FR26 · Conformation and stress

Coil–Stretch Transition: Polymers in Shear and Extension

Why does the same polymer solution barely change in shear yet stiffen enormously when stretched? Shear keeps rotating each polymer coil away from the direction it stretches, so the coils lengthen only a little and the viscosity barely moves. Extension has no such rotation. Once the stretching rate exceeds half the coil relaxation rate, the chain unravels until its finite length stops it, and the extensional viscosity can climb a hundredfold above the shear viscosity.

Start shear and planar extension at the same rate. Past a critical rate the extending coils unravel and resist a hundred times harder, while shear barely changes.

Coil stretch in extension $6.45\times$

Coil stretch in shear $1.19\times$

Extension over shear viscosity $114$

Extension rate

\mathrm{Wi}

$0.8$

Shear it, stretch it, compare

Weissenberg number

0.8

\mathrm{Wi}=\lambda\dot\gamma=\lambda\dot\varepsilon

Solvent share of viscosity

0.5

\beta=\eta_s/\eta_0

Maximum extension squared

400

L^2

Drag the top plate to set the rate of both flows. Stress mode holds the stress that would drive a plain liquid at that rate.

Model, units and sources

$\eta_0=\eta_s+\eta_p,\quad \beta=\eta_s/\eta_0,\quad G=\eta_p/\lambda,\quad \operatorname{tr}\mathbf A<L^2$

Time is in relaxation times, viscosity in the zero-shear viscosity. Both flows are homogeneous and inertialess, so the conformation obeys ordinary differential equations, integrated with fourth-order Runge–Kutta and a step that shrinks near the finite-extension bound. In stress mode the rate solves the force balance between solvent and polymer stress at every step.

Course chapter: why extension unravels coils and shear does not

Morozov & Spagnolie (2015), Eqs. 1.72–1.87 · Schroeder (2018), Sec. III

Model assumptions

The model is a dilute polymer solution in a homogeneous, inertialess flow. Each polymer is an elastic dumbbell described by the conformation tensor: Oldroyd-B with a linear spring, or FENE-P with a Peterlin finite-extension spring. Hydrodynamic interaction, conformation-dependent drag (the cause of coil–stretch hysteresis), polydispersity, chain-length distributions and any spatial flow structure are omitted. Every curve is a model simulation; no measured data are shown.

Stress growth after startup

Steady viscosity versus rate

Energy ledger in extension

More numbers and how they are computed

Time shown on the stage: $0$
Shear rate times relaxation time: $0$
Transient shear viscosity: $0$
Transient planar extensional viscosity: $0$
Shear: normal-stress difference over shear stress: $0$
Extension stretch at the end of the run: $0$
Largest share of the finite-extension bound used: $0$
Smallest conformation eigenvalue in the run: $0$
Largest mismatch of work minus stored minus dissipated: $0$
Runge–Kutta steps in both runs: $0$

Steady curves solve the algebraic steady state by bisection on the spring factor; the Oldroyd-B extension curve ends where no steady state exists. The dashed lines are the same run with an infinitely extensible spring, shown as a reference, not as data.

Physics tutorial

Rotation saves a coil; pure stretching does not

BackgroundA dissolved polymer is a random coil that thermal motion keeps pulling back to its rest size. A flow drags its two ends apart. Which wins depends on how fast the flow stretches compared with how fast the coil relaxes.

Why it mattersShear and planar extension at the same rate stretch fluid elements equally fast at first. In shear the flow also rotates every coil away from the stretching direction; in extension nothing rotates it away.

Start with the essentials

Focus question: Why does the same polymer solution barely change in shear yet stiffen enormously when stretched?
One-sentence intuition: Above $\mathrm{Wi}=1/2$ planar extension stretches a linear dumbbell faster than it relaxes, so the stretch grows exponentially. Only the finite length of a real chain stops it, and the extensional viscosity then sits orders of magnitude above the shear viscosity.

Core mathematical model

Conformation tensor and polymer stress

\overset{\triangledown}{\mathbf A}=-\frac{f\mathbf A-\mathbf I}{\lambda},\qquad \boldsymbol\tau_p=G\,(f\mathbf A-\mathbf I),\qquad f=\frac{L^2-3}{L^2-\operatorname{tr}\mathbf A}

$\mathbf A$ is the average end-to-end tensor scaled so that it equals the identity at rest. The upper-convected derivative carries it with the flow. Oldroyd-B is the limit $f=1$ .

Startup of shear, Oldroyd-B

\eta^+(t)=\eta_s+\eta_p\left(1-e^{-t/\lambda}\right),\qquad N_1^+(t)=2\eta_p\lambda\dot\gamma^2\left[1-\left(1+\tfrac{t}{\lambda}\right)e^{-t/\lambda}\right]

Shear viscosity never exceeds its zero-rate value, whatever the rate. The normal-stress difference grows with the square of the rate instead.

Steady planar extension, Oldroyd-B

A_{xx}=\frac{1}{1-2\mathrm{Wi}},\qquad \eta_P=4\eta_s+\frac{4\eta_p}{1-4\mathrm{Wi}^2}

The steady state exists only below $\mathrm{Wi}=1/2$ . Above it the startup grows like $e^{(2\mathrm{Wi}-1)t/\lambda}$ without bound.

Energy ledger of the polymer

\int_0^t\boldsymbol\tau_p:\nabla\mathbf u\,dt'=\psi(t)+\int_0^t\frac{G}{2\lambda}\sum_i\frac{(f a_i-1)^2}{a_i}\,dt'

Work done on the polymers is either stored as elastic free energy $\psi$ or dissipated; $a_i$ are the eigenvalues of $\mathbf A$ . The dissipation is never negative.

Common difficulties

Same rate, same stretch?

Typical misconceptionShear at a given rate stretches polymers as much as extension at that rate.

Better mental modelShear is half stretching and half rotation. The rotation tips a stretched coil toward the flow direction, where the stretching component no longer pulls on it, so the shear stretch saturates while the extension stretch keeps growing.

A diverging viscosity

Typical misconceptionOldroyd-B predicts a real liquid with infinite extensional viscosity.

Better mental modelThe divergence comes from a spring that can stretch forever. A finite chain caps the stretch, which FENE-P captures, and real chains also show hysteresis that neither model contains.

Run the experiment

01
Watch both cells
Play Coils unravel and watch the coils in each cell.
What to observe: Shear coils tilt and lengthen a little; extension coils keep stretching along the outflow axis, and the stress-growth curve for extension climbs by two orders of magnitude.
02
Drag the top plate
Drag the top-plate handle left until the extension rate reads below one half.
What to observe: The extension coils stop growing and settle at a modest size, and the steady-viscosity marker slides back below the coil–stretch line.
03
Hold the stress instead
Choose Stress puts on the brakes.
What to observe: The extension rate starts high, then the growing polymer stress takes over the load and the rate falls toward one half, while shear creeps to a steady rate.