G Bassignana
email: giulia.bassignana@gmail.com.

J Fransson

O Colliot

V Zujovic

F De

Vico Fallani

Identification of Driver Nodes in Genetic Networks Regulating Macrophage Activation

Macrophage cells play an important role in the Multiple Sclerosis disease. They are known to participate both to the degenerative process, myelin destruction, and to the regenerative one, coordinating remyelination. The correct genetic activation of macrophage phenotypes permits a correct remyelinating response [1], thus the possibility to steer it towards an healthy state while acting on a limited number of genes (drivers) would be greatly advantageous.

We modeled macrophage activation as a network (Figure 1.a), where nodes correspond to genes involved in phosphorylation and directed links corresponded to significant activations as retrieved from the STRING Database [2]. We adopted the structural controllability framework [3], mapping the Kalman controllability criterion into the maximum cardinality matching on a graph, to identify the driver nodes. Because different configuration of driver nodes are in general possible, we repeated the analysis R=60000 times and shuffled the order of the nodes in the adjacency matrix in order to explore different configurations. We defined the node driverness as the frequency of times a nodes is selected as driver over R iterations.

Our work is a preliminary step towards the identification of the genes influencing the inflammatory process of macrophages, a crucial mechanism in multiple sclerosis' disease.

Figure 1 :

 1 Figure 1: (a) Inferred network of genes involved in phosphorylation, 46 nodes and 168 directed edges. Size of the nodes corresponds to their degree, color to percentage of outgoing links. (b) Driverness of the nodes over their in-degree. (c) Genes ranked by driverness.


Bobillier Chaumon
email: marc-eric.bobillier-chaumon@univ-lyon2.fr

Study of conditions of use of E-services accessible to visually disabled persons

Keywords: Accessibility, Visually disabled persons, E-Government

Study of conditions of use of E-services accessible to

visually disabled persons.

INTRODUCTION & CONTEXT

The development of new technologies may prove to be a tremendous springboard for the integration of disabled persons (DP) provided that these environments are accessible, usable, and useful; in other words that they take into consideration the various characteristics of the activity and the needs and particularities (cognitive, perceptive, or motive) related to the disability of the users [START_REF] Paciello | M Web accessibility for people with disabilities[END_REF][START_REF] Wai | Web Content Accessibility Guidelines 2.0 Checklist[END_REF]. This question is even more pertinent in the context of quasigeneralized media coverage of the service relationship (Eadministration, E-banking, E-commerce, etc.). Various studies worldwide have shown the very weak respect of accessibility criteria despite the numerous standards (section 508 in the USA, the law concerning digital accessibility of administrative services in France, etc.) or labels (Blindsurfeur in Belgium, See it Right in England, Accessiweb in France, etc.) required during the conception of these online services [START_REF] Jaeger | Telecommunication's policy and individuals with disabilities: Issues of accessibility and social inclusion. In the policy and research agenda[END_REF]: more than 75% of the assessed sites present level 1 WAI guideline accessibility flaws [START_REF] Wai | Web Content Accessibility Guidelines 2.0 Checklist[END_REF], meaning that accessibility to these sites is impossible for DP [START_REF] Beckett | 30% accessible -a survey of the UK Wide Web[END_REF][START_REF] Braillenet | Pour une meilleure accessibilité des sites publics aux personnes handicapées[END_REF][START_REF] Fagan | An accessibility study of state legislative Web sites Government[END_REF]. This is becoming a serious problem insomuch as accessibility seems to be one of the social and political levers playing a role in the amelioration of the quality of life of people with disabilities [START_REF] Accessibilité | Enjeux de mots : regards multiculturels sur les sociétés de l'information[END_REF][START_REF] Bastalaer | Améliorer l'accessibilité des sites Web publics pour les personnes handicapées de la vue[END_REF][START_REF]WAI Accessibility is a Social[END_REF]. Indeed, if on the one hand, accessible Internet sites can allow DP greater autonomy by giving them the possibility to complete various activities by themselves; on the other hand, these technologies are also the source of a new type of social stigmatism because of their lack of technological accessibility. The DP must first ask for help to use the system and perform the act.

The objective of our communication is to determine the real contributions of accessible E-services for visually disabled persons as well as evaluate the repercussions of the lack of digital accessibility to these E-services on this population 1 . This is based on the hypothesis that inaccessible technologies will only confirm the inequalities of access to information and services between able-bodied persons and disabled persons, and could even reinforce and intensify them.

In this perspective, we studied the conditions of use of accessible electronic services.

In this perspective, we propose an original approach to study the conditions of use of electronic services accessible to disabled persons. The methodological approach is indeed both:

-Multidimensional: by diagnosing their utility (adaptation to user expectations), usability (ease of use), accessibility (respect of standards and principles), and acceptability (meaning and stakes attributed to the technologies).

-And comparative: since carried out on two user samples (able-bodied and visually impaired) with various levels of E-service experience (novice to expert).

METHODS

Our approach draws on three complementary studies:

• The utility of the sites was studied using an online questionnaire on 439 DP with motive, perceptive, and cognitive disabilities in order to determine what the Eservices bring to the DP and what the DP expect from them.

• The usability and accessibility of the sites was evaluated 2 with user tests based on 3 scenarios (specified below) and two populations: 10 visually disabled participants (VDP) and 10 sighted participants. The participants had comparable sociobiographical characteristics (age, sex, education, etc.), only the mastery of the Internet varied equally in each group (5 novices and 5 experts). For this confrontation, we wanted to know if the problems encountered by the blind were the same as those of the sighted (general problems of usability), or if the problems were amplified by a choice of technology incompatible with their perceptive limits (problems of accessibility).

The data collection tools used were simultaneous verbalisation, observations and a satisfaction questionnaire (adapted from the Wammi grid 3 ). The indicators measured were the efficiency (time, frequency and nature of errors, omissions, number of selections/strategies to perform a scenario), satisfaction (score out of 5 on the Wammi scale) and efficacy (pass/fail test).

• The acceptability of E-services was analysed using semidirective interviews of 8 blind participants. The objective was to determine to what extent these services could transform the practices, contacts, and status of the blind. These interviews were recorded and entirely transcribed.

A thematic content analysis was performed on this corpus.

MAIN RESULTS

Study of utility of the sites

Of the 439 DP who answered the online questionnaire, 52% indicated having help with their classic administrative processes. This is due to difficulties in mobility (33.5%) physical accessibility to the building or administrative hours (30.5%), the complexity of forms (23%), or difficult contact with agents (feelings of "being different") (13%). E-administration thus seems like an alternative solution that, incidentally, 52.4% of participants declared to have already used and 32.4% would like to use. These users benefited from them. The role of these E-services as a facilitating tool (finding information, avoiding going out to fill out forms, etc.) is thus confirmed by 90%. The fact that these electronic services allow the DP to avoid requesting someone's help to perform tasks that are often intimate and personal and that they favour the social integration of the DP by providing the same access as an able-bodied person 2 Ergonomic inspections of accessibility were also performed during the research but won't be presented here due to lack of room.

3 http://www.wammi.com/using.html is underlined by, respectively, 90% and 96% of participants.

For the 40% who refuse to use E-services, this position is principally due to technical and ergonomic causes (lack of reliability and accessibility of environments, data protection, delay of data processing, etc.) informational causes (services not complying with the users' needs, unawareness of services offered) and personal reasons (preference for classic modes of access, fear of social isolation, entry errors, etc.). DP support (sensitisation, education, etc.) in the acquisition of E-services would certainly help breakdown these barriers at least in part. Finally, even though 46% were opposed to transforming classic services into E-services, and this despite the benefits indicated above, this position should not be seen as a rejection of innovation, but rather as concern and worry, shared by 60% of participants that their specific needs and profiles would not be sufficiently taken into account in the conception of these technologies.

Evaluation of the usability and accessibility

Three scenarios were used for these tests: information retrieval from the ANPE (French national employment agency) site (Scenario 1: informational), participation in a public forum (Scenario 2: interactive) and filling out an online form on the Nancy les Vandoeuvre municipal site (Scenario 3: transactional). From these analyses (Cf. Table 1), large divergences between the two populations emerge concerning the usability of E-services, as would be expected. The efficacy and efficiency are thus lower for the blind participants than for the sighted participants (with the performances, notably the time, that are up to seven times superior to those of the sighted). However, the satisfaction is globally the same for both groups. We even note a surprising result concerning scenario 3 where the efficiency (for the strategy and selection) is almost advantageous for the blind participants. This piece of data could be explained by a learning effect since the users performed scenarios 2 and 3 on the same site. So, it is the blind expert participants who exploited this learning the best, undoubtedly being used to taking advantage of each action to compensate for their disability.

We note moreover that the usage difficulties penalize mainly the blind the least habituated. The novice blind users seem, in fact, extremely resourceless in dealing with the problem of accessibility of the interface whereas the expert blind users, from their practice and their experience, solicit mental models to compensate for the ergonomic deficiencies of the tool. We can therefore observe a recourse to such schemas when certain blind users anticipate the display of information or interpret inexplicit or polysemous wording by calling on their navigation habits: "Normally, we should find this information by clicking here…" On the level of navigation strategies, we can observe that novices opted more often to use search engines to enter the key words of the scenarios to perform (on average 4 of 5 novices) whereas the experts preferred going to the home page to systematically read the proposed links with a voice synthesiser (3 experts of 5). The results show that the failures are more frequent for novices because the key words entered in the search engine are often vague and imprecise. This strategy, which we could qualify as heuristic, is less efficient than the experts' more systematic and general strategy: their mastery of the Jaws system allowed them indeed to consult different the different links very quickly and their experience with E-services also gives them the possibility to promptly locate the most pertinent elements to reach their goal.

