# An attempt to categorize yield stress fluid behaviour

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Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 Phil. Trans. R. Soc. A (2009) 367, 5139–5155 doi:10.1098/rsta.2009.0194 An attempt to categorize yield stress ﬂuid behaviour BY PEDER MOLLER1 , ABDOULAYE FALL2 , VIJAYAKUMAR CHIKKADI2 , DIDI DERKS1 AND DANIEL BONN1,2, * 1 Laboratoire de Physique Statistique de l’ENS, 24 Rue Lhomond, 75231 Paris Cedex 05, France 2 van der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 65, 1018 Amsterdam, The Netherlands 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. Keywords: yield stress; ageing; shear banding; glassy dynamics; viscosity bifurcation; rheology 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 *Author for correspondence (bonn@science.uva.nl). One contribution of 12 to a Discussion Meeting Issue ‘Colloids, grains and dense suspensions: under ﬂow and under arrest’. 5139 This journal is © 2009 The Royal Society

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 5140 P. Moller et al. 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: σ = σy + Aγ̇ n , with σ being the stress, σy the yield stress and γ̇ 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. 2002a,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 Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 Categorizing yield stress ﬂuids 5141 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. 2002a,b), which has recently been observed for thixotropic yield stress ﬂuids and leads to the so-called ‘viscosity bifurcation’ (Coussot et al. 2002b; 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. 2002a). 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 Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 5142 P. Moller et al. (a) 100 80 shear stress (Pa) 60 40 20 10–3 10–2 10–1 1 10 (b) 50 40 shear stress (Pa) 30 20 10 0 10–4 10–3 10–2 10–1 1 10 102 103 shear rate (s–1) Figure 1. (a) The behaviour of 0.1% carbopol under increasing and decreasing shear stresses clearly shows that this material is non-thixotropic (ﬁlled circle, up; open circle, down). (b) Thixotropy of a 10% bentonite solution under an increasing and then decreasing stress ramp. 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: ρgh sin(θ ) > σy , 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. 2002a). 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 Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 Categorizing yield stress ﬂuids 5143 (a) (b) 104 stress (Pa) 103 viscosity (Pa s) 102 10 1 –1 10 10–2 1 10 100 1000 time (s) Figure 2. (a) Avalanche ﬂow of a clay suspension over an inclined plane covered with sandpaper. The suspension was presheared and poured onto the plane, after which it was left at rest for 1 h. The pictures are taken at the critical angle for which the suspension just starts to ﬂow visibly. (b) Bifurcation in the rheological behaviour: viscosity as a function of time for a bentonite suspension (solid fraction: 4%) in water. 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. 2002a,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, σc , 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 σc , 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 Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 5144 P. Moller et al. 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. 2002a,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. (i) There exists a structural parameter λ that describes the local degree of interconnection of the microstructure. (ii) The viscosity increases with an increase of λ. (iii) 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. 2002a,b) dλ 1 = − α γ̇ λ dt τ and η = η0 (1 + βλ)n , where τ is the characteristic ageing time for buildup of the microstructure, α determines the rate at which the microstructure is broken down under shear, η0 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. 2002a,b). In steady state, dλ/dt = 0 and the resulting steady-state ﬂow curve is easily found from solving the above equations in steady state: dλ 1 = 0 ⇒ λss = dt ατ γ̇ and η0 β σss = η(λss )γ̇ = + η0 γ̇ . (ατ )n γ̇ n−1 There are now three physically different situations we can distinguish: (i) 0 < n < 1: the material is a simple shear thinning ﬂuid, with no yield stress; (ii) n = 1: the material behaves as a simple yield stress ﬂuid, η0 β/(ατ )n being the yield stress. In this case, the viscosity diverges continuously when the stress is lowered towards the yield stress; and (iii) 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. Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 Categorizing yield stress ﬂuids 5145 (a) 6 σ (Pa) 4 . γ = 2 × 10–7s–1 2 10 100 1000 . γglobal (s–1) (b) 100 80 shear stress (Pa) 60 40 20 10–2 10–1 1 10 shear rate (s–1) Figure 3. (a) Thixotropic yield stress ﬂuid (colloidal particle gel, as described in Moller et al. (2008)). For an imposed stress, one cannot measure the whole ﬂow curve; the stress plateau signalling shear banding manifests itself only under an imposed shear rate (dashed line, critical shear rate from the MRI measurements; ﬁlled diamond, imposed shear rate; ﬁlled circle, imposed shear stress). (b) Simple yield stress ﬂuid (carbopol): imposed stress and imposed shear rate measurements give identical results (ﬁlled circle, shear stress rheometer controller; open circle, shear strain rheometer controller). It should be veriﬁed experimentally that such measurements correspond to steady states, of course. 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 3b 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 Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 5146 P. Moller et al. 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: σ = σy + η0 γ̇ , which is a special case of the Herschel–Bulkley model. Still, the material can age and shear rejuvenate: if not in steady state, dλ/dt 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. Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 Categorizing yield stress ﬂuids 5147 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 “π αντ αρει”—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 Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 5148 P. Moller et al. 107 106 apparent viscosity (Pa s) 105 107 viscosity plateau (Pa s) 104 103 106 102 ηapp~t0.60 10 105 1 101 102 103 104 measurement time (s) 10–1 2 4 8 16 32 64 128 256 imposed stress (Pa) Figure 4. Viscosity versus stress for a carbopol system for different durations of application of the different stresses. Beyond the yield stress of roughly 30 Pa, all curves superimpose. For lower stresses, the value of the viscosity plateau shows a clear time dependence: consequently these data points do not correspond to steady states and thus should not be considered as indicative of a ‘viscosity’ of the material. The inset shows the power-law dependence of the plateau value of the ‘viscosity’ as a function of time. Open circle, 10 s; open square, 30 s; open diamond, 100 s; open triangle, 300 s; plus, 1000 s; cross, 3000 s. 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 Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 Categorizing yield stress ﬂuids 5149 described in the previous section. The difference between the carbopol and the bentonite is the following. (i) At rest, the (complex) viscosity of the bentonite increases, whereas that of the carbopol does not. (ii) The response of the carbopol is in fact an elastic response, since the viscosities are so high (typically 106 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 5a). 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 5b). 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 Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 5150 P. Moller et al. (a) 10 shear stress (Pa) 1 10–1 10–4 10–3 10–2 10–1 1 10 (b) 10 shear stress (Pa) 1 10–1 10–3 10–2 10–1 1 10 shear rate (s–1) Figure 5. (a) Confocal image and ﬂow curve of a colloidal glass: concentrated (60%) suspension of 1.03 μm hard-sphere colloids (ﬂuorescent polymethylmethacrylate (PMMA) particles) in a density- and index-matched mixture of decaline and cyclohexyl bromide. The ﬂow curve was obtained on a rheometer coupled to the confocal microscope, and shows no hysteresis upon increasing and decreasing the stress; hence the system is a simple yield stress material (ﬁlled circle, up; open circle, down). (b) Confocal image of an ‘attractive glass’: the same particles to which a polystyrene polymer is added with a mass of 30 million a.m.u. The ﬂow curve is made on a slightly different attractive glass: silica spheres and cellulose as a polymer. It shows an important hysteresis effect; hence the system is a thixotropic yield stress ﬂuid (ﬁlled square, up; open circle, down). Colloids: PMMA colloids PDMS stabilized, coumarin labelled, diameter 1.1 μm, polydispersity 6%. Polymer: polystyrene Mw = 30 × 106 g mol−1 . Solvent: tetrachloroethylene/decaline mixture (w/w) = 0.56/0.44. Ludox-cellulose for rheology colloids: 22 nm (Ludox™, 40 wt%) polymers: cellulose Mw = 90 000 (Acros) in water. 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. Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 Categorizing yield stress ﬂuids 5151 Table 1. Distinguishing different systems. thixotropic systems simple yield stress ﬂuids Brownian motion yes no attraction yes no gravity yes no In conclusion, table 1 may be presented. This helps us to distinguish the different systems in terms of their physical properties. (i) Thixotropic yield stress ﬂuids: particle and polymer gels (Moller et al. 2008), attractive glasses (this paper), ‘soft’ colloidal glasses (Bonn et al. 2002a), 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). (ii) 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. 2002b), 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 5a. 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 σy , 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. Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 5152 P. Moller et al. 1.0 0.9 0.8 viscosity (Pa s) 0.7 0.6 0.5 0.4 0.3 0.2 0 20 40 60 80 100 120 time (s) Figure 6. Viscosity as a function of time demonstrating the thixotropy of a hard-sphere colloidal suspension. The suspensions are left to rest for 2 h, after which a constant stress is applied and the viscosity is followed in time. The particles are 1.3 μm PMMA in a solvent that matches both the density and the refractive index: decaline and cyclohexyl bromide. The volume fraction of particles is 62%. Black square, 8 Pa; grey circle, 16 Pa. (i) During imposed shear rate measurements, if the imposed shear rate is lower than some critical shear rate, the material shear bands. (ii) The material inside the sheared band is sheared exactly at the critical shear rate. (iii) 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 Phil. Trans. R. Soc. A (2009)

