Wicking is slower along a crimped thread

Two strips are cut from one piece of shirting, one along the warp and one along the weft, and each is hung with its end in water. Thirty minutes later the two damp lines are at different heights, and the difference is about a tenth.

That is a real effect and it has a real cause, and the cause is not the one a laboratory report usually gives. It is not the fibre, which is the same fibre in both strips. It is not the finish, the twist, the spinning system or the direction of the surface hairs. It is that one of the two threads is longer than the cloth it lies in by more than the other, and the water has to go along the thread.

The thread is longer than the cloth, and that is the whole of it #

Crimp is the oldest number on this site after the float. A thread in cloth goes over and under, so it is longer than the fabric it crosses, and how much longer is the crimp: a thread of length l lies over a cloth length l/(1 + c). Eight per cent crimp means 108 millimetres of yarn to a hundred millimetres of cloth.

The previous rung established where the liquid actually goes. Not through the holes between the threads, which are twenty to a hundred times too coarse to lift anything and which fill and stop within seconds, but along the channels between the fibres inside a yarn — 2.33 µm across in a ring-spun cotton, and the reason a candle wick is a loose bundle rather than a rod.

Those channels run along the yarn. They have no choice: they are the spaces between fibres that are themselves lying along the yarn, wound at an angle the twist decides and following the thread wherever it goes. So a front climbing a warp end climbs a path that wanders up and down through the cloth, and a ruler laid on the fabric measures the projection of that path.

The identity, and it has no free parameter #

Washburn’s law says the wetted length along a uniform capillary grows as the square root of time: L = √(C t), with C = γ r cos θ / 2µ. Everything in C is a property of the liquid and the pore.

The front has gone L along the thread. In cloth coordinates it has gone

x = L / (1 + c) = √( [ C / (1 + c)² ] · t )

which is Washburn’s law again with a different coefficient. So the apparent wicking coefficient of the fabric is the yarn’s own divided by exactly (1 + c)².

Nothing else appears. The surface tension cancels, the viscosity cancels, the contact angle cancels, the pore radius cancels, the fibre cancels. What is left is a path length over a projected length, squared, and it is an identity rather than a measurement — the sort of statement this site asserts rather than reports.

The numbers are not small. At two per cent crimp the loss is 3.9 per cent, which nobody would notice. At five per cent it is 9.3. At the eight per cent a plain-woven cotton shirting typically carries in its warp it is 14.3 per cent, and at the sixteen per cent of a heavy cloth off a hard-tensioned loom it is 25.7 per cent. A quarter of a fabric’s measured wicking performance can be geometry.

What was measured, and how #

An identity checked against itself proves nothing, so the ratio is measured off a drawn path.

The machinery builds a sinusoid whose slope amplitude is solved until its arc length carries the stated crimp, then accumulates that arc length by cumulative trapezoid over twenty thousand samples — deliberately a different quadrature from the Simpson rule this site uses for an undulating tow, so that an error in either shows up as a disagreement rather than cancelling. The front’s position on the cloth is then read off the accumulated path by inverse interpolation, and the assertion is that the path is longer than its projection by 1 + c to nine decimal places.

The measurement also returns something the identity does not, and it is worth having. The front’s position within one repeat wobbles. A thread climbing over a crossing advances less along the cloth per unit of its own length than one running flat between two crossings, so the front runs ahead and behind by 0.58 per cent of a repeat at eight per cent crimp and 1.06 per cent at sixteen. The identity is exact per whole repeat and approximate inside one, and on a repeat a fifth of a millimetre wide that is a wobble nobody will ever see — but it is a property of the model rather than an assumption about it, and it was found by integrating rather than by thinking.

A cloth wicks anisotropically, and no fibre is involved #

Now put the two thread systems side by side.

