The paste that forgets it was stirred

The fluid that answers back found that for paint and ketchup viscosity is not a number but a function of the shear rate. The liquid that remembers found that for pitch whether a material flows or breaks depends on how long it is watched. The paste that holds up its own hill found materials that do not flow at all below a threshold stress, and the liquid that climbs the rod, the stretch a chain cannot outrun and the suspension that seizes when pushed hard found stresses and transitions that no Newtonian liquid has.

Each of those treated the material as having a state that depends on what is being done to it now: a shear rate, a stress, an observation time. Several of them noted in passing that pastes also depend on what was done to them before — the yield stress of a sample left standing is higher than that of one just stirred. That dependence on history is thixotropy, and taken seriously it changes more than a number. It makes a paste’s flow unable to settle on slow steady states, it turns a smooth increase of stress into an abrupt choice between stopping and running, and it makes a gently sheared paste divide itself in two.

A structure that builds and breaks #

Clay slurries, drilling muds, cement pastes, mayonnaise, many paints and printing inks share a microscopic picture. Their particles attract one another weakly and, left alone, link into a loose network — flocs of clay platelets touching edge to face, droplets bridged by other droplets. The network resists flow. Shearing breaks the links faster than they form, and the material thins. Stop shearing and the links reform, slowly, by the random motion of the particles; the material thickens again over seconds, hours or days.

The simplest model that captures this, used by Philippe Coussot and colleagues to describe bentonite suspensions in 2002, gives the material one extra variable, a degree of structure $\lambda$. Its viscosity is $\eta_0(1 + \lambda^n)$, so a fully broken material ($\lambda = 0$) flows like a thin liquid and a well-structured one is far thicker. The structure evolves by

$$\frac{d\lambda}{dt} = \frac{1}{\theta} - \alpha\dot\gamma\lambda,$$

building at a steady rate $1/\theta$ at rest and being destroyed at a rate proportional to the shear rate and to how much structure there is to destroy. Nothing in these rules is exotic, and the figures here use them with illustrative numbers of the size fitted to bentonite. Everything strange follows from one feature: the viscosity feeds back on the flow that sets it.

Stopping or running #

Hold such a paste, freshly stirred, under a constant stress, and watch its viscosity.

The figure shows four runs. At 2.0 and 2.4 pascals the paste accelerates: its flow breaks structure faster than rest builds it, the viscosity drops, the flow speeds up further, and it settles at a steady, fast flow with little structure left. At 1.8 and 1.5 pascals the opposite happens: the flow is too slow to break structure as fast as it forms, the viscosity rises, the flow slows further, and the process runs away in the other direction. The viscosity grows without limit — in this model, eventually as the 1.5 power of time — and the paste creeps to a stop.

The runs at 1.8 and 2.0 pascals differ by ten per cent in stress and end, after a thousand seconds, a factor of a hundred apart in viscosity and going opposite ways. This is a bifurcation: two outcomes, separated by a critical stress, with no intermediate one. Coussot’s group called it a viscosity bifurcation and demonstrated it by letting balls sink through bentonite or rolling fluids down inclines, which either stopped or accelerated as an avalanche with nothing in between. Close to the critical stress the paste hesitates, creeping at an almost constant rate for a long time before deciding which way to go, and the hesitation lengthens the closer the stress is to the threshold.

The flow curve with a hole in it #

The origin of the bifurcation is visible in the steady states themselves.

If the paste flows steadily at shear rate $\dot\gamma$, its structure settles where building and breaking balance, $\lambda = 1/\alpha\theta\dot\gamma$, and the stress needed is the viscosity at that structure times the rate. At high rates the structure is broken and the stress rises with the rate as for any liquid. At low rates the structure has time to build, and it builds more the slower the flow, so the stress needed rises again as the rate falls. The figure plots the result: a curve with a minimum, here 1.89 pascals at 0.63 per second.

The left-hand branch cannot be occupied. Suppose a layer on it flows a little faster than its neighbour. It breaks a little more structure, its viscosity falls, and at the same stress it speeds up further — the perturbation grows. On the right-hand branch the same perturbation would raise the stress needed and restore the layer. So the only steady flows are on the right, faster than the critical rate, and the minimum of the curve is the critical stress: below it no steady flow exists, which is what the creep experiments showed. The paste has an effective yield stress, but it did not come from the paste being solid. It came from the flow curve having a stretch no stable flow can use.

