## Abstract

We propose a new view on yield stress materials. Dense suspensions and many other materials have a yield stress—they flow only if a large enough shear stress is exerted on them. There has been an ongoing debate in the literature on whether true yield stress fluids exist, and even whether the concept is useful. This is mainly due to the experimental difficulties in determining the yield stress. We show that most if not all of these difficulties disappear when a clear distinction is made between two types of yield stress fluids: thixotropic and simple ones. For the former, adequate experimental protocols need to be employed that take into account the time evolution of these materials: ageing and shear rejuvenation. This solves the problem of experimental determination of the yield stress. Also, we show that true yield stress materials indeed exist, and in addition, we account for shear banding that is generically observed in yield stress fluids.

## 1. Introduction: the yield stress problem

Many of the materials we encounter on a daily basis are neither elastic solids nor Newtonian fluids, and attempts to describe these materials as either fluid or solid fail; try, for instance, to determine which material has the higher viscosity, whipped cream or thick syrup: when moving a spoon through the materials, we clearly conclude that syrup is the more viscous fluid, but if we leave the fluids at rest, the syrup will readily flatten and become horizontal under the force of gravity, while whipped cream will keep its shape and we are forced to conclude that whipped cream is more viscous than syrup. The problem is that while the syrup is a Newtonian fluid, whipped cream is not, and its flow properties cannot simply be reduced to a viscosity. To quantify the steady-state flow properties of non-Newtonian fluids, since the viscosity is not defined, one typically measures the flow curve, which is a plot of the shear stress versus the shear. If one does so, one observes that while syrup is Newtonian, whipped cream is not at all Newtonian: it hardly flows if the imposed stress is below some critical value, but it flows at very high shear rates at stresses above this value. A material with this property is called a yield stress fluid and the stress value that marks this transition is called the yield stress. The most usual rheological model for these materials is the Herschel–Bulkley model: , with *σ* being the stress, *σ*_{y} the yield stress and the shear rate (velocity gradient); *A* and *n* are adjustable model parameters.

Maybe the most ubiquitous problem encountered by scientists and engineers dealing with everyday materials such as food products, powders, cosmetics, crude oils, concrete, etc. is that the yield stress of a given material has turned out to be very difficult to determine (Barnes 1999; Mujumdar *et al*. 2002; Moller *et al*. 2006). In the concrete industry, the yield stress is very important since it determines whether air bubbles will rise to the surface or remain trapped in the wet cement and weaken the resulting hardened material. Consequently, a large number of tests have been developed to determine the yield stress of cement and similar materials (Asaga & Roy 1980; Tremblay *et al*. 2001; Roussel *et al*. 2005). However, the different tests often give very different results and even in controlled rheology experiments the same problem is well documented: depending on the measurement geometry and the detailed experimental protocol, very different values of the yield stress can be found (James *et al*. 1987; Nguyen & Boger 1992; Barnes 1997, 1999; Barnes & Nguyen 2001). Indeed it has been demonstrated that a variation of the yield stress of more than one order of magnitude can be obtained depending on the way it is measured (James *et al*. 1987). The huge variation in the value for the yield stress obtained cannot be attributed to different resolution powers of different measurement techniques, but hinges on more fundamental problems with the applicability of the picture of simple yield stress fluids to many real-world yield stress fluids. This is of course well known to rheologists, but since no reasonable and easy way of introducing a variable yield stress is generally accepted, researchers and engineers often choose to work with the yield stress nonetheless and treat it as if it is a material constant that is just tricky to determine, or as Nguyen & Boger (1992) have put it: ‘Despite the controversial concept of the yield stress as a true material property … there is generally acceptance of its practical usefulness in engineering design and operation of processes where handling and transport of industrial suspensions are involved’. Even worse, almost unrelated to the exact definition and method used, yield stresses obtained from experiments generally are not adequate for determining the conditions under which a yield stress fluid will flow and how exactly it will flow, since generally the yield stress measured in one situation is different from the yield stress measured in a different situation (Coussot *et al*. 2002*a*,*b*).