These usage problems come specifically from the choice of conception that does not take into account the perceptive limits of the level of participants, and more generally the principles of accessibility: for example, we can cite newly opening contextual menus remaining unsignalled the appearance of contextual menus not signalled, the density of information presented (over 84 links on a single opening page of a municipal site), the absence of textual alternatives to images, the incoherent structure of pages organised in table format, the use of javascript which makes the screen reader used (Jaws) obsolete, insufficiently explicit links (with do not consider the remaining text content), the opening of new windows not signalled, etc.

Other difficulties common to both groups show, instead, a lack of ergonomics of the sites (according to [START_REF] Bastien | L'ergonomie des sites web[END_REF]). It is mainly a matter of certain polysemic terms (Téléprocédures~Téléservices), of confusing visited and non-visited links, of the non-deactivation of links on the current page, of unclear error messages, of the dynamic reorganisation of the menus from one page to another, etc.

In the end, these results prove that these sites do not take into consideration the inabilities of VDP, and specifically for E-services novices. The accessibility to certain content is very difficult, short of impossible; but moreover, the use of E-services generates a greater mental load that hinders all involvement in the process (shown by the mediocre level of efficiency and by the efforts made to overcome the obstacles to use).

Analysis of acceptability

The thematic analysis performed on these data brought out several themes grouped into contributions and risks related to the use of E-services (cf. Overall, the visually disabled persons questioned felt that the benefits of the E-services far outweighed the difficulties posed (63% to 37%). E-services thus open "spaces of possibility" that allow DP not only to avoid the cognitive and operative constraints (spatial and temporal) due to their disability, but also to regain a certain autonomy and freedom of action.

These new perspectives contribute to their psychological stability and personal fulfilment (self-esteem). Nevertheless, these people do not idealise these new services either since they are well aware of the stakes linked to the lack of accessibility. Therefore, if administrations are not able to better organize their electronic services, there is a great risk of marginalizing people with specific needs even further. From this point of view, the lack of accessibility represents an additional factor of exclusion and an obstacle to the integration of disabled persons.

Conversely, an exclusive and excessive use of these tools could also turn out to be dangerous since leading to a social isolation (doing everything remotely from home) and the loss of a know-how of physical mobility combined with a loss of autonomy of the VDP. "The problem is that staying home, not moving much, that can create a certain isolation but also a small decline in my mobility comfort If I stay at home for a long time and do everything on the Internet, at a certain point, there will be certain things I wouldn't have the courage to do anymore".

CONCLUSION AND DISCUSSION

Our study enables us to show that the conditions of use of E-services depend on three principal factors:

Utility factors in such that the proposed E-services must meet the expectations of the visually disabled persons and bring them a real added value through their use (by augmenting their ability to act, interact, and be informed).

Ergonomic factors (usability and accessibility) in which the specificities of visually disabled persons as well as their level of expertise (with the internet and screen readers) are taken into consideration from the conception.

Psychosocial factors of where the proposed Eservices offer the possibility to truly compensate, assist, and valorise visually disabled persons. These technologies can, in fact, give value to the individual and confirm/reinforce his place in society by providing him autonomy. The mastery of these ICT could therefore result in the modification of his own perception, the redefinition of his relationship with his entourage and the amelioration of his capacity for social integration. However, these contributions could be limited by the choice of conception. We have, in fact, shown that the quality of ergonomics and the insufficient level of accessibility of the interfaces risk frustrating the user's interaction with the administrative sites and in the end hindering their appropriation and acceptance.

Also, in opting for environments that do not take into account the specific needs and aptitudes of disabled persons, the site creators risk establishing a sort of "technological glass ceiling" that prevents their disabled users from using the E-services offered naturally, whereas able-bodied persons do so with no apparent difficulty. This digital exclusion would be amplified by a social exclusion if the services could only be accessed by the technological channel --as is planned in the law concerning administrative modernisation.

In sum, the digital chasm resulting from the lack of technological accessibility can be addressed as an additional dimension that adds to the social chasms that disabled persons are subjected to and as a factor contributing to their exclusion and their social isolation.

Table 1 : Main results of user tests

 1

	Efficiency

  Table 2 below).

	Impact of E-	References to	E-services perceived	E-services perceived
	services on	the theme in 8	more as a source of	more as a source of
	the lives of			interviews		improvement	deterioration
	VDP
	Social			21		13 (62%)	8 (38%)
	dimension				Autonomy,	social	Disembodied relation to
					integration (through	machines (absence of
					equal access),	personalised attention
					Social	recognition	and consideration)
					(acting without help	Risk of social isolation
					like	able-bodied	and fear of social
					persons)	exclusion brought on by
							a digital exclusion
	Psychological	12		7 (58%)	5 (42%)
	dimension				Self-esteem,	Loss of "know-how" of
					evaluation	(being	mobility
					able to fend for one-	Fear of losing control of
					self)	information transmitted
					Conservation	of	(hacking), or increased
							and	control	(cross
					privacy of personal	referencing	of
					information	information.)
							Feeling of helplessness
							when confronted with
							an	environment
							perceived as complex
	Cognitive			11		8 (73%)	3 (27%)
	dimension				Ability	to	read,	Entropy	phenomena:
					classify and collect	Sorting through the
					information in a	mass of information
					virtual environment	presented
					"Demystification" of	Standardised content of
					the	administrative	E-services	and
					process through a	inadequacies to the
					simplified access	needs and profiles of the
					Acquisition of an	VDP
					administrative culture
	Instrumental			13		8 (61%)	5 (39%)
	and operational dimension				Comfort of life: more Possibility for tenfold mobility	Insufficient accessibility	digital
					increase in action,
					interaction,	and
					information
	Total			57		36 (63%)	21 (37%)

Table 2 : Main results of thematic analyses of acceptability interviews

 2

These results are extracted from research on the digital accessibility of electronic administration (ADELA project) financed by the Minister of Research and New Technology (Ministère Délégué à la Recherche et aux Nouvelles Technologies) (Nov.

to Dec.

2005).


Laurent Lobry

Elisabeth Lemaire

Frédéric Blanc

Stany Gallier

François Peters

Shear thinning in non-Brownian suspensions explained by variable friction between particles

We propose to explain shear thinning behaviour observed in most concentrated non-Brownian suspensions by variable friction between particles. Considering the low magnitude of the forces experienced by the particles of suspensions under shear flow, it is first argued that rough particles come into solid contact through one or a few asperities. In such a few-asperity elastic-plastic contact, the friction coefficient is expected not to be constant but to decrease with increasing normal load. Simulations based on the Force Coupling Method and including such a load-dependent friction coefficient are performed for various particle volume fractions. The results of the numerical simulations are compared to viscosity measurements carried out on suspensions of polystyrene particles (40 µm in diameter) dispersed in a Newtonian silicon oil. The agreement is shown to be satisfactory. Furthermore, the comparison between the simulations conducted either with a constant or a load-dependent friction coefficient provides a model for the shearthinning viscosity. In this model the effective friction coefficient µ ef f is specified by the effective normal contact force which is simply proportional to the bulk shear stress. As the shear stress increases, µ ef f decreases and the jamming volume fraction increases, leading to the reduction of the viscosity. At last, using this model, we show that it is possible to evaluate the microscopic friction coefficient for each applied shear stress from the rheometric measurements.

Introduction

Suspensions of rigid particles in low Reynolds number flows are ubiquitous in industry (food transport, cosmetic products, civil engineering, etc.) or natural flows (such as mud or lava flows) to mention but a few. This wide occurrence of suspensions has fostered significant research in the past years that has revealed a great complexity in the behaviour of these systems. Concentrated suspensions may present shear-thinning or shear-thickening behaviours, anisotropic normal stresses, irreversibility under oscillating shear and many other complex behaviours. Some of these features come from the possible complexity of the suspensions themselves. For instance, the suspending liquid may be non-Newtonian, particles may have irregular shape, be elongated or faceted, be deformable or polydisperse, they may interact through colloidal forces, Brownian or buoyancy forces may exist... But, it is quite fascinating that even very simple "model" suspensions made of non-Brownian, non-colloidal, monodisperse spherical particles dispersed in a Newtonian fluid exhibit non-Newtonian behaviours. Most of these behaviours may be explained by the coupling between flow, contact interactions and microstructure. To mention just a few examples, [START_REF] Pham | Particle dispersion in sheared suspensions: Crucial role of solid-solid contacts[END_REF] showed that irreversible dynamics of non-Brownian neutrally buoyant particles subjected to a periodic shear flow originates in particle solid-solid contacts. [START_REF] Gadala-Maria | Shear-induced structure in a concentrated suspension of solid spheres[END_REF] proposed to explain the transient response of the viscosity under shear reversal by the presence of an asymmetric shear-induced microstructure that is broken and re-built along the new direction of the flow. Later [START_REF] Blanc | Experimental signature of the pair trajectories of rough spheres in the shear-induced microstructure in noncolloidal suspensions[END_REF] showed that the asymmetry of the microstructure comes from solid contact between particles through the asperities they have at their surface. The involvement of contact forces has also made it possible to explain discontinuous shearthickening often observed in very concentrated suspensions of non-Brownian spheres [START_REF] Seto | Discontinuous shear thickening of frictional hard-sphere suspensions[END_REF][START_REF] Mari | Shear thickening, frictionless and frictional rheologies in non-brownian suspensions[END_REF]) by a transition from frictionless to frictional contacts. The effect of solid friction between particles on both normal stress differences and viscosity of concentrated suspensions is also very significant. Numerical simulations of [START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF] showed that changing the friction coefficient, µ, from 0 to 0.5 for a 0.45 volume fraction suspension leads to an increase of the viscosity by a factor of 1.5 while the first normal stress difference magnitude-which is negative-is divided by almost 2 and the second normal stress is two-fold-increased. As the particle volume fraction, φ, increases, the contribution of contact forces to the viscosity becomes more and more important compared to the contribution of hydrodynamic forces. For rough frictional particles with µ = 0.5, hydrodynamic and contact contributions are almost the same for φ = 0.4 but, at φ = 0.46, the contact contribution is around twice the hydrodynamic contribution. So one has to keep in mind that solid contacts between particles is a key ingredient to explain non-Brownian suspension rheology.

