The wall a melt really does slip on

The squeeze-film measurement establishes that the no-slip condition is a boundary condition rather than a law, that Navier’s alternative is equally admissible to the equations, and that for a wetting liquid on a smooth solid the slip length is under two nanometres. Two nanometres is a few molecular diameters, so an ordinary calculation that uses no-slip is safe by three orders of magnitude.

That conclusion is about simple liquids, and the essay says so in one line without following it up. Follow it up and the picture changes completely: for an entangled polymer melt the slip length is not molecular. It is microns to millimetres, it is often larger than the channel the melt is being pushed through, and the flow in an extrusion die is mostly plug.

That is not a small correction to a boundary condition. It is the statement that for one enormous class of industrially important fluids — every thermoplastic, processed by every extruder, injection moulder and blow-moulding machine in the world — the no-slip condition is simply false, by four to seven orders of magnitude in the one length that measures it.

It is also, unlike the three other well-known exits from no-slip, not a matter of the fluid stopping being a continuum or of the surface stopping being a surface. A rarefied gas slips because the gas is not a continuum at the wall; a superhydrophobic surface slips because part of the wall is not a wall; a moving contact line has no slip length so much as a singularity that has to be cut off. A melt slips because it is a liquid whose bulk and whose first molecular layer have completely different viscosities, and there is nothing pathological about either.

One line of de Gennes #

The reason a melt is different is not chemistry and not the surface. It is length.

A melt slides over a solid by the friction of the monomers actually touching it, and those monomers have no idea they belong to a long chain — the friction is that of an unentangled liquid of the same chemistry, with a viscosity $\eta_s$ of order a tenth of a pascal second. The melt resists being sheared, on the other hand, with the viscosity of an entangled melt, $\eta$, which is larger by four to seven decades — and which is already a quantity that depends on the question asked of it.

A slip length is the ratio of a bulk resistance to a wall resistance, so

$$b = a\,\frac{\eta}{\eta_s},$$

with $a$ a monomer size. Nothing exotic is required: only that the molecules be long.

The physical picture behind the ratio is worth one sentence. The slip length is the distance into the fluid at which the bulk velocity profile, extrapolated, would reach the wall’s own velocity — so it is large when the bulk is stiff and the interface is soft. An entangled melt is the extreme case of exactly that: it is stiff because its chains cannot pass through one another, and its interface is soft because the monomers touching the wall are not entangled with anything.

A commodity polyethylene at $10^4$ pascal seconds gives thirty microns. A high-molecular-weight melt at $10^6$ gives three millimetres. An extrusion die’s bore is a fraction of a millimetre, so the melt is being pushed down a channel narrower than its own slip length — which is the regime in which the boundary condition stops being a correction and becomes the flow.

It is worth checking that the argument really contains no chemistry, because the conclusion is strong. What goes into it is a monomer size, which is the same for every liquid made of the same repeat unit; a monomeric viscosity, which is a property of that unit; and a melt viscosity, which for an entangled melt scales as the cube of the molecular weight. So the slip length scales as the cube of the chain length, and a polymer twice as long slips eight times as far.

That is a very fast dependence on the one variable a chemist controls most easily, and it is why the effect is invisible in a laboratory working with short chains and dominant in a plant extruding long ones. It is also why the phenomenon was contested for so long: two laboratories using the same polymer at different molecular weights are not doing the same experiment, and the one at low molecular weight is entitled to report no slip.

The other thing the scaling says is what would break it. Below the entanglement molecular weight the melt viscosity scales as the first power rather than the cube, the ratio $\eta/\eta_s$ is of order the chain length rather than its cube, and the slip length collapses. Entanglement is the mechanism, not molecular weight as such, and a long unentangled chain slips no more than a short one.

What that does to a measurement #

A capillary rheometer pushes melt through a die at a known volumetric rate and measures the pressure needed — an instrument that takes a derivative of what it is shown — reporting an apparent shear rate $\dot\gamma_a = 4Q/\pi R^3$. With slip,

$$\dot\gamma_a = \dot\gamma_{\text{true}}(\sigma_w) + \frac{4 V_s(\sigma_w)}{R},$$

so the reported curve is the material’s curve plus a term that depends on the die. Three dies give three curves, and none of them is the material.