Both are the same yarn: same fibre, same count, same twist, same finish, therefore the same C. The only thing that differs is the crimp, and the two systems never carry the same crimp. A cloth comes off the loom with most of its crimp in the warp, because the warp is what was held under tension for the whole of weaving while the weft was laid in slack and pushed home. So

warp coefficient ÷ weft coefficient = ( (1 + c_weft) / (1 + c_warp) )²

A cloth with ten per cent warp crimp and four per cent weft crimp has a ratio of 0.894 — it wicks 11.9 per cent faster across than along. At fourteen and three it is 0.816, and the difference is 22.5 per cent. Every one of those numbers is two path lengths and nothing else.

That is a genuinely awkward result for a laboratory. Directional wicking is real, it is measured, and it is routinely attributed to fibre orientation, to surface hairiness, or to a difference between the warp and weft yarns. In a balanced cloth woven from one yarn in both directions there is no such difference, and the anisotropy is still there, and it is exactly the crimp ratio squared.

Stretch it and it changes its mind #

Crimp interchange is this site’s oldest mechanical result. No thread stretches, so extending a cloth along its warp takes the extension out of the warp crimp, and the length that leaves the warp reappears in the weft: the fabric gets longer and narrower and no fibre has moved relative to itself.

Applied to wicking it says something a person could check on a shirt. As the cloth is stretched warpwise the warp path shortens and the weft path lengthens, so the ratio above rises through one. The faster direction reverses, and the crossing is where the two crimps are equal:

pull* = (1 + c_warp) / (1 + (c_warp + c_weft)/2) − 1

For ten and four that is 2.80 per cent extension, which is well within what a woven shirting gives under a hand pull. For fourteen and three it is 5.07 per cent. The closed form is checked against a bisection over the same bookkeeping rather than trusted, and at the crossing the machinery asserts that the ratio is one to nine decimal places — a test that fails if either half of the algebra is wrong.

The consequence for a garment, which is not small #

A woven fabric in wear is not at its relaxed state. A shirt across the shoulders, a sleeve at the elbow, a sheet on a mattress and a bandage on a limb are all extended, in one direction, by a few per cent — which is the range this whole reversal happens in.

So the direction a cloth moves moisture fastest is not a fixed property of the fabric. It depends on how the fabric is being held, it changes sign at an extension a person routinely applies, and it does so for a reason that has nothing whatever to do with the fibre, the finish or the yarn. A moisture-management claim quoted for a fabric flat on a bench is a claim about a state the garment is not in.

The gap between the two strips widens as the test runs #

The identity is a statement about a coefficient, so the ratio of the two directions is a constant of the fabric and does not change with time. What a person actually looks at is a distance, and distances behave differently.

Both fronts go as the square root of time, so the difference between them does too. The warp strip at 14.3 per cent below its yarn’s rate and the weft strip at, say, 3.9 per cent below stay in fixed proportion — and the number of millimetres between the two damp lines grows without bound as √t, which is why the observation at the top of this essay is quoted at thirty minutes rather than at thirty seconds.

That has a practical edge for anybody running the test. A short test understates a real effect and a long one exaggerates how obvious it was, because the ratio is what is constant and the eye reads the gap. Two strips a millimetre apart at one minute are ten millimetres apart at a hundred, with nothing about either strip having changed. The right thing to report is the ratio of the squares of the two heights, which is the ratio of the coefficients and is the quantity the geometry predicts; reporting the difference in heights makes the finding a function of when somebody looked.

It also says which end of the test is worth trusting. The square-root law is at its best once the front is well past the wetted-in region at the bottom of the strip and before evaporation and gravity have begun to flatten it, and the gap is largest in exactly that middle stretch — so the interval where the mechanism is cleanest is the interval where it is most visible, which is a convenience that does not always come free.

The exponent depends on what is travelling #

One power of the crimp comes from the path being longer and the second from the gradient along it being gentler, and the second is a property of capillary flow rather than of cloth. Any transport confined to the thread picks up the first factor; only some of them pick up the second.

A steady conduction along the yarn falls by one power, not two. Put a potential difference across a strip of a conductive-yarn fabric and the current follows the thread; the thread is (1 + c) times as long as the strip, so the resistance is (1 + c) times what a straight yarn of the same length would give, and the effective conductivity is the yarn’s divided by one plus the crimp. At eight per cent that is a 7.4 per cent loss where the wicking loss is 14.3 — the same geometry, half the exponent, because a steady current has no diffusive square root in it.