Among non-Newtonian behaviours that still raise question, shear-thinning occupies a special place because it is quite ubiquitous and can be observed in most experimental results on shear rate-shear stress dependence but is rarely discussed. A first attempt to explain shear-thinning has been made by [START_REF] Acrivos | On the measurement of the relative viscosity of suspensions[END_REF] who showed that the apparent shear-thinning that they observed when the viscosity of a suspension of non-Brownian particles is measured in a Couette device is due to a mismatch in the densities of the liquid and of the particles. But whereas this explanation seems compelling in the case of the Couette flow where the larger and larger resuspension of the denser particles as the shear rate increases should lead to a decrease of the measured apparent viscosity, it cannot explain shear-tinning observed in other flow geometries such as for example in torsional parallel-plate flow. Shear-thinning, however, has been reported in rotating parallel plate measurements by several authors [START_REF] Dbouk | Rheology of concentrated suspensions and shear-induced migration[END_REF][START_REF] Zarraga | The characterization of the total stress of concentrated suspensions of noncolloidal spheres in newtonian fluids[END_REF][START_REF] Vázquez-Quesada | Shear thinning of noncolloidal suspensions[END_REF]. [START_REF] Vázquez-Quesada | Shear thinning of noncolloidal suspensions[END_REF] explains the shearthinning they observed by the shear-thinning behaviour of the suspending liquid (silicon oil) itself. More precisely, the suspending liquid is Newtonian at the macroscopic shear rates involved in the measurement of the viscosity but may be shear-thinning in the range of the higher shear rates experienced by the host liquid in the interparticle gap. Later, the authors showed that this mechanism cannot apply for all systems and suggested that variable contact friction between particles may also be responsible for shear-thinning [START_REF] Vazquez-Quesada | Investigating the causes of shear-thinning in non-colloidal suspensions: Experiments and simulations[END_REF]. Recently, [START_REF] Tanner | A bootstrap mechanism for non-colloidal suspension viscosity[END_REF] proposed a model based on a bootstrap mechanism of friction that introduces shear-thinning by assuming that the friction coefficient is a decreasing function of the sliding speed between particles. Here, we explore the same kind of idea and propose to explain shear-thinning by a decrease of the friction coefficient between particles as the contact force between them, which is controlled by the suspension shear stress, increases. Prior to describe the contact law between particles, we would like to highlight that particle contacts are enabled by the asperities that are present at particle surface. For perfectly smooth particles, solid contact is prevented by lubrication forces that diverge as the particle separation tends to zero. On the opposite, for rough particles, contact between particles is made possible through isolated asperities that maintain finite lubrication forces between particles. In other words, roughness is able to screen lubrication divergence. In such a scenario, only one or a few asperities may be involved in contact, and the macroscopic Amontons-Coulomb law that is used to describe contact between macroscopic surfaces and according to which the friction coefficient does not depend on the magnitude of the tangential or normal load is expected not to hold anymore. Indeed, this macroscopic contact law is usually understood in the frame of theories where a large number of asperities are compressed [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF]. Instead, in the case of a small number of elastically contacting asperities, the friction coefficient is expected to decrease with load [START_REF] Archard | Elastic deformation and the laws of friction[END_REF]).

In the present paper, we modify the contact model presented in [START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF]) in order to account for a variable friction between particles. The first part is dedicated to the presentation of the numerical method and of the contact model that has been chosen to introduce a load-dependent friction coefficient in the numerical simulations. The second section presents the simulation results and proposes a viscosity model that makes it possible to estimate the influence of any particular friction law on the viscosity without performing new simulations. In the third section, experiments carried out on suspensions of monodisperse polystyrene particles, 40 µm in diameter, dispersed in a silicone oil for five values of the particle volume fraction are presented. And the last section is devoted to a comparison of the numerical model to the experimental results.

Numerical method

The Force Coupling Method

We use the Force Coupling Method to study the rheology of non-Brownian suspensions made of rigid rough particles in a Newtonian liquid, η 0 in viscosity. The main features of the method are recalled in [START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF]) and the interested reader will find more detail in (Yeo & Maxey 2010b). In this method, the motion of the fluid is computed in the whole meshed domain, where the presence of the particles is taken into account using a specified force density that enforces rigid body motion in the particles. Small scale flows between the particles are not resolved by the direct computation, so that the hydrodynamic force and other moments are corrected using theoretical expressions from lubrication approximation. Thus, this method is able to catch both long range and short range hydrodynamic interactions. Inertia is neglected in computing both the liquid and the particle motion, and the particle density is matched to that of the liquid. As a consequence, the total force and torque on each particle vanish.

F h + F c = 0
(2.1)

T h + T c = 0 (2.2)
where the superscript h stands for the hydrodynamic moments (both direct computation and lubrication corrections) and c for the contact moments. The later will be described in the following section, and the former are computed as explained in [START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF].

The hydrodynamic force moment (stresslet) S h as well as the contact force moment S c of each particle are computed too, yielding the effective stress, and then the relative viscosity η S defined as the ratio of the suspension viscosity to the host liquid viscosity.

The particles are contained in a cubic domain, where the shear rate γ is imposed. The flow is made periodic in all three dimensions using Lee Edwards conditions. The total solid volume fraction is denoted by φ. We consider a bidisperse collection of spheres, of radius a 1 and a 2 =1.4 a 1 , with the same volume fraction φ/2, in order to avoid crystallisation. The particular value a 1 /a 2 = 1.4 that was chosen is widely used in the literature [START_REF] Seto | Discontinuous shear thickening of frictional hard-sphere suspensions[END_REF][START_REF] Mari | Shear thickening, frictionless and frictional rheologies in non-brownian suspensions[END_REF][START_REF] Ness | Flow regime transitions in dense non-brownian suspensions: Rheology, microstructural characterization, and constitutive modeling[END_REF][START_REF] Seto | Microstructure and thickening of dense suspensions under extensional and shear flows[END_REF]. We note here that a recent study was published, that explored the influence of bidispersity on the suspension rheology, and compared it to polydisperse system rheology [START_REF] Pednekar | Bidisperse and polydisperse suspension rheology at large solid fraction[END_REF]. From this study, it is possible to infer that, at least for the volume fraction range that is probed in the present article, the viscosity is nearly the same as for a monodisperse system without crystallisation. The size of the computation cell is L x × L y × L z = 16a 1 × 16a 1 × 16a 1 corresponding to approximately 300 particles for a solid volume fraction φ=0.5. The suspension is sheared during a total strain ranging between 40 and 100, depending on the solid volume fraction, in order to obtain correct averaging.

Contact model

Introduction

Up to now, most simulations of sheared frictional suspensions [START_REF] Mari | Shear thickening, frictionless and frictional rheologies in non-brownian suspensions[END_REF][START_REF] Seto | Discontinuous shear thickening of frictional hard-sphere suspensions[END_REF][START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF][START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF] have considered a simple model of frictional contact with a constant static friction coefficient, as defined by the ratio of the largest tangential force that is allowed before sliding motion occurs to the normal force. This simple Amontons-Coulomb model is a simplified description of the contact between macroscopic surfaces for which the real area of contact is far smaller than the apparent macroscopic contact area, meaning that contact occurs between surface asperities [START_REF] Tabor | Friction-the present state of our understanding[END_REF]. Considering a macroscopic contact, it is usually assumed that a statistically large number of asperities are deformed as in [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF]. Even though the contact area for each single asperity is not proportional to the corresponding load, the total area is approximately proportional to the total load, leading to a static friction coefficient that does not depend on the load, in agreement with Amontons-Coulomb law.

Contrarily, when a contact between a single asperity and a smooth surface is considered, the contact law is not linear anymore and a decrease of the friction coefficient with increasing normal load is expected. Many models have been proposed, where the deformation remains elastic at low applied load, and enters a plastic regime as the load is increased (see for example [START_REF] Chang | An elastic-plastic model for the contact of rough surfaces[END_REF]Brizmer et al. 2006b[START_REF] Brizmer | Elastic-plastic spherical contact under combined normal and tangential loading in full stick[END_REF]), and some experimental validations have been performed for the contact of a smooth sphere against a flat substrate, leading to a load-dependent static friction coefficient [START_REF] Ovcharenko | Experimental study of adhesive static friction in a spherical elastic-plastic contact[END_REF].