One method that has been frequently used for characterizing yield stress materials is to work with two yield stresses—one static and one dynamic—or even a whole range of yield stresses (Mujumdar *et al*. 2002 and references therein). The static yield stress is the stress above which the material turns from a solid state to a liquid one, while the dynamic yield stress is the stress where the material turns from a liquid state to a solid one. This suggests that for these materials the flow itself affects the viscosity and yield stress of the material, i.e. yield stress materials may be *thixotropic* (Mewis 1979; Barnes 1997). Thixotropy means that at a fixed stress or shear rate, the material shows a time-dependent change in viscosity; the longer the duration for which the material flows, the lower is its viscosity. Wikipedia even asserts that ‘many gels and colloids are thixotropic materials, exhibiting a stable form at rest but becoming fluid when agitated’. Thus, to correctly predict their behaviour, one should take the flow history of these materials into account.

These (and other) difficulties have resulted in lengthy discussions about whether the concept of the yield stress is useful for thixotropic fluids and how it should be defined and subsequently determined experimentally if the model is to be as close to reality as possible. The problem is that in spite of the fact that it is the microstructure that gives rise to both the yield stress and thixotropy, the two phenomena are hardly ever considered together (Moller *et al*. 2006; see also Pignon *et al*. (1996, 1998) for early studies on thixotropy). For instance, Barnes wrote two different reviews on the yield stress (Barnes 1999) and thixotropy (Barnes 1997), each without considering the other.

## 2. A possible solution to the problem: differentiate between thixotropic and normal yield stress fluids

Many of the problems mentioned above disappear if we try and make a simple distinction between two types of yield stress behaviour. Yield stress fluids are then classified into two distinct types: thixotropic and non-thixotropic (or simple) yield stress fluids. A simple yield stress fluid is one for which the shear stress (and hence the viscosity) depends only on the shear rate, while for thixotropic fluids the viscosity depends also on the shear history of the sample. The distinction is simple to make, in principle: one can, for instance, measure the flow curve by using an up-and-down stress ramp. This is shown in figure 1; if the material is a ‘normal’ yield stress fluid, the data for increasing and decreasing shear stresses coincide: the material parameters do not depend on flow history. On the other hand, if the material is thixotropic, in general the flow will have significantly ‘liquefied’ the material at high stresses, and the branch obtained upon decreasing the stress is significantly below the one obtained while increasing the stress.

Thus, for normal yield stress fluids, there is no problem in determining the yield stress. However, for thixotropic materials there is, since any flow liquefies the material and thus decreases the yield stress. As thixotropy is, both by definition and in practice, reversible, this also means that when left at rest after shearing, the yield stress will increase again.

### (a) Thixotropic yield stress fluids

A very striking demonstration of how the simple yield stress fluid picture often fails in predicting even qualitatively the flow of actual yield stress fluids is the ‘avalanche behaviour’ (Coussot *et al*. 2002*a*,*b*), which has recently been observed for thixotropic yield stress fluids and leads to the so-called ‘viscosity bifurcation’ (Coussot *et al*. 2002*b*; Da Cruz *et al*. 2002). One of the simplest tests to determine the yield stress of a given fluid is the so-called inclined plane test (Coussot & Boyer 1995). In figure 2, photos from an inclined plane experiment on a bentonite suspension are shown (Coussot *et al*. 2002*a*). A large amount of the material is deposited on a plane, which is subsequently slowly tilted to some angle, *θ*, when the fluid starts flowing. According to the Herschel–Bulkley and Bingham models, the material will start flowing when an angle is reached for which the tangential gravitational force per unit area at the bottom of the pile is larger than the yield stress: , with *ρ* being the density of the material, *g* the gravitational acceleration and *h* the height of the deposited material. In reality however, inclined plane tests on a bentonite clay suspension reveal that for a given pile height, there is a critical slope above which the sample starts flowing, and once it does, the thixotropy leads to a decrease in viscosity, which accelerates the flow since fixing the slope corresponds to fixing the stress (Coussot *et al*. 2002*a*). This in turn leads to an even more pronounced viscosity decrease and so on; an avalanche results, transporting the fluid over large distances, i.e. when the system is made to flow, the flow destroys the structure, and hence the viscosity can decrease tremendously. Here, a simple yield stress fluid model predicts that the fluid moves only infinitesimally when the critical angle is slightly exceeded, since the pile needs to only flatten slightly for the tangential gravitational stress at the bottom of the pile to drop below the yield stress. It is interesting to compare the results of the inclined plane tests with experiments showing avalanches in granular materials—a situation for which there is general agreement that avalanches exist. Exactly the same experiment had in fact been done earlier for a heap of dry sand, with results that are strikingly similar to those observed for the bentonite—notably identical horse-shoe-shaped piles are seen to be left behind the avalanche in both experiments.

