Illustrated Physics
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Fluids

The suspension that seizes when pushed hard

Cornflour stirred into half its volume of water runs off a spoon like cream, and a fist driven into it bounces off as if it had hit a floor. The particles have not changed and neither has the water. What changes, above a threshold stress, is the kind of contact between neighbouring grains: lubricated, they slide past one another and the suspension can pack to 64 per cent before it locks; pressed into frictional contact, they lock at 58. A suspension sitting between those two numbers is a liquid when handled gently and a solid when handled hard.
15 min read 5 figures Approximations that lieOrder out of the random

Assumes: The fluid that answers back · The heap that becomes a solid

The fluid that answers back sorted liquids by the shape of their flow curves — the stress needed to shear them at a given rate — and found that the straight line of a Newtonian liquid is the exception. Paint and blood thin under shear; a paste holds a yield stress; and a suspension of cornflour in water thickens, stiffening as it is stirred faster. That essay described the curves and gave the thickening a line. This one asks what happens in the cornflour, because its thickening is not a gentle curvature of the flow curve. It is violent enough to walk on, and a model that explains it has to explain a stress that jumps by orders of magnitude at a single rate, and a liquid that stops flowing altogether when pushed hard enough.

The model is due to Matthieu Wyart and Michael Cates, from 2014, and its ingredients are few: a short-range repulsion that keeps neighbouring grains apart, a threshold stress above which it is overcome, and the fact, known from the heap that becomes a solid, that frictional grains lock at a looser packing than frictionless ones. Every figure here is those three statements.

Two packing limits and a switch

Dense suspensions of hard particles have a viscosity that diverges as their volume fraction approaches a jamming point — the fraction at which the particles can no longer move past one another at all. Near it, a small increase in concentration produces a large increase in viscosity, roughly as the inverse square of the distance to jamming. For smooth spheres with nothing between them but liquid the jamming point is near random close packing, about 64 per cent by volume.

That number assumes the particles slide freely past each other where they meet. Real particles in a liquid are held a little apart by something — a layer of adsorbed polymer, a surface charge whose repulsion extends a few nanometres, a film of liquid that must be squeezed out before the surfaces touch. As long as that barrier holds, contacts are lubricated and frictionless and the jamming point is 0.64. Push the particles together hard enough and the barrier fails: the surfaces touch, and where rough surfaces touch there is friction. Frictional spheres cannot pack as densely before they lock, because a contact can now hold a tangential force and a network of such contacts can bear load with fewer neighbours per particle. Their jamming point is lower, 0.58 or so, depending on the coefficient of friction.

The switch between the two is a stress. The stress at which the typical contact is pushed through its barrier is an onset stress, σ∗\sigma^*, set by the force the barrier can resist divided by the square of the particle size. The model takes the fraction of contacts that are frictional to rise smoothly with stress, as e−σ∗/σe^{-\sigma^*/\sigma}, and lets the jamming point slide between the two limits in proportion to that fraction.

Why friction lowers the limit: a count of constraints

The two jamming numbers are not arbitrary, and the reason one is lower than the other is a piece of counting worth doing. A packing of particles is rigid when the forces at its contacts can balance every particle without any freedom left over. Each particle in three dimensions has three ways to move if it cannot turn, and each contact between two frictionless spheres supplies one number, the normal force. Counting forces against equations of balance, a frictionless packing needs on average six contacts per particle to be rigid — each contact shared between two particles supplies half a force to each. Random close packing, at 0.64, is where a frictionless packing first has that many.

Friction changes the count. A frictional contact can carry a tangential force as well as a normal one, three numbers in all, and the particles can now also rotate, adding three more equations of balance each. The count comes out at four contacts per particle. A frictional packing becomes rigid with fewer neighbours, and so at a looser, lower volume fraction — around 0.58 for grains of ordinary roughness. It is the same counting that decides whether a table on four legs has determinate leg forces: a structure with exactly as many unknown forces as equations is just rigid, one with fewer can move, and one with more has load paths nobody can compute from statics alone.

So the switch from lubricated to frictional contacts is a switch in how many contacts a particle needs before the whole assembly locks. Nothing in the particles’ size, shape or number changes; the threshold for rigidity moves, and the suspension, sitting where it always sat, finds itself on the other side of it.