Ninety-nine per cent of the throughput never sheared at all. A rheometer run in that condition is not measuring a viscosity; it is measuring a wall.

The die dependence has a form worth remembering, because it is what makes the effect so easy to miss and so easy to find once looked for. The slip contribution goes as $1/R$ and the shear contribution does not, so the ratio of the two goes as $1/R$ as well — halving the bore doubles the slip’s share of the throughput. There is no threshold and nothing switches on; a wide die has the same slip and simply passes more shear alongside it.

So a laboratory that has always used one die size has a consistent, reproducible, precisely measured flow curve that is wrong by a factor it cannot detect. Consistency is not a check on this, and that is the reason the phenomenon survived being disbelieved for decades: every individual measurement was good.

What is actually being misread #

The consequence for anybody using a rheometer is worth setting out plainly, because the error is not that a number comes out slightly wrong.

A melt measured in a narrow die reports an apparent viscosity far below the true one, because the throughput is being carried by sliding rather than by shearing and the pressure needed is therefore small. The apparent curve also looks more shear-thinning than the material is, because the slip velocity rises with stress faster than the shear rate does — so the reported power-law index is too low.

Both errors are in the direction that flatters the material, and both get worse as the die gets narrower. A laboratory characterising a melt in a small die and a plant extruding it through a large one are therefore not disagreeing about the polymer; they are measuring two different mixtures of the same two mechanisms, and the laboratory’s number will over-predict the plant’s throughput.

The signature that distinguishes it from shear thinning is the die. Shear thinning is a property of the material and produces the same curve in every geometry; slip does not. A single measurement cannot separate them, and no amount of care with one die will.

There is a second, cruder signature that is worth knowing because it needs no second die. A melt slipping in a die produces a flat velocity profile, so a tracer particle introduced at the wall arrives at the exit at the same time as one on the axis. A melt shearing produces a profile whose centre arrives first by a factor of two or more. That is directly visible in the residence-time distribution of a coloured pulse, and it is how the phenomenon was first argued for against people who did not believe it.

A flow with no shear in it is a flow with no mixing in it, which matters for a quite different reason: a plug flow does not blend, so an extruder relying on the die to homogenise its melt is relying on something that is not happening.

The construction that separates them #

The separation needs no assumption about either quantity, and it is the same trick played plays on a squeeze film — vary a geometry and read the two contributions off their different dependence on it.

At a fixed wall shear stress the true rate is fixed, because it is a material property; the slip term goes as $1/R$. So plotting the apparent rate against the reciprocal of the die radius gives a straight line whose slope is four times the slip velocity and whose intercept is the true rate.

That is Mooney’s construction, of 1931, and it is the reason a rheometer is quoted with a die geometry. The check that matters is in the figure below: with the wall condition set to no slip, the three dies agree exactly. The disagreement is the slip and not the arithmetic.

Two practical cautions go with it. The first is that three dies must differ in radius and not in anything else: the entrance geometry, the length-to-diameter ratio and the surface finish all have to be held, because each of them changes the pressure drop and would be fitted as a slip velocity. That is why the construction is done with dies of the same length-to-diameter ratio rather than the same length.

The second is that the fit is a straight line through three points, so its precision is the precision of the pressure measurements divided by the range of $1/R$ available — and the range is limited at the narrow end by the pressure the machine can supply and at the wide end by the flow rate it can meter. A factor of four in radius, which is what is drawn here, is a realistic span, and it makes the slope about as well determined as the intercept. Neither is a measurement to be made casually, which is the honest reason a single-die flow curve is still what most laboratories report.

The transition, and the defect it makes #

The slip velocity does not rise smoothly with stress. Above a critical wall shear stress of order a tenth of a megapascal the chains adsorbed at the wall disentangle from the bulk, and the slip velocity jumps by more than an order of magnitude.

