Guillaume
Chatté
^{a},
Jean
Comtet
^{b},
Antoine
Niguès
^{b},
Lydéric
Bocquet
^{b},
Alessandro
Siria
^{b},
Guylaine
Ducouret
^{a},
François
Lequeux
^{a},
Nicolas
Lenoir
^{cd},
Guillaume
Ovarlez
^{e} and
Annie
Colin
*^{af}
^{a}ESPCI Paris, PSL Research University, Sciences et Ingénierie de la Matière Molle, CNRS UMR 7615, 10 rue Vauquelin, F-75231 Paris Cedex 05, France. E-mail: annie.colin@espci.fr
^{b}Laboratoire de Physique Statistique, Ecole Normale Supérieure, UMR CNRS 8550, PSL Research University, 24 rue Lhomond, 75005 Paris Cedex 05, France
^{c}PLACAMAT, UMS 3626-CNRS/Université de Bordeaux, 33608 Pessac, France
^{d}Grenoble-INP/UJF-Grenoble 1/CNRS UMR 5521, Laboratoire 3SR, Grenoble, France
^{e}Univ. Bordeaux, CNRS, Solvay, LOF, UMR 5258, F-33608 Pessac, France
^{f}Université de Bordeaux, Centre de Recherche Paul Pascal, UPR-8641 CNRS, 115 avenue Schweitzer, 33600 Pessac, France

Received
30th September 2017
, Accepted 26th November 2017

First published on 27th November 2017

We study the flow of suspensions of non-Brownian particles dispersed into a Newtonian solvent. Combining capillary rheometry and conventional rheometry, we evidence a succession of two shear thinning regimes separated by a shear thickening one. Through X-ray radiography measurements, we show that during each of those regimes, the flow remains homogeneous and does not involve particle migration. Using a quartz-tuning fork based atomic force microscope, we measure the repulsive force profile and the microscopic friction coefficient μ between two particles immersed into the solvent, as a function of normal load. Coupling measurements from those three techniques, we propose that (1) the first shear-thinning regime at low shear rates occurs for a lubricated rheology and can be interpreted as a decrease of the effective volume fraction under increasing particle pressures, due to short-ranged repulsive forces and (2) the second shear thinning regime after the shear-thickening transition occurs for a frictional rheology and can be interpreted as stemming from a decrease of the microscopic friction coefficient at large normal load.

Let us consider the simple case of an assembly of hard spheres suspended in a fluid of viscosity η_{f} sheared with shear rate , under a constant particle pressure P^{p}. In the absence of external force scales coming from inertia or thermal forces, these dispersions have to be Newtonian. Indeed, in these systems, there is only one single control parameter, , governing the dynamics of the flow. Following the approaches used in the studies of granular material,^{3} the shear stress τ and the solid fraction ϕ are given by two constitutive equations: τ = μ_{eff}(I_{V})P^{p} and ϕ = ϕ(I_{V}), where μ_{eff}(I_{V}) is an effective frictional coefficient. These relations led to τ = g(ϕ)η_{f}. In the situation of a homogeneous controlled solid fraction dispersion, solid particles immersed in a Newtonian fluid are viscous. Both shear stress and normal stress differences are proportional to the shear rate. Strikingly, this analysis does not describe the reality. Dispersions exhibit a wide range of rheological behavior including shear thinning and shear thickening.

Shear thickening corresponds to an increase of the viscosity as a function of the shear rate. This behavior is one of the most striking phenomena occurring in complex fluids. In cornflour suspensions, the formation of a dynamic jamming front under impact makes the fluid so resistant that a person can run on it.^{4,5} Industrially, shear thickening can have disastrous effects by enhancing the amount of energy required to pump suspensions at high shear rates, damaging mixer blades or clogging pipes.^{6} Shear thinning corresponds to a decrease of the viscosity as a function of the shear rate. These non-linear phenomena are essential from the perspective of applications and materials, and a better understanding of controlling the flow of dispersions in industrial processes is required.