Viscosity between two plateaus

Viscosity climbs from one plateau to another as friction switches on. The viscosity of a dense suspension, relative to the liquid's, against the shear stress in units of the onset stress σ* at which particles start to be pressed into frictional contact, for particle volume fractions 0.4, 0.5, 0.54, 0.56, 0.57. At low stress every contact is lubricated and the suspension jams only at 0.64; at high stress every contact is frictional and it jams at 0.58. Between the two plateaus the viscosity climbs: from 7.1 to 10 at φ = 0.4, from 20.9 to 52 at φ = 0.5, from 41.0 to 210 at φ = 0.54, from 64.0 to 836 at φ = 0.56, from 83.6 to 3325 at φ = 0.57. Near the frictional jamming point the climb is by orders of magnitude over a factor of a few in stress, which is what a spoon pushed into cornflour meets.
Fig. 1 Viscosity relative to the liquid’s against shear stress over the onset stress σ*, for volume fractions 0.40 to 0.57. Each curve climbs from a plateau set by the frictionless jamming point, 0.64, to one set by the frictional one, 0.58: from 7.1 to 10 at 0.40, from 64 to 836 at 0.56, from 84 to 3,325 at 0.57.

The drawing shows what that does to the viscosity. At low stress every contact is lubricated and the suspension’s viscosity is set by its distance from 0.64. At high stress every contact is frictional and the viscosity is set by its distance from 0.58. Between them the viscosity climbs from one plateau to the other, and the climb happens over a range of stresses around σ∗\sigma^*.

How large the climb is depends entirely on how close the suspension is to 0.58. A dilute suspension, at 40 per cent, is far from both jamming points and its viscosity changes from about seven times the liquid’s to ten — a thickening nobody would notice. At 56 per cent it rises thirteen-fold, from 64 to 836. At 57 per cent, forty-fold. Each extra percentage point of particles near the frictional limit multiplies the effect, because the upper plateau is diverging while the lower one barely moves. The physics is the same at every concentration; what changes is how near the suspension sits to the number that friction makes critical.

This is why the thickening of cornflour is so sensitive to the recipe. Cornflour grains are irregular, around ten micrometres across, and a mixture of roughly two parts starch to one of water by weight puts them close to their frictional jamming fraction. A little more water and the mixture merely thickens; a little less and it will not flow at all.

A flow curve that bends back

Flow curves that bend back, and one that stops. Shear stress against shear rate, both on logarithmic axes, for volume fractions 0.5, 0.55, 0.565, 0.575, 0.6, from the same model: stress in units of σ, rate in units of σ over the liquid's viscosity. Below φ = 0.569 every curve rises steadily — the suspension thickens but continuously. Above it and below the frictional jamming point 0.58 the curve bends back into an S: over a range of stress, more stress gives less flow, and a suspension driven at a fixed rate must jump between branches. At φ = 0.6 the curve returns to zero rate at 2.47 σ*: pushed that hard, the suspension stops flowing altogether and holds the stress like a solid.
Fig. 2 Shear stress against shear rate on logarithmic axes for volume fractions 0.50 to 0.60. Below φ = 0.569 each curve rises monotonically. Between 0.569 and 0.58 the curve bends back into an S, with a range of stress in which more stress gives less flow. At 0.60 it returns to zero rate at 2.47 σ*: pushed that hard, the suspension stops flowing.

The same model drawn the other way round — stress against rate, which is what a rheometer measures — shows something the viscosity curves only hint at. The shear rate is the stress divided by the viscosity. At moderate concentrations the viscosity rises more slowly than the stress, so the rate rises with the stress and the flow curve is a steepening line: continuous shear thickening. At a volume fraction near 0.569 the viscosity starts to climb faster than the stress in the middle of its range, so over that range a higher stress produces a lower rate. The flow curve folds back on itself into an S.

An S-shaped flow curve cannot be followed in steady flow, for the same reason that a friction law which weakens with sliding speed produces the chatter a stiffer holder removes: any branch on which a force falls as the speed rises feeds on its own fluctuations. On its backward-bending branch the suspension is mechanically unstable: a small fluctuation towards higher stress slows the flow, which in a rate-controlled experiment raises the stress further, and the suspension runs away to the upper branch. This is discontinuous shear thickening, and the discontinuity is the fold.

Above 0.58 the curve does something more drastic. The frictional jamming point is now below the suspension’s own concentration, so once enough contacts become frictional the suspension is jammed: its viscosity is infinite and its flow rate is zero. The curve for 0.60 rises from the origin, bends back, and returns to zero rate at a finite stress — about two and a half times the onset stress. Pushed harder than that, it does not flow at all. It is a solid for as long as the stress is maintained, and a liquid again the moment it is released.

