FR26 · Extensional rheometry

Capillary Thinning: Reading a Polymer Relaxation Time

Why does a drop of polymer solution pulled between two plates form a long thread instead of snapping like water? Surface tension squeezes the stretched bridge, and in plain water only viscosity resists, so it pinches off quickly. Dissolved polymers are stretched by the thinning flow, and their stress takes over the load. The neck then thins exponentially on a time scale of three relaxation times, which is why the thread lives long and why its slope measures the polymer relaxation time.

Stretch a liquid bridge and let surface tension squeeze it. Polymers make the neck thin exponentially, and the slope of that line gives their relaxation time.

Neck radius now $0.5\,R_p$

Relaxation time read from the slope $0.891\,\lambda$

Breakup time $9.47\,\lambda$

Polymer share of the squeeze $0\,\%$

Let surface tension do the pulling

Plate separation after the stretch

2

h_f/R_p

Polymer stiffness against surface tension

0.05

\mathrm{Ec}=GR_p/\sigma

Solvent viscosity against surface tension

0.25

S=\eta_sR_p/(\sigma\lambda)

Maximum extension squared

2000

L^2

Drag the top plate to stretch the bridge. Where the log radius falls in a straight line, polymers hold the thread and the slope gives their relaxation time.

Model, units and sources

$\frac{\sigma}{R}=3\eta_s\dot\varepsilon+G\,f\,(A_{zz}-A_{rr}),\quad \dot\varepsilon=-\frac{2}{R}\frac{dR}{dt},\quad f=\frac{L^2-3}{L^2-\operatorname{tr}\mathbf A}$

Lengths are in plate radii, time in polymer relaxation times. The plates move apart instantly from a gap of half a plate radius, so the neck starts as a cylinder of the same volume with relaxed polymers. The neck radius and the conformation are integrated with an L-stable second-order implicit Runge–Kutta step; Oldroyd-B runs stop at a radius of one ten-thousandth of the plate radius without breaking.

Course chapter: reading a relaxation time from a thinning thread

McKinley (2005), Eq. 15 and Sec. 4 · Morozov & Spagnolie (2015), Eqs. 1.80–1.87

Model assumptions

The model is the zero-dimensional neck balance of capillary breakup extensional rheometry: surface tension squeezes a uniform thread, and the solvent viscosity and the polymer stress resist. Polymers are FENE-P or Oldroyd-B dumbbells that start relaxed after an instantaneous step stretch. Gravity, inertia, drainage, axial curvature, end effects, evaporation and the real filament shape are omitted; the drawn profile is schematic and only the neck radius comes from the model. For a solvent-only thread this balance gives a thinning speed of one sixth of surface tension over viscosity, more than twice the 0.0709 of the slender-thread similarity solution. Every curve is a model simulation; no measured data are shown.

Neck radius, log scale

Resistance to stretching

Who resists surface tension

More numbers and how they are computed

Time shown on the stage: $0$
Strain rate at the neck: $0$
Hencky strain at the neck: $0$
Apparent extensional over zero-shear viscosity: $0$
Axial conformation component: $0$
Window used for the exponential fit: $0$
Zero-shear viscosity: $0$
Breakup time of the solvent-only reference: $0$
Largest share of the finite-extension bound used: $0$
Implicit Runge–Kutta steps: $0$

The fit uses only the simulated radius, as an experiment would: it takes the longest stretch where the local decay rate stays within 10 % of its mean and fits a straight line to the log radius there. The dashed curves are reference runs with the same numbers, one with an infinitely extensible spring and one with the solvent alone; they are not data.

Physics tutorial

A thread that measures its own polymers

BackgroundPull a drop apart between two plates and stop. Surface tension then squeezes the liquid bridge, and nothing else drives the flow. A plain liquid pinches off quickly. Add a little long-chain polymer and a thin thread forms that can last many times longer.

Why it mattersThe thinning neck is a pure extensional flow whose rate the fluid chooses for itself. That makes the thread a rheometer: watch only the neck radius, and the polymers write their relaxation time into its decay.

Start with the essentials

Focus question: Why does a drop of polymer solution pulled between two plates form a long thread instead of snapping like water?
One-sentence intuition: Once the polymers are stretched, their stress balances surface tension and the neck thins as $R\propto e^{-t/3\lambda}$ . The strain rate locks at $2/(3\lambda)$ , just above the coil–stretch rate, so the slope of $\ln R$ gives the relaxation time.

Core mathematical model

Force balance at the neck

\frac{\sigma}{R}=3\eta_s\dot\varepsilon+\left(\tau_{p,zz}-\tau_{p,rr}\right),\qquad \dot\varepsilon=-\frac{2}{R}\frac{dR}{dt}

Capillary pressure squeezes the thread; the solvent resists with three times its viscosity times the strain rate, and the stretched polymers resist with their normal-stress difference.

Solvent alone

R(t)=R_0-\frac{\sigma}{6\eta_s}\,t

Linear thinning to breakup at $t=6\eta_sR_0/\sigma$ . This simple balance overestimates the speed; the slender-thread similarity solution gives $0.0709\,\sigma/\eta_s$ instead of $\sigma/6\eta_s$ .

Elasto-capillary balance

\frac{R(t)}{R_0}\propto\exp\!\left(-\frac{t}{3\lambda}\right),\qquad \lambda\dot\varepsilon=\frac{2}{3}

When the polymer stress carries the load, the solvent drops out and the neck follows the polymers alone. An Oldroyd-B thread keeps this forever and never breaks.

Fully stretched chains

\eta_E\approx3\eta_s+2\eta_pL^2,\qquad \frac{dR}{dt}\approx-\frac{\sigma}{2\eta_E}

Finite chains run out of length. The thread then behaves like a very viscous liquid and thins linearly to breakup.

Common difficulties

Thicker means more viscous?

Typical misconceptionThe polymer thread lasts longer because the polymers make the liquid more viscous.

Better mental modelThe zero-shear viscosity changes little. What changes is the stress in extension: stretched chains resist far more than their shear viscosity suggests, and that stress grows with strain, not with rate.

The slope is the relaxation time

Typical misconceptionAny straight part of the log-radius plot gives the exact relaxation time.

Better mental modelOnly the window where polymer stress dominates counts. Finite extensibility bends the line before breakup, and in this model it makes the slope steeper, so the fitted value comes out below $\lambda$ .

Run the experiment

01
Watch the neck
Play Polymers hold the thread and watch the neck radius on the log plot.
What to observe: After a short viscous start, the radius falls along a straight line, the pink fit, until it bends down and the thread breaks.
02
Remove the polymers
Choose Solvent alone snaps.
What to observe: The same liquid bridge pinches off more than ten times sooner, and the radius falls linearly instead of exponentially.
03
Shorten the chains
Choose Short chains break early, then drag the extensibility slider up and down.
What to observe: Shorter chains run out of length sooner, so the straight window ends early and the thread breaks sooner.