# Rheology of non-Brownian suspensions: a rough contact story

Elisabeth Lemaire, Frédéric Blanc, Cyrille Claudet, Stany Gallier, Laurent

Lobry, François Peters

► To cite this version:

Elisabeth Lemaire, Frédéric Blanc, Cyrille Claudet, Stany Gallier, Laurent Lobry, et al.. Rheology of non-Brownian suspensions: a rough contact story. Rheologica Acta, 2023, 62 (5-6), pp.253-268. 10.1007/s00397-023-01394-z. hal-05182440

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# Rheology of non-Brownian suspensions: a
rough contact story

Elisabeth Lemaire¹*, Frédéric Blanc¹, Cyrille Claudet¹, Stany Gallier², Laurent Lobry¹ and François Peters¹

¹*Institut de Physique de Nice, CNRS Université Côte d'Azur, Parc Valrose, Nice, 06100, France.

²ArianeGroup, 9, Rue Lavoisier, Vert le Petit, 91710, France.

*Corresponding author(s). E-mail(s): elemaire@unice.fr;
Contributing authors: frederic.blanc@univ-cotedazur.fr;
cyrille.claudet@univ-cotedazur.fr; stany.gallier@ariane.group;
laurent.lobry@univ-cotedazur.fr;
francois.peters@univ-cotedazur.fr;

## Abstract

This paper presents a compilation of experimental and numerical results that both show that particles in a suspension under shear flow come into contact through surface roughness. Accounting for those frictional contacts captures well the measured viscosity values as well as the shear-thinning observed in many concentrated non-Brownian suspensions. More specifically, the precise analysis of the asymmetric microstructure of the suspension clearly shows the role played by surface asperities in the interparticle contact. Numerical simulation results are then recalled and show the major impact of solid friction between particles on the viscosity. It is then pointed out that shear reversal experiments are a good way to assess this impact and to deduce useful information on the frictional properties of contacts. Finally, when considering together contacts via asperities and the close link between friction and rheology, a natural explanation for the shear-thinning observed in most suspensions can be proposed. According to it, as the shear stress increases, the friction coefficient decreases which leads to a viscosity decrease. This scenario first implemented in numerical simulations is verified experimentally through coupled AFM and rheometry measurements.
# 1 Introduction

The study of non-Brownian concentrated suspensions has undergone a remarkable revival in the last ten years, in particular thanks to the understanding of the role played by solid contact forces between particles in the rheology of suspensions (see Guazzelli and Pouliquen (2018) for a review). Concentrated suspensions are used in many different fields, from natural flows (landslides, debris flows, blood circulation...) to industrial flows (concrete or solid propellant casting, composite shaping...). Those "real life" suspensions often display a wide variety of complex rheological behaviors (flow-induced migration, thixotropy, shear-thinning, shear-thickening, etc.) that might be attributed to their complex formulation. However, most of those behaviors are also found in model suspensions where the suspending fluid is Newtonian and the particles spherical, quasi-monodisperse and non-buoyant. For a long time, the study of suspension rheology has focused on the effect of short-range (lubrication) and long-range hydrodynamic forces on the flow properties. Batchelor (1970) extended Einstein's work to take into account the contribution of two-particle hydrodynamic interactions and then numerical works (Brady and Bossis, 1988; Yeo and Maxey, 2010c) focused on more concentrated regimes in which pairwise interactions were not sufficient to describe the rheology. Nevertheless, the viscosity values obtained with those numerical simulations were much lower than what is measured experimentally, especially when suspensions are concentrated. This discrepancy has been explained by the role played by the frictional contacts that were not accounted for in the previous numerical simulations (Gallier et al, 2014). Despite lubrication, these contacts are promoted
by particle roughness and are responsible for fore-aft asymmetry of the parti-
cle pair trajectory in a simple shear flow. They also explain the asymmetry of
the shear-induced microstructure in dilute suspensions.

As early as the 90’s, several microstructure studies (Parsi and Gadala-
Maria, 1987; Rampall et al, 1997) have shown, at least in a qualitative way,
that particles of a suspension subjected to a simple shear flow come into
contact through the asperities present on their surface. Then, with technical
advances in suspension imaging (Gao et al, 2010; Wiederseiner et al, 2011),
quantitative links between microstructure and interparticle contacts have been
established (Blanc et al, 2011). Solid contacts between particles play a crucial
role in the flow dynamics of suspensions (Arp and Mason, 1977a; Drazer et al,
2002; Pham et al, 2015) and rheology. A vivid illustration of the role played
by contacts in rheology is the discontinuous shear-thickening (DST) explained
by a transition from a flow where particles do not contact to a flow where par-
ticles exert frictional forces on each other (Seto et al, 2013; Mari et al, 2014;
Wyart and Cates, 2014). These numerical and theoretical models were con-
firmed by several experimental studies that have succeeded in showing the link
between the repulsive forces between particles and the DST transition stress
(Guy et al, 2015; Bossis et al, 2017; Comtet et al, 2017; Clavaud et al, 2017;
Hsu et al, 2018). As stressed by Guy et al (2015), a shear-thickening transi-
tion is expected at an onset stress σc ≈ f*/d², where f* is the repulsive force
and d the particle diameter, that has to be overcome to push particles into
frictional contact. This means that the onset stress for shear-thickening can
be very low for particles of several tens of microns in diameter. This is the
reason why shear-thickening transition is rarely observed with such large par-
ticles since it would require repulsive forces hardly attainable. This also means
that the rheology of non-Brownian suspensions and of suspensions exhibiting
4 *Article Title*

shear-thickening, beyond the shear-thickening transition, is the same and is
dominated by frictional forces between particles. The influence of the pairwise
friction coefficient has been exhaustively studied through numerical simula-
tions by Gallier et al (2014), Singh et al (2018) and Chèvremont et al (2019).
Peters et al (2016) showed, also by numerical simulations, that shear reversal
tests, which consist in abruptly reversing the shear direction, provide accurate
information on inter-particle friction.