A diffusive front picks up two, for the same reason a capillary one does: the quantity that scales linearly with the coefficient is a squared length, so a path-length factor enters squared. A dye front travelling along a wetted yarn, a heat pulse spreading along a thread, and the water in this essay are all in that class.

So the crimp identity is not one correction but a family, indexed by whether the quantity of interest goes as a length or as a length squared. Reading the exponent off the transport is the whole of applying it, and getting it wrong is a factor of (1 + c) — small enough to be absorbed into a fitted constant, which is exactly what makes it worth stating.

Where the model stops #

Washburn’s law is a tube. It assumes Poiseuille flow, fully developed, in a uniform circular capillary with gravity left out. A yarn is a tortuous connected bed of channels that are not uniform, that swell as they fill, and whose front is ragged. What is used here is the scaling — a wetted length going as √t — and the identity is about how a path length enters that scaling, not about whether the scaling is exact.

The interchange rule is first order and says so. Holding the total crimp fixed in percentage points is a bookkeeping rule, not a conservation law. Peirce’s geometry cannot supply a better one: its closure condition h₁ + h₂ = D is a statement about a cloth in equilibrium with itself, and with both thread lengths and the thickness fixed the system is fully determined — so the exact geometry has no stretch path in it at all. A stretched cloth is not in equilibrium with itself and Peirce’s equations do not describe it. The rule gets the sign right at any extension and the magnitude right near the reference state, and the reversal is near the reference state.

The fine channels are not exactly along the thread axis. A fibre in a twisted yarn lies on a helix at α to the axis, so the channel between two fibres does too, and its true length exceeds the yarn’s by 1/cos α. That is a second path-length factor sitting inside the first, it is a property of the twist rather than of the weave, and nothing here computes it. It would multiply both directions equally in a cloth woven from one yarn, so it cancels out of the anisotropy and does not cancel out of the absolute coefficient.

Nothing here says what the front does at a crossing. Two systems of channels meet at every intersection, they are pressed together, and liquid can pass from warp to weft. If it does, the cloth is not two independent one-dimensional wicks and the anisotropy is smaller than computed. That transfer needs a contact mechanics this site does not have, and the honest statement is that the identity bounds the anisotropy rather than predicting it.

And every argument here needs a cloth. The (1 + c)² factor is a statement about a thread that somebody wove at some tension into a fabric at some sett. Take the cloth away and the identity says nothing — there is no path to be longer than anything.

Who found it, and when #

The path-length correction is old in a neighbouring subject and has a name there: tortuosity. Flow through any porous medium travels further than the sample is thick, and the ratio is what a permeability has to be divided by. Kozeny’s 1927 treatment carries it explicitly, and the factor is squared there for exactly the reason it is squared here — one power for the longer path, one for the gentler pressure gradient along it.

What is different in cloth is that the tortuosity is not a fitted parameter. In a soil or a sinter, or in the fibre bed of a preform, tortuosity is measured because nobody can see the paths. In a woven fabric the path is the thread, its length is the crimp, and the crimp is a number the trade has measured routinely since the nineteenth century by unravelling a specimen and straightening it. The correction that everywhere else is an unknown is here a quantity printed on a fabric analysis sheet.

The textile literature on wicking generally quotes an apparent coefficient and leaves it there. Where crimp is mentioned it is as one of several reasons a fabric’s coefficient is lower than its yarn’s — which is true and is far weaker than what the geometry supports, because the geometry gives the factor exactly and does not need the others to be estimated first.

Where the ladder goes next #

Both rungs so far have taken a cloth as given and asked what it does. The setting ladder’s question is the other way round: the sett is a decision, so what does the decision buy? The answer for wicking turns out to be the opposite of what every other rung of that ladder has found, and the negative result is sharp enough to be worth its own essay.