Measuring the thixotropy of a 10 wt% bentonite suspension under a constant shear stress, viscosity is seen to decrease more than four orders of magnitude within 500 s. Since the shear stress is constant, this leads to a 10 000-fold increase in the shear rate within 500 s—avalanche behaviour! In the more quantitative experiment accompanying the inclined plane test (Coussot *et al*. 2002*a*,*b*), a sample of bentonite solution was brought to the same initial state by a controlled history of shear and rest. Starting from identical initial conditions, different levels of shear stress were imposed on the samples and the viscosity was measured as a function of time. The result is shown in figure 2 and deserves some discussion. For stresses smaller than a critical stress, *σ*_{c}, the resulting shear rate is so low that buildup of structure wins over the destruction of it, and the viscosity of the sample increases in time until the flow is halted altogether. On the other hand, for a stress only slightly above *σ*_{c}, destruction of the microstructure wins, and the viscosity decreases with time towards a low steady-state value. The important point here is that the transition between these two states is discontinuous as a function of the stress. This phenomenon is now called viscosity bifurcation.

Since the increase of viscosity with time is also seen in glasses where it is called ageing, the same term is used to describe the same phenomenon in yield stress fluids, while the opposite phenomenon—that of the viscosity decreasing under high shear rates—is consequently called *shear rejuvenation*. Thixotropy is thus the phenomena of reversible ageing and shear rejuvenation. It is generally perceived that what causes thixotropic behaviour is the individual particles in the material assembling into a flow-resisting microstructure when the fluid is at rest and that the microstructure is torn apart to give a lower viscosity under shear (Moller *et al*. 2006; see also Pignon *et al*. (1996, 1998) for early studies on thixotropy). A simple toy model based on this feature of a thixotropic fluid can be used (Coussot *et al*. 2002*a*,*b*) in order to qualitatively understand the avalanche and viscosity bifurcation data. There is nothing profound or microscopic about the model; it merely seeks to establish what the minimal ingredients are for the viscosity bifurcation. The basic assumptions of the model are the following.

There exists a structural parameter

*λ*that describes the local degree of interconnection of the microstructure.The viscosity increases with an increase of

*λ*.For an ageing system at low or zero shear rate,

*λ*increases, whereas the flow at sufficiently high shear rates breaks down the structure and*λ*decreases to a low steady-state value.

These assumptions are quantified into a toy model for the evolution of the microstructure and the viscosity as (Coussot *et al*. 2002*a*,*b*)andwhere *τ* is the characteristic ageing time for buildup of the microstructure, *α* determines the rate at which the microstructure is broken down under shear, *η*_{0} is the limiting viscosity at high shear rates, and *β* and *n* are parameters designating how strongly the microstructure influences the viscosity. Since the symbol *λ* is used to designate the structural parameter of the material, this model is called the *λ*-model (Coussot *et al*. 2002*a*,*b*). In steady state, d*λ*/d*t*=0 and the resulting steady-state flow curve is easily found from solving the above equations in steady state:andThere are now three physically different situations we can distinguish:

0<

*n*<1: the material is a simple shear thinning fluid, with no yield stress;*n*=1: the material behaves as a simple yield stress fluid,*η*_{0}*β*/(*ατ*)^{n}being the yield stress. In this case, the viscosity diverges continuously when the stress is lowered towards the yield stress; and*n*>1: the shear stress diverges both at zero and infinite shear rates, so there exists a finite shear rate at which the shear stress has a minimum. Once the stress is decreased below this minimum, the steady-state shear rate drops abruptly from the value corresponding to the minimum in the flow curve to zero, so the steady-state viscosity jumps discontinuously from some low value to infinity—the viscosity bifurcation.