The present paper deals with contacting micro-particles immersed in a host liquid, the whole suspension experiencing a shear flow. This particular problem may be considered quite specific for several reasons. First, the contact forces that are controlled by the sheared suspension bulk stress may be very weak, depending on the shear rate and the volume fraction. As a consequence, contact occurs between a small number of asperities and not between a large distribution of asperities as in Greenwood-Williamson model (see Sec. 2.2.4 for details on the model parameters and the estimation of the number of contacting asperities). Hence a load-dependent friction coefficient is expected, instead of the Amontons-Coulomb law.

During the last 2-3 decades, experimental studies have been performed by means of the so-called colloidal probe technique, where a single bead, a few microns in diameter typically, is attached to the cantilever of an atomic force microscope (AFM) (see for example the review by [START_REF] Butt | Force measurements with the atomic force microscope: Technique, interpretation and applications[END_REF]). Some of these studies deal with the adhesion and/or friction coefficient between a particle and a hard flat substrate. When experiments are performed in dry atmosphere, a significant adhesion is measured [START_REF] Schaefer | Surface roughness and its influence on particle adhesion using atomic force techniques[END_REF][START_REF] Biggs | Atomic force microscopy investigation of the adhesion between a single polymer sphere and a flat surface[END_REF], that may depend on the applied load [START_REF] Biggs | Atomic force microscopy investigation of the adhesion between a single polymer sphere and a flat surface[END_REF] possibly due to plastic deformation of the surface asperities [START_REF] Reitsma | Elasto-plastic and visco-elastic deformations of a polymer sphere measured using colloid probe and scanning electron microscopy[END_REF]. The static friction coefficient is affected by the adhesion forces, so that a finite friction force is observed with vanishing normal load [START_REF] Biggs | Atomic force microscopy investigation of the adhesion between a single polymer sphere and a flat surface[END_REF][START_REF] Reitsma | Elasto-plastic and visco-elastic deformations of a polymer sphere measured using colloid probe and scanning electron microscopy[END_REF][START_REF] Ecke | Friction between individual microcontacts[END_REF]. Adhesion forces between particles have been also measured for smooth particles in air, again leading to a friction coefficient that decreases as the load increases.

The intensity of the adhesion depends on the nature of the particles and on the surface roughness [START_REF] Heim | Adhesion forces between individual gold and polystyrene particles[END_REF][START_REF] Ling | Quantitative measurement of friction between single microspheres by friction force microscopy[END_REF]. When particles are immersed in water, the interaction forces are more complicated due to electric double layer [START_REF] Ducker | Measurement of forces in liquids using a force microscope[END_REF][START_REF] Li | Direct measurement of interaction forces between colloidal particles using the scanning force microscope[END_REF][START_REF] Zou | Influence of nanoscale surface roughness on colloidal force measurements[END_REF]. In the case of weak adhesion, the friction force between silica particles in a salty alkaline solution has been shown to follow Amontons-Coulomb law [START_REF] Fernandez | Direct, robust technique for the measurement of friction between microspheres[END_REF]. This contrasts with the results of the article of [START_REF] Chatté | Shear thinning in non-brownian suspensions[END_REF] who used a quartz-tuning fork based AFM to measure the friction force between PVC particles immersed in a plasticizer (Dinch). They observed that the friction coefficient decreased with increasing normal load. To conclude this overview of experimental studies, the friction law between microparticles in a liquid strongly depends on the particular system that is considered, and is affected by many factors such as adhesion, roughness, plastic deformation and surface coating that may introduce discrepancies with the Amontons-Coulomb law and yield an apparent friction coefficient that decreases as the load increases. In the following subsection, we focus on the choice of a contact model that would account for such load-dependent friction properties and we discuss the main underlying assumptions.

Choice of a contact model

As discussed above, the sliding friction between two rough micro-spheres is expected to involve mono-or few-asperity contacts which would result in load-dependent friction coefficient. Among the relevant models available in the literature, we select one of the simplest that has been proposed by [START_REF] Brizmer | Elastic-plastic spherical contact under combined normal and tangential loading in full stick[END_REF] who investigated the elasticplastic contact between a single asperity and a rigid flat under full stick conditions. Using the finite element method, they show that, at low normal loads, the contact is elastic and the friction coefficient decreases with increasing normal load while, for higher normal loads, the contact becomes more and more plastic and the friction coefficient approaches a constant value. The decrease of the friction coefficient in the elastic regime can be explained by the variation of the contact area. According to the Hertz theory, the area of the elastic contact zone varies as F 2/3 N (F N being the normal force), which leads to a decrease of the friction coefficient with increasing normal force: µ ∝ F -1/3 N . When the normal force is high enough for the contact to enter the plastic regime, the friction coefficient levels off at a constant value of about 0.3. In the following, we are going to use a simplified version of this model. Before explaining it in more detail, we have to make a few comments.

(i) the decrease of µ with increasing normal force is expected in the elastic regime i.e. at low normal loads. Thus, shear-thinning behaviour is anticipated for low shear stresses.

(ii) in the plastic regime, at higher normal loads, µ is expected to level off and the viscosity to reach a plateau.

(iii) the existence of an elastoplastic transition introduces a force scale in the system, L c , which is the critical normal force at yield inception.

(iv) a single asperity contact is considered but the contact model can be straightforwardly refined to account for a few contacting asperities and would display the same qualitative trends.

(v) the contact is hysteretic at high normal load. This means that, in the plastic regime, the deformation is not totally recovered upon unloading. In the following the irreversibility associated with plasticity will not be considered. The main reason behind this simplification is the following: during the contact phase, the primary effect of plasticity on the particle dynamics would be depicted as a reduction of the asperity height and so of the interparticle distance. Several authors showed that the viscosity weakly increases with decreasing particle separation distance, i.e. typically less than 20% for a hundredfold decrease of the distance between particle surfaces [START_REF] Sierou | Rheology and microstructure in concentrated noncolloidal suspensions[END_REF]Yeo & Maxey 2010a;[START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF]. Thus, considering the weakness of the interparticle distance effects on viscosity, contact irreversibility will be neglected.

To conclude this discussion on the choice of the contact model, we would like to stress out that the aim of the present paper is not to elucidate the micro-mechanisms at play in contact between particles but to test the influence of a variable friction coefficient on the rheological behaviour of non-Brownian suspensions. Notwithstanding this remark, we will show in Sec. 5 that the numerical simulations conducted with such an elasticplastic contact model yield flow curves in quite good agreement with experimental data. The realistic values of the physical parameters (Young's modulus, Poisson's ratio, yield strength, roughness height) that have to be introduced in the model to get a good agreement with the experimental results are a strong argument in favour of the presence of elastic-plastic contacts between particles. However, we do not pretend that other mechanisms cannot contribute to the decrease of the friction coefficient with increasing shear stress and to its saturation at high shear stresses. In particular, it is likely that the number of contacting asperities increases with load which would lead also to a saturation of µ.

An elastic-plastic mono-asperity contact model

We consider rough spherical particles of radius a 1 or a 2 that come into contact, through their surface asperities. It is assumed that the whole contact between particles occurs between one single asperity, modelled by a spherical surface with radius h r , and a smooth patch of the second particle (figure 1). In the original work of [START_REF] Brizmer | Elastic-plastic spherical contact under combined normal and tangential loading in full stick[END_REF], the sphere was made of an elastic linear isotropic hardening material, where the Hooke's and Prandtl-Reuss constitutive law governed the elastic and plastic regime, respectively. A normal preload P was applied first and kept constant as the tangential load Q was increased until sliding inception, which was treated as a plastic yield failure. The friction coefficient was defined as the ratio of the largest value of the tangential load Q max to the normal preload P , and a correlation law was fitted to the variation of Q max with P . Two different regimes were identified: when the preload is smaller than a critical value, L c (equation 2.4), the deformation of the sphere keeps fully elastic during normal loading, and the friction coefficient decreases with the applied preload approximately as (P/L c ) 1/3 . As P is increased, the fraction of the material that experiences plastic deformation increases and the friction coefficient gradually tends to a constant value. The correlation between the preload P and the surface overlap is also provided, that includes the transition from elastic to plastic regime.

In the present article, the contact force exerted on the smooth patch (figure 1) is split into the force normal to the surfaces, F n and the tangential force, F t . The normal force

F n = -L c δ δ c 3/2 n (2.3)
where L c and δ c are the critical normal load and surface overlap at yield inception, respectively:

L c = Lc π 3 Y 0 6 C 3 ν h r 2(1 -ν 2 ) Y 0 E 2 (2.4) δ c = δc h r πC ν (1 -ν 2 ) Y 0 E 2 (2.5)
E, ν and Y 0 are the Young's modulus , Poisson's ratio and yield strength of the material respectively, and C ν = 1.234 + 1.256 ν. Equations 2.3, 2.4 and 2.5 differ from the usual Hertz contact law between a sphere and a plane half-space made of the same material by the over-bar quantities 2.6 and 2.7, that are characteristic of the full stick contact condition: Lc = 8.88ν -10.13 ν 2 + 0.089 (2.6) δc = 6.82ν -7.83 ν 2 + 0.0586 (2.7)

The case that was considered in the original articles of Brizmer et al. (2006aBrizmer et al. ( ,b, 2007) ) of a rigid flat surface was transposed to the present case of an elastic surface by considering that the relevant parameters for the contact between two spheres (1) and ( 2) are the mean modulus

1/E * = (1 -ν 2 1 )/E 1 + (1 -ν 2 1 )/E 2 = 2(1 -ν 2
)/E and the mean curvature 1/R * = 1/R 1 + 1/R 2 . This procedure is rigorously exact only for the elastic contact. In the present article, we only consider a contact between a spherical asperity with a radius h r and a flat surface, so that 1/R * ≈ 1/h r .