The jamming point is a function of stress

The jamming point moves when friction switches on. The volume fraction at which the suspension jams, against the shear stress: 0.64 while contacts are lubricated, 0.58 once they are all frictional, with the fraction of frictional contacts rising as exp(−σ/σ). At σ it has fallen to 0.6179. A suspension at a fixed volume fraction between the two jamming points is a liquid at low stress and a jammed solid at high stress, and the stress at which the falling line crosses it is where it seizes: never for φ = 0.55, never for φ = 0.57, 2.47 σ for φ = 0.6, 0.91 σ for φ = 0.62. Nothing about the particles or their number changes; only the kind of contact between them.
Fig. 3 The jamming volume fraction against shear stress: 0.64 with lubricated contacts, 0.58 with frictional ones, 0.618 at the onset stress. A suspension at φ = 0.60 is crossed at 2.47 σ* and one at 0.62 at 0.91 σ*; suspensions at 0.55 and 0.57 are never crossed, because they sit below the frictional limit.

The central idea is visible in a single curve. The volume fraction at which the suspension jams is not a property of the particles alone; it is a property of the particles and the stress they are under, sliding from 0.64 down to 0.58 as the stress passes through σ∗\sigma^*. A suspension at a fixed concentration is a horizontal line on this drawing. If the line lies below 0.58 it is never crossed, and the suspension flows at any stress, though with a viscosity that may become enormous. If it lies between 0.58 and 0.64, the falling curve crosses it at some stress, and beyond that stress the suspension is jammed.

So a jar of cornflour at the right concentration holds two materials. Stirred slowly, its particles slide past one another on their lubricating films and it pours. Hit hard, the films fail — a coefficient of friction being, as the grip that is not a coefficient found, a property of what is actually touching — the particles grip one another, the jamming point drops below the jar’s concentration, and the contents lock into a solid network spanning the jar. Nothing has been added or removed and the particles have not moved closer together. Only the rule for what a contact can bear has changed. The friction that decides this is the same friction that takes whatever force it needs up to a limit between a block and a slope, and it matters here for the same reason: a frictional contact can resist a sideways push, and a network of contacts that can each resist sideways pushes needs fewer of them to stand up.

Three states on one map

Three states of one suspension. Volume fraction across, shear stress up (units of σ). Below φ = 0.569 the suspension flows at every stress, thickening continuously. Between 0.569 and 0.58 there is a band of stress, shaded, in which the flow curve bends backwards and steady flow is unstable — discontinuous thickening. Above 0.58 there is a stress, the lower line, beyond which it cannot flow at all: shear jamming, 5.36 σ at φ = 0.59, 2.42 σ at φ = 0.60, 0.93 σ at φ = 0.62, falling towards zero as the frictionless jamming point is approached. At 0.64 it is jammed at any stress. The whole map comes from one onset stress and two jamming points.
Fig. 4 Volume fraction across, shear stress up. Below 0.569 the suspension flows at every stress. Between 0.569 and 0.58 a band of stress, shaded, is where the flow curve bends back. Above 0.58 a line of stress bounds a region where it cannot flow at all — 5.4 σ* at 0.59, 2.4 at 0.60, 0.93 at 0.62 — falling towards zero at 0.64.

The model’s predictions can be gathered on a single map of concentration against stress. At low concentrations the suspension flows everywhere, thickening continuously. In a narrow window just below the frictional jamming point the flow curve bends back over a band of stresses, and steady flow within that band is impossible. Above the frictional jamming point there is a stress beyond which the suspension cannot flow, a shear-jammed state, and that stress falls towards zero as the concentration approaches the frictionless limit, where the suspension is jammed even at rest.

The window of discontinuous thickening is narrow in the model with these numbers — about one percentage point of concentration — and narrow in experiments too, which is why the phenomenon was hard to pin down for decades and why its appearance is so sensitive to preparation. The shear-jammed region is wide, and it is where the dramatic demonstrations live: a suspension that is liquid at rest and solid under a hammer blow is one sitting between 0.58 and 0.64, and the blow has driven it above the line. Unlike a paste that holds up its own hill, which has its yield stress at rest and loses it when sheared hard, the shear-jammed suspension has no yield stress at rest and acquires one only while it is being pushed.

