A melt's slip length has an exponent it cannot choose

The wall a melt really does slip on found the one common liquid for which the no-slip condition — the rule a spreading drop already has to break at its edge — fails by a large margin. A polymer melt, long molecules entangled with each other, slides over a solid wall by the friction of the few monomers touching it, while it resists shear with the viscosity of the whole entangled network, and de Gennes’ ratio of the two makes the slip length not nanometres, as for water, but tenths of a millimetre or more. Pushed through a capillary die the melt is partly plug: three dies of different radius report three different flow curves for one melt, and Mooney’s construction — plot the apparent shear rate against the inverse radius at a fixed wall stress, and read the slip velocity off the slope — separates the sliding from the shearing exactly.

That essay ended on a question the practice of rheometry does not usually ask. There are two ways to state how much a melt slips. A slip length $b$ says the slip velocity is $b$ times the wall’s shear rate, as Navier wrote it for simple liquids and as a length measured by squeezing measures it. A slip law says the slip velocity is a function of the wall’s shear stress, usually a power, $V_s = \beta\sigma^m$, because that is what a Mooney analysis naturally produces: slip velocities at a set of stresses. For a Newtonian liquid the two are the same statement, since stress and shear rate are proportional. For a melt they are not, and the question is which of them is the material’s property — which one would transfer from one die, one rate, one temperature to another.

What a slip length is, and why it suits water #

The slip length has a picture that makes it attractive. Continue the velocity profile in the liquid straight past the wall, into the solid, and it reaches zero at a depth below the surface: that depth is the slip length. For water on a smooth solid it is a nanometre or two — a length that can only be measured by squeezing — and for a Newtonian liquid it is the whole of the boundary condition, one number, independent of how fast the liquid is sheared. In a rarefied gas it is the same kind of number, set by the mean free path, and it is how a fluid that stops being one first announces it at a wall. It is also what twice as slippery along as across computed for a patterned surface: an effective length, the same at every rate, because the liquid’s viscosity is.

For a shear-thinning melt the picture still exists but the depth moves. The profile’s slope at the wall is the shear rate there, the slip velocity depends on the stress, and stress and rate are not proportional; the extrapolation depth is the ratio of the two and changes when either does. The picture that makes a slip length intuitive for water is the one that hides its rate dependence for a melt.

Two descriptions, matched at one stress #

The melt is the one the essay before used: a power-law fluid whose true shear rate is $(\sigma/K)^{1/n}$ with a flow index $n$ of 0.4, the shear thinning of an ordinary commodity polymer. Two versions of its slip are set up to agree exactly at one wall stress, 100 kPa, where both slip at 2 millimetres per second: description A has a constant slip length of 0.099 mm; description B has a slip law of exponent 2, the value a rheometer typically reports. At 100 kPa they are indistinguishable — the same flow curves, the same Mooney plot. Everywhere else they differ, and the essay is about how.

A slip law implies a slip length that moves #

The first figure converts the slip law into the language of the slip length. Divide the slip velocity at each stress by the true shear rate at that stress, and the result is the slip length the law implies there. For an exponent of 2 it is not constant: 0.22 mm at a shear rate of 0.4 per second, 0.16 at 2, 0.10 at 20 and 0.08 at 47. Across the hundredfold range of rates a capillary rheometer covers in one session, the implied length changes by a factor of nearly three. For an exponent of 1.5 it changes by a factor of seven; for 3, it grows instead of falling.

Only one exponent gives a constant length, and that exponent is not a property of the slip at all. The true shear rate goes as the stress to the power $1/n$, so a slip velocity proportional to it goes as the stress to the power $1/n$ too. A constant slip length is a slip law with exponent exactly $1/n$, 2.5 for this melt, and a slip law with any other exponent is a slip length that changes with the rate. The two descriptions are therefore not two ways of saying the same thing about a melt; they are two different claims, and a measured exponent is the evidence between them.