All the above mentioned numerical simulations of non-Brownian frictional
suspensions deal with rate-independent suspensions while most concentrated
frictional suspensions display shear-thinning behavior (Papadopoulou et al,
2020) whose origin has long been debated. It has been proposed to explain
shear-thinning behavior by the slip of solvent molecules on the particle surface
(Kroupa et al, 2017). The shear-thinning character of the suspending fluid itself
has been suspected. More precisely, it can happen that the suspending liquid
behaves like a Newtonian liquid at the macroscopic shear rate range but may
exhibit a shear-thinning behavior for the higher effective shear rate experienced
in the interparticle gap (Vázquez-Quesada et al, 2016; Kumar et al, 2020).
Adhesion between articles is also a possible explanation for shear-thinning.
In the frame of the model proposed by Richards et al (2020), the presence
of adhesive forces introduces an additional constraint which hinders particle
rotation and consequently enhances the suspension viscosity. Alternatively,
adhesive forces may induce the formation of aggregates, which in turn increase
the viscosity of the suspension (Zhou et al, 1995; Koumakis et al, 2015). In
addition, as the shear stress increases, adhesion plays a less important role and
the suspension viscosity decreases. Variable friction is another possible cause
of shear-thinning as suggested by Vázquez-Quesada et al (2017) and Tanner
et al (2018) and demonstrated experimentally by Chatté et al (2018) and
Arshad et al (2021). Chatté et al (2018) have studied the thinning behavior observed in a suspension of micron-size PVC (PolyVinyl Chloride) particles dispersed in a Newtonian plasticizer beyond the DST transition and correlated the viscosity decrease to the reduction of the friction coefficient with increasing contact force between particles. Using a quartz-tuning-fork-based atomic force microscope, the authors also measured the relation between the tangential and normal force between two contacting particles. They showed that the friction coefficient, as defined as the ratio of the dissipative friction force to the normal force, decreased with increasing load and that the decrease of the friction coefficient $\mu$ was correlated to the decrease of the viscosity with increasing shear stress. They attributed the decrease of $\mu$ with the normal load to the presence of polymer brushes forming at the surface of the PVC particles due to the effect of the plasticizer solvent. This was the first experimental proof that shear-thinning and friction could be related. Around the same time, Tanner et al (2018) proposed a model based on a bootstrap mechanism of friction where the friction coefficient is a decreasing function of the sliding velocity between particles. Lobry et al (2019) performed numerical simulations based on the force coupling method where a load-dependent pairwise friction coefficient adapted from a tribological model proposed by Brizmer et al (2007) is introduced to mimic the elasto-plastic contact between an asperity on the surface of a particle and another particle.

This idea was then used by More & Ardekani to study the influence of roughness size on the rheology of non-Brownian suspensions More and Ardekani (2020, 2021). Then, the model proposed by Lobry et al (2019) was tested against experimental measurements by Arshad et al (2021) with the results that, on the one hand, the friction coefficient decreases with the contact force according to a law close to that of Brizmer et al (2007) and that, on
# 6 Article Title

the other hand, the introduction of the experimental friction law in the model enables to capture well the variation of the viscosity with shear stress.

In this paper we present a set of results that tend to show that it would be possible to construct a general framework that links rheology to particle contact forces and microstructure in concentrated non-Brownian suspensions. In a first section, we recall how the scrutiny of the spatial distribution of particles demonstrates that particles in a suspension under a simple shear flow experience contacts through the asperities present on their surfaces. In the second section, we discuss the effect of these contacts on the viscosity of non-Brownian suspensions. The third part points out the interest of performing shear reversal experiments to discriminate the respective contributions of hydrodynamic forces and contact forces to suspension viscosity. Finally, we show that the variation of the viscosity with the interparticle friction coefficient provides an explanation for the shear-thinning behavior of concentrated non-Brownian frictional suspensions.

# 2 Contacts and microstructure

The first direct experimental determinations of the microstructure of a suspension date back to the works of Husband and Gadala-Maria (1987) in the dilute regime ($\phi \le 0.05$) and of Parsi and Gadala-Maria (1987) in the concentrated regime. These were the first measurements of pair distribution functions in non-Brownian suspensions that established, at least qualitatively but without any ambiguity, the anisotropy of the microstructure of non-Brownian suspensions of spherical particles subjected to a steady shear flow. In dilute suspensions, the measured pair distribution functions are such that the probability of finding contacting pairs is maximum in a direction close to—but distinct from—the
velocity direction. On the contrary, in concentrated suspensions, the asymmetry of the pair distribution function is more pronounced with an excess of contacting particle pairs aligned in the compression quadrant and a depletion in the dilatation quadrant. These experiments have been repeated later and lead to the same conclusions, see for example (Rampall et al, 1997; Gao et al, 2010; Blanc et al, 2013). Figure 1, extracted from (Blanc et al, 2013), is an illustration of this shear-induced asymmetry of the microstructure.

<figure><img src="image_2.png" /><figcaption>**Fig. 1** Pair distribution function in the plane of shear measured with a suspension of PMMA spherical particles ($\phi = 0.40$) with a diameter of $170 \pm 12 \mu m$ dispersed in a Newtonian mineral oil (Cargile, *Immersion liquid Code 11295101160*). The neutrally buoyant suspension is sheared in a wide-gap Couette rheometer and the shear plane is illuminated with a thin laser sheet, which allows the accurate determination of the particle positions. It is observed that the microstructure is neither isotropic nor symmetric with respect to the velocity gradient direction and that close contact pairs are mainly aligned along the compression axis direction of the flow.</figcaption></figure>

Leighton and Acrivos (1986) and Drazer et al (2002) proposed to examine two possible reasons that would explain the asymmetry of the trajectories or, equivalently, of the microstructure at low Reynolds numbers. The first one is the chaotic nature of the dynamics of concentrated suspensions caused by the large number of interacting particles (Eckstein et al, 1977; Breedveld et al, 2001a,b). If we consider only two perfectly smooth, non-Brownian and non
colloidal particles interacting (colliding) in a simple shear flow, the pair tra-
jectory remains fore-aft symmetric. But, during the collision, each of the two
particles undergoes a large transient displacement with respect to its original
streamline and may simultaneously interact with a third particle. Those three
(or more) particle interactions result in a loss of memory of the initial state
and in the emergence of a chaotic behavior of the suspensions. On the other
hand, it is well-known that the presence of repulsive forces between particles
breaks the fore-aft symmetry of the microstructure (Bossis and Brady, 1984;
Da Cunha and Hinch, 1996; Rampall et al, 1997; Drazer et al, 2002; Davis
et al, 2003). In the presence of a repulsive force, particle pairs tend to align
along the compression axis direction while the probability of finding a contact-
ing particle pair along the dilatation axis decreases. The angular asymmetry
of the pair distribution function is explained by the fact that during a collision
the particles spend a longer time when they are oriented along the compression
axis, where the repulsive force is balanced by the shear force pushing the two
particles together, than in the dilatation direction (Drazer et al, 2002). Later,
Metzger and coworkers showed, through numerical simulations of oscillating
flows, that this second mechanism was indeed responsible for the trajectory
irreversibility. First, by comparing simulations carried out with and without
long-range hydrodynamic forces, they showed that the presence of the latter
was neither a source nor a magnifier of irreversibility (Metzger and Butler,
2010). Second, they found that N-body dynamics occurring through the lubri-
cation interactions may lead to chaotic trajectories. Nevertheless, this chaos
is too weak to produce a significant trajectory irreversibility. So they con-
clude that, in periodically sheared suspensions, irreversibility is dominated by
contact interactions even though lubrication interactions must be included to
recover experimental observations (Metzger et al, 2013). Finally, it should be
stressed that the microstructure asymmetry for non-Brownian suspensions has also been predicted theoretically by Brady and Morris (1997) who studied the influence of Brownian motion and particle interactions on the shear-induced microstructure and non-Newtonian behavior such as normal stress differences. Following Batchelor, they solved Smoluchowski equation in dilute regime, yielding the pair distribution function and the stress for large Péclet numbers ($Pe = 6\pi\eta_0a^3\dot{\gamma}/kT$ varying from 10 to $5.10^4$). In particular, for interacting particles, they found that the intensity of the pair distribution function at contact diverges as $Pe$ tends to infinity, while the volume of the boundary layer vanishes, with the result that the second normal stress difference levels off to a $Pe$-independent value.

