Nonwovens, and what holds them together instead

Every fabric on this site so far has had a repeat. A weave is a small matrix that tiles the plane, a knit is a course of loops repeated, a braid is a crossing sequence that comes back to where it started. The repeat is what makes the subject exact: a question about a cloth of any size collapses into a question about a handful of integers, and the answer is a count.

A nonwoven has no repeat at all. Fibres are laid onto a moving belt from a spinneret or a carding machine, they land where they land at whatever angle they happen to have, and the sheet that results is held together by nothing but the fact that there are a great many of them and they touch.

So the question this ladder is built on — does it hang together — has to change character. It cannot be answered by a linear-time pass over a digraph, because there is no periodic structure to build the digraph from. What replaces the count is a probability, and what makes the subject tractable again is that the probability does not drift gently: it turns over from nearly zero to nearly one across a narrow band of fibre density.

Why the exact criterion has nothing to bite on #

It is worth being precise about what fails, because “the fabric is random” is not on its own a reason.

The connectivity criterion needs two things. It needs a finite set of strands, and it needs the whole cloth to be described by what happens among them. In a weave both hold: the repeat has a few ends and a few picks, and the infinite cloth is those, tiled.

In a web neither holds. There are as many fibres as there are, the sheet is not a tiling of anything, and asking whether “the fabric” separates is a question about a particular sample rather than about a structure. Two webs made on the same machine at the same settings are different objects, and a statement true of one need not be true of the other.

What survives is a weaker and still useful question: for a given fibre density, how likely is it that the sheet holds together? That is a question about the process rather than about any sheet, and it has a definite answer.

What the fibres do when they land #

The model here is the plainest one that can support the question. Straight fibres of a single length, midpoints uniformly distributed over an area, orientations uniform over a half turn, and a contact wherever two fibres cross.

That model is not a nonwoven. Real fibres are curled, are not all the same length, are laid with a strong preference for the machine direction, have thickness, and are usually bonded to one another by heat or adhesive rather than merely touching. Every one of those omissions matters to a real product. What the model is for is the structure of the answer — that there is a threshold, roughly where it sits, and what controls it — and for that the omissions are affordable in a way that has to be stated rather than assumed.

The density that matters is not fibres per square centimetre. It is the dimensionless quantity

$$\rho = \frac{n \ell^2}{A}$$

the number of fibres times the square of their length, over the area. Two webs with the same $\rho$ behave alike whatever their absolute scale, which is why the same number describes a spun-bond fabric and a simulation on a unit square.

The contact rate, which is not measured but derived #

Before any threshold can be trusted, the simulation has to be shown to be simulating the right thing, and there is a clean way to do it.

Take two straight needles of length $\ell$, with midpoints independently uniform over an area $A$ and orientations independent and uniform. The set of positions of the second needle for which the two cross, at a fixed relative angle $\varphi$, is a parallelogram of area $\ell^2 |\sin\varphi|$. Average $|\sin\varphi|$ over a uniform half turn and it is $2/\pi$. So

$$P(\text{cross}) = \frac{2\ell^2}{\pi A}$$

and a fibre far enough from the edges to see the full distribution expects $(n-1)$ times that many contacts. It is Buffon’s needle wearing a different hat, and it is a closed form: nothing in it was measured.

The simulation is required to reproduce it. Averaged over two dozen independent webs the measured contact rate lands within about one per cent of the prediction, and that agreement is the licence to believe anything else the simulation says. The errors it catches are not subtle in their effect and are very subtle in their appearance: an orientation drawn over a full turn instead of a half, a length used as a radius, a crossing test that misses a case. Each of those shifts the rate by tens of per cent and none of them makes a picture look wrong.

The averaging is not politeness. A single web of two hundred fibres has a contact rate that scatters by about eight per cent from realisation to realisation, so one web can sit four standard errors from the mean and be entirely correct. A check written against one picture would fail on honest samples and pass on broken ones.

The transition #

With the model verified, the actual question can be asked: at what density does a heap become a sheet?

The measurement is straightforward and slow. Build many webs at each density, find the connected components, and ask whether any one component reaches both the left and the right edge. The fraction that do is the spanning probability.

Two things in that figure are worth separating.

The largest component grows smoothly. At low density it holds a fifth of the fibres, at high density essentially all of them, and in between it climbs without any obvious feature. That curve alone would not suggest a transition.

The spanning probability does not grow smoothly. It sits at zero through densities two, three and four, and by density eight it is one. Everything happens between five and seven, and the sharpness is what makes the idea of a threshold worth having at all.