So while the Herschel–Bulkley model and other simple yield stress models fail to describe qualitatively avalanche behaviour and viscosity bifurcation, the *λ*-model can at least qualitatively capture the essence of the behaviour when *n*>1. In addition, ‘simple’ yield stress behaviour is recovered in steady state when *n*=1. However, quantitatively it is clear from, for example, figure 3*b* that even so-called simple yield stress fluids can, for instance, have strongly nonlinear rheology above the yield stress, which the model with *n*=1 does not account for. But on the whole, the model can capture both simple and thixotropic yield stress materials, depending on the parameter *n*. This parameter describes how strongly the viscosity changes when changing the structural parameter *λ*. Thus, for a system such as bentonite for which the viscosity changes dramatically under flow, one would expect *n*>1, whereas for systems that have no structure that is destroyed by the flow (e.g. foams, emulsions, colloids) one may anticipate that *n* approaches unity. For a detailed analysis of a similar model, see Picard *et al*. (2002). The question is then how to make a clear distinction between the two types of yield stress materials (thixotropic and non-thixotropic) and how to understand this in terms of their microscopic structure.

Apart from numerous purely phenomenological descriptions of complex fluid rheology such as the Bingham model, the Herschel–Bulkley model, and (to some extent) the *λ*-model, there exist a very large number of models that take the micro- or mesoscopic physical properties of the material as a starting point for describing the fluid macroscopic properties. These range from doing molecular dynamics simulations on glassy systems (e.g. Varnik *et al*. 2003) and mode-coupling theory for the glass transition (Bengtzelius *et al*. 1984), to mesoscopic approaches such as the soft glassy rheology (SGR) model for soft glassy materials (Sollich *et al*. 1997; Sollich 1998). The latter is interesting to compare to the phenomenological *λ*-model. In fact, as remarked by P. Sollich (2009, personal communication), the flow curve of the SGR model can be calculated analytically and shows typical non-thixotropic yield stress fluid behaviour. This is not in contradiction to the fact that the systems described by the SGR model show ageing, since we are interested only in that steady state for which ageing and shear rejuvenation exactly balance each other. In a very recent paper, Fielding *et al*. (2009) show in fact that the SGR model can be modified and can also predict non-monotonic flow curves accompanied by a viscosity bifurcation and shear banding, describing thixotropic yield stress fluids. It is perhaps insightful to see that the two different types of yield stress fluids can even be described within the simplest of models: the *λ*-model. Namely, for *n*=1 the steady-state flow curve is that of a Bingham fluid: , which is a special case of the Herschel–Bulkley model. Still, the material can age and shear rejuvenate: if not in steady state, d*λ*/d*t* may be non-zero, so that the system indeed ages. The main dissimilarity with the case *n*>1 is that, for *n*=1, the effects of ageing and shear rejuvenation exactly cancel each other for any finite shear rate, meaning that all shear rates are possible.

### (b) Simple yield stress fluids

There are a few systems that do not show marked ageing (and hence shear rejuvenation) and that almost behave like perfect Herschel–Bulkley materials. The three pertinent examples we have been able to find are foams, emulsions and carbopol ‘gels’. Carbopol is in fact not a gel in the sense that there is a percolating network of polymers connected by chemical bonds. Rather, when it shows yield stress behaviour, it is a very concentrated suspension of very soft, sponge-like particles that are jammed together. In its microscopic structure, it hence resembles foams and emulsions. The absence of ageing is easy to detect experimentally: if one does an up-and-down stress sweep, the flow curves are identical upon going up and down, as is shown in figure 1 for carbopol.