The repulsive forces responsible for the microstructure asymmetry may be of different nature. They can be of electrostatic origin, including for particles of several tens of microns as shown by Clavaud et al (2017) or arise from steric interactions (Comtet et al, 2017; Bossis et al, 2017). They can also be due to direct solid contacts between particles (Da Cunha and Hinch, 1996; Drazer et al, 2002). Particle roughness is a key ingredient in explaining direct solid contact between particles since, for perfectly smooth spheres, lubrication forces which diverge as the separation distance between particles tends to zero prevent any true contact between particles. But the presence of even a small amount of surface roughness results in the possibility for the particles to come into contact without diverging lubrication forces. Arp and Mason (1977a) showed that surface roughness as small as $h_r = 3.10^{-4} \times a$ (with $a$ the particle radius) was sufficient to eliminate the closed orbits and to induce asymmetry in pair trajectories. More precisely, a contact between particles through the surface asperities is possible and takes place as soon as the size of the roughness is higher than the minimum distance of approach of two particles
interacting hydrodynamically in a simple shear flow: $h_r^* = 4.2 \cdot 10^{-5} \times a$ (Arp and Mason, 1977b). The typical non-dimensional roughness sizes, identified in various experimental works on model suspensions (using Teflon, PMMA, polystyrene, glass,...), generally range between $5 \cdot 10^{-4}$ and $10^{-2}$ (Arp and Mason, 1977a; Smart and Leighton Jr, 1989; Davis et al, 2003; Vinogradova and Yakubov, 2006; Blanc et al, 2011; Tanner and Dai, 2016). Because of their small size, asperities have little impact on particle drag but play an important role in promoting solid contact between particles.

The role of asperities on the fore-aft asymmetry of the pair distribution function has been clearly showed by Blanc et al (2011). In this study, the pair distribution function $g(r, \theta)$ is measured experimentally and compared to a theoretical pair distribution function computed from the trajectories of a pair of rough particles in a shear flow (Da Cunha and Hinch, 1996). In the latter model, the asperities are assumed to exert a normal force between the surfaces that prevent the particles from approaching further but do not change the tangential motion. As a consequence, the effect of the asperities is only to cancel the relative radial velocity when the distance between the particle centers equals $2 \times (a + h_r)$. This results in a break of the fore-aft symmetry of the trajectories, the spheres separating on streamlines further apart than on their approach. In experiments by Blanc et al (2011), pair distributions are measured using a suspension of PMMA particles (Arkema BS510) with a diameter of $170 \pm 12 \mu m$ dispersed in a Newtonian liquid (viscosity 0.85 Pa.s) designed by Cargille Laboratories (Immersion liquid Code 11295101160) in order to match both the density and the refractive index of PMMA particles. The suspension is sheared in a vertical Couette cell with a shear rate of about $0.5 s^{-1}$. The roughness surface is measured using an atomic force microscope. The mean roughness height is of about 20 nm with big humps of about 200 nm with a
<figure><img src="image_3.png" /><figcaption>**Fig. 2** Pair distribution functions for a suspension of spherical particles with a solid fraction of 5%. a) pdf measured experimentally with a suspension of PMMA spherical particles with a diameter of 170 ± 12 µm and a surface roughness of 200 nm in height dispersed in a Newtonian mineral oil (Cargile, *Immersion liquid Code* 11295101160). b) pdf computed from the trajectories of rough particles deduced from the model of Da Cunha and Hinch (1996). The better agreement between the computed and measured pdf is obtained for a roughness of 255 nm, in good agreement with the roughness of the particles measured by AFM. Data extracted from Blanc et al (2011).</figcaption></figure>

surface density of about 0.1 µm⁻². The best agreement between experiment and model is found for a roughness size of $h_r = 3.10^{-3}a$ or $h_r = 255$ nm, a value very close to the one measured by AFM (Blanc et al, 2011). Such measured and expected pair distribution functions are presented in Fig. 2. Thus the asymmetry of the trajectories and hence of the microstructure seems to be quantitatively related to the size of the surface roughness. This result has been confirmed by Pham et al (2015) who measured the trajectory of pairs of particles of different roughness in a simple shear flow and showed that the pair trajectory asymmetry is quantitatively controlled by the roughness height.

## 3 Impact of frictional contacts on rheology

As seen in the previous section, the irreversible nature of particle-particle contacts, which are capable of compressive but not tensile forces, modifies the suspension microstructure and affects the rheological properties. To account for the presence of surface asperities, a short-range repulsive force can be
introduced with its range mimicking the roughness height. Several numerical studies have shown a strong impact of the repulsive interparticle force on the microstructure (Singh and Nott, 2000; Drazer et al, 2002; Sierou and Brady, 2002) while its impact on the value of the shear viscosity has been found to be limited. In the dilute regime, theoretical studies have shown that roughness decreases the viscosity and increases the magnitude of normal stress differences (Wilson and Davis, 2002; Davis et al, 2003; Wilson, 2005). In more concentrated regimes, Stokesian dynamics simulations also demonstrate this decrease in viscosity as roughness increases (Wilson and Davis, 2002). The slight increase of the viscosity when the range of the repulsive force (or the roughness size) decreases can be interpreted by the enhancement of the lubrication forces caused by shorter separation distances between particles.

Until the 2010s, most studies were interested in the influence of the normal contact force on rheology while the impact of its tangential component (related to friction) has been little studied. By theory as well as by simulation of a particle monolayer, Wilson and Davis (2002) and Davis et al (2003) report fairly moderate effects of friction with a small increase in viscosity. Sierou and Brady (2002) have developed in Stokesian Dynamics a simplified modeling of friction and have also shown an increase in viscosity with increasing friction coefficient, $\mu$. According to those first results, the viscosity of a suspension at volume fraction $\phi = 0.4$ increases by about 10% when $\mu$ increases from 0 to 0.5.

In 2013, Seto et al (2013) and Mari et al (2014) showed that friction is a key element to explain the shear-thickening observed in a variety of concentrated non-Brownian suspensions. They proposed to explain the shear-thickening observed in concentrated suspensions of repulsive particles by a transition from a lubricated flow to a friction-dominated flow. When the shear stress is raised,
the number of frictional contacts resulting from a competition between applied shear forces and short-range repulsive forces increases, leading to an increase in the viscosity (Wyart and Cates, 2014). This very promising scenario was then confirmed by several experimental studies (Bossis et al, 2017; Comtet et al, 2017; Clavaud et al, 2017) and highlighted the crucial role of friction in rheology of non-Brownian suspensions. In 2014, Gallier et al (2014) carried out a comprehensive numerical study using a fictitious domain method on the role of friction in suspension flow properties, including viscosity, normal stress differences, particle normal stresses, microstructure, and $\mu(I)$ rheology.

Figure 3 shows the relative viscosity $\eta_r$ of a suspension as a function of the volume fraction $\phi$ for different friction coefficients $\mu$. Some experimental measurements by Zarraga et al (2000) and Dbouk et al (2013) have also been added. The computed viscosities obtained for frictionless particles ($\mu = 0$) are significantly lower than the viscosity values measured experimentally. This effect is observed in a recurrent way in all other simulations in Stokesian Dynamics and Force Coupling Method, e.g. in Sierou and Brady (2002) or Yeo and Maxey (2010a). Taking friction into account substantially increases the viscosity and the agreement with viscosity measurements is significantly improved.

