Illustrated Physics
physics, drawn
Concept

Polymer solution — where it appears

A liquid with long chain molecules dissolved in it, whose flow depends on how far the chains are stretched and how quickly they recoil. A few parts per million can make a liquid climb a rotating rod, or resist stretching thousands of times more strongly than it resists shear.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The same liquid climbs a thin rod and is thrown off a thick one. The height of the free surface against distance from the axis, for rods of 10.0, 25.0, 60.0 millimetres radius turning at one revolution a second in the same polymer solution, with the Newtonian answer dashed. Two effects compete and they fall off at different rates: a hoop tension along the curved streamlines pulls the fluid inward and lifts it, falling as the fourth power of the distance, while the centrifugal term pushes it outward and lowers it, falling as the second. Near a thin rod the fourth power wins and the liquid climbs — 2.21 millimetres at the thinnest. Beyond a radius of 34.64 millimetres it does not, and the same fluid at the same speed is thrown outward exactly as water would be. The changeover is a property of the fluid and the rod together, so a demonstration that works on a glass stirring rod fails on a spoon handle.

The liquid that climbs the rod

Stir water with a rod and it is flung outward, leaving a dip at the centre. Stir a polymer solution and it climbs the rod instead. Nothing about the viscosity can produce that, however large it is made — what produces it is a second stress that appears in shear and has no Newtonian counterpart at all, growing as the square of the rate where the familiar one grows in proportion.

fluids · Rheology Above a half, the chains stop letting go. The extensional viscosity of a dilute polymer solution, as a multiple of its zero-shear viscosity, against the accumulated stretch (Hencky strain, the stretch rate times the time), on a logarithmic axis, for stretch rates whose product with the relaxation time is 0.1, 0.4, 0.6, 1. The polymer carries 90 per cent of the viscosity. Dashed curves are the Oldroyd-B dumbbell, integrated from its conformation equation and matching its closed form; solid curves are the same chains with a finite length, FENE-P with L² = 400. At 0.1 the Oldroyd-B viscosity reaches 3.37 and the finite chain 3.37 by a strain of 6; at 0.4 the Oldroyd-B viscosity reaches 9.49 and the finite chain 9.25 by a strain of 6; at 0.6 the Oldroyd-B viscosity reaches 58 and the finite chain 49 by a strain of 6; at 1 the Oldroyd-B viscosity reaches 725 and the finite chain 329 by a strain of 6. Below one half the viscosity settles at a few times the shear value. Above it the Oldroyd-B chain stretches without limit and its resistance grows exponentially, while the finite chain grows until it is nearly fully extended and then stops, hundreds of times higher than it began.

The stretch a chain cannot outrun

A Newtonian liquid pulled into a thread resists exactly three times as hard as it resists being sheared, whatever it is made of. A polymer solution resists three times as hard only until the stretch rate passes one over twice its relaxation time. Past that, its chains can no longer recoil as fast as they are pulled apart, and the same liquid that is barely thicker than water in a stirred beaker becomes hundreds of times stiffer in a thread.

fluids · Rheology Chains counted, and then blobs. The osmotic pressure of a polymer solution in a good solvent against concentration, on logarithmic axes, for chains of 1,000, 10,000, 100,000 segments. Dilute, each chain is one particle and the pressure is van 't Hoff's, proportional to the number of chains: a line of slope one, lower for longer chains by the ratio of their lengths. Past the overlap concentration of each, marked, the coils interpenetrate and the curves bend up onto a single line of slope 2.31 that does not depend on chain length at all. The overlap concentration falls as the chains lengthen, by a factor of 5.8 for each tenfold increase in length, which is why the longest chains leave the counting of molecules first.

The pressure that stops counting molecules

Van 't Hoff's law counts dissolved particles, and for a dilute polymer each chain is one particle, however many thousand segments it has. That is how osmometers weigh polymers. Concentrate the solution until the coils overlap and the count changes. The pressure stops depending on how long the chains are and grows as the 2.3 power of the concentration, the same for chains a thousand segments long and a hundred thousand. The unit being counted is no longer a molecule. It is a stretch of chain as long as the mesh of its neighbours allows, and it has no chemical identity at all.

fluids · Osmosis

Named alongside it

The objects these essays reach for when they reach for this one.

Relaxation timeViscoelasticityWeissenberg numberCoil stretch transitionConstitutive lawCorrelation lengthCreeping flowDumbbell modelExtensional viscosityFree surfaceHoop stressHysteresis

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