The stretch a chain cannot outrun

The liquid that climbs the rod found a tension along the streamlines of a sheared polymer solution, and it ended on a list of things that tension does outside a rheometer: the swell of a stream leaving a die, the thread that follows a drop of shampoo, the jet that will not break. Every item on that list has something in common that a rheometer’s shear cell does not. In each, the liquid is being stretched — pulled apart along one direction and drawn in along the others — rather than sheared, and a polymer solution responds to stretching in a way its behaviour in shear gives almost no hint of.

The difference is not one of degree. A dilute solution of long chains can be barely distinguishable from water when stirred and several hundred times stiffer than water when pulled into a thread, and there is a sharp rate that separates the two behaviours. It is set by one comparison, between how fast a flow pulls a molecule apart and how fast the molecule can pull itself back together.

Three ways to stretch a liquid #

Before a polymer is added, stretching already differs from shearing, and the difference is pure geometry.

Pull a thread of Newtonian liquid from its ends at a stretch rate $\dot\varepsilon$. The liquid is lengthening at $\dot\varepsilon$ and, being incompressible, narrowing at $\dot\varepsilon/2$ in both of the other directions. The viscous stress in each direction is twice the viscosity times the rate in that direction — the same coefficient momentum going sideways measures in shear — and the thread’s side surface is free, so the pressure has to cancel the stress there. What remains along the thread is $2\eta\dot\varepsilon + 2\eta\cdot\tfrac12\dot\varepsilon = 3\eta\dot\varepsilon$. The same bookkeeping gives four for a sheet, which narrows in only one direction, and six for a film stretched in two directions, which has to thin twice as fast.

Frederick Trouton measured the first of these in 1906 by pulling threads of pitch, and the factor of three carries his name. Nothing about the liquid enters it. That makes it a useful instrument: a thread of any liquid whose extensional viscosity is not three times its shear viscosity is reporting a property of the material, because the flow has already been accounted for.

Shear rotates a chain, and stretching does not #

The polymer changes the picture because a polymer chain is a spring, and whether a flow can stretch a spring depends on whether the flow also turns it.

A long chain in a solvent at rest is a random coil, its size set by the entropy of its possible configurations: pulling its ends apart reduces the number of shapes it can take, and the resulting restoring force is entropic, proportional to temperature, and relaxes over a time λ — the time the liquid that remembers weighs against an observation. The simplest honest model of such a chain is a dumbbell — two beads carried by the flow, joined by that spring — and it contains the whole argument.

Simple shear is a rotation added to a stretch. A chain aligned with the direction that stretches it is carried round, within a fraction of a turn, into a direction that compresses it, so the flow undoes its own work. At a shear rate equal to one over the relaxation time the dumbbell’s stretch settles at $1 + \tfrac23(\lambda\dot\gamma)^2$ times its resting value — 1.67 — and at a hundred times that rate it still grows only as the square of the rate. That is the tension the liquid that climbs the rod climbs on: real, and bounded.

Uniaxial extension has no rotation. A chain aligned with the stretching axis stays aligned with it, and its ends separate exponentially at a rate set by the flow. The spring relaxes at a rate set by the chain. Whichever rate is larger wins outright, and at the same numerical rate that gave a modest, settled stretch in shear, the dumbbell in extension is two thousand times its resting stretch after eight relaxation times and has not begun to level off.

The kinematic fact underneath is simple and has nothing to do with polymers: two points in a flow without rotation separate exponentially, and two points in a shear flow separate linearly. The polymer is what turns that difference into a stress.

A rate at which the spring loses #

The threshold comes from one line. In uniaxial extension the square of the dumbbell’s length grows at twice the stretch rate, because each of the two dimensions along the axis grows at the rate itself. The spring relaxes it at one over the relaxation time. So the stretch grows without limit whenever $2\dot\varepsilon > 1/\lambda$, which is to say $\lambda\dot\varepsilon > \tfrac12$.

This is the coil–stretch transition, identified by Lumley for turbulent drag reduction in 1969 and by de Gennes in 1974. Below it, a chain in an extensional flow is a somewhat elongated coil and the solution’s extensional viscosity is a few times its shear viscosity — Trouton’s three, plus a little. Above it, the chain unravels, and the stress it carries grows as fast as it does.

The curves show both sides. At $\lambda\dot\varepsilon = 0.1$ the extensional viscosity settles at 3.37 times the zero-shear value, hardly distinguishable from Trouton’s Newtonian three with a polymer correction. At 0.4 it settles, more slowly, near 9.5. At 0.6 — just past the threshold — it has not settled by a Hencky strain of 6, a stretch of $e^6$, or four hundred times; and at 1 the unbounded dumbbell has reached 725 times the shear viscosity and is still climbing exponentially. The chains of finite length, drawn solid, follow the unbounded ones until their extension becomes a substantial fraction of their contour length, and then fall behind.