The suspension that seizes when pushed hard met a flow curve that bent back too, at high stress, where friction between grains switched on and the suspension jammed. The mechanism here is the opposite one. A shear-thickening suspension stiffens because it is pushed hard; a thixotropic paste stiffens because it is left alone, and the unstable stretch is at the slow end rather than the fast one.

Stiffer for every minute on the shelf #

Because the structure builds for as long as the paste rests, the paste’s resistance when flow restarts depends on how long it stood.

The figure plots it. Ten seconds after stirring the paste is barely thicker than when stirred; after a hundred seconds its viscosity has risen seventeen-fold, and after a thousand, five hundred-fold. In this model there is no resting time after which it stops stiffening, because the structure grows at a constant rate for ever. Real pastes eventually saturate, as flocs fill the space, but many go on stiffening measurably for days. Cement paste is the extreme case: its thixotropy is followed by chemical setting, and the practical question of how long fresh concrete can stand in a pump before it will no longer move is a question about this curve.

That is why the yield stress of a thixotropic material is not a number the material has. The paste that holds up its own hill found yield stresses measured by different methods disagreeing, and noted that waiting time was one reason. The ageing curve says how much: a laboratory that waits ten minutes after loading a sample and one that waits an hour are measuring different materials. The same shape of dependence appears far from rheology. Nothing keeps a magnetisation for ever found a coercive field that depends on how long the measurement lasts, and static friction between rocks grows with the logarithm of how long they have been in contact, which is part of why earthquakes recur. Any threshold set by a structure that heals or builds over time carries the time in it.

A history the steady state forgets #

The history shows most clearly when flow is restarted at a fixed rate.

The figure shears three samples at the same rate after different rests. Each begins at a stress set by how much structure it accumulated, from twice the steady value to five hundred times it, and each decays towards the same steady stress as the flow breaks the structure down. After about a minute and a half the three are indistinguishable. The steady state has forgotten the history; the transient to it is a record of it.

This is the measurement a rheologist makes to tell thixotropy from simple shear thinning, and the reason the distinction matters in practice. A shear-thinning paint is thin while brushed and thick at rest; a thixotropic one also takes time to become thick again, which is what lets brush marks level out before the paint sets up and stops it running down a wall. Too little recovery time and the paint sags; too much and the brush marks stay. The number that matters is the time constant of rebuilding, $\theta$, which a flow curve measured at steady state does not contain at all.

A gap that splits in two #

The last consequence is the strangest to watch. Put the paste between two plates and drive them apart at a relative speed that corresponds to a mean shear rate below the critical one.

No uniform flow at that rate is stable, so the paste does not flow uniformly. It divides into two bands, both at the critical stress: one sheared at the critical rate and one not sheared at all. The widths are fixed by the requirement that the flowing band carry the whole of the plates’ relative motion, so the flowing fraction is the imposed rate divided by the critical rate — 48 per cent here. Such bands have been seen directly, by magnetic resonance imaging of pastes and clays in rheometers, with the boundary between them sharp to a few particle diameters.

The construction is the same one the part of the curve no fluid follows found for a liquid and its vapour. A van der Waals isotherm has a stretch of the pressure–volume curve no uniform fluid can occupy, and a fluid held at an average volume inside that stretch divides into liquid and vapour, in proportions fixed by the average, at one pressure. Here the shear rate plays the part of the volume and the stress the part of the pressure: a flow held at an average rate inside the forbidden range divides into a flowing phase and a solid one, in proportions fixed by the average rate, at one stress. The analogy is exact enough that shear banding is often described as a non-equilibrium phase transition, with the band boundary as the interface.

Where the bifurcation is used, and where it kills #

Drilling for oil depends on thixotropy working as designed. The mud pumped down a drill pipe carries rock cuttings back up the annulus outside it, and whenever the pumps stop — to add a length of pipe, every half hour or so — the cuttings must not sink back down onto the bit. A thixotropic mud gels within seconds of stopping and holds the cuttings suspended, and breaks down again when pumping resumes. The engineering specification is a pair of gel strengths, measured after ten seconds and after ten minutes of rest: the ageing curve read at two points. Too little gel and the cuttings settle; too much and restarting the pumps demands a pressure that can fracture the rock around the well.