From Fig. 3, it is also clear that the value of the jamming fraction, $\phi_m$, i.e. the value for which the viscosity diverges, should vary with friction coefficient. The value of $\phi_m$, at a fixed $\mu$, can be deduced from the variation of the viscosity with the particle volume fraction and a Maron-Pierce type fitting law:

$$ \eta_r = \frac{\alpha(\mu)}{\left(1 - \frac{\phi}{\phi_m(\mu)}\right)^2} \quad (1) $$

where $\alpha$ is a fitting parameter of order 1.

By performing numerical simulations for different values of the interparticle friction coefficient, ranging from 0 to 10, Lobry et al (2019) have proposed the
<figure><img src="image_4.png" /><figcaption>**Fig. 3** Relative viscosity as a function of particle volume fraction for different friction coefficients. The reduced roughness is $h_r/a = 5.10^{-3}$. Measurements from Zarraga et al (2000) and Dbouk et al (2013) are also shown. Data extracted from Gallier et al (2014).</figcaption></figure>

following expressions for $\alpha(\mu)$ and $\phi_m(\mu)$:

$$ \alpha(\mu) = \alpha^{\infty} + (\alpha^{0} - \alpha^{\infty}) \frac{\exp(-X^{a}\operatorname{atan}(\mu)) - \exp(-\pi X^{a}/2)}{1 - \exp(-\pi X^{a}/2)} \quad (2) $$

$$ \phi_m(\mu) = \phi_m^{\infty} + (\phi_m^0 - \phi_m^{\infty}) \frac{\exp(-X^p\operatorname{atan}(\mu)) - \exp(-\pi X^p/2)}{1 - \exp(-\pi X^p/2)} \quad (3) $$

with parameters $\alpha^0$, $\alpha^\infty$, $X^a$, $\phi_m^0$, $\phi_m^\infty$, and $X^p$ given in Table 1

<table><caption>**Table 1** Fitting parameters of Eq. (2)-(3)</caption><thead><tr><th>α<sup>0</sup></th><th>α<sup>∞</sup></th><th>X<sup>a</sup></th><th>φ<sub>m</sub><sup>0</sup></th><th>φ<sub>m</sub><sup>∞</sup></th><th>X<sup>p</sup></th></tr></thead><tbody><tr><td>1.00</td><td>0.640</td><td>1.85</td><td>0.700</td><td>0.546</td><td>2.43</td></tr></tbody></table>

Other expressions relating the jamming fraction to the friction coefficient have been proposed in the literature. For instance, the variation of $\phi_m$ with $\mu$ proposed by Singh et al (2018) and Chèvremont et al (2019) together with the relation Eq. (3) are represented in Fig. 4. All three studies agree to show
<figure><img src="image_5.png" /><figcaption>**Fig. 4** Evolution of maximum packing fraction $\phi_m$ with particle friction coefficient $\mu$, extracted from three numerical studies (Singh et al, 2018; Chèvremont et al, 2019; Lobry et al, 2019). The insert is a zoom for $\mu \le 1$, extracted from Peters et al (2016).</figcaption></figure>

a decrease of $\phi_m$ when $\mu$ increases, with most of the variation occurring in the range $\mu \in [0, 2]$. However a rather significant difference in the values of $\phi_m$ can be observed. But it should be kept in mind that $\phi_m$ is deduced by fitting the variation of the viscosity with particle volume fraction to Eq. (1). Therefore the values of $\phi_m$ may depend on the range of volume fractions that have been explored. This difference in the ranges explored by the different authors can probably explain some of the disagreements noted between the results from different numerical simulations. Furthermore, Chèvremont et al (2019), who studied a wide range of particle volume fractions ($10^{-3} < 1 - \frac{\phi}{\phi_m} < 4.10^{-1}$) also stressed the difficulty of determining $\phi_m$. In particular, they show that for sufficiently low solid fractions, the exponent of the divergence of the viscosity with $1 - \phi/\phi_m$ is close to 2 but tends to decrease when $1 - \phi/\phi_m$ becomes smaller than typically 0.1. The conclusion on this point is presumably that the value of $\phi_m$ determined from the fit of experimental or numerical data may not be the exact value of the jamming fraction—for which the viscosity
diverges—but should rather be considered as a fitting parameter that enables
to account for the variation of the viscosity in a particular range of particle
volume fraction.

The role of friction can be better understood by studying separately the
hydrodynamic and contact contributions to the viscosity. Figure 5 shows the
hydrodynamic (solid circles) and contact (solid squares) components of the
relative viscosity η_r as a function of particle volume fraction, in the frictionless
(μ = 0) and frictional (μ = 0.5) cases. It is observed that the hydrodynamic
contribution in the frictional and frictionless cases are superimposed on the
same single curve, meaning that the hydrodynamic part η^H of the viscosity is
not sensitive to friction, with weak variation close to the jamming volume frac-
tion though. Friction only affects the contact contribution η^C which increases
significantly with μ.

<figure><img src="image_6.png" /><figcaption>**Fig. 5** Hydrodynamic (circles) and contact (squares) contribution to the relative viscosity as a function of the volume fraction and for friction coefficients μ = 0 (green) and μ = 0.5 (orange). The reduced roughness is h_r/a = 5.10<sup>-3</sup>. Data extracted from the numerical study of Gallier et al (2018)</figcaption></figure>
It is also worth mentioning that, for frictional suspensions (here, $\mu = 0.5$), the contact contribution to the viscosity is predominant as soon as the particle volume fraction becomes higher than 0.4, which means that the rheology of concentrated suspensions is controlled by the inter-particle contact forces.

# 4 Shear reversal tests

In this section, we would like to discuss how the contribution of contact forces to viscosity can be estimated through shear reversal experiments. These experiments consist in reversing the shear flow direction and have been first proposed by Gadala-Maria and Acrivos (1980). A typical transient response of the viscosity following shear reversal is shown in Fig. 6.

<figure><img src="image_7.png" /><figcaption>**Fig. 6** Variation of the viscosity with shear strain during a shear reversal performed at $\sigma = 10$ Pa. The suspensions are made of polystyrene particles (Dynoseeds TS 80), 80 µm in diameter dispersed in silicone oil (Bluesil, 47V1000). Experiments are performed at $T = 30^{\circ}$C. The viscosity of the suspending fluid is 0.90Pa.s. Just after shear reversal, the viscosity suddenly drops to $\eta_{init}$, passes through a minimum, $\eta_{min}$ and then increases to return to its steady value $\eta_S$.</figcaption></figure>
Just after the shear reversal, the viscosity jumps down to a value, $\eta_{init}$, lower than its steady value $\eta_S$, decreases to $\eta_{min}$ and then rises to reach the value $\eta_S$ it had before the reversal. Along the lines of the work of Gadala-Maria and Acrivos (1980), Parsi and Gadala-Maria (1987) interpreted this kind of transient response of the viscosity by a transition between one microstructure to the mirror microstructure when the flow is reversed. The large decrease in viscosity observed during the transition is associated with the destruction of the microstructure which must be reconstructed along the new flow axes. Blanc (2011) conducted a combination of microstructure and viscosity measurements during shear reversal that confirmed the findings of Gadala-Maria and Acrivos (1980) and of Parsi and Gadala-Maria (1987). As displayed in Fig. 7, during shear reversal, the asymmetric microstructure with an excess of contacting particles in the compression axis direction is first destroyed before it is rebuilt along the new direction of the compression axis.