The rate that matters is the product of a stretch rate and a relaxation time, and relaxation times of dilute solutions of long polymers are milliseconds to seconds. A thread thinning under surface tension, a jet breaking up, a fibre being drawn and the flow into a narrow contraction all reach stretch rates of hundreds per second, so for almost any polymer long enough to matter the transition is not a laboratory curiosity but the normal state of those flows.

How stiff a stretched solution becomes #

For an unbounded dumbbell the steady extensional viscosity has a closed form, $3\eta/(1 - 2\lambda\dot\varepsilon)(1 + \lambda\dot\varepsilon)$ for the polymer’s share, and it has a pole at one half. A real chain cannot be stretched beyond its contour length, and a finite spring — one whose force diverges as the chain approaches full extension — replaces the pole with a smooth rise to a plateau. For a chain whose fully stretched length squared is 400 times its resting size squared, the plateau is near 664 times the shear viscosity.

The plateau scales with that extensibility, and the extensibility scales with the molecular weight. A chain of polyethylene oxide with a molecular weight of a few million has an extensibility in the thousands, which is why a few parts per million of it — too little to change the shear viscosity measurably — can raise the extensional viscosity of water by orders of magnitude. The Trouton ratio of a dilute polymer solution is a measure of how long its molecules are, and it is the property that sets how far a sprayed droplet drifts and how thin a fibre can be drawn.

The figure hides a fact about the transition itself that the steady curve cannot show. Near $\lambda\dot\varepsilon = \tfrac12$ the time a solution takes to reach its steady stretch grows without limit, in the same way the time to settle diverges at every continuous transition. The steady curve for the finite chains was computed from the steady equations directly for that reason, and checked against a time integration at rates where the integration converges in a reasonable time.

Ten parts per million #

The size of the effect is easiest to believe with a real molecule in it. Polyethylene oxide with a molecular weight of four million is sold by the kilogram. In water its intrinsic viscosity is about 1,800 millilitres per gram, so ten parts per million of it — a hundredth of a gram in a litre — raises the shear viscosity by about two per cent. The solution pours, stirs and runs through a pipe like water, and an ordinary viscometer can barely tell the two apart. A liquid’s whole shear curve, the subject of the fluid that answers back, is not where this additive shows at all.

The same molecule, fully extended, is a chain about thirty micrometres long, coiled in water to a size of a few hundred nanometres. The square of the ratio of those two lengths is its extensibility, of the order of ten thousand. Once the stretch rate passes the transition, each chain’s contribution to the stress is amplified by roughly that factor over its contribution in shear, so a two per cent addition to the shear viscosity becomes an addition of some hundreds of times water’s viscosity in extension. A shear rheometer and a thread measure the same molecules, at the same concentration, and disagree by four orders of magnitude. The estimate is rough in its details and not in its size, and it is why a spray nozzle, an inkjet head and a fire hose can each be changed completely by an additive that no ordinary measurement of the liquid detects.

The contraction that costs more than its viscosity says #

Flows that are not threads contain extension too, and one of them became a long-running puzzle. Push a polymer solution through a sudden contraction — a wide pipe stepping down to a narrow one — and the fluid accelerating into the narrow section is stretched along its axis as it goes. For a Newtonian liquid the extra pressure the contraction costs is a modest, known amount. For solutions made deliberately to have a constant shear viscosity, so that shear-thinning cannot muddy the comparison, the measured extra pressure drop rises well above the Newtonian value once the Weissenberg number passes one, and large vortices grow in the corners upstream of the step.

Numerical solutions of the unbounded-dumbbell equations for the same geometry predicted a slight fall instead, and for decades the disagreement stood as one of the standing embarrassments of computational rheology. Its partial resolution runs straight through this essay. A chain stretches without limit only if it stays in the extensional region long enough, and a fluid passing through a contraction spends a fraction of a second there; computations that gave the chains a finite length, so that their stress grows and then saturates instead of diverging, began to reproduce at least part of the measured increase, though no model yet matches the experiments quantitatively. What a contraction costs is set by how far the chains stretch during their brief passage, which is the transient extensional viscosity of the second figure rather than the steady one.

The thread that picks its own rate #

The thinning thread that the liquid that climbs the rod followed to its beads-on-a-string ending is where the transition shows up without being set.

A Newtonian thread pinches off because surface tension drives the neck to zero faster and faster. A polymer thread thins exponentially instead, its diameter falling as $e^{-t/3\lambda}$, and the exponential rate is a balance between the capillary pressure squeezing the thread and the elastic stress of the chains resisting. What is striking is the stretch rate that balance produces. A thread thinning as $e^{-t/3\lambda}$ is stretching at $2/3\lambda$, so its $\lambda\dot\varepsilon$ is exactly two thirds — just past the coil–stretch threshold of one half.

That is not a coincidence. Below one half the chains would relax and offer no resistance, the thread would thin faster, and the stretch rate would rise; far above it the chains would stiffen so much that the thinning would slow. The thread settles itself just on the stretched side of the transition, which is why the thinning rate measures the relaxation time and nothing else, and why the same few parts per million of polymer can turn a jet that breaks in milliseconds into one that holds together for seconds.