The same behaviour, in a natural material, is one of the most dangerous in geology. Quick clays, found in Norway, Sweden and eastern Canada, are marine clays deposited in salt water and later leached by fresh groundwater, which left their flocs in an open, fragile arrangement. Undisturbed, they stand in slopes; disturbed, they lose almost all their strength and flow like a liquid. In 1978 at Rissa in Norway a small excavation at the edge of a lake set off a slide that liquefied some 330,000 square metres of farmland within minutes, carrying away houses, and it was filmed. The failure spread backwards from the lake shore in a sequence of collapses, each one liquefying the clay behind it — a viscosity bifurcation on the scale of a landscape, where the push that the disturbed clay could no longer resist was its own weight on a slope.

What connects the two is the shape of the flow curve at low rates. A material that stiffens at rest and softens in motion has two stable conditions under a moderate load, stopped and running, and a disturbance can carry it from one to the other without any change in the load. A drilling engineer wants that switch and designs for it; a slope of quick clay has it by accident. The angle that does not know the size of the heap found dry sand switching between two angles of repose in the same way, standing steeper than it can flow and avalanching down to a shallower slope when it goes.

What rheometers do not show #

The consequences above are nearly invisible in a standard measurement. A rheometer ramping the rate up and down and reporting stress against rate will show a loop — higher stresses on the way up than on the way down, because the structure lags — and that hysteresis loop was for decades the definition of thixotropy. It depends on how fast the ramp is run, it mixes the time dependence with the rate dependence, and it hides the banding entirely, because the instrument reports an average shear rate across a gap that is not being sheared uniformly. A paste that has banded looks, to the instrument, like a liquid with a flat flow curve at the critical stress, and it was the imaging that showed half of it was not flowing at all.

Slip at the walls adds a further layer: pastes tend to leave a thin layer of liquid at a smooth wall and slide on it, which looks like a low viscosity. Rheometers for pastes therefore use roughened surfaces or vanes, and the history-dependence then makes the order of measurements part of the result. None of this is exotic laboratory lore. It is the reason the same drilling mud behaves differently in a borehole and in a quality-control laboratory an hour later.

Where the model stops #

The model used here has one structural variable, one rebuilding time and one breaking rate, and real pastes have many. A clay suspension’s flocs break into pieces of many sizes and reform on many timescales, so the rebuilding is not a single exponential but spread over decades of time; the model’s steady linear growth at rest is the simplest caricature of that. The model also has no elasticity, so it cannot show the stress rising before it falls when flow starts, or the paste springing back when stress is removed, both of which real pastes do. Models with an elastic network added, or with a distribution of relaxation times, describe measurements better and share the features argued here: a flow curve with a minimum, a critical stress set by it, banding below it, and a yield stress that grows with resting time.

There is also a question of whether the bifurcation is truly sharp. In the model it is; in experiments the transition is sharp to within a few per cent in stress, and whether any finite window of slow steady flow survives in real pastes, for example because of the particles’ own thermal motion, is a question the imaging experiments have addressed only for a handful of materials.

Still open: whether ageing and yielding are one phenomenon #

Some materials that yield do not age visibly, and some that age have no yield stress in the usual sense; attempts to classify pastes into “simple” yield-stress fluids and thixotropic ones have run into materials that change category depending on how long they are tested. Whether every yield stress eventually proves to be a thixotropic one, set by a structure that forms at rest on some timescale however short, is argued over. So is the microscopic origin of the rebuilding: whether it is flocs colliding and sticking by diffusion, a glass-like slowing of particle rearrangements, or chemistry at the contacts between particles, and whether one description can serve clays, emulsions and colloidal gels at once.

The habit worth carrying away is to ask whether a material’s state depends on what is happening to it or on what has happened. A paste whose structure builds at rest and breaks in flow has a steady viscosity at each rate but no path to it that forgets how long it stood, and its flow curve has a stretch that no steady flow can occupy — so it stops or runs under a steady push, with nothing between, and splits into a flowing band and a still one when made to go slowly. A yield stress measured on such a material is a record of how long the sample waited on the bench.