Downloaded from http://rsta.royalsocietypublishing.org/ on September 11, 2015 Categorizing yield stress ﬂuids 5153 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. References Asaga, K. & Roy, D. M. 1980 Rheological properties of cement mixes: IV. Effects of superplasticizers on viscosity and yield stress. Cement Concrete Res. 10, 287–295. (doi:10.1016/ 0008-8846(80)90085-X) Barnes, H. A. 1997 Thixotropy—a review. J. Non-Newtonian Fluid Mech. 70, 1–33. (doi:10.1016/ S0377-0257(97)00004-9) Barnes, H. A. 1999 The yield stress—a review or ‘π αντ αρει’—everything ﬂows? J. Non-Newtonian Fluid Mech. 81, 133–178. (doi:10.1016/S0377-0257(98)00094-9) Barnes, H. A. 2007 The ‘Yield stress myth?’ paper—21 years on. Appl. Rheol. 17, 43110. (doi:10.3933/ApplRheol-17-43110) Barnes, H. A. & Nguyen, Q. D. 2001 Rotating vane rheometry—a review. J. Non-Newtonian Fluid Mech. 98, 1–14. (doi:10.1016/S0377-0257(01)00095-7) Barnes, H. A. & Walters, K. 1985 The yield stress myth? Rheol. Acta 24, 323–326. (doi:10.1007/BF01333960) Bengtzelius, U., Gotze, W. & Sjolander, A. 1984 Dynamics of supercooled liquids and the glass transition. J. Phys. C 17, 5915–5934. (doi:10.1088/0022-3719/17/33/005) Bertola, V., Bertrand, F., Tabuteau, H., Bonn, D. & Coussot, P. 2003 Wall slip and yielding in pasty materials. J. Rheol. 47, 1211–1226. (doi:10.1122/1.1595098) Phil. Trans. R. Soc. A (2009)

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