L. Lobry, E. Lemaire, F. Blanc, S. Gallier and F. Peters In the plastic regime (δ > δ c , L > L c ), the normal force is softer :

F n = -L c δ δ c 3/2   1 -exp    1 1 -δ δc β       n (2.8)
where β=0.174+0.08 ν. In addition, due to the softness of the normal force, and the high level of contact forces at large shear rate and high solid volume fraction, particles may overlap, meaning that the asperity is fully flattened out, and the distance s may become negative. To avoid this, a multiplying function is included, that goes to infinity for s = 0.2 h r (δ= 0.8 h r ), so that the distance between particle surfaces, s, is kept larger than 0.2 h r . This particular value was chosen arbitrarily but has only a quite weak influence on the lubrication stress. Figure 2 displays the interparticular force normalised by critical force L c as a function of the the dimensionless overlap, δ/h r . The dashed line is the dimensionless force from equation 2.8, while the solid line displays the force used in the simulations, in order to keep δ/h r <0.8 as mentioned above. Its expression differs from the original one only for δ/h r 0.7. The tangential force is modelled as a linear spring-like force with a threshold as in [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF][START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF][START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF]:

F t = -κ t Y for |F t | < µ|F n | (stick-phase) µ|F n | Ft |Ft| otherwise (slip-phase) (2.9)
where µ is the (static and dynamic) friction coefficient. Y is the relative tangential displacement of the two particle surfaces and it is calculated as the integral of the slip velocity during contact [START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF]. The value of the stiffness of the tangential spring, κ t , is linked to the normal force intensity |F n | through equation 2.10 so that it depends on the surface overlap δ (Eq. 2.8). The friction coefficient has been computed by [START_REF] Brizmer | Elastic-plastic spherical contact under combined normal and tangential loading in full stick[END_REF] as :

κ t = 2 7 |F n | δ (2.10)
µ = 0.27coth 0.27 |F n | L c 0.35
(2.11)

The last equations deserve a few comments. Firstly, the tangential force in the stick-phase as defined in equations 2.9 and 2.10 is an approximate version of the original model by [START_REF] Brizmer | Elastic-plastic spherical contact under combined normal and tangential loading in full stick[END_REF]. In the present paper, the tangential force (equation 2.9) is assumed to be linear with the tangential displacement, while in the original work of [START_REF] Brizmer | Elastic-plastic spherical contact under combined normal and tangential loading in full stick[END_REF], the material undergoes plastic deformation, and the force does not depend on the tangential displacement in a strict linear way. In addition, the coefficient 2/7 is smaller than the value 3(1 -ν)/(2 -ν) that may be estimated from [START_REF] Brizmer | Elastic-plastic spherical contact under combined normal and tangential loading in full stick[END_REF]. It is chosen here mostly to be consistent with earlier work [START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF][START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF]. This particular value was originally chosen to fit experimental data from impact experiments [START_REF] Shäfer | Force schemes in simulations of granular materials[END_REF].

The most important feature of the model, at least for the present work, is displayed in equation 2.11, i.e. the friction coefficient decreases as the normal force increases and reaches a plateau at high load. The variation of µ is plotted in figure 3. In the elastic regime (F n /L c < 1), equation 2.11 yields µ ∝ (L c /F n ) 0.35 in approximate agreement with Hertz theory according to which the contact area is determined by the normal force A ∝ F 2/3 n . Indeed, since sliding inception is modelled as a plastic yield failure, it occurs as the tangential force exceeds

F max t ∝ Y 0 A ∝ Y 0 F 2/3 n , leading to µ = F max t /F n ∝ F -1/3 n ∼ F -0.35 n .
As the dimensionless normal force F n /L c increases far beyond 1, the material in the whole contact zone undergoes plastic deformation, and the contact area is proportional to the normal force, giving the constant friction coefficient µ=0.27. This particular value depends weakly on the material properties [START_REF] Brizmer | Elastic-plastic spherical contact under combined normal and tangential loading in full stick[END_REF]).

Dimensionless parameters

From the microscopic model presented in the last sections, several dimensionless parameters can be identified from the relevant length scales a 1 , a 2 , h r , δ c , and force scales L c , 6πη 0 a 2 1 γ. The latter is approximately the largest interparticle force experienced by two contacting particles of radius a 1 in a sheared dilute suspension. The value that we chose for the ratio of the roughness height to the smallest radius h r /a 1 =5.10 -3 is typical for polymer spheres model suspensions [START_REF] Blanc | Experimental signature of the pair trajectories of rough spheres in the shear-induced microstructure in noncolloidal suspensions[END_REF][START_REF] Moon | The effect of sphere roughness on the rheology of concentrated suspensions[END_REF]. The interparticle force is completely specified by the critical force L c and the critical overlap δ c , so that the relevant dimensionless parameters are the dimensionless shear rate defined as:

Γ = 6πη 0 a 2 1 γ L c
(2.12)

and the ratio of the critical overlap to the roughness height δ c /h r .

In addition, in dilute suspensions, the interparticle force level is F n ∼ 6πη 0 a 2 1 γ, so that the value of the reduced shear rate Γ determines the typical overlap (figure 2) and the typical friction coefficient µ (figure 3). When Γ 1, the contact is mainly elastic (δ < δ c ) and the typical friction coefficient decreases with shear-rate, while the contact is mainly plastic and the typical friction coefficient tends to a plateau as Γ 1 (δ δ c ). Table 1 gathers numerical values for different parameters computed for polystyrene for usual values of the shear rate γ. Most importantly, the reduced shear rate Γ ranges from 4.10 -3 to 4.10 1 , meaning that the contacting material behaviour is expected to switch from elastic to plastic as the shear rate is increased. In addition, the initially decreasing friction coefficient will level off at high shear rate.

The small value of the ratio δ c /h r indicates that the roughness deformation at yield inception is very low, and supports the idea of a mono-or few-asperity contact, at least for Γ 1. It is also possible to evaluate the number of asperities that come into contact as a first asperity is compressed with δ = δ c . The asperities likely to come into contact are those which are on the surface of a spherical cap of radius d (see figure 4) where d is the radius of the circle defined by the intersection of the two spheres of radius a 1 and a 1 + h r whose centres are separated by a distance 2a 1 + h r -δ c . We then obtain that d 2 ≈ δ c a 1 or d/h r ≈ δc hr a1 h r ≈ 1.4 suggesting again a small number of compressed asperities.

The Reynolds number Re = γa 2 1 ρ/η 0 is estimated from the values in table 1, using also the typical values of the viscosity η 0 = 0.5 Pa.s and density ρ = 1g. cm -3 , which are close to the values in our experimental system (Sec. 4). Re < 8.10 -5 , so that inertia can be safely neglected.

Simulation parameters

From a practical point of view, the variety of the length scales [δ c ; h r ; a 1 ] is quite difficult to deal with in computations, since very small time steps are required to accurately compute the contact dynamics, and quite long deformation is needed for correct averaging. To keep the time step reasonably large (i.e. larger than 10 -6 in strain units), we set the critical length ratio δ c /h r =5.10 -2 . Finally, table 2 gathers the simulation parameters. Besides suspensions of particles displaying load-dependent friction coefficient, suspensions made of particles with constant friction coefficient have been considered as well in another set of simulations for comparison. The microscopic model is strictly the same, except for equation 2.11.

Simulation results

In this section, the shear stress and the viscosity are investigated. The normal stress differences are shortly studied in the first appendix at the end of the article, and some elements concerning contact force statistics are given in the second appendix.

Constant friction coefficient

Before we address the issue of particles with load-dependent friction coefficient, we first present simulations where µ is kept constant. This will be the opportunity to recall the great influence of this microscopic parameter on the suspension viscosity, as well as to derive correlation laws from the simulations that will help quantitatively understanding the more original case of load-dependent friction.

Quite recent numerical studies concerning frictional non-Brownian suspensions demonstrated a strong influence of the friction coefficient µ, taken as a constant, on the effective viscosity [START_REF] Mari | Shear thickening, frictionless and frictional rheologies in non-brownian suspensions[END_REF][START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF][START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF][START_REF] Singh | A constitutive model for simple shear of dense frictional suspensions[END_REF][START_REF] Gallier | Simulations of sheared dense non-colloidal suspensions: evaluation of the role of long-range hydrodynamics[END_REF]. The key feature here is that the jamming volume fraction significantly depends on the friction coefficient. For instance, [START_REF] Mari | Shear thickening, frictionless and frictional rheologies in non-brownian suspensions[END_REF], in their study of concentrated discontinuous shear-thickening suspensions, computed the high shear rate jamming volume fraction φ m for three values of the microscopic friction explained the variation of the viscosity of non-colloidal suspensions with the microscopic particle friction coefficient by a smooth decrease of the jamming volume fraction for microscopic friction coefficient ranging from 0 to 1. We note that such a correlation between jamming volume fraction and microscopic friction coefficient was evidenced in simulations of granular flows [START_REF] Silbert | Jamming of frictional spheres and random loose packing[END_REF] and in experiments concerning discontinuous shear-thickening suspensions [START_REF] Fernandez | Microscopic mechanism for shear thickening of non-brownian suspensions[END_REF][START_REF] Chatté | Shear thinning in non-brownian suspensions[END_REF].