The threshold’s location is the part that can be checked against something outside this site. Percolation of randomly placed zero-width sticks in two dimensions is a well-studied problem, and the critical dimensionless density is known to several decimal places: 5.637. The measurement here crosses a half at 5.4, with the value moving between about 5.4 and 5.8 as the box size is varied — scatter that is consistent with sampling noise at forty replicates and too large to resolve the finite-size shift that certainly exists.

That last sentence is the honest form of the result. The measurement brackets the published constant; it does not confirm it to any precision, and claiming that it did would be reading the noise as a signal.

What the threshold means on a production line #

A number like 5.637 sounds academic and is not.

Basis weight — grams per square metre — is the quantity a nonwoven line controls, and it is proportional to $n$. Fibre length and fineness set $\ell$. So the dimensionless density is a combination of three things a machine operator can actually turn, and the threshold says there is a basis weight below which no amount of care produces a fabric.

The practical consequence runs the other way as well, and is the more interesting one. A line running near the threshold produces a sheet whose strength is wildly variable, because at the threshold the largest component’s size is at its most sensitive to chance. Running comfortably above it is not merely a matter of making the fabric stronger; it is a matter of making it predictable. That is why the useful range of basis weights for a given fibre is narrower than the strength requirement alone would suggest.

And it says where bonding comes in. Thermal or adhesive bonding does not add fibres; it makes each contact into a joint. In percolation terms it does not move the threshold — the geometry of who touches whom is unchanged — but it changes what a spanning cluster is worth, from a set of fibres leaning on each other to a connected solid. A bonded web below the threshold is still not a fabric, which is a prediction the model makes and which happens to be true.

Where the threshold sits in grams per square metre #

The threshold is a dimensionless density and a production line runs in grams per square metre, so the conversion is worth doing — and it lands a long way from where the previous section implies.

Take an ordinary spun-bond fibre: twenty micrometres of polypropylene, which is 0.29 tex, cut or laid at forty millimetres. The critical density ρ = 5.64 needs n = 5.64/ℓ² fibres per square metre, which is 3,525. Each carries forty millimetres of a 0.29 tex fibre, so the mass is

0.04 grams per square metre.

Commercial nonwovens start at about ten grams per square metre and run to hundreds. The lightest product anybody makes is two hundred and fifty times above the percolation threshold, and an ordinary one is thousands of times above it.

So the threshold is real, it is where the theory says, and no nonwoven line has ever run near it. The basis weight below which a web will not hold — which every operator knows and which the previous section attributes to this arithmetic — is not the percolation threshold. It is somewhere else entirely.

Which says what the real floor is #

That is a correction rather than a demolition, and it sharpens what the model is for.

If a commercial web is three orders of magnitude above the connectivity threshold, then its fibres are connected many times over and the question of whether a path exists is settled long before anything else is. What limits a light nonwoven must therefore be one of the things the model explicitly does not have:

Bonding. A spanning cluster is a set of fibres leaning on one another, and the essay is right that percolation says whether there is a route and not whether the route holds. At a hundred times the threshold there are plenty of routes and the question is entirely what each contact carries — which is a bonding and a friction question.

Uniformity. A web at ten grams per square metre has enormous local variation in basis weight, and a light spot is light by a factor rather than by a per cent. What fails is a place rather than a sheet, and the arithmetic here is about a mean.

And handling. A very light web has to survive being carried from the laydown to the bonder, and that is a strength requirement at a moment when nothing is bonded at all.

All three of those bite at basis weights hundreds of times above where connectivity does, which is why the operator’s floor is where it is and why it moves with the bonding process rather than with the fibre geometry.

And it says what the threshold is good for #

Two uses survive, and they are the ones where the density really is low.

The moment of laydown. A web on the belt before consolidation is at its own local density, and the model’s honest domain — the unbonded web, as the essay says — is exactly there. What it says is that connectivity is never the problem at that moment either.

And the diluted case. The same arithmetic with the fibre count reduced describes a scrim, a very light reinforcing veil, or the fibre phase of a composite before the matrix is added. Those genuinely run at densities where a factor of two in basis weight decides whether anything holds together, and the threshold is the right tool for them.

So the model’s reach is narrower than the production paragraph claims and it is not empty. Percolation decides whether a handful of fibres is a sheet; it does not decide anything about a fabric anybody sells, and knowing which of the two is being asked about is the whole of using it correctly.

Against the woven case #

Putting the two structures side by side is the point of the ladder, and the contrast is sharper than expected.