For a number of years now there has been a controversy about whether or not the yield stress marks a transition between a solid and a liquid state, or a transition between two liquid states with very different viscosities. Numerous papers have been published that apparently demonstrate that these materials flow as very viscous Newtonian liquids at low stresses (Macosko 1994; Barnes 1999), as well as many replotted datasets shown in Barnes (1999). Possibly the earliest work that seriously questions the solidity of yield stress fluids below the yield stress is a 1985 paper by Barnes & Walters (1985), where they show data on carbopol samples apparently demonstrating the existence of a finite viscosity plateau at very low shear stresses—rather than an infinite viscosity below the yield stress. In 1999 Barnes published another paper on the subject, titled ‘The yield stress—a review or “*πανταρει*”—everything flows?’, where he presents numerous viscosity versus shear stress curves with viscosity plateaus at low stresses (Barnes 1999). Following the initial publication by Barnes, a number of papers appeared that discuss the definition of yield stress fluids, whether such things existed or not, and how to demonstrate them either way (Hartnett & Hu 1989; Schurz 1990; Evans 1992; Spaans & Williams 1995; Barnes 1999, 2007). The outcome of this debate has been that the rheology community at present holds two coexisting and conflicting views: (i) the yield stress marks a transition between a liquid state and a solid state and (ii) the yield stress marks a transition between two fluid states that are not fundamentally different—but with very different viscosities.

Here we reproduce the experiments used to demonstrate Newtonian limits at low stresses and also find the apparent viscosity plateaus at low stresses; for a more complete account, see Moller *et al.* (2009). However, we also show that such curves can be very misleading and that extreme caution must be taken before concluding that a true viscosity plateau exists. We examine some typical simple yield stress fluids and show that the apparent viscosity plateau can be an artefact arising from falsely concluding that a steady state has been reached. For measurement times as long as 10 000 s, we find that viscosities for stresses below the yield stress increase in time and show no signs of nearing a steady value. This extrapolates to a steady-state material that is solid, and does not flow below the yield stress. For the experimental examination of the nature of the yield stress transition for simple yield stress, we discuss the results for a 0.2 wt% aqueous carbopol sample (neutralized to a pH of 8 by NaOH), which are representative also of the measurements reported on other concentrations of carbopol, and foams and emulsions reported in Moller *et al*. (2009), where also all the experimental details are given.

We performed so-called creep tests: measurements where the shear stress is imposed and the resulting shear rate is recorded. The resulting viscosity curves are shown in figure 4, showing the apparent viscosity as a function of the imposed stress. Measurements are shown where the viscosity is determined at several different times after each stress has begun to get imposed. The results demonstrate that a low-stress viscosity plateau is found. However, while all measurements collapse at high stresses, they do not collapse below some critical stress. Below this stress the apparent viscosity values depend on the delay time between beginning the stress and measuring the viscosity: the inset shows that the viscosity value of the low-stress viscosity plateau increases with the delay time as *t*^{0.6}. It is clear that each of the several curves when seen individually greatly resembles the curves of Barnes and others, and that such curves can be misleading since one assumes that the data represent measurements in steady state, whereas in fact the flows may well be changing with time as is the case here. If we remove all the points from the flow curve that do not correspond to steady states, we are left with a simple Herschel–Bulkley material with a yield stress corresponding to the critical stress mentioned above (Moller *et al*. 2009)

Note that this cannot be an evaporation or ageing effect since all measurements were done on the same sample after it had been allowed to relax after the previous experiment. Since evidently no steady-state shear is observed, one should not, contrary to what is suggested by Barnes, take the instantaneous shear rate at any arbitrary point in time to be proof of a high-viscosity Newtonian limit at low stresses for these materials.

The behaviour of carbopol below the yield stress, at first glance, resembles the behaviour of ageing, glassy systems. However, carbopol is non-thixotropic, as shown in figure 1. In fact, the strange ‘ageing’ seems to happen only when the sample is under load—and not at rest where it seems to be ‘rejuvenating’—which is the exact opposite of thixotropic materials that show shear rejuvenation and ageing at rest. This phenomenon has been observed before and was dubbed ‘overageing’ in the literature (Viasnoff & Lequeux 2002); so far no detailed microscopic explanation for this phenomenon exists. In any case, if in the flow curve only the steady-state viscosities are plotted, carbopol, foams and emulsions behave as simple yield stress fluids, and their rheology can be well described by Herschel–Bulkley-type models. This is therefore a different class of yield stress fluids from the thixotropic materials described in the previous section. The difference between the carbopol and the bentonite is the following.

At rest, the (complex) viscosity of the bentonite increases, whereas that of the carbopol does not.

The response of the carbopol is in fact an elastic response, since the viscosities are so high (typically 10

^{6}Pa s) and the stress low (10 Pa) that even if we measure for 1000 s, the deformation is only 1 per cent. On the other hand, for the bentonite, typical viscosities are 1 Pa s for a stress of 10 Pa, and thus already for 1 s, the deformation is of the order of 1000 per cent. Thus, the viscosity increase of the bentonite must be due to a structural change that occurs within the material in time.

## 3. Which is which? An attempt to categorize yield stress fluids

After having made this distinction between these two classes, it is perhaps useful to ask what the origin of the difference is, and how to determine to which class it belongs for a given system. From the above examples, a number of perhaps interesting observations emerge.

First, one of the most obvious ones is that if one observes that a system ages spontaneously at rest, this implies that it is Brownian in the sense that temperature is important. This is the case for instance for the very thixotropic system of bentonite. As a rule of thumb, the crossover between Brownian and non-Brownian particle systems is often taken to be around 1–5 μm. Smaller particles remain in suspension by Brownian motion, whereas large systems are practically immobile; granular systems are a good example of the latter case. For the Brownian systems, most of the time, a measurement of the shear elastic modulus as a function of time clearly shows an increase in the modulus due to the ageing. On the contrary, for systems made up of large entities such as the drops or bubbles in emulsions and foams, no ageing is observed; for the foams some irreversible ageing may occur due to draining, but this is negligible on the time scale of our experiments.

A second important point is the existence of attractive interactions between the entities that make up the yield stress materials. If we take again the bentonite system as an example: bentonite is a particle gel, and consequently there has to be an attraction between the particles in order to form the percolated structure, which in turn is the origin of the thixotropy. Measuring the flow curve of concentrated hard-sphere colloids (Pusey & van Megen 1986) in an up-and-down stress sweep, no clear indication of thixotropy is found: it looks like a simple yield stress material (figure 5*a*). However, hard-sphere colloids, when mixed with similar-sized polymers, can also form ‘attractive glasses’: the polymers mediate an effective attraction between the colloids influencing also the rheology (Pham *et al*. 2008). If an up-and-down stress sweep of such a system is done, we immediately see clear evidence for thixotropy in the sense that the viscosity depends on the flow history (figure 5*b*). Thus, attraction between the entities leads to or enhances the thixotropy.

Third, gravity may play a rather subtle role in some systems. Granular avalanches and thus also the viscosity bifurcation observed in granular systems (Da Cruz *et al*. 2002) occur through a dilatancy of the system: gravity compacts the system, whereas the flow dilates it. This is in fact a rather peculiar type of thixotropy, but does manifest itself macroscopically in a similar way: the viscosity bifurcation in a dry granular system is not very different from that of bentonite. Thus, even a non-Brownian system may ‘age’ and ‘shear rejuvenate’ (and thus show thixotropy) in this way.

In conclusion, table 1 may be presented. This helps us to distinguish the different systems in terms of their physical properties.

Thixotropic yield stress fluids: particle and polymer gels (Moller

*et al*. 2008), attractive glasses (this paper), ‘soft’ colloidal glasses (Bonn*et al*. 2002*a*), adhesive emulsions (Ragouilliaux*et al*. 2007), dry granular systems (Da Cruz*et al*. 2002), pastes (Huang*et al*. 2005), hard-sphere colloidal glasses (this paper).Simple yield stress fluids: emulsions and foams (Bertola

*et al*. 2003; Moller*et al*. 2009), hair gel (Moller*et al*. 2009), carbopol (Moller*et al*. 2009).

Here, the very interesting case of hard-sphere colloids deserves further discussion. According to the classification above, it should be a thixotropic system, since it shows Brownian motion. In agreement with this idea, light scattering (Martinez *et al*. 2008) and confocal microscopy (Simeonova & Kegel 2004) have clearly shown that these systems indeed age in the sense that their diffusional relaxation times grow with waiting time. Since an increase in the relaxation time also implies an increase in viscosity (Bonn *et al*. 2002*b*), the system should be thixotropic. However, if it is, the effect is so small that it hardly shows up in the up-and-down stress sweep of figure 5*a*. We therefore do a ‘real’ thixotropy test: we let the system age for a few hours, and then impose a constant stress, to see whether the viscosity varies in time. Figure 6 convincingly shows that it does and thus that the system is indeed thixotropic, although less so than the thixotropic systems mentioned above.