We show here similar simulations where the friction coefficient is kept constant (table 2). For each value of µ, the following simple correlation law is fitted to the variation of the relative viscosity against the volume fraction :

η S (µ, φ) = α 0 (µ) 1 -φ φm(µ) 2 (3.1)
We note that such a fitting function has already been used in recent works concerning concentrated suspensions [START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF][START_REF] Singh | A constitutive model for simple shear of dense frictional suspensions[END_REF]. Obviously, the correlation 3.1 is not suitable at low volume fraction, for which the value α 0 = 1 is expected. The data from the simulation together with the respective correlation laws are displayed in figure 5 for two values of the friction coefficient.

The fitting parameters α 0 and φ m are displayed in figure 6 against the friction coefficient µ, together with the best fit to equations 3.2, where the φ x m , α x 0 and X x are again fitting parameters, that are gathered in table 3.

φ m = φ ∞ m + (φ 0 m -φ ∞ m ) exp(-X p atan(µ))-exp(-πX p /2) 1-exp(-πX p /2) α 0 = α ∞ 0 + (α 0 0 -α ∞ 0 ) exp(-X a atan(µ))-exp(-πX a /2) 1-exp(-πX a /2) (3.2)
In particular, the jamming volume fraction seems to approach the limit φ ∞ m = 0.546 as the friction coefficient goes to ∞ and is equal to φ 0 m = 0.700 for µ=0. These values

φ 0 m φ ∞ m X p α 0 m α ∞ m
X a 0.700 0.546 2.43 1.00 0.640 1.85 Table 3. Fitting parameters of equation 3.2 are somewhat different from the values that can be found in the literature and that have been obtained from simulations performed at higher volume fraction. It is especially true for frictionless particles where the jamming fraction was estimated around 0.64 for µ = 0 [START_REF] Mari | Shear thickening, frictionless and frictional rheologies in non-brownian suspensions[END_REF][START_REF] Gallier | Simulations of sheared dense non-colloidal suspensions: evaluation of the role of long-range hydrodynamics[END_REF][START_REF] Singh | A constitutive model for simple shear of dense frictional suspensions[END_REF]. This is not really a problem in the present article, since the fitting functions in equations 3.1 and 3.2 properly describe the simulation data in the volume fraction range that is kept throughout the study. In addition, our data are in quite good agreement with measurements performed with discontinuous shear-thickening suspensions in frictional regime [START_REF] Fernandez | Microscopic mechanism for shear thickening of non-brownian suspensions[END_REF]. Concerning α 0 , inspection of figure 6 shows that its variation with the friction coefficient is correlated to the variation of φ m . It can be easily understood, noting that the viscosity from the simulations for any volume fraction lower than 0.3 hardly depends on the friction coefficient, so that the variation of φ m has to be balanced by the variation of α 0 in this volume fraction range. Finally, even though α 0 significantly decreases with the friction coefficient, the viscosity variations for sufficiently high volume fraction are mostly due to the variation of φ m . recently shown for monodisperse suspensions [START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF] and in more detail in the following, the friction coefficient has a strong influence on the viscosity only in a range of roughly [0;1-2]. In the present model, the friction coefficient may vary in the range [0.27;+∞] depending on the normal force between particles. For moderately concentrated suspensions, it is thus expected from figure 3 that the viscosity mostly varies when the reduced shear rate Γ (Eq. 2.12) ranges from 10 -1 to 10. This is approximately the case for the lowest volume fraction φ=40%, and as the volume fraction increases, the curves shift toward lower reduced shear rate. This is expected, since the viscosity and more generally the stress level increase as φ increases, yielding a stronger contact force between particles at the same value of Γ . It is also instructive to split the viscosity into the hydrodynamic and contact contributions, as displayed in figure 8. While the contact contribution features shear-thinning behaviour, the hydrodynamic part weakly increases as the reduced shear rate increases. This can be qualitatively understood, considering that the typical surface separation between contacting particles decreases, due to contact elasticity, leading to stronger lubrication interactions. Such a feature has already been noted for constant friction coefficient particles as the surface roughness height was varied [START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF]. In figure 9, the contact contribution is further split into the normal and tangential force contributions. The tangential force contribution is shown to undergo stronger decrease as the reduced shear rate increases compared to the normal force contribution. Again, this can be understood based on previous simulations of suspensions with constant friction coefficient [START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF], where the decrease of the friction coefficient leads to the decrease of the ratio between the tangential to normal force contributions. As a consequence, in the present study, as the reduced shear rate increases, the ratio of the tangential to normal force contribution decreases so that the decrease of the tangential force contribution is amplified compared to the normal force contribution.

Load-dependent friction coefficient

To get a deeper insight on this transition from high to low viscosity, the mean friction coefficient, µ , over all contacting particle pairs is displayed in figure 10 by two contacting particles. A striking feature is that the data collapse on a single curve whatever the particle volume fraction. This feature is not completely obvious and deserves a comment. The microscopic friction coefficient between contacting particles is indeed controlled by the actual normal contact force. However, the mean force is only qualitatively connected to the bulk shear stress. In addition, the contact force distribution is quite broad, with a distribution width which is comparable with the mean force (Gallier 2014) and may depend on the volume fraction. Since the relation between microscopic friction coefficient and force is highly non-linear, even the relation between mean friction coefficient and mean force is not easy to anticipate. On the whole, it means that the reduced shear stress can be understood as the reduced effective normal force between contacting particles that controls the mean friction coefficient. The model built in section 3.3 is based on this idea.

Viscosity model

In this section, we show that a simple model based on the correlation laws from the simulations with constant friction coefficient described in section 3.1 allows to precisely reproduce the viscosity that is simulated when the load-dependent friction is taken into account. Returning to the particles with load-dependent friction coefficient, we make the assumption that the viscosity can be computed from equations 3.1 and 3.2, provided that the relevant effective friction coefficient, µ ef f , is controlled by the reduced shear stress as proposed in section 3.2 and supported by the collapse of the mean friction coefficient with η S Γ (figure 10). We note that this approach is quite similar to that recently followed by [START_REF] Chatté | Shear thinning in non-brownian suspensions[END_REF] to explain their experimental results. Therefore, we obtain the following self-consistent expression for the shear-thinning suspension viscosity:

η S ( Γ , φ) = α 0 (µ ef f (η S Γ )) 1 - φ φm(µ ef f (η S Γ )) 2 (3.3)
A very good prediction of the viscosity-shear stress behaviour is obtained if the following expression of the effective friction coefficient is introduced in equation 3.3:

µ ef f = 0.27coth   0.27 η S Γ 1.69 0.35   (3.4)
where the scaling factor 1.69 has been tuned in order the model to fit to the simulation data. The computed viscosity from equations 3.2, 3.3 and 3.4 is displayed in figure 11, in close agreement with the data from the simulation. It follows from this phenomenological model that the characteristic reduced force that controls friction, and consequently the suspension viscosity, is proportional to the reduced shear stress:

Fn

L c = 6πη 0 η S a 2 1 γ 1.69 L c = η S Γ 1.69 (3.5)
It should be noted (figure 10), that the effective friction coefficient, µ ef f , which has been tuned in order the modelled viscosity to fit the simulated viscosity -shear stress behaviour, is quite close to the mean friction coefficient, µ , that is a direct output of the simulations (Section 3.2).

Using equations 3.2 and 3.4, we can compute the variation of α 0 (solid line in figure 12) and φ m (solid line in figure 13) with η S Γ . At low shear stress, where the friction coefficient diverges, the jamming volume fraction approaches the asymptotic value φ ∞ m = 0.546 for high friction coefficient (equation 3.2 and text thereafter). At high shear stress, φ m levels off at the value 0.625 computed from equation 3.2 for the lowest friction coefficient 0.27 corresponding to infinite shear stress (equation 3.4).

To conclude this numerical section, it should be noted that the very good agreement between the data from the load-dependent friction simulations and the viscosity from equation 3.3 allows to check the influence of any particular friction law on the viscosity without need to perform new simulations. We follow this idea at the end of the experimental section.

Experiments

Suspensions

The suspensions are made of polystyrene (PS) particles (Dynoseed TS40, Microbeads), dispersed in silicon oil (M 500, Roth). The particles are 40 µm in diameter with a relative standard deviation lower than 5 %. Their roughness has been measured using AFM (see figure 14) and has been found to be around 36 nm (RMS value) which agrees closely with the results of [START_REF] Moon | The effect of sphere roughness on the rheology of concentrated suspensions[END_REF] who measured 32 nm and is not so far from the estimation of [START_REF] Garland | Normal stress measurements in sheared non-brownian suspensions[END_REF] who measured a characteristic roughness height around 100 nm. Before adding the particles to the suspending liquid, they are carefully washed with clear water and dried under vacuum at 60 • C for 12 hours. The particle density has been measured to be 1.044 ± 0.005 g/cm 3 and the density of the silicon oil to 0.972 g/cm 3 at T = 25 • C. The viscosity of the suspending liquid, η 0 , has been measured in cone-plate geometry. At T = 25 • C, η 0 is measured to be 0.487 P a.s and shows no variation with shear rate in the tested range 5.10 -1 s -1 < γ < 250 s -1 . Five particle volume fractions have been tested : 0.4, 0.43, 0.45, 0.47 and 0.49, all at T = 25 • C. Due to density mismatch between particles and silicon oil, it is necessary to measure the viscosity for stresses that are high enough for particles to be resuspended. The settling of the particles is controlled by the Shields parameter, Sh, that is the ratio of fluid force on the particle to the weight of the particle.