What a rheometer sees

Driven at a rate, the stress jumps. The flow curve of a suspension at φ = 0.574, and what a rheometer that sets the shear rate and measures the stress sees as the rate is raised and then lowered. Rising, it follows the lower branch until the curve turns back at a rate of 5.08 × 10⁻³, then jumps from 1.15 to 24.4 σ — 21 times — at the same rate. Falling, it stays on the upper branch down to 3.95 × 10⁻³, a rate 1.3 times lower, before dropping back from 7.8 to 0.447 σ. The dashed middle branch, where more stress gives less flow, is never seen in steady flow; a controlled-rate measurement jumps across it, and in a real sample the flow splits into bands or fluctuates instead.
Fig. 5 The flow curve at φ = 0.574 as a rate-controlled rheometer sees it. Raising the rate, the stress follows the lower branch to the turning point and jumps from 1.15 to 24.4 σ*, twenty-one-fold, at a single rate. Lowering it, the stress stays on the upper branch down to a rate 1.3 times lower before dropping back from 7.8 to 0.45 σ*. The dashed backward branch is never seen in steady flow.

An instrument that imposes a shear rate and measures the stress needed cannot sit on the backward-bending branch. As the rate is raised the stress follows the lower branch until the curve turns back, and then, at that same rate, it jumps to the upper branch — in the drawing, from just over the onset stress to twenty-four times it. Lowered again, the stress stays on the upper branch until that branch turns back, at a lower rate, and then drops. The two jumps happen at different rates, so the measurement shows hysteresis: a loop, traced one way going up and another coming down.

An instrument that imposes the stress and measures the rate can, in principle, sit anywhere on the S, since each stress has one rate. In practice the backward branch is unstable under either kind of control once the sample is large enough to split, and what is measured there is flow that fluctuates violently or divides into bands — a layer near one wall flowing on the lower branch and a layer near the other jammed on the upper, the average rate lying on the missing part of the curve. The steady curve drawn is a model’s; the time-averaged curve an experiment reports has a flat or nearly vertical section where the S has been cut through.

Why a person can run across it

The demonstration that made shear thickening famous — people running across a pool of cornflour and water, and sinking when they stop — is more than a fold in a flow curve, and the extra ingredient is worth stating because it shows what the steady model leaves out.

A foot striking the surface drives the suspension beneath it past the onset stress. The jammed region does not stay under the foot. It spreads downward and outward as a front, much faster than the foot is moving, converting liquid into solid as it goes, because each newly jammed layer transmits the stress on to the next. Scott Waitukaitis and Heinrich Jaeger measured this in 2012 with a rod driven into a deep suspension: the solid plug grew ahead of the rod about ten times faster than the rod moved, and when it reached the bottom of the container the rod met a sudden, large resistance. A running foot is supported by a column of transiently jammed suspension reaching down to the pool’s floor. A standing foot, pressing gently, never jams it, and sinks.

The same transient jamming is used on purpose. Fabrics impregnated with a shear-thickening suspension of silica nanoparticles in a polymer liquid stay flexible when bent slowly and stiffen under the impact of a fragment or a blade, spreading the load across more of the fabric. The design goal is to put the onset stress between the stresses of wearing the fabric and the stresses of being struck by something, which means choosing the particle size, since the onset stress goes roughly as one over the square of it.

Where the model’s two numbers come from, and where they stop

The two jamming points in the drawings, 0.64 and 0.58, are those of monodisperse spheres; the frictional one depends on the coefficient of friction, falling towards 0.55 for very rough particles, and both shift for particles of mixed sizes or irregular shapes. Cornflour grains are neither spherical nor uniform, and its thickening happens at a weight fraction that corresponds to a volume fraction of less than a half, because the effective packing limits of angular grains are lower. The model’s structure survives; its numbers must be measured for each material.

The exponential switch-on of friction is a convenient form rather than a derivation. What is really needed is the distribution of forces at contacts in a sheared suspension, which is broad, so that as the stress rises a growing share of contacts exceeds the barrier’s strength. Simulations that model the barrier explicitly and compute the contact forces reproduce the S-shaped curves, which is the strongest evidence that the mechanism is right. The inverse-square divergence of viscosity is also an approximation, adequate near jamming and not exact.

And the model leaves out inertia, which matters for large particles or fast flows and adds a thickening of its own, and the confining walls, which matter because a jammed suspension pushes outward on its container. In the experiments that first established discontinuous thickening, the maximum stress the suspension could carry was set by the surface tension at its free boundary, which held the particles in — a reminder that “the suspension jams” means “the suspension jams against whatever is containing it.”

What the curves do not show

A flow curve reports a stress and a rate averaged over a sample. The drawings do not show where in the sample the frictional contacts are. In a thickening suspension they form chains of strongly loaded contacts running along the compressive direction of the shear, a network that forms and breaks continually, much like the force chains in a heap of dry grains and like the chains of polarised particles in a liquid a field turns solid, where an applied field rather than friction decides which contacts carry load. It is that network, not any single contact, that bears the load, and its fragility is why the jammed state disappears the moment the stress is released.

