past half the add-on that flattens the face.

That account left out surface tension, and said so. A liquid can bridge what a depth cannot. At what sett does a gap between ends close to a meniscus, below which the twill’s staircase has its risers back? The answer has a closed form, it is about cover and not about sett, and it comes out where almost no cloth lies.

What it takes to hold liquid across a gap #

The question can be taken with the cloth reduced to two neighbouring ends. They are parallel, round and a little apart, and at a step of the ridge the pick is under both, in contact with each. That puts its crown a whole end’s diameter below their crowns, since it touches their undersides, and for a film that means a long way down. So a web of liquid across a gap at crown level has air beneath it, and needs a second free surface along with its first.

The energy is simple to set out. Take a full column of liquid between the two threads, along their length. Its top and bottom surfaces are nearly flat, and they meet a thread at the place where the thread’s tangent makes the contact angle with them, an angle $\theta$ from the crown. The column wets an arc $r(\pi - 2\theta)$ of each thread, which earns $\gamma\cos\theta$ per unit area. It makes two free surfaces of width $p - 2r\sin\theta$, which cost $\gamma$ per unit area. It pays when

$$2r(\pi - 2\theta)\cos\theta \;\geq\; 2\,(p - 2r\sin\theta).$$

This is Princen’s condition for liquid between two parallel cylinders, published in 1969 and borrowed here whole. Rearranged with $d = 2r$, it is a statement about the ratio of a thread to its spacing and about nothing else:

$$\frac{d}{p} \;\geq\; \frac{2}{(\pi - 2\theta)\cos\theta + 2\sin\theta}.$$

That ratio is the warp cover. For a liquid that wets perfectly it is $2/\pi$, 0.637: a meniscus can close the gap only when the threads fill just under two thirds of the cloth’s width. No thread diameter appears, so a fine cloth and a coarse one have the same limit, and a gap’s width in micrometres, which is what the earlier essay asked about, was not the right quantity.

Eight cloths and one curve #

The limit rises with the contact angle, and reaches one at ninety degrees, where a wet thread earns nothing from being wet.

The eight named cloths here have warp covers from 0.20 for a cheesecloth to 0.52 for a sheeting. Not one is near the curve, even at the angle most favourable to the liquid. A sheeting’s ends fill a little over half its width, with a clear gap of 0.17 millimetres between threads of 0.19, so a gap of about nine tenths of a thread. The limit asks for about 0.57 of a thread.

So for every cloth in this table, the earlier result stands as it was. A gap between ends at a sheeting’s sett cannot hold a meniscus at all, in equilibrium, for any liquid. The ridge the census counted stays a staircase with empty risers. The prediction of a window opened by capillarity, on those cloths, is wrong.

Two things soften that, and they go in opposite directions. A thread is not a smooth cylinder. A spun yarn is rough, and roughness on a wetting liquid lowers the contact angle it presents, so nought is the fair angle for a low-viscosity finish on clean cotton. It also has hairs, and a light touch never reaches the crowns because of them: a hair standing across a gap gives liquid a bridge the two-cylinder condition does not know about. On a singed cloth that matters little. On a raised cloth a film is whole for other reasons, as the earlier essay said.

A cloth runs out of room about where the limit begins #

The limit would matter more if cloth could go there freely. How close a warp can be set is its own problem, settled by the jam and reached with the crimp shared in some ratio between warp and weft.

With the crimp shared evenly and threads round, a cloth stops being buildable at warp covers of 0.61 to 0.75, depending on its weft. A poplin runs out at 0.64, and a sheeting at 0.61. They run out, near enough, where a meniscus would begin. That is a coincidence of two pieces of geometry, one of circles touching across a crossing and one of a wetted arc against a free surface. But it means balanced cloth has essentially no access to this regime. To get well past 0.637 a cloth has to take more of its crimp in the warp, ending up warp-faced, which is what a poplin at higher setts and a warp sateen are.