At this stage, a global picture describing the mechanisms at the origin of shear thickening starts to emerge. In the absence of inertia, recent theoretical studies have put forward the role of contact forces.^{7,8} At low pressure, the particles do not touch. The suspension is a Newtonian fluid. At high pressure, repulsive forces are overcome, and frictional contacts are the norm. Depending upon the value of the solid fraction, different behaviors are predicted at high shear rates. At low solid fraction, a gradually denser frictional contact network is observed when the shear rate is increased. This regime corresponds to continuous shear thickening (CST). For higher values of solid fraction ϕ, the theoretical flow curve displays an S-shape allowing discontinuous shear thickening (DST) between a lower (sparse contact network) and an upper (dense contact network) branch at equal strain rates. Both branches correspond to a Newtonian behavior. Finally, for very high solid fraction, the system transits from a liquid state to a solid state unable to flow without fracture. This situation corresponds to shear jamming or shear-induced jamming. This picture and the role of frictional forces have been validated through direct experimental measurements.^{9,10} The pairwise force profile and the frictional interactions between pairs of particles have been measured using quartz-tuning fork based atomic force microscopy. The normal load required to transit from lubricated to frictional interactions is quantitatively related to the critical shear stress at the shear thickening transition as predicted by numerical simulations.

Many studies report shear thinning for various suspensions: PMMA particles in polyethylene glycol (PEG),^{11} fumed silica particles in polypropylene glycol (PPG),^{12} cornstarch particles in water,^{13} glass spheres in mineral oil,^{14} cementitious pastes,^{15} polystyrene particles dispersed in PEG^{16} and also PVC particles dispersed in a plasticizer.^{17–20}

For Brownian suspensions, shear thinning occurs due to a competition between diffusion and convection. At low shear rates, particle diffusion is significant and particles occupy a larger effective volume than at high shear rates, leading to a larger viscosity. Explanations concerning shear thinning in non-Brownian suspensions are more vague. At low shear rates, in a non-Brownian suspension, shear thinning may occur due to the presence of short-ranged stabilizing repulsive forces between particles. In this situation, the apparent size of the particles includes the hard sphere contribution and a part of the surrounding soft repulsive potential. The apparent size value will decrease with increasing shear rate. Indeed, higher shear rates correspond to higher pressures and thus to a decrease of the minimum possible distance between particles as they flow. The apparent size of the particles and thus the suspension viscosity would then decrease as a function of the shear rate. Such thinning has already been observed in a charge stabilized suspension^{21,22} and predicted numerically.^{23}

At high shear rates, few other mechanisms have been proposed. Using cornflour suspensions, Ovarlez and coworkers^{13} showed that the flow at high shear rates after DST is inhomogeneous. The system separates into two phases: a dilute phase and a concentrated phase. This separation is concomitent with a shear thinning behavior. The shear thinning variation seems to be due to the particular rheological properties of the two shear-induced phases and the evolution of their respective size under shear. However, this explanation may not be universal, as some dispersions display shear thinning by flowing homogeneously. Inspired by the old order–disorder theory of Hoffman et al.,^{17} Nakajima et al.^{24} explained the shear thinning at high shear rates by the breakdown of spanning clusters to smaller sizes, releasing the trapped plasticizer and increasing the maximum packing density; thus causing a decrease of viscosity at high shear rates. More recently, elastohydrodynamic interactions have been proposed to explain the shear thinning behavior.^{11,25} Under high normal load and shear rates, the particles may deform via a lubricating liquid film opposing contact between particles. Last but not least, Vazquez-Queseda^{26} and coworkers have proposed recently that shear thinning might be related to the non-Newtonian properties of the solvent. They point out that hidden shear-thinning effects of the suspending medium, which occur at shear rates at orders of magnitude larger than the range investigated experimentally in dispersions, lead to significant shear thinning of the overall suspension at much smaller shear rates. They consider the behavior of the solvent at ultra-high shear rates by assuming that the shear rate in the film between the particles might be much greater than the applied one.

In this article, we revisit the question of shear thinning in non-Brownian suspensions. We take advantage of a system of particles previously characterized. Our subject of study is the dispersion of polyvinyl chloride (PVC) particles suspended in a Newtonian plasticizer (Dinch) (see Fig. 1). This dispersion is an assembly of lubricated grains that becomes frictional at a high shear rate.^{9} Using capillary rheometry, X-ray radiography and quartz-tuning fork based atomic force microscopy, we characterize the system in the shear thinning regimes in DST materials. We restrict our study to moderate solid volume fraction values to avoid the shear jamming zone. The main originality of our work is to couple these techniques to get a comprehensive picture of the flow. The first part of the article deals with the description of the sample. We report the rheological study in the second part. The third part is devoted to X-ray radiography analysis. In the fourth part, we report measurements of the solid friction coefficient between two PVