The paste that holds up its own hill

Squeeze a ribbon of toothpaste onto a vertical brush and it stays. Tip a bottle of ketchup and nothing happens for a while, and then rather too much happens. Both behaviours belong to one number.

The number, and what it is not #

A Newtonian liquid answers a stress with a rate: apply a shear stress and it shears, at a rate proportional to the stress, however small the stress is. There is no threshold, and the constant of proportionality is the viscosity.

A yield-stress fluid does not. Below $\tau_y$ it behaves as a solid — it deforms elastically by a small amount and then stops — and above $\tau_y$ it flows. The simplest description is Bingham’s,

$$\tau = \tau_y + \mu_p \dot\gamma$$

for $\tau > \tau_y$ and $\dot\gamma = 0$ otherwise, with $\mu_p$ called the plastic viscosity to distinguish it from the thing a Newtonian liquid has.

The distinction from “a very large viscosity” is not a fine one, and the way to see it is to ask what happens when the driving is halved. A thick Newtonian liquid flows half as fast. A yield-stress fluid below its threshold was not flowing and continues not to; halved, doubled or multiplied by a thousand, the answer is the same until the threshold is crossed and then it is not.

The consequence for the apparent viscosity — the ratio of stress to rate — is that it diverges as the rate goes to zero rather than approaching a constant. Anybody measuring a paste with an instrument that reports a viscosity gets a number that depends entirely on the rate the instrument happened to use, which is the standard way this property is misreported.

The stress that is a length #

The most useful thing about a yield stress is that it converts into a thickness, and the conversion has nothing in it but the material.

A layer of thickness $h$ resting on a slope at angle $\alpha$ puts a shear stress $\rho g h \sin\alpha$ on its own base. It stays while that is below $\tau_y$ and flows when it is above, so the greatest thickness that can rest there is

$$h_c = \frac{\tau_y}{\rho g \sin\alpha}$$

with no viscosity in it and no time. A thicker layer flows until it is this thick and then stops.

That is why the hero figure exists and why a lava flow is worth looking at. A basaltic flow that came to rest thirty centimetres thick on a gentle slope had a yield stress of a few hundred pascals, and the rheology of an event nobody was standing next to is recoverable from its shape a thousand years afterwards. Field volcanologists do exactly this, and the numbers agree with laboratory measurements on remelted samples to within a factor of two, which for this subject is good.

The same arithmetic runs through drilling mud — which must hold rock cuttings in suspension when the pumps stop, so its yield stress has to exceed what a cutting’s weight imposes — through paint, whose sag on a vertical wall is set by exactly the balance above, and through the design of anything that must be thick enough to stay and thin enough to spread.

The plug down the middle #

Put a yield-stress fluid in a pipe and the picture is quite unlike a Newtonian one.

The shear stress at radius $r$ is $\Delta p\, r/2L$, which is zero on the axis whatever the driving. So there is always a core in which the stress is below the yield value, and that core is not sheared at all: it moves as a solid, at one speed, with a perfectly flat velocity profile.

The plug’s radius is the pipe’s radius times the ratio of the yield stress to the wall stress, so it is 67 per cent of the pipe at one and a half times the threshold, 40 per cent at two and a half, and 20 per cent at five. It shrinks in exact proportion to the overdrive and never reaches zero.

The flow rate is Buckingham and Reiner’s,

$$Q = \frac{\pi R^4 \Delta p}{8\mu_p L}\left[1 - \tfrac43 x + \tfrac13 x^4\right], \qquad x = \frac{\tau_y}{\tau_w}$$

which reduces to the fourth-power law when the yield stress is negligible and to nothing at all when the wall stress falls to the yield stress. The figure integrates the flow rate off its own drawn profiles and agrees with that expression to better than a fifth of a per cent, which is the check that the plug has been put where the stress balance puts it.

Consequences that follow from the plug #

Mixing in the plug is nil. Material in the core is never sheared, so nothing in it is stirred, and a reactor or a heat exchanger carrying a yield-stress fluid has a core that exchanges nothing with the wall except by diffusion. That is a serious problem in food processing, where the core of a plug can be under-pasteurised while the wall is overcooked.

The residence time distribution is bimodal. A Newtonian pipe flow has a smooth spread of transit times; a plug flow has a lump of material all arriving together and a sheared annulus spread out behind it.

A bubble in a paste does not rise. Buoyancy on a bubble of radius $a$ produces a stress of order $\rho g a$ on the surrounding material, and if that is below the yield stress the bubble stays where it is for ever. The condition gives a critical bubble size, which is why aerated food products hold their bubbles, why gas pockets in drilling mud are a hazard, and why the ordinary buoyancy argument needs the surrounding fluid to be a fluid.

And the pressure needed to start a line is not the pressure needed to run it. Restarting a pipeline full of gelled crude after a shutdown is a yield-stress problem, and the pressure required can exceed the pipe’s rating — which is why cold-restart calculations are a standard part of pipeline design and why lines are sometimes kept moving at a loss rather than stopped.

A Newtonian liquid in the same pipe takes a parabolic profile, sheared everywhere — including on the axis, where the shear rate is zero and the material is still a fluid. That is the contrast the plug depends on: a yield-stress material has a region where the stress is below the yield value and nothing shears at all, and it moves as a solid plug carried by the sheared annulus around it.

What the number is made of #

A yield stress is not a fundamental property in the way a viscosity is. It is what a structure costs to break, and every material that has one has a structure.

In a clay suspension the particles are plate-like and stick to one another edge-to-face, forming a network that spans the sample; the yield stress is the stress needed to break enough of those bonds for the network to fail. In an emulsion — mayonnaise is the standard example — the droplets are packed above their random close-packing fraction and are deformed against one another, so flowing requires them to squeeze past, and the yield stress is set by the interfacial tension and the droplet size. In a foam it is the same argument with gas instead of oil.

That origin explains a family of behaviours the Bingham model does not contain. The structure rebuilds at rest, so a sample left alone develops a higher yield stress than one just sheared — thixotropy — and the “yield stress” measured depends on how long the sample stood. And it explains why the property is so often reported inconsistently: a stress ramp, a rate ramp and a creep test interrogate the structure on different timescales and get different answers, sometimes differing by a factor of several.

How it is measured, and why the answers differ #

Three standard measurements, three definitions, and the spread between them is not experimental error.

A stress ramp raises the applied stress slowly and watches for the rate to become non-zero. The answer depends on how slowly, because a material below its nominal yield stress still creeps, and “non-zero” is whatever the instrument can resolve during the time spent at each step. Ramp faster and the reported yield stress rises.

A rate ramp imposes a shear rate and reads the stress, then extrapolates the flow curve back to zero rate. This gives the intercept of a fitted line rather than an observed threshold, so the answer depends on which model was fitted and over what range — Bingham’s straight line and Herschel–Bulkley’s curve extrapolate to different intercepts through the same data.

A creep test applies a fixed stress and watches the strain for a long time. Below the threshold the strain approaches a constant; above it, it grows without limit. This is the closest to the definition and the slowest, and it is the one that most often finds that the threshold is not sharp.

The three disagree because they interrogate the structure on different timescales, and the material’s structure changes on those timescales. The honest report is not a number but a number with a method attached, which is the same discipline a relaxation time demands and for the same reason: a material with an internal clock has properties that are functions of the observation.

The measurement they were already making #

A construction site has tested concrete the same way since 1918. Fill a truncated metal cone with the fresh mix, lift the cone off, and measure how far the top of the heap settles. That is the slump test, it requires no instrument beyond a ruler, and it was devised as a consistency check with no rheology behind it at all.

It is a yield stress measurement. A heap of paste on a flat surface cannot stay heaped, and it cannot flatten completely either: it spreads until the stress its own weight imposes falls to $\tau_y$ everywhere, and then stops. Balancing those two gives a deposit whose thickness at a distance $\ell$ from its edge is

$$h(\ell) = \sqrt{\frac{2\tau_y \ell}{\rho g}},$$

which is the same statement as the slope condition earlier, written for a surface that supplies its own slope as it goes.

Conserving the volume then fixes how far it spreads. For a circular pancake the radius comes out proportional to $(\rho g V^2/\tau_y)^{1/5}$ — and the fifth root is the interesting part. Doubling the yield stress reduces the spread by only thirteen per cent, so the test is remarkably insensitive, which is exactly why it survives on a windy site with a wooden ruler and why it cannot distinguish two mixes that a rheometer separates easily.

The relation was worked out properly in the 2000s, three generations after the test became a standard, and what it established was that a number the industry had been using as an arbitrary index of workability had been a material property all along. The self-compacting mixes now measure the final diameter instead of the settlement, which is the same arithmetic read from the other end of the same pancake.

Printing with a paste #

The clearest place the two sides of a yield stress are designed against each other is a printer that extrudes material rather than melting it — a ceramic slurry, a hydrogel with cells in it, or concrete.

Two requirements pull opposite ways. In the nozzle the wall stress $\Delta p R/2L$ must exceed $\tau_y$ by enough to give a usable flow rate, so a low yield stress is wanted. The instant the bead leaves the nozzle it must hold its own shape, so the stress its own weight imposes, of order $\rho g h$ for a bead of height $h$, must fall below $\tau_y$ — and a high yield stress is wanted.

The window between them is what makes a material printable, and it is narrow. A one-centimetre bead of cement mortar imposes about two hundred pascals, so a mix below that slumps on deposition and one far above it needs a pressure the pump cannot supply.

The severe version of the problem is a printed wall, because the bottom bead has to carry everything printed above it. Thirty layers is three kilopascals, which is an order of magnitude beyond any yield stress that would still extrude. Nothing in this essay’s model permits that, and the resolution is that the yield stress of a fresh cement paste is not a constant: it climbs steadily in the minutes after it is disturbed. A printed structure is therefore racing its own material, and the print rate is bounded above by the geometry and below by how fast the paste stiffens. What sets that rate is a later rung on this ladder.

The neighbouring case with no liquid in it #

A heap of dry sand also has a threshold, holds a slope, and refuses to flow below a stress — and the mechanism is different in a way worth stating, because the two are constantly conflated.

A granular pile’s threshold is frictional: the stress it can bear is proportional to the pressure holding the grains together, so a deeper layer can sustain more shear. A yield-stress fluid’s threshold is a fixed stress independent of pressure. The consequences separate immediately: a sand pile has an angle of repose independent of its size, while a paste has a critical thickness that depends on the slope.

A granular column’s stress is screened by friction against the walls, so the floor never feels the full weight — and a yield-stress fluid in the same column would transmit it. That is the neighbouring case with no liquid in it: both materials support a shear stress indefinitely, and they do it by different mechanisms, so the two look alike from outside and behave differently the moment the geometry changes.

Which is why a silo’s floor does not feel what the silo holds and a tank of mayonnaise’s does.

The length that decides whether it matters #

Every situation above is a comparison between a stress the geometry imposes and a stress the material can bear, and the comparison collapses to a length.

Gravity imposes $\rho g h$ over a depth $h$; the material bears $\tau_y$; so the yield stress is worth a height $\tau_y/\rho g$, and anything much smaller than that height is held rigidly while anything much larger flows. For mayonnaise that height is nine millimetres, for toothpaste sixteen, for a stiff drilling mud a metre or two.

That number is the yield-stress fluid’s counterpart of the capillary length, and it plays the same role: it says at what size the property stops being a curiosity and starts deciding the shape of things. Below it, structures made of the material hold themselves up; above it, they slump to that thickness and stop. A blob of toothpaste is the size it is because that is the size at which the two stresses balance, and so is a lava flow, four orders of magnitude larger and governed by the same ratio.

It also says where a laboratory measurement is relevant. A rheometer applies stresses of a few pascals to a few hundred; a lava flow’s base carries several thousand; and a paste squeezed through a nozzle sees tens of thousands. Whether the same yield stress applies across that range is a question about the structure rather than about the equation, and the answer is often no.

What the picture cannot show #

The yield stress is drawn as a sharp threshold and it never is one. Measured flow curves show a rapid but continuous rise, and whether there is a true stress below which the rate is exactly zero, or merely a viscosity that grows by many orders of magnitude, is an open question for most materials and probably has different answers for different ones. The Bingham model is an idealisation whose value is that it produces the plug, and a material with a very steep but finite low-rate viscosity produces something almost indistinguishable from one.

Elasticity below yield is left out entirely. A real paste below its threshold is an elastic solid: it deforms, stores energy, and springs back. The model here says it does not move at all, which is right about the steady state and wrong about everything transient — and the transient is what a rheometer measures first.

Thixotropy is absent. All the curves treat the yield stress as a constant of the material, and for many pastes it is a function of the shear history over the preceding minutes. Where that matters, a single flow curve is not a description of the material but a description of one experiment.

And the pipe calculation assumes the plug is rigid and the flow is steady and fully developed. Near an entrance, a bend or a valve none of those hold, and the plug — which is a region defined by a stress falling below a value — reshapes itself in ways that need the full stress field rather than the one-dimensional balance used here.

The ladder from here #

Later rungs on this anchor: thixotropy and the structural kinetics that produce it, where the yield stress becomes a state variable with its own equation; the Herschel–Bulkley law, which adds shear-thinning above the threshold and describes most real pastes better than Bingham’s straight line; viscoelastic yielding, where the transition is followed in time rather than assumed instantaneous; the jamming transition, which unifies the yield stress of an emulsion with the rigidity of a granular pile and a glass under one phase diagram; and the flow of yield-stress fluids around obstacles, where a stationary region of unsheared material forms and the boundary between flowing and static is itself unknown.

The neighbouring ladders are the Newtonian response, which is the case with no threshold, and viscoelasticity, which is the case where the material’s answer depends on how long it is asked. The silo is the frictional version of the same solid-like behaviour and behaves quite differently.