They also do not show time. Real suspensions take a finite strain to jam and a finite time to relax, so a rapid oscillation can see a different response from a steady shear, and the stress a suspension can support depends on how the load was applied. The time dependence of a liquid that remembers its past deformation is a separate phenomenon, set by molecular relaxation; here the memory is in the contact network and is much shorter.

Still open: controlling friction to control the flow

If shear thickening is friction switching on, it should be possible to switch it off — and experiments since 2015 have done so, by superposing a small oscillation across the main flow that disrupts the contact network before it can jam, lowering the viscosity of a thickened suspension by orders of magnitude, or by changing particle surface chemistry to strengthen the barrier. Whether frictional contacts can be controlled well enough to design suspensions that thicken exactly when wanted — cement pastes that pump easily and set strong, battery electrode slurries that coat without clogging — is being explored, as is how much of the thickening in suspensions of irregular grains is friction and how much is the interlocking of rough shapes, which is a related mechanism with different numbers.

The habit worth carrying away is to ask what a limit depends on. A packing limit that looks like a property of the particles can be a property of their contacts, and a contact can change its nature under stress — so a suspension below its jamming point at rest can find itself above it when pushed, and a liquid can become a solid without anything in it getting closer together.

Part 6 of 6

This essay is one argument about Rheology. The others:

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 Part 5 — 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.

Where this goes next

Four answers to one push. Shear stress against shear rate for four fluids. The straight line through the origin is the Newtonian definition and is the only one of the four for which the word viscosity names a number. The Bingham fluid does not move at all until the stress passes 0.4, which is why toothpaste holds a shape on a brush and why wet concrete can be stood in a heap. The fluid that answers back For water, stress is proportional to how fast it is sheared, and the constant of proportionality is its viscosity. For paint, blood, ketchup and cornflour in water, it is not — and once the proportionality goes, so does the idea that viscosity is a number a substance has. Where a heap stops flowing. The number of motions that cost nothing, against the mean number of contacts each grain has, for a patch of 37 grains. Every point is the rank of an actual rigidity matrix — one row per contact, one column per coordinate — subtracted from the number of coordinates, so it is a measurement of the network rather than a formula about it. The dashed line is Maxwell's count, two coordinates a grain minus half a contact each, which is what the counting alone predicts. The measured curve reaches its floor of three free motions — the rigid-body ones — at a coordination of 4.11, and Maxwell's line reaches zero at exactly four, which is twice the dimension of the plane. That is the isostatic point and the difference between the two numbers is the boundary of a finite patch, whose outer grains have fewer neighbours than they would in an infinite one. Above the threshold the two curves separate: the measured floor stays at three while the count keeps falling, and the gap is the redundancy — contacts whose forces no equilibrium equation can determine. Below it the pack has genuine mechanisms and will rearrange under any load at all. The transition is in the count and not in the material, which is why a fluid and a solid here are made of exactly the same grains. The heap that becomes a solid Sand poured into a jar flows; the same sand shaken down and pressed does not. Nothing about the grains has changed — not their size, their hardness, their friction or their density by more than a per cent — and the thing that changed is a count of contacts, which crosses a threshold set by the dimension of space and by nothing else. The thickness a paste can hold on a slope. The greatest thickness a yield-stress fluid can rest at without flowing, against the angle of the surface it is resting on, on a logarithmic vertical axis, for four materials. A layer of thickness h puts a shear stress ρgh sin α on its own base; it stays put while that is below the yield stress and flows when it is not, so the critical thickness is τ_y divided by ρg sin α and it depends on nothing else — not on the viscosity, not on how long it is left, not on how it was put there. On a vertical wall the numbers are 1.4 mm of ketchup, 9.2 mm of mayonnaise, 15.7 mm of toothpaste, 15.1 mm of basaltic lava, which is why toothpaste stays on a brush and ketchup does not stay on a plate held up. Read the other way it is a measurement: a lava flow that came to rest 3 centimetres thick on a 30° slope had a yield stress of about 400 pascals, and that is how the rheology of a flow nobody was standing next to is recovered from its shape a thousand years later. The model stops where the layer is thin enough for surface tension to matter and where the material's yield stress depends on how long it has been left alone, which for most of these it does. The paste that holds up its own hill Toothpaste stands on a brush and ketchup does not stand on a plate, and the difference is a single number with the units of a pressure. Below it a material does not flow slowly — it does not flow. That threshold turns a rheological property into a length, and the length is why a lava flow's thickness says what the lava was made of.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Flow curveFrictionJammingNon-newtonianShear thickeningSuspensionViscosityVolume fraction