η S Γ, Σ 12 /Σ c
Sh = τ 2a ∆ρ g >> 1 (4.1)
Sh = 1 for τ = 0.03 P a and in the experiments only stresses larger than 0.1 P a are applied so that particle settling is not expected to take place.

Rheometric measurements

The viscosity has been measured in rotating parallel plate geometry. The radius of the disks, R, is 30 mm and the gap height is set to 1 mm, i.e. 25 particle diameters. Rotating plate geometry provides the advantage that no migration [START_REF] Chow | Shear-induced particle migration in couette and parallel-plate viscometers: NMR imaging and stress measurements[END_REF] or very slow migration [START_REF] Merhi | Particle migration in a concentrated suspension flowing between rotating parallel plates: Investigation of diffusion flux coefficients[END_REF] takes place. The drawback of using such a geometry to study non-Newtonian materials is obviously the variation of the shear rate magnitude over the gap. To account for shear rate variation from 0 at the centre to γR = ΩR/h at the rim, the Mooney-Rabinovitch correction is used :

η( γR ) = η mes 1 + 1 4 d ln(η mes ) d ln( γR ) (4.2)
where η mes = σ R / γR is the apparent viscosity delivered by the rheometer, and the apparent stress σ R is given as a function of the torque Γ as σ R = Γ × 2/(πR 3 ).

The shear stress ranges from 0.1 to 600 P a and each measurement is preceded by a pre-shear of 100 P a for a period of 20 s.

Experimental results

Figure 15 displays the shear rate dependence of the relative viscosity, η S = η/η 0 , corrected using the Mooney-Rabinovitch correction (Eq. 4.2) for the five particle volume fractions: 0.4, 0.43, 0.45, 0.47 and 0.49. The measurement uncertainties have been estimated by reloading five times the suspensions in the rheometer.

The shear-thinning behaviour extends over a range of more than the three decades that have been tested and is more pronounced as the particle volume fraction increases. These results are very consistent with those of [START_REF] Tanner | Particle roughness and rheology in noncolloidal suspensions[END_REF] obtained with the same particles (Microbeads, TS40 ) dispersed in a silicon oil (viscosity 1.1 P a.s.) in a slightly narrower range of shear stress.

Discussion: comparison between experiments and simulations

No simulations were performed for φ = 0.43 and φ = 0.49. However, since the model defined in equation 3.3 accurately fits the simulation data, the experimental data are compared to the model. In addition, in order to draw the experimental and numerical data on the same figure, a single value of the critical stress Σ c has to be stated for all values of the volume fraction so that η S Γ = Σ 12 /Σ c . We determine the value that fits the best: Σ c = 9.53 P a. Figure 16 displays the suspension viscosity as a function of reduced shear stress, both experiments and model.

The experimental data are in quite good qualitative agreement with the model. The viscosity range for a given volume fraction is roughly the same in both cases, and is spanned over the same reduced stress range too. From the value of the fitted critical stress Σ c =9.53 Pa , we can deduce the value of the critical load L c : L c = Σ c × (6πa 2 1 ) = 46 nN. As for the radius a 1 , that is 0.8 times the mean radius of the bidisperse suspension in the simulations, we take a 1 = 16 µm. The value of L c yields an estimation of the asperity radius of curvature R exp -h r in the contact law-using equation 2.4 together with the material constant in table 1 : R exp ≈ 156 nm. We recall here that in the contact model, the asperity was considered as half a sphere of radius h r . However, the typical extension of an asperity in the direction parallel to the surface L may be different from the asperity height h r ∼ 36 nm. This extension may be estimated from the curvature: L ∼ 2R exp h r ≈ 3.4 µm. We did not perform extensive study of the surface roughness statistics. However, this value for the extension of an asperity is quite consistent with the AFM image displayed in figure 14. We note also that contact may occur through a few contact points instead of a single point as assumed in the model. The consequence would be qualitatively the same as increasing the radius of curvature, i.e. to increase the critical load. For instance, if the contact is supposed to involve two asperities, the curvature radius can be computed from L c = 1/2 × Σ c × (6πa suggests that the physical mechanism of the shear-thinning behaviour is satisfactorily captured by the load-dependent friction model. In more detail, the viscosity from the model (or the simulations) presents some discrepancies with the experimental values. At low shear stress, the experimental viscosity does not really show any plateau, contrarily to what is expected from the model or the simulations. This difference may be explained by residual adhesion between particles regardless the great care that has been taken in the choice and in the preparation of the suspensions. Such adhesive interactions, even very small, should influence the viscosity at small enough shear stress. In the high stress range, the experimental measurement could not be performed at such high stresses that the viscosity plateau would be reached, due to the usual edge fracture in the parallel plate geometry.

To go further in the comparison between the model and the experiments, it is possible to compute for each stress the parameters of the viscosity model α 0 (Σ 12 ) and φ m (Σ 12 ) from the experimental data. These experimental parameters are displayed in figures 12 and 13, respectively, against the reduced shear stress, together with their numerical counterpart. We chose the same value of the critical stress Σ c = 9.53 Pa as in figure 16. The variations of φ m from the experiments and from the model are in quite good agreement, with approximately the same volume fraction range, and the same reduced stress range. However, the experimental jamming fraction takes one value under φ ∞ m at low shear stress, possibly due to residual adhesion mentioned above. The experimental and numerical data for α 0 are in quite good agreement too, even though the experimental values are above the model. It should be stressed here that the values of α 0 and φ m from the experiment significantly depend on the precise adjustment procedure, apparently due to the dispersion of the experimental data, especially in the low stress range, where the discrepancies are larger. It does not affect the qualitative shape of the φ m curve, but leads to variations of ∆φ m ∼ 0.01.

Finally the effective friction coefficient µ ef f (Σ 12 ) can be computed from the experimental jamming volume fraction (figure 13) reversing equation 3.2. It is then assumed, as previously in equation 3.4, that µ ef f is actually the microscopic friction coefficient for the applied load Fn = 6πa 2 1 Σ 12 /1.69. The variation of resulting experimental loaddependent friction coefficient with the normal load is displayed in figure 17, together with the mono-asperity contact model (equation 3.3). As expected, the experimental data are in quite satisfactory agreement with the mono-asperity contact model. We note that it is possible to deduce µ ef f from the viscosity measurements only if the values of the experimental jamming fraction belong to the variation range of φ m deduced from the numerical simulations at constant friction coefficient. These values are comprised between φ ∞ m = 0.546 for µ → ∞ and φ 0 m = 0.70 for µ = 0. Thus the inversion procedure presented above can be used only for values of the force corresponding to values of φ m larger than the minimum value φ ∞ m , i.e. F n > 3nN. The values of the friction coefficient deduced from the viscosity measurements must be considered an estimation rather than a precise quantification. As shown in figure6, for µ 2, φ m hardly varies with µ so that even a low error on φ m may have a significant impact on the value of µ.

Conclusion

In this paper it is shown that it is possible to capture the shear-thinning behaviour that is observed in most concentrated non-Brownian frictional suspensions by considering variable friction interactions between particles. Owing to the low interparticle forces involved in sheared suspensions, it is expected that particles come into contact through one or a few asperities. For such a contact, it is anticipated that the value of the friction coefficient varies with normal load and thus with shear stress. To test the effect of variable friction on suspension rheology, we introduce in a numerical simulation code based on the Force Coupling Method a contact law adapted from the numerical work of [START_REF] Brizmer | Elastic-plastic spherical contact under combined normal and tangential loading in full stick[END_REF]. This model considers an elastic-plastic contact and results in a decrease of the friction coefficient at low normal loads followed by its saturation at high normal loads. The results of the simulations show a decrease of the viscosity with the shear stress that is in the same range as that observed experimentally. The comparison between simulations conducted either with constant friction coefficient or load-dependent friction coefficient makes it possible to propose a model for the shearthinning viscosity. As already shown by [START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF] and [START_REF] Singh | A constitutive model for simple shear of dense frictional suspensions[END_REF], the viscosity in the former case (constant friction coefficient) increases with the friction coefficient, mainly due to the decrease of the jamming volume fraction. In the proposed viscosity model for the particles with load-dependent friction coefficient, the effective friction coefficient µ ef f is specified by the effective normal contact force which is simply proportional to the shear stress. As the shear stress increases, µ ef f decreases and the jamming volume fraction increases, leading to the reduction of the viscosity. A very good agreement between the model and the simulations is found. These results are compared to experimental measurements carried out on suspensions of polystyrene particles (40 µm in diameter) dispersed in silicone oil for five particle volume fractions between 0.40 an 0.49. The overall agreement between the model (or the simulations) and the experiments is satisfactory. The viscosity reduction as the shear stress increases is well captured by the model and the variation of the jamming volume fraction with the shear stress from the experiments is in quite good agreement with the predictions. The closeness between the experiments and the predictions of the model enabled us to evaluate the friction coefficient and its variation with the applied shear stress from rheometric measurements. Finally, the simplistic nature of the contact law deserves a few words. The essential feature is the variation of the friction coefficient with load, that is connected to the small number of asperities in contact -here a single asperity-in contrast to the usual model of a statistical distribution of contacting asperities. Such a mono-asperity contact model could be refined at will, to take into account a finite number of asperities, possibly growing with the bore load. This may also have an influence on the way the friction coefficient levels off at large load, possibly supplementing or even replacing plasticity. More generally, due to the complexity of contact at microscopic scale, and to the number of parameters that are potentially relevant in the surface physical chemistry, the next step to get a more quantitative understanding of the physical mechanism leading to this shear-thinning behaviour would be to determine the friction law between particles from colloidal probe AFM measurements. the probability of a plastic contact takes a finite value, leading to a high value of the characteristic strain, that decreases as the probability of plastic contact increases. For reduced stress larger that typically one, the characteristic strain levels off to a plateau. We note significant noise for the largest value of the reduced stress. The strain range of the simulation may be to short to reach completely satisfying statistical accuracy.