## 4. Shear banding

There is an interesting connection between the distinction made between the two types of yield stress fluids, and the occurrence or not of shear banding. Hitherto, shear banding in yield stress fluids has always been viewed as being a direct consequence of the existence of a stress heterogeneity in the flowing material. If in some parts of the flow the stress is smaller than the yield stress, and in some other parts the stress is larger than *σ*_{y}, it follows immediately from the definition of the yield stress that the former part of the fluid will not move, whereas the latter will. However, this is only part of the story, as is evidenced by the observation of shear banding of a (thixotropic) yield stress material in a cone-plate geometry (Bonn *et al*. 2008; Moller *et al*. 2008). In summary, the observations are as follows.

During imposed shear rate measurements, if the imposed shear rate is lower than some critical shear rate, the material shear bands.

The material inside the sheared band is sheared exactly at the critical shear rate.

The amount of sheared material is given by the ratio of the macroscopically imposed shear rate to the critical shear rate. There is thus a

*lever rule*that can be used to calculate the fraction of material that flows.

All this disagrees with the idea that shear banding can only be due to a stress heterogeneity, and strongly suggests that thixotropic materials have a critical shear rate that is intrinsic to the material. This is in fact exactly what is predicted by the *λ*-model for *n*>1 (corresponding again to thixotropic systems). If within the model one calculates the steady-state flow curve, one finds that if one plots the stress as a function of shear rate, for small enough shear rates the stress is a *decreasing* function of the shear rate, corresponding to unstable flows. This therefore not only shows that a critical shear rate exists, but also shows that the flows with a global shear rate below that are unstable. That is exactly what was observed experimentally: the unstable flows are in fact shear banding flows.

The main distinction in the steady-state rheology between the two cases is therefore that, for a simple yield stress fluid, the viscosity diverges continuously when the yield stress is approached from above. This can easily be seen from, for instance, the Herschel–Bulkley model. On the contrary, for thixotropic yield stress materials, the viscosity jumps discontinuously to infinity at the critical stress, due to the viscosity bifurcation. Although for very thixotropic systems this distinction is easily made, the difference between a slightly thixotropic and a simple yield stress fluid is less clear. However, there is a simple experimental protocol that allows distinguishing between the two. Namely, it follows from the above considerations that imposing the stress on a thixotropic yield stress material will either lead to ageing (small stresses) or to avalanche behaviour (large stresses). This means that in steady state, the viscosity is either infinite (small stresses) or very small (for large stresses). This in turn implies that there is a range of shear rates between zero flow and the post-avalanche rapid flows that are not accessible under applied stress. However, a rheometer can also impose a shear rate, and the following question arises: What happens when a shear rate is imposed that is in between no flow and the rapid post-avalanche flows? The answer is of course shear banding, and together with the lever rule mentioned above, this implies that the flow curve (stress versus shear rate) exhibits a stress plateau that is only accessible under imposed shear rate. The experimental distinction between a thixotropic and a simple yield stress fluid is shown in figure 3; for the latter, there is no difference between imposing the stress and the shear rate, and this thus suffices to distinguish the two.

## 5. Conclusion

We have proposed a new view on yield stress materials. We show that most if not all of the difficulties people have with accounting for yield stress behaviour disappear when a clear distinction is made between two types of yield stress fluids: thixotropic and simple ones. For the former, adequate experimental protocols need to be employed that take into account the time evolution of these materials: ageing and shear rejuvenation. This solves the problem of the experimental determination of the yield stress. We also show that simple yield stress materials in fact do have a true yield stress, contrary to many reports in the literature. We also discuss shear banding that is generically observed in yield stress fluids and need not be a consequence of the existence of a stress heterogeneity.

## Footnotes

One contribution of 12 to a Discussion Meeting Issue ‘Colloids, grains and dense suspensions: under flow and under arrest’.

- © 2009 The Royal Society