<figure><img src="image_8.png" /><figcaption>**Fig. 7** Pair distribution function in the shear plane during a shear reversal. The suspension is made of PMMA particles, 170 ± 12 µm in diameter dispersed in a Newtonian mineral oil (Cargile, Immersion liquid Code 11295101160). The solid fraction is equal to 0.35. The right and left images represent the steady microstructures measured for a flow in one direction and in the opposite direction. The center image was obtained for strains between 0.5 and 1.3, close to the strain for which the viscosity is minimum, $\eta_{min} \approx 1$.</figcaption></figure>

As a consequence, the minimum viscosity—measured when the microstructure is the most isotropic and when there are few contacts—should be of the order of the hydrodynamic viscosity. It is therefore expected that the difference $\Delta\eta = \eta_S - \eta_{min}$ corresponds to the contribution of the contact forces to the viscosity.
In order to clarify this idea, Peters et al (2016) performed numerical sim-
ulations for particle volume fraction between 0.2 and 0.5, friction coefficient
between 0 and 1 and roughness height ranging from 5.10⁻³ a and 2.10⁻² a.
Consistent with Fig. 7, they found that most pre-existing contacts between
particles are broken right after shear reversal, meaning that the contact contri-
bution to the viscosity jumps down to zero, before it increases quite slowly to
its steady value. On the other hand, the hydrodynamic contribution increases
first at shear reversal, due to enhanced lubrication, before it levels off to the
steady value. This latter decrease is faster that the increase of the contact
contribution, resulting in the observed viscosity minimum. In addition, as the
minimum is reached, the hydrodynamic contribution is quite close to the steady
value, while the contact contribution is still weak, so that the total viscosity is
closer to the steady hydrodynamic viscosity. It was quantitatively shown that:

$$
\Delta\eta \approx 0.85 \eta^C \qquad (4)
$$

which happens to hold for values of the viscosity difference ranging over nearly
three orders of magnitude (Peters et al, 2016).

Thus, shear reversal experiments provide useful information on the fric-
tional properties of the particles. Shear reversal tests have been actually used
to show that shear-thickening—and even continuous shear-thickening— does
not originate from hydrodynamic interactions, as proposed by Gurnon and
Wagner (2015), but from the formation of frictional contacts (Lin et al, 2015).
Those shear reversal experiments can be used to assess the contribution of con-
tact forces to viscosity and are useful for suspension design when it is desired
to evaluate the influence of a change in the surface properties of the particles
on the rheology of a suspension (Moratille et al, 2022).
Another striking feature is that the reduced viscosity difference $\Delta\eta/\eta_S$ only depends on the reduced volume fraction $\phi/\phi_m$ (Peters et al, 2016). This has been shown to also hold in experiments, for suspensions of quite monodisperse spherical polymer particles, as well as for polydisperse suspensions of irregular particles, with aspect ratio close to 1 (Blanc et al, 2018). It means in particular that one single shear reversal experiment is required to get an approximate value of the jamming volume fraction, which in turn is highly dependent on the microscopic contact law between particles, as shown in the previous section. For instance, from Fig. 6 and from Fig. 11 in Peters et al (2016) which displays the variation of $(\eta_S - \eta_{min})/\eta_S$ with the ratio $\phi/\phi_m$, we obtain $\phi_m \approx 0.61 \pm 0.01$ (this value is averaged over the shear reversal curves obtained for the three particle volume fractions, 0.5, 0.45 and 0.4). Then, we can obtain an estimate of $\mu$ from the variation of $\phi_m$ with $\mu$ (Fig. 4). Here, the correlation proposed by Lobry et al (2019) (Eq. 3) would yield $\mu \approx 0.35$.

In addition, the numerical simulations performed by Peters et al (2016) showed that the "instantaneous" viscosity jump $\eta_s - \eta_{init}$ is highly sensitive to the value of the friction coefficient. This is shown in the graph of Fig. 8 where it is noticeable that the viscosity jump—at fixed volume fraction—is much smaller for a suspension of frictionless particles than for a suspension of frictional particles. In connection with the discussion above, it means that, for frictionless particles, the enhanced lubrication contribution at shear reversal roughly counterbalances the vanishing of the contact contribution, which is not true for frictional particles. On the whole, shear reversal tests are therefore good tests to rule on the frictional or non-frictional character of a suspension. Using this method, we can again evaluate the value of the friction coefficient of the TS80 polystyrene particles from the shear reversal curves displayed in
Fig. 6 and we obtain $\mu = 0.4 \pm 0.05$, which is in good agreement with the value deduced from the difference between $\eta_S$ and $\eta_{min}$.

<figure><img src="image_9.png" /><figcaption>**Fig. 8** Variation of the initial viscosity drop, normalized by the steady viscosity, as a function of the particle volume fraction for different friction coefficients. Data extracted from the numerical study of Peters et al (2016).</figcaption></figure>

We finally note that this powerful experimental method has recently been upgraded to cross-shear experiments (Blanc et al, 2023). In such experiments, after the suspension is first pre-sheared, the direction of shear is rotated in any direction in the plane normal to the velocity gradient, allowing a more precise insight into the relation between shear-induced microstructure and stress tensor.

# 5 Shear-thinning

In Sec. 3, it has been shown that viscosity increases with increasing friction coefficient between particles or, equivalently, that the jamming fraction
decreases with increasing friction coefficient. This relationship between pair-
wise friction and suspension rheology opens a way to explain the shear-thinning
observed in frictional suspensions by a decrease of the friction coefficient when
the shear stress increases. This has already been demonstrated by Chatté
et al (2018) who studied both the shear-thinning behavior of several suspen-
sions of PVC particles dispersed in a plasticizer, above the DST transition,
and measured the variation of the friction coefficient with the normal contact
force between two particles. Their result show a clear correlation between the
decrease of the viscosity with increasing shear stress and the decrease of the
friction coefficient with increasing load. According to Chatté et al (2018), the
latter is due to the peculiar nature of the friction between polymer brushes for
which the shear strength, defined by the authors as the tangential frictional
dissipative force per unit area, decreases with increasing normal load (Grest,
1999). Here we present another mechanism to explain the decrease of the fric-
tion coefficient with the contact force, which is based on the modeling of the
elastoplastic contact between surface asperities existing on most stiff particles
such as polymeric, glass, metallic or mineral particles (Hsiao and Pradeep,
2019).