A deeper understanding can be achieved plotting the fraction of plastic contacts f (F N > L c ) against η S Γ . As shown in figure 22, the data obtained for the four particle volume fractions collapse on a single curve. Figure 22 is very similar to what has been obtained by [START_REF] Mari | Shear thickening, frictionless and frictional rheologies in non-brownian suspensions[END_REF] who studied the fraction of frictional contacts in a shear-thickening suspension. They showed that the number of contacts was controlled by a competition between the applied shear forces and the short-range repulsive forces between particles. It is quite remarkable that similar results on contact statistics are obtained for discontinuous shear-thickening and shear-thinning suspensions, these two cases being different in many respects. In particular, in the description of discontinuous shear-thickening proposed by [START_REF] Mari | Shear thickening, frictionless and frictional rheologies in non-brownian suspensions[END_REF], the repulsive force between particles controls the transition from a frictionless (µ = 0) rheology to a contact dominated (µ ≈ 1) rheology. In contrast, in the model of shear-thinning that we propose here, the characteristic plastic force L c marks a soft transition between high to low friction coefficient that is continuously crossed by the sheared suspension.

Figure 1 .

 1 Figure 1. Sketch of the roughness modelling. s is the hydrodynamic separation distance, hr is the roughness height

Figure 2 .

 2 Figure 2. Dimensionless normal contact force as a function of dimensionless overlap. Dashed line : original force from equation 2.8. Solid line : corrected force to keep δ/hr <0.8.

FFigure 3 .

 3 Figure 3. Friction coefficient as a function of dimensionless contact normal force (Eq. 2.11).

Figure 4 .

 4 Figure 4. Sketch of the contact. d is the distance from a compressed asperity to the furthest compressed asperity.

Figure 5 .

 5 Figure 5. Inverse of the square root of the shear viscosity against volume fraction for two values of the friction coefficient. (•) µ = 0.27 ( ) µ = 2. Solid lines : best fit of equation 3.1 to the data. The variation of φmax and α0 with µ is clearly displayed.

Figure 6 .

 6 Figure 6. Fitting parameters of equation 3.1 as a function of the friction coefficient. • Simulation data. Solid line : best fit (equation 3.2).

Figure 7 Figure 7 .

 77 Figure7displays the computed relative viscosity as a function of the reduced shear rate Γ . As expected, as the reduced shear rate increases, the viscosity decreases, due to friction coefficient reduction. The viscosity exhibits two plateaus, one at low reduced shear rate and the other at high reduced shear rate. The latter is due to the saturation of the friction coefficient as F n /L c increases. The low shear plateau originates in the saturation of the viscosity as the friction coefficient increases. More quantitatively, as

Figure 8 .Figure 9 .

 89 Figure 8. Hydrodynamic and contact contributions to the viscosity. Open symbols : contact contribution. Closed symbols : hydrodynamic contribution. Same volume fraction values as in figure 7.

Figure 10 .

 10 Figure 10. Symbols : Friction coefficient averaged over all contacting particle pairs. (•) φ = 40% (•) φ = 45% φ = 47% φ = 50%. Solid line : Effective friction coefficient from equation 3.4

Figure 11 .

 11 Figure 11. Viscosity as a function of reduced shear stress. Symbols : same data as in figure 7. Solid lines : equation 3.3.

Figure 12 .Figure 13 .

 1213 Figure 12. Second fitting parameter in the viscosity correlation (equation 3.1). Solid line : computed from equations 3.2 and 3.4. ( ) Experiments.

Figure 14 .

 14 Figure 14. AFM image of a polystyrene particle.

Figure 15 .

 15 Figure 15. relative viscosity as a function of shear stress. Symbol : 40µm PS particles (•) φ = 0.4 . ( ) φ = 0.43. ( ) φ = 0.45 ( ) φ = 0.47 ( ) φ = 0.49

Figure 16 .

 16 Figure 16. Viscosity as a function of reduced shear stress. φ = 0.4, φ = 0.43, φ = 0.45, φ = 0.47, φ = 0.49 . Symbols : experiments. Solid lines : model (equation 3.3).

FFigure 17 .

 17 Figure 17. Microscopic friction coefficient. Symbol : experimental data. Solid line : mono-asperity model from equation 3.4 with Lc = 46 nN.

Figure 19 .Figure 20 .

 1920 Figure 19. (a) Reduced first normal stress difference. (b) Reduced second normal stress difference. Open symbols : contact contribution. Closed symbols : hydrodynamic contribution. Same volume fractions as in figure 18.

Figure 21 .Figure 22 .

 2122 Figure 21. Characteristic strain for a particle to undergo a plastic contact

Table 1 .

 1 Typical values of physical parameters

	Young's	Yield Poisson's Roughness Part.	Shear	Critical Length	Reduced
	Modulus strength	ratio	ratio	radius	rate	load	ratio	shear rate
	E (Pa)	Y0 (Pa)	ν	hr/a1	a1 (µm)	γ (s -1 )	Lc (nN) δc/hr	Γ
	3.10 9	60.10 6	0.4	5.10 -3	20	[10 -2 -10 2 ] ∼ 20 ∼ 10 -2 [4.10 -3 -4.10 1 ]

Table 2 .

 2 Simulation parameters

† Email address for correspondence: francois.peters@unice.fr

Acknowledgements

The authors thank Cyril Claudet and Olga Volkova for the AFM roughness measurements. This work was granted access to the HPC and visualisation resources of "Centre de Calcul Interactif" hosted by "Université Nice Sophia Antipolis".

Appendix A. Normal stress differences

Figure 18 presents the variation of the two normal stress differences normalised by the shear stress, N 1 /Σ 12 and N 2 /Σ 12 , with the non-dimensional stress η S Γ . These results are consistent with previous numerical results performed with a constant friction coefficient and present the same characteristics [START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF]): both N 1 and N 2 are negative, N 1 is much smaller than N 2 . The values of the reduced normal stress differences are also of the same order of magnitude as those obtained with a friction coefficient in the range [0.3 -0.5]. At last, N 2 /Σ 12 appears not to vary much with φ for φ 0.45. This saturation of N 2 /Σ 12 has already been observed in experiments [START_REF] Zarraga | The characterization of the total stress of concentrated suspensions of noncolloidal spheres in newtonian fluids[END_REF][START_REF] Boyer | Dense suspensions in rotating-rod flows: normal stresses and particle migration[END_REF][START_REF] Dbouk | Normal stresses in concentrated non-brownian suspensions[END_REF] and in numerical simulations conducted at constant friction coefficient [START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF]. Concerning the variation of N 1 /Σ 12 and N 2 /Σ 12 with η S Γ , the trends are also consistent with the numerical results that have been obtained with different constant friction coefficient. According to the findings of Sec. 3.2 and3.3 increasing η S Γ leads to a decrease of the friction coefficient. Figure 18 shows that the magnitude of N 1 /Σ 12 increases and that of N 2 /Σ 12 decreases with increasing η S Γ , just as the magnitude of N 1 /Σ 12 increases and that of N 2 /Σ 12 decreases when the friction coefficient decreases, as shown by [START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF].

The hydrodynamic and contact contributions to the normal stress differences are displayed on figure 19. It is difficult to compare the trends displayed on figure 19(a) with the results of [START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF]. The latter seem to indicate that for particle volume fraction larger than 0.4 N H 1 /Σ 12 hardly varies with µ while the magnitude of N C

1 /Σ 12 tends to decrease weakly with increasing µ. The behaviour of the shear-thinning suspensions considered here seems different with a trend of decreasing N H 1 /Σ 12 and of increasing N C 1 /Σ 12 with increasing η S Γ . But we have to keep in mind that both N H 1 and N C 1 are much smaller than the total stresses, Σ 11 and Σ 22 from which they are deduced and can be very sensitive to the details of the suspension microstructure that may be a little different from a non-shear-thinning suspension. Concerning the results obtained for N H 2 and N C 2 , they are consistent with those of [START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF] with a small N H 2 compared to N C 2 and a trend of N C 2 to decrease in magnitude with increasing η S Γ .

Appendix B. Some elements on contact statistics

Figure 20 displays the fraction of particles that have experienced plastic contact versus strain for 3 values of η S Γ at φ = 0.45. The initial value of N P lastic /N T otal is the average fraction of particles that undergo plastic contact. As the strain increases, N P lastic /N T otal increases with an exponential-like relaxation. From this variation, we deduce a characteristic strain γ c which is the average strain necessary for a particle to undergo a plastic contact. The variation of γ c with η S Γ is displayed in figure 21 for the four values of the particle volume fraction. As η S Γ increases, γ c decreases and reaches for η S Γ ≈ 1 a plateau whose value depends on the volume fraction. A qualitative understanding is quite straightforward. For reduced stress η S Γ smaller than 2 10 -1 no particle was found to undergo plastic deformation. As the reduced stress increases,