A chain that remembers being stretched #

The dumbbell treats a chain’s drag as fixed, and a real chain’s is not. A coiled chain is compact, and the solvent streams around it as around a small sphere; an unravelled chain is long, and the solvent has to be dragged past the whole of it. So the flow grips a stretched chain harder than a coiled one, and that feedback does something no fixed-drag model can.

The coiled chain is stable until the flow’s pull exceeds its spring at the threshold rate. The stretched chain, once formed, is held by a flow that grips it hard, and stays stretched at rates well below the one that stretched it. For a drag that grows fourfold between coil and full extension, the stretched state survives down to 0.44 of the unfolding rate. Between 0.44 and 1 the same molecule, in the same flow, can be either coiled or stretched, and which one it is depends on what the flow was doing earlier — a molecule with a history, in the way a magnetised material has one.

De Gennes predicted this in 1974 and it waited nearly thirty years for an experiment that could see it. In 2003 Schroeder, Babcock, Shaqfeh and Chu held single fluorescently labelled DNA molecules at the stagnation point of a cross-slot, where the flow is a steady extension and a molecule stays put, and found exactly the loop: molecules prepared coiled stayed coiled, and molecules prepared stretched stayed stretched, at the same flow rate, for as long as they were watched.

The shape of the figure is familiar from somewhere quite different. A curve that folds back on itself, with two stable branches joined by an unstable one and a region in which the state depends on history, is the part of the curve no fluid follows — the van der Waals loop of a liquid and its vapour — drawn for a single molecule. The coil and the stretched chain are two phases of one polymer, the unstable branch is the spinodal, and switching between the branches inside the window needs a fluctuation large enough to carry the chain over a barrier, which is the barrier a new phase has to climb in a different coordinate. With thermal noise — the agitation the jiggle that proved atoms made visible — included, a chain in the window does eventually switch, and the time it takes depends exponentially on how deep into the window the rate sits.

Where the dumbbell stops describing a polymer #

A dumbbell is one relaxation mode. A real chain has a spectrum of them, from the whole chain down to a few monomers, and the longest mode sets the transition while the shorter ones set how the stress grows past it. The curves here are the single-mode idealisation, correct in shape and approximate in detail.

The finite-chain model is averaged before it is solved. The spring force here depends on the average stretch of the population rather than on each chain’s own, an approximation that makes the equations closed and that smooths the transition: individual chains unravel at different times, and the population’s stress rises less sharply than any one of them.

The solution is dilute. Every result assumes chains too far apart to interact. At concentrations where the chains overlap, they entangle, their relaxation is set by their neighbours, and the extensional behaviour of a polymer melt is a different subject.

The drag feedback is a model too. The fourfold growth used for the hysteresis figure is a parameter chosen to show the window clearly. Measured DNA molecules show a window, but its width depends on the molecule’s length and on hydrodynamic interactions that a single drag coefficient compresses.

And the flow is taken to be pure extension. Real flows mix extension with shear and rotation, and a chain that spends part of its time in a region of rotation relaxes there. Whether the transition is reached in a given flow depends on how long a chain stays in its extensional part, which for a jet or a contraction is a strain of a few units — often not enough to reach steady state.

What the curves cannot show #

The curves are averages over a population, and the population is not uniform. Watching individual DNA molecules unravel in an extensional flow showed that identical chains, started from similar coils, unfold by entirely different routes — some as dumbbells with their ends pulled out first, some folded in half, some kinked — at different rates, and that the average hides that variety completely. The spread matters, because the stress a solution carries is dominated by its most stretched chains.

Nor do the curves show a flow. Measuring a steady extensional viscosity of a mobile liquid is one of the hardest measurements in rheology, because keeping a thread in pure extension long enough to reach a Hencky strain of seven means stretching it eleven hundred times, and most instruments run out of length first. The steady curve is a calculation that experiments approach from below.

Still open: whether the loop survives in a bulk flow #

The coil–stretch hysteresis is established for single molecules held at a stagnation point. Whether it matters for a solution flowing through a real geometry is disputed. If it does, a polymer solution could carry two different extensional stresses at the same flow rate, depending on the history of the fluid that arrived there, and flows through contractions and around cylinders would show a corresponding bistability. Some experiments on concentrated DNA and on very long synthetic polymers report history-dependent stresses consistent with that; others attribute them to incomplete stretching and to the averaging that smooths any population’s transition.

The difficulty is that a bulk flow exposes each chain to the transition only briefly, and a window that a single held molecule can occupy indefinitely may be crossed too quickly in a flowing liquid for its two branches to separate. How long a chain must spend inside the window for the solution’s stress to remember which branch it was on is the quantity that would settle the question, and it has not been measured.

The habit worth carrying away is to ask whether a flow rotates what it deforms. A flow that stretches without turning can win against any restoring force slower than itself, and the rate at which it starts to win is set by the thing being stretched rather than by the flow’s strength.