The cloths that do get there are specific. A poplin at 48 ends a centimetre has to put twice the crimp in its warp as in its weft, at 56 three times, at 60 five times. A close warp-faced cloth is the only ordinary construction where the question arises. And on it, the answer is abrupt.

On the far side, the census is right #

Past the limit the arithmetic changes shape, because a web does not have to be a full column.

A web can thin. Its two surfaces bow in toward each other as an arc of a single radius, since the pressure inside is one number, and they can bow until they meet at the threads’ waist, where it can hold no less. The contact points move down the threads as it thins. That least web is a small object: on a 15 tex warp at a cover of 0.64 it is about a thousandth of a square millimetre in section, a sliver between threads of 145 micrometres.

So the volume to span a gap falls away, and the two weaves separate. A 2/2 twill’s ridge crosses one gap per pick for every four ends. A web across each of those steps and nothing else is a volume per unit area of the web’s section over four spacings. At the limit, on the poplin of the first figure, that is 1.45 grams a square metre at a density of 1.2. With every gap on the cloth filled along its whole length, the most a step could take, it is 5.81.

The film by depth needs 108. So at a cover of 0.64 the twill’s ridge joins at between a seventy-fifth and a nineteenth of the liquid a depth-only film needs, and at higher covers that margin widens: with every gap filled, 4.0 at 0.70, 2.6 at 0.75, 1.5 at 0.81, 0.7 at 0.87.

What the hopsack does on the same cloth is the point. A hopsack’s face cells are two-by-two squares, and two squares in neighbouring columns never share an edge: there is no pick at which an end of one square and the next square’s nearest end are both on the face. So no step exists for a meniscus to fill, and a gap’s web leaves its islands as they were. Its film still joins by depth at 120. Between about 6 and 120 grams, a twill on such a cloth has a continuous film along its diagonal and a hopsack has patches.

That window is the one the census predicted and the earlier essay could not find, a barrier against a sieve, at the same surface volume, by weave alone. The earlier window of 8 to 18 grams came from picks; this one is an order of magnitude wider, and it starts as a cliff.

So the matrix’s edge and the cloth’s edge do agree, on the far side of a cover limit, for a liquid of the right kind. The census was not wrong about the matrix, and the height field was not wrong about the cloth: they were describing two regimes separated by one number.

A worse wetter closes later, and the web is thin all the same #

At thirty degrees the limit moves to 0.711, beyond a poplin at 48 ends and short of one at 52, and at sixty it moves to 0.887, closer than any sett computed here. Contact angle therefore moves the limit a great deal and the volume above it very little. Above whichever limit applies, every web sits in the same few grams.

The first web a cloth can hold is also its largest. At the limit at nought degrees it is 5.6 grams a square metre, for a 15 tex warp at a density of 1.2, and it thins toward nothing as the threads close. So a closer warp makes the meniscus both possible and cheaper, which is the opposite of the depth account, where a closer warp left the picks so far down that a film had further to sink. That reverses the instinct from the earlier essay’s sweep: a poplin’s depth-only film barely cared about sett, and the web cares a great deal.

The practical reading #

For a coater of ordinary cloth, the result is plain and it repeats the earlier one. On a sheeting, a poplin at its usual sett, a duck or a filter fabric, a liquid cannot hold across the gaps between ends, so a thin finish is patches whatever its surface tension, and a coated cloth’s failure at its holes begins where the film first spans them, by depth.

The new reading is for a cloth at a warp cover past two thirds, a tight warp-faced poplin, a warp sateen, a gaberdine-like twill, finished with something thin that wets well. There a first few grams can run along the diagonals of a twill and make a continuous film, and the same grams on a hopsack of the same yarn are islands. It predicts something measurable. The same low-viscosity repellent or fluorocarbon finish at a few grams a square metre, on a tight twill and a tight hopsack, should show a larger difference in air permeability or spray rating at low add-on than the earlier arithmetic allows. On an open cloth it should show none.