The exponent a slip length cannot choose #

The second figure shows both descriptions run through the measurement a rheologist would actually make. At five stresses, the apparent shear rate in each of three dies — a quarter, a half and one millimetre in radius — goes into Mooney’s construction, and out comes the slip velocity. For description B it comes out with the exponent it was given, 2. For description A it comes out with exponent 2.5, to fifteen figures, because the melt’s own flow curve fixes it. The two lines cross at the stress where they were matched and part by a factor of two across the range.

This is the sharpest thing that can be said about the question, and it costs nothing to test: every Mooney analysis that reports a slip exponent also reports, from its intercepts, the melt’s flow index. If the exponent equals one over the index, the data are consistent with a slip length and a slip length can be quoted. If it does not, a quoted slip length is a number that belongs to one rate only. Measured slip exponents for polymer melts in their weak-slip regime cluster between about one and three, and they are not, in general, one over the flow index.

What de Gennes’ own argument says #

The slip length most often quoted for melts is de Gennes’ ratio, $b = a\,\eta/\eta_s$ — a monomer size times the melt’s viscosity over the viscosity of an unentangled liquid of the same monomers. It is worth asking what that argument actually derives. Its physics is a friction: the monomers touching the wall slide over it with the friction of a simple liquid, so the wall stress is a friction coefficient $k = \eta_s/a$ times the slip velocity. That is a slip law, with exponent one. The slip length appears only when the stress is rewritten through the melt’s viscosity, $\sigma = \eta\dot\gamma$, and for a melt whose viscosity thins with rate, the $\eta$ in the ratio is the thinned one: the slip length it defines falls as the rate rises.

So even the simplest physical account says the material property is the friction — the relation between wall stress and slip velocity — and that the slip length is derived from it and moves with the rate unless the melt is Newtonian. A constant slip length for a shear-thinning melt would need a wall friction that thins exactly as the bulk does, which is what the exponent $1/n$ would be evidence of. Measured exponents above one reflect something the linear friction leaves out, chains at the wall stretched by the flow and sliding more easily as they are pulled, and they are one more reason the slip length cannot be the constant.

Temperature separates them outright #

The exponent test uses the rate. The essay before pointed to a second test that uses temperature, and the third figure carries it out. Heat or cool a melt and its viscosity changes by a shift factor $a_T$, the same factor at every rate — the time–temperature superposition that polymer rheology rests on. What happens to the slip at a fixed stress depends on which description is the material’s.

If the slip length is the property, it is de Gennes’ ratio of two viscosities: the entangled melt’s and the unentangled liquid’s of the same chemistry at the wall. Both are set by the same monomer friction and shift with temperature together, so the slip length does not move; the slip velocity is that length times the true shear rate, and the true shear rate at a fixed stress goes as $a_T^{-1/n}$. A tenfold rise in viscosity cuts the slip velocity 316-fold. If the slip law is the property, the slip velocity at a fixed stress is the stress over the friction coefficient of the chains at the wall, which shifts like the monomer liquid’s viscosity, and the same tenfold rise cuts the slip velocity tenfold.

Three hundred against ten, from a change of temperature of a few tens of kelvin, is not a subtle difference. It is the experiment the essay before described — dies of several radii at several temperatures, the slip velocity compared at a fixed stress — and it would settle the question for a given melt in an afternoon.

Why the practice does not notice #

The confusion survives because each description, used within the range it was fitted in, reproduces the data. A slip law fitted to one temperature and one range of stresses predicts those flow curves perfectly; so does a slip length chosen at the middle of that range, to within the factor the first figure shows, which is inside the scatter of many measurements. The trouble arrives when the number is carried somewhere else — to a die of a different size, a faster extrusion, a hotter melt — which is exactly what a material property is for. A viscosity that depends on the question made the same point about a shear-thinning viscosity quoted without its rate; a slip length quoted without its rate, or its temperature, has the same defect with a different name.

The damage is largest in simulation. A die-design simulation that imposes a Navier slip condition needs one number, a slip length, and takes it from a rheometer’s slip law evaluated at some convenient rate. If the melt’s slip is really a law of exponent 2, the simulated wall slips a factor of two too little at its lowest rates and too much at its highest, and the extrudate’s predicted swell and pressure drop inherit the error.

