An attempt to categorize yield stress fluid behaviour

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                                                   Phil. Trans. R. Soc. A (2009) 367, 5139–5155

             An attempt to categorize yield stress
                      fluid behaviour
                    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 (

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
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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

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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

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5142                                                          P. Moller et al.



                      shear stress (Pa)   60



                                                 10–3        10–2       10–1        1        10

                      (b) 50

                      shear stress (Pa)




                                               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

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(a)                                                           (b) 104 stress (Pa)

                                                              viscosity (Pa s)
                                                                                       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

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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
                                         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
                                dλ                 1
                                   = 0 ⇒ λss =
                                dt               ατ γ̇
                                                       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.

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                       (a) 6

                       σ (Pa)

                                                  γ = 2 × 10–7s–1

                                                 10                      100                1000
                                                                     γglobal (s–1)


                        shear stress (Pa)




                                                      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

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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
   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.

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   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

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5148                                                                                           P. Moller et al.



                 apparent viscosity (Pa s)

                                                        viscosity plateau (Pa s)

                                               1                                     101    102    103     104
                                                                                       measurement time (s)
                                                    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

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                                  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

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5150                                      P. Moller et al.


                                                 shear stress (Pa)


                                                                      10–4       10–3     10–2   10–1         1    10


                                                 shear stress (Pa)


                                                                          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.

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                                   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.

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5152                                               P. Moller et al.



                 viscosity (Pa s)





                                          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

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                                  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.

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