It is also a limit on how a cloth wicks along its surface. The coarse pore between ends, computed there from the gap, is taken as a channel a liquid can occupy. Below 0.637 a channel with no floor at crown level cannot hold a column at all, and it has to run in the wedges against the picks. The fine pores inside the yarn lift the liquid whichever way, so wicking height is untouched, and what changes is whether the coarse channel is a channel.

How the number was reached #

Princen’s condition is used as published, for two parallel cylinders at a contact angle. The least web is a pair of arcs of one radius, each leaving a thread at the contact angle at a point $\varphi$ from its crown, with $\varphi$ solved by bisection so that the arcs just meet at the waist. Its section is the area under the arcs with the threads’ own area taken out, and gravity is neglected, which at these sizes it can be.

The cloths are the height-field cloths of the earlier essay, the depth-only joins taken from it directly. Warp setts above balanced crimp take the first ratio from a list of 1, 2, 3, 5, 8, 13 and on that the round-thread geometry will solve. Add-ons use the density of 1.2 the coating account uses. Ten warp setts from 24 to 60, on a poplin, at nought and thirty degrees, with the two named cloths and the eight covers read from the same state.

Required of it, not shown: a limit of exactly $2/\pi$ at nought degrees and one at ninety; a limit that rises at every step of the angle; no web found below the limit; and a web that shrinks as the threads close.

Where the two cylinders stop #

The threads are round and straight here. A warp at two-thirds cover is usually flattened, and flat threads change both the wetted arc and the free surface. A racetrack section has a flat flank to wet, and the limit should move, probably downward. Nothing here computes it.

This is equilibrium with no time in it. Whether a viscous paste ever reaches these webs in a coating head’s dwell time is another matter. The condition says what lowers the energy, not how soon, so all of this is about thin liquids, and a paste is outside it altogether. A coating paste at the viscosities of a coated cloth lies on the crowns as the earlier essay had it.

A yarn is porous. A real wetting liquid enters the yarn and the web competes with the pores for it, and the pores are much finer and they win. Some of the few grams above goes into the yarn first. So those figures are the least that must be available at the surface, not the least that must be applied.

The web’s length is assumed. A web across one step stops somewhere near the crossing on either side, and how it ends there is a three-dimensional problem. The figure between 1.45 and 5.81 grams brackets the answer rather than finding it. Whether the hopsack’s between-pair gaps, which do fill as channels, then join its islands through their undersides is a question this reasoning sets aside by treating a film as a crown-level object.

And it does not include the weft’s own gaps. A weft-faced cloth has the same question the other way round, and a twill’s ridge in the weft system is a separate staircase.

Who worked out which part #

The condition for liquid between parallel cylinders is Princen’s, from a 1969 series on capillarity in assemblies of parallel cylinders, and capillary bridges between fibres have a long literature in wet webs and textiles. The height field, the census and the depth-only join are the earlier essays’.

What is added here is the reading of Princen’s condition as a warp cover, with no size in it; the comparison with the covers real cloths have and with the cover where balanced crimp runs out; the least web’s volume against the depth-only join; and the finding that, across that one limit, a hopsack and a twill go from indistinguishable to about twenty-fold apart.

Still open: whether a flat yarn reaches the limit sooner #

Every thread here is round, and the cloths that actually sit near this limit are the ones whose threads are not. A close warp is flattened by its own crowding, by a calender’s nip, or just by being a low-twist yarn. A flat section at the same mass presents a longer wetted flank and a narrower gap at once.

So a calendered cloth might cross the limit at a sett where a round one cannot, and a light calender before a thin finish would then be a way of getting a continuous film at a few grams on a cloth that could not otherwise hold one. The racetrack section is already in the height field. What is missing is the same energy balance on that section, and the same arcs meeting at its flat waist, and it would say at which flattening a sheeting, which cannot reach the limit round, gets there.