A machine asked for a rate inside that band finds no steady state, which is the same shape as a channel that has two flows for one flux. The pressure builds until the stress reaches the critical value, the melt lets go, the throughput leaps and the pressure falls, the chains re-adsorb, and it repeats — at a frequency set by the compressibility of the melt in the barrel. The extrudate comes out with a periodic surface, and the trade calls the result sharkskin at the onset and melt fracture further in.

It is a production limit rather than a curiosity. An extrusion line is run below the critical stress, which caps its output; raising the temperature lowers the viscosity and the stress with it, at the cost of degrading the polymer; and a substantial part of polymer processing is arranging to stay on the left-hand branch.

The shape of the instability is worth one more sentence, because it is a shape fluid mechanics has met before. A multivalued constitutive curve — one stress, several rates — with a machine imposing the rate rather than the stress is exactly the structure that produces a relaxation oscillation, and it is the same structure as the stick-slip of a brake, the jerking of a rusty valve and the note of a bowed string. In every case the oscillation’s frequency belongs to the machine and its existence belongs to the material, and in every case the cure is to operate off the unstable branch rather than to damp the machine.

What the picture cannot show #

The slip law’s constants are stated rather than measured. The critical stress, the exponent and the size of the jump are chosen to be of the right order for a commodity melt; a real material’s are measured, and vary between grades of the same polymer. What does not depend on them is the shape of the argument and the recovery by Mooney’s construction.

The flow is isothermal and it is not. Viscous heating in a die at these stresses raises the temperature by tens of kelvin, the viscosity falls with it, and the slip length — which is a ratio of two viscosities with different temperature dependences — moves as well.

The melt is taken as inelastic apart from its shear thinning. A melt is viscoelastic; it stores stress, remembers its history, and swells when it leaves the die. None of that is here, and the entry pressure loss it causes is a large part of what a capillary rheometer actually measures.

And the wall is clean and smooth. Adsorbed chains, fluoropolymer processing aids, and the roughness of a worn die all change the slip law substantially — which is the industry’s main lever on the problem and is a statement about chemistry rather than about fluid mechanics.

Who found it, and when #

Mooney published the construction in 1931, for rubber. De Gennes gave the slip-length argument in 1979 and Brochard and de Gennes the two-branch transition in 1992; Wang’s and Denn’s reviews through the 1990s and 2000s established that the melt-fracture instabilities and the slip transition are the same phenomenon. The experimental separation of slip from shear thinning took until careful multiple-die work in the 1980s, because the two have similar signatures on a single die.

The surprising connection is with the squeeze film, and it is the same experiment. Both measure a boundary condition by varying a geometry: a squeeze film varies the gap and reads the slip length off the deviation from the inverse-cube force, and a capillary rheometer varies the bore and reads the slip velocity off the deviation from a single flow curve. Neither can be done with one measurement, because a boundary condition is a statement about a surface and every measurement is about a volume — so the only route is to change the ratio of the two and watch what moves. That is the whole methodology for measuring any boundary condition, and it is why the answer for a simple liquid took a hundred and fifty years.

Still open: whether the slip length or the slip velocity is the material property #

The two descriptions in use are not the same, and the literature moves between them without saying which is meant.

A Navier slip length is a fixed length, so the slip velocity is proportional to the wall shear rate. A slip velocity law, which is what a rheometer fits, makes the slip velocity a function of the wall stress with a power that is measured and is usually not one. Those agree only for a Newtonian fluid, and a melt is not one: with a power-law index of 0.4 and a slip exponent of 2, the implied slip length varies by a decade across an ordinary range of rates.

Which of them is the material property matters because it decides what transfers. A slip length measured on one die predicts the behaviour of another; a slip velocity law fitted against stress does too; but a slip length inferred from a velocity law is a different number at every rate, and using it outside the range it was fitted in is the commonest error in the practice.

The measurement that would settle it is a rheometry programme in which the die radius and the temperature are varied independently — since the slip length is a ratio of two viscosities and the slip velocity is not, the two descriptions predict different temperature dependences at fixed stress. That experiment is not difficult and does not appear to have been done as a test between the two pictures rather than as a fit to one of them.