As discussed in Sec. 2, in a sheared non-Brownian suspension, particles
come into contact through their surface asperities whose typical size is a few
thousandths of the particle radius (h<sub>r</sub> ≈ 5.10<sup>-3</sup> a). The typical normal contact
force induced by the flow can be written as F<sub>n</sub> ~ 6πa<sup>2</sup>σ. Then, assum-
ing an elastic contact between a particle and an asperity—or between two
asperities—and considering that h<sub>r</sub> is the radius of curvature for the asperi-
ties, the order of magnitude of an asperity flattening during a contact can be
evaluated by using the Hertz contact theory: δ ~ (F<sub>n</sub><sup>2</sup>/h<sub>r</sub>E<sup>2</sup>)<sup>1/3</sup>, where E is the
Young modulus of the particles. From this expression, we can deduce that,
for $\sigma \approx 100$ Pa, $E \approx 3$ GPa and $h_r \approx 5.10^{-3}$ a, the typical deformation of the asperity, $\delta/h_r$ is of the order of 10%. In addition, most glass or polymeric (PMMA, polystyrene...) particles have low roughness density ($\sim 10^{-1} - 10^{-2}$ roughness /$\mu m^2$) (Blanc et al, 2011; Pham et al, 2015; Hoyle et al, 2020; Lobry et al, 2019). Therefore, taking into account the rather moderate asperity deformation, the curvature of the particles, and the quite low roughness density, it is anticipated that the contact between particles is likely to take place through one or a few asperities. As a consequence, the interparticle contact cannot be modeled by the Amontons-Coulomb friction law which holds only for either macroscopic contacts where surfaces touch through a large number of small asperities (Greenwood and Williamson, 1966) or for plastic contact of smooth surfaces (Bowden and Tabor, 1939). Instead, in the case of a small number of elastically contacting asperities, the friction coefficient is expected to decrease with load (Archard, 1957). The case of two contacting particles in a sheared suspension can be idealized by a single contact between an asperity, characterized by its radius of curvature that is, for the sake of simplicity, taken to the asperity height $h_r$, and the surface of the other particle which can be represented by a smooth flat surface (see Fig. 9). For fully elastic materials, the radius of the contact area is given by the Hertz theory:

$$r_c = \left( \frac{3(1 - \nu^2) F_n}{2 E} h_r \right)^{1/3} \quad (5)$$

where $\nu$ is the Poisson coefficient. The frictional force $F_t$ is expected to be proportional to the contact area and thus to scale as $F_n^{2/3}$ which leads to a decrease of the friction coefficient with increasing normal force as $F_n^{-1/3}$. Furthermore, for sufficiently high loads, the contact becomes plastic. In the fully plastic regime, the contact area is proportional to the normal load and thus the
<figure><img src="image_10.png" /><figcaption>**Fig. 9** Sketch of two contacting rough particles. Successive zooms of the contact between one asperity, modeled by a hemisphere, and the surface of the other particle, modeled by a smooth plane.</figcaption></figure>

friction coefficient is expected to level off. Thus, to sum up, it is anticipated that with increasing normal load, the friction coefficient first decreases in the elastic regime, at low normal loads, and approaches a constant value as the contact becomes more plastic, for higher loads. This is precisely what has been shown by Brizmer et al (2007) who studied with the finite element method the elastic-plastic contact between a smooth sphere and a rigid flat surface. By fitting their numerical results, the authors proposed the following expression for the variation of the static friction coefficient with the normal force:

$$ \mu = 0.27 \coth \left[ 0.27 \left( \frac{F_n}{L_C} \right)^{0.35} \right] \quad (6) $$

where $L_C$ is the critical load at yielding inception<fnref n="1" />:

$$ L_C = \frac{\pi^3}{6} \bar{L}_C Y_0 C_\nu^3 \left[ 2 h_r (1 - \nu^2) \left( \frac{Y_0}{E} \right) \right]^2 \quad (7) $$

with $\bar{L}_C = 8.88\nu - 10.13(\nu^2 + 0.089)$ and $C_\nu = 1.234 + 1.256\nu$, $\nu$ being the Poisson's ratio and $Y_0$ and $E$ the yield strength and the Young modulus, respectively.

From this expression, both the elastic behavior, expected at small normal forces ($F_n \ll L_c$), and the plastic behavior, anticipated at large forces ($F_n \gg L_c$)

<footnote n="1">¹In the original model of Brizmer et al (2007), the contact is between a soft sphere and a rigid flat plane while here we consider the contact between two soft materials with the same Young modulus $E$. This is the reason why a factor 2 is introduced into the brackets.</footnote>
$L_C$), are recovered. In the elastic deformation regime, the Hertz contact law predicts a decrease of the friction coefficient with $F^{-1/3}$ while expression (6) leads to a decrease of $\mu$ with the normal force to the power $-0.35 \approx -1/3$. For $F_n \gg L_C$, the constant friction coefficient levels off at the value 0.27, in agreement with what is expected in the fully plastic regime.

Such a decrease of the friction coefficient has been measured experimentally for rough particle pairwise both in air (Jones, 2003; Le et al, 2023) and in liquids (Ecke and Butt, 2001; Arshad et al, 2021; Le et al, 2023).

Lobry et al (2019) introduced this load-dependent friction coefficient in numerical simulations based on the Force Coupling Method (Yeo and Maxey, 2011; Peters et al, 2016) under 3D periodic flow conditions, at imposed shear rate, $\dot{\gamma}$. To avoid crystallization, an assembly of rough ($h_r = 5.10^{-3} a_1$) bimodal particles (radii $a_1$ and $a_2 = 1.4 a_1$), in equal proportions, are considered. The force coupling method enables the computation of both hydrodynamic and contact force moments in order to get the effective stress and to deduce the viscosity.

The symbols in Fig. 10 display the variation of the computed relative viscosity with the reduced shear stress, $\eta_S \dot{\Gamma}$ with $\dot{\Gamma} = 6\pi\eta_0 a_1^2 \dot{\gamma} / L_C$. As expected, as the reduced shear stress increases, the viscosity decreases due to friction coefficient reduction.

Lobry et al (2019) then showed that these results could be reproduced using a correlation law based on simulations carried out with a constant friction coefficient (Eq. (1-3)) by introducing in the latter correlation law an effective stress-dependent friction coefficient, $\mu^{eff}(\eta_S \dot{\Gamma})$, given by (Lobry et al, 2019):

$$ \mu^{eff} = 0.27 \coth \left[ 0.27 \left( \frac{\eta_S \dot{\Gamma}}{1.69} \right)^{0.35} \right] \quad (8) $$
<figure><img src="image_11.png" /><figcaption>**Fig. 10** Variation of the reduced viscosity as a function of the normalized shear stress. Symbols: results of the numerical simulations performed at variable friction coefficient. Solid lines: model where the viscosity is computed from equations (1-3) and (8). Data extracted from Lobry et al (2019).</figcaption></figure>

The solid lines in Fig. 10 represent the results of this simple heuristic model, in good agreement with the simulation data.

Figure 11 displays the variation with the normalized shear stress of both the mean friction coefficient $<\mu>$ from the simulations, averaged over all the interparticle contacts, and the effective friction coefficient that has to be introduced—at a given shear stress—to recover the viscosity value. It is quite remarkable that $<\mu>$ and $\mu^{eff}$ are so close. It is also noticeable that the mean friction coefficient does not depend on the particle volume fraction but only on the shear stress.