A woven cloth holds together exactly or not at all. The criterion returns an integer; there is no sense in which a weave is seventy per cent connected. Whether a draft describes one cloth is decided by arithmetic on a small matrix, and two drafts differing in a single square can differ in the answer.

A web holds together with a probability, and the probability is a smooth function of a density. There is no draft, no square to change, and no yes-or-no.

That difference propagates into everything. A woven cloth frays at a cut edge because the threads it is made of are long and can be withdrawn; a nonwoven does not fray, because withdrawing one fibre from a spanning cluster changes nothing. A woven cloth’s strength is anisotropic in a way its draft predicts; a web’s is isotropic in the mean and erratic in the sample. A woven cloth can be taken apart into its threads and reconstructed from a description; a web cannot be described at all, only characterised.

Entangling, which puts the topology back #

There is a third possibility between touching and bonding, and it is the one that makes the best nonwovens.

Needling drives barbed needles through the web and drags fibres from the surface down through its thickness, so that a fibre threaded through the plane holds the layers together mechanically. Hydroentangling does the same thing with fine high-pressure water jets, which is gentler and gives a softer fabric. In both cases nothing is glued and nothing is melted: fibres are simply wrapped around one another.

What that does, in the terms of this ladder, is restore a topological criterion where there had been only a statistical one. A fibre that passes over one neighbour and under the next is held in the same way a warp end is held by the picks crossing it, and it cannot be withdrawn without straightening — which is a mechanical event rather than a free one. An entangled web is not a periodic structure and still has no draft, but the reason it holds is no longer that there happen to be enough contacts. It is that the fibres are, in a weak and disorderly way, interlaced.

That is why hydroentangled fabrics feel like cloth and thermally bonded ones feel like paper, and it is also why the two fail differently: a bonded web tears along a line of bond points, and an entangled one pulls apart fibre by fibre. The model in this essay describes neither, and it is worth saying which of the three kinds of nonwoven it is a model of. It is a model of the unbonded web — the state the fibres are in immediately after laydown and before anything is done to them, which is the state in which the question “is this a fabric yet” has a genuine answer.

What the picture cannot show #

Every figure on this page draws fibres as straight line segments of equal length, lying in a plane, touching where they cross.

Not one of those is true of a nonwoven. Fibres are crimped and curled, which shortens their effective reach and raises the density needed to span. They vary in length. They lie in a sheet of finite thickness, so two fibres crossing in plan need not touch at all — and three-dimensional percolation has a different threshold from two-dimensional. They are laid with a machine-direction bias, which makes spanning easier along the belt and harder across it, so a single threshold is already a simplification of two.

The figures also draw contacts as dots at the midpoint between two fibres’ centres, which is close to where the crossing is at this scale and is not exactly it. That is a drawing convenience with no effect on any number here, and it is mentioned because a reader measuring a dot’s position would be measuring the wrong thing.

And the model has no mechanics whatever. A spanning cluster is a connected set of fibres, not a load path. Whether the sheet can carry a force depends on friction at the contacts, on the fibres’ bending stiffness, and on bonding — none of which is present. Percolation says whether there is a route; it does not say the route holds.

Where this came from #

Percolation theory began in 1957 with Broadbent and Hammersley, who were thinking about fluid moving through a porous medium and about gas-mask filters. Continuum percolation of sticks — as opposed to bonds on a lattice — came later and was pushed hardest by people interested in conductive composites, where the question is when randomly dispersed conducting fibres first form a path across an insulating matrix. That literature is where the 5.637 comes from, and its authors were not thinking about fabric.

The nonwoven industry arrived at the same place from the other direction and rather earlier, empirically: everyone who has made a web knows there is a basis weight below which it will not hold, and the number was established by finding it. The theory did not tell the industry anything it did not know. What it added is the shape of the knowledge — that the failure is a threshold rather than a gradual weakening, and that its position is set by a dimensionless group rather than by a weight.

Where the ladder goes next #

This is the end of the integrity ladder in one direction: the structure with no repeat, where the exact criterion has nothing to work on. The other direction runs back through the braid, where the criterion applies unchanged to a single oblique thread system, to the woven case it was built for.

The ladder’s other branch goes to the construction where more than one component is the intended answer: backed and stitched cloths, where a double cloth must be verified as two and a stitched one as one, and where the number of mistakes it takes to move between them turns out to be one.

And for the question of what all these structures have in common, which is less than it appears, the base of the whole site is a fabric is a structure, not a material.