How much slip a measurement needs #

Mooney’s construction subtracts: the slip velocity is the difference between the apparent rates of dies of different sizes, divided by the difference in their inverse radii. When slip is a large share of the flow, the difference is large and the errors in each rate matter little. When slip is small, the construction subtracts two nearly equal numbers and amplifies their errors. The fourth figure measures how much.

With each apparent rate carrying a two per cent error — good capillary rheometry — three dies and five stresses give an exponent to within ±0.06 when slip is a quarter of the widest die’s apparent rate, as it is for this melt. The two descriptions’ bands are then far apart. Weaken the slip and the bands widen: at 4 per cent of the flow the spread is ±0.3 and the bands touch; at 1 per cent four fits in ten return a negative slip velocity and no exponent at all. The error formula for the least-squares slope predicts the scatter to within a per cent, so the figure is the statistics of the construction, not of the simulation.

That sets where the test can be run. A melt that slips strongly — high molecular weight, a clean metal die, a stress below the slip transition — can be tested with three dies. A melt that slips weakly needs more dies, smaller ones, or much better rates, and the exponent it reports otherwise is mostly noise.

What was checked #

The fifth figure is the ledger. From clean data, Mooney’s construction returns each description’s slip velocity to two parts in $10^{16}$ at three stresses. The fitted exponent of a constant slip length is 2.5 to fifteen figures, and of the slip law 2, as given. And four thousand trials of the construction with two per cent errors give a scatter of the slip velocity within 0.7 per cent of the least-squares error formula, which is an independent route to the same number.

What the two descriptions leave out #

The slip transition. Above a critical stress a melt’s chains disentangle from those adsorbed on the wall and the slip jumps by orders of magnitude — the spurt and the sharkskin of the essay before. Both descriptions here are of the weak-slip branch below it.

Pressure. A capillary’s pressure varies along its length by hundreds of bar, and the stress a pipe knows is the wall stress at each point of it, and both the melt’s viscosity and its wall friction depend on pressure. Mooney’s construction assumes they do not.

Wall chemistry and history. The monomer friction at the wall depends on what the wall is made of and on what has adsorbed on it. A die conditioned by previous runs is not the same wall as a fresh one, and a slip property of either description is a property of the pair, melt and wall.

The activation energies are assumed equal. The temperature test above takes the melt’s and the wall friction’s temperature dependence to be the same. If they differ, the slip length is not constant in temperature either, and the test measures that difference as well; it still distinguishes the descriptions, but less cleanly.

The convention the numbers depend on #

Stresses are wall shear stresses; shear rates are true rates, $(\sigma/K)^{1/n}$ with $K = 3\cdot10^4\ \mathrm{Pa\,s}^n$ and $n = 0.4$; apparent rates are $4Q/\pi R^3$. The slip length is the slip velocity over the true wall shear rate. The descriptions are matched at 100 kPa. The shift factor $a_T$ multiplies the melt’s viscosity at every rate. Slip’s share is of the 1 mm die’s apparent rate at the matching stress.

Who found it, and when #

Navier proposed the slip length in 1823. Mooney gave the construction that separates slip from shear in capillary data in 1931. De Gennes derived the melt’s large slip length from the ratio of two viscosities in 1979, and Brochard and de Gennes the transition from weak to strong slip in 1992. Hatzikiriakos and Dealy, among others, measured melt slip laws of the power form through the 1990s, and the observation that a power-law slip law and a constant slip length are compatible only when the exponent is one over the flow index follows from putting the two definitions side by side.

Still open: what the wall remembers #

Both descriptions treat the slip as instantaneous — the slip velocity set by the present stress or shear rate. A melt’s chains at the wall relax on the time of the melt’s own relaxation, seconds for a high polymer, and a stress that changes faster than that finds the wall’s chains still arranged for the old one. The next calculation gives the slip a relaxation time — the slip velocity approaching its steady value at the rate the adsorbed chains can rearrange — and asks what a rheometer’s step in piston speed then records, whether the lag distinguishes the two descriptions as the temperature test does, and whether the oscillating extrusion of the slip transition needs the lag to exist at all.