The simulations and the model lead to a very simple physical picture, where the viscosity is driven by the variation of the jamming volume fraction with an effective friction coefficient, which is controlled by the mean force between contacting particles or, equivalently, by the mean suspension stress. In the low-stress range, the friction coefficient $\mu^{eff}$ diverges, and the jamming volume fraction goes to the high-$\mu$ value $\phi_m^\infty \approx 0.55$ (Tab. 1), which drives
<figure><img src="image_12.png" /><figcaption>**Fig. 11** Effective friction coefficient ($\mu^{eff}$) and mean friction coefficient ($<\mu>$), averaged over all the interparticle contacts, as a function of the normalized shear stress. The different symbols correspond to different particle volume fractions (see Fig. 10). Data extracted from Peters et al (2016).</figcaption></figure>

the suspension to the low-stress high-viscosity plateau, while in the high-stress range, $\mu$ levels off to its value in the plastic regime 0.27, driving the jamming volume fraction to the corresponding higher value (according to Eq. 3, $\phi_m(\mu = 0.27) \approx 0.625$) and resulting in the high-stress low-viscosity plateau.

Finally, that the viscosity can be captured by simply introducing an effective friction coefficient is a puzzling result. In particular, as displayed in Fig. 12, the contact force distribution is quite large which means that the friction coefficient distribution is also broad. We note that this force distribution is similar to what is observed in DST suspensions in frictional regime (Edens et al, 2021). Despite this broad distribution, it seems that the introduction of an effective friction coefficient is sufficient to capture the variation of viscosity with stress.

The results of this numerical study were then experimentally tested by Arshad et al (2021) who, like Chatté et al (2018), combined measurement of pairwise friction coefficient and rheological characterization. Arshad et al
<figure><img src="image_13.png" /><figcaption>**Fig. 12** Probability density function for the normal contact force intensity, deduced from the numerical simulations of Lobry et al (2019) carried out with a load-dependent friction coefficient for $\phi = 0.5$.</figcaption></figure>

(2021) measured the pairwise friction of polystyrene particles (Dynoseeds TS40 from Microbeads) immersed in a 10% vol. aqueous solution of ucon oil (75-H-90,000, Dow). They showed that introducing the experimental frictional law into the viscosity model proposed by Lobry et al (2019) enabled to capture the shear-thinning behavior exhibited by concentrated suspensions made of the same particles dispersed into the same liquid. The friction coefficient is measured by Lateral Force Microscopy in liquid environment. A polystyrene microsphere of radius $a = 20 \ \mu m$ is glued to the end of an AFM cantilever. Another similar particle is glued on a glass substrate mounted on a 3-axis piezoelectric actuation stage. The substrate is moved vertically at a constant velocity of 0.1$\mu m/s$ and is simultaneously oscillated in the perpendicular plane with an amplitude of about 1$\mu m$ at a frequency comprised between 25 and 100 Hz. The normal and tangential forces are deduced from the measurement of the vertical and lateral deflections of the cantilever. The variation of the friction coefficient with the normal load is displayed in Fig. 13 where are collected
<figure><img src="image_14.png" /><figcaption>**Fig. 13** Measurement of the friction coefficient as a function of the normal load for polystyrene particles (TS 40, Microbeads) immersed in a 10% vol. aqueous solution of ucon oil (75-H-90,000, Dow). The symbols correspond to the measurements carried out for six different particle pairs, different frequencies (25, 50 and 100 Hz) and different lateral displacement amplitudes (ranging from 600 to 2000 nm). The pink line is the mean friction coefficient averaged over all the realizations and the dashed blue line is the variation of μ deduced from the model proposed by Brizmer et al (2007) with $L_C = 20$ nN. Data extracted from Arshad et al (2021).</figcaption></figure>

12 measurements carried out on 6 different particle pairs. The pink line is the average of all the experiments and this average value of the friction coefficient can be fitted with:

$$ \mu = 0.18 \coth \left[ 0.18 \left( \frac{F_n}{L_C} \right)^{0.54} \right] \quad \text{with} \quad L_C = 33.2 \, \text{nN} \qquad (9) $$

In Fig. 13, the expression proposed by Brizmer is also plotted (dashed blue line) with $L_C = 20$ nN, the value proposed by Lobry et al (2019) and deduced from the characteristic material properties of the particles ($\nu \approx 0.4$, $E \approx 3$ GPa and $Y_0 \approx 60$ MPa), with $h_r/a = 5 \times 10^{-3}$. It is quite remarkable that the model of Brizmer written to describe the very idealized situation of a perfectly smooth sphere contacting a perfect smooth flat is able to give a variation of
<figure><img src="image_15.png" /><figcaption>**Fig. 14** Left: AFM micrograph of the surface of a polystyrene particle (*Dynoseeds* TS 40). Right: distribution of radii of curvature determined from the analysis of 100 linear roughness profiles measured on 4 different particles.</figcaption></figure>

the friction coefficient that is close to what is measured even though the shape of the surface asperities is far from spherical (see Fig. 14).

From the AFM micrograph, the curvature radius of the asperities can be evaluated. A mean value of about 280 nm is obtained but the distribution is broad (see Fig. 14, right) which probably explains why the friction coefficient measurements shown in Fig. 13 are scattered: from one measurement to another, the contacting asperity is not the same and the radius of curvature is different as well. The non-spherical shape of the asperities, together with the possibility of contacts taking place through a few asperities rather than through a single one, are likely to explain the differences between the experimental data and the law proposed by Brizmer et al (2007). Based on the measured surface density and mean curvature of asperities, and the maximum normal force involved in the flows considered (~1000 nN), some geometrical calculations suggest that two nearby asperities can simultaneously contact if they are separated by a distance smaller than ~2 µm. Here the estimated average distance between asperities is about 3 µm, meaning that, statistically, contact takes place mostly through one single asperity at a time. Even though those are only orders of magnitude, using oversimplified assumptions (homo geneously distributed spherical asperities), they show that it is relevant to use a mono-asperity contact model. Furthermore, the predicted and measured
values of $L_C$ are quite close. To go further into the comparison between the predictions of Brizmer and the value of $L_C = 33.2 \ nN$ deduced from the variation of $\mu$ with $F_n$, it is necessary to know the exact value of $\nu$, $E$ and $Y_0$. The first two can be found in literature: $\nu \approx 0.4$ and $E \approx 3 \ MPa$ but it is much more difficult to propose a value for $Y_0$ since the values of $Y_0$ found in the literature are around 50 $MPa$ (Martienssen and Warlimont, 2005) but typically span from 20 $MPa$ (Tomari et al, 1990) to 200 $MPa$ (Du et al, 2001). Taking $h_r = 280 \ nm$, $L_C = 33.2 \ nN$ is obtained for $Y_0 = 35.8 \ MPa$, which is in the range of reported $Y_0$ values. The good qualitative agreement between the Brizmer model and the friction coefficient measurements (overall decrease of the friction coefficient with increasing normal force, order of magnitude of $L_C$) strongly supports that the contact between particles occurs through surface asperities and confirms the findings from the pair distribution function analysis (see Sec. 2).

The rheological behavior of the suspensions made of the same particles dispersed in the same liquid is also studied in torsional parallel plate geometry with a controlled stress rheometer (Mars II, Thermofisher). The gap height is set to 1.2 mm (30 particle diameters) in order to avoid confinement effects (Yeo and Maxey, 2010b; Gallier et al, 2016). The viscosity of the suspended liquid is measured as 55 $mPa.s$. The viscosity of suspensions at 5 different values of particle volume fraction is measured and a Mooney-Rabinovitch correction is applied in order to account for the radial variation of the shear rate across the gap. The symbols in Fig. 15 display the relative viscosity as a function of the shear stress. These experimental results can be compared with the result of the model proposed by Lobry et al (2019) where the experimental friction law has been introduced (dashed blue lines). The agreement is only qualitative, in particular in the high-stress range (i.e., the low-friction coefficient range)
for which the model predicts a viscosity significantly lower than the one mea-
sured experimentally. However, a slight modification of the fitting parameters
involved in the equation which relates the jamming volume fraction to the
friction coefficient (Eq. 3) allows to obtain a very good agreement between
the experimental data and the model results, for all the particle volume frac-
tions. The best agreement is obtained for $\phi_m^0 = 0.55$ (instead of 0.546) and
$\phi_m^\infty = 0.65$ (instead of 0.70) with all the other four parameters kept the same.
On the whole, these results confirm the strong links between pairwise friction

<figure><img src="image_16.png" /><figcaption>**Fig. 15** Variation of the reduced viscosity with the shear stress. Symbols: experimental data; Blue dashed lines: expected results from the model of Lobry et al (2019) together with the experimental frictional law (Eq. (9)) with the original values included in the model (see Tab. 1); purple solid lines: results of the model with $\phi_m^0 = 0.55$ (instead of 0.546) and $\phi_m^\infty = 0.65$ (instead of 0.70) and all the other four parameters unchanged.</figcaption></figure>

and suspension rheology. They also show that the viscosity model proposed
by Lobry et al (2019), according to which the influence of a variable friction
between particles can be captured by taking into account an average friction
coefficient despite the large width of the contact force distribution, is reliable.
The possibility to relate the rheology to the microscopic friction coefficient
between particles through the model proposed by Lobry et al (2019) has been confirmed by Le et al (2023). Another important result stressed by the authors of the latter article is the influence of the solvent nature on both the frictional properties of the particles (which are also different from those measured in air) and the rheological behavior of the suspensions. The studied suspensions are made of the same polystyrene (Microbeads TS40) as those used by Arshad et al (2021) dispersed in either silicone oil, an aqueous solution of NaI or polyethylene glycol. In silicon oil, the decrease of the friction coefficient is close to that measured by Arshad et al (2021) and can be well represented by a Brizmer-like law: $\mu = 0.15 \coth (0.15 (F_n/L_C)^{0.4})$ with $L_C = 32 nN$. However, the latter authors question the idea that the particles come into contact by their surface asperities and suggest that the decrease of the friction coefficient originates in particle swelling. Further experimental investigations on particle friction together with microstructure and shear reversal measurements would be highly valuable to assess the microscopic origin of the friction coefficient decrease.

## 6 Conclusion

By recalling the results of experimental and numerical studies concerning the microstructure of non-Brownian suspensions under simple shear flow and the links between frictional contact and rheology, we have tried to demonstrate that a consistent framework can be built. Microstructure measurements in dilute suspensions show that particles come into contact due to surface roughness. Indeed the asymmetry of the pair distribution function shows that non-hydrodynamic interactions exist but, more interestingly, the precise analysis of the asymmetry measured in dilute suspensions quantitatively
demonstrates that particles come into contact via their surface asperities. Fur-
thermore, numerical simulations have shown that the viscosity of a suspension
increases as the coefficient of friction between particles increases (Gallier et al,
2014; Chèvremont et al, 2019; Singh et al, 2018). These features, which have
been broadly used to understand and describe the DST transition (Seto et al,
2013; Mari et al, 2014; Wyart and Cates, 2014), are also very useful in under-
standing the high values of the relative viscosity of concentrated non-Brownian
suspensions. When splitting the hydrodynamic contribution from the contact
contribution to the total viscosity of a suspension, it turns out that the hydro-
dynamic part of the viscosity is not sensitive to friction. Friction only affects
the contact contribution, which significantly increases with μ.

Shear reversal experiments provide an easy way to evaluate indepen-
dently these two contributions. Indeed, during a shear reversal the asymmetric
microstructure of the suspension is broken and must be reconstructed along the
new compression axis of the flow. In the meantime, the suspension is weakly
structured and the contacts between particles are few. This explains the sig-
nificant decrease in viscosity after a shear reversal. These experiments provide
quantitative information on the frictional contact interactions between parti-
cles. For example, the viscosity jump that appears just after the reversal is
strongly dependent on the value of the interparticle friction coefficient and its
measurement is meant to discriminate between a frictional and a non-frictional
suspension. It would be interesting to complement these measurements with
large-amplitude oscillatory experiments which have already shown that the
rheological response is sensitive to both the microstructure induced by the
oscillatory flow and the contact forces between particles (Bricker and Butler,
2007; Park et al, 2011).
Finally, the close link between interparticle friction and rheology is likely
to explain the shear-thinning observed in many concentrated non-Brownian
suspensions, even in the absence of adhesive forces between particles. Since
microstructure measurements show that the contact between particles takes
place by means of asperities present on the particle surface, the geometry of
the contact, which can be modeled by the contact between a sphere (mimicking
an asperity) and a smooth plane (representing the surface of the other parti-
cle), directly leads to a decrease of the friction coefficient as the interparticle
force, or equivalently the suspension stress, increases. This scenario was first
explored through numerical simulations and then experimentally tested. The
numerical simulations carried out using the Force Coupling Method consisted
in computing the viscosity of suspensions for various particle volume fractions
by injecting a contact law where the friction coefficient decreases with the
normal force. The simulation results were rationalized in the form of a model
based on the introduction of an effective friction coefficient, which depends on
the shear stress, and controls the suspension jamming volume fraction. The
experiments consisted in measuring by AFM the variation of the friction coeffi-
cient with the normal contact load between two particles immersed in a liquid
and to measure the variation of the viscosity of a suspension made of the
same particles and the same liquid. The good agreement between the experi-
mental results and the predictions of the model resulting from the numerical
simulations points out the relevance of a scenario based on a load-dependent
friction coefficient proposed to explain shear-thinning in non-Brownian and
non-adhesive suspensions. Finally, it should be recalled that the decrease of
the friction coefficient with load can have other origins than the geometry
of the single asperity contact. Such a decrease has been measured by Chatté
et al (2018) for swollen particles and, in that case, is a peculiar feature of
the friction of polymer brushes. Nonetheless, we are quite confident that the scenario described above should apply to more standard systems such as suspensions of mineral or polymeric particles dispersed in an inert solvent. In the future, it will probably be useful to further test the load-dependent friction coefficient scenario depicted above on other kinds of suspensions using all the available tools, including microstructure measurements, shear reversal experiments, steady viscosity measurements and determination of the pairwise friction coefficient.

**Acknowledgments.** We are grateful to Duncan Gilbert for fruitful discussions and for providing the shear reversal data displayed in Fig. 6. This work was supported by the French National Agency (ANR) under the program Blanc AMARHEO (ANR-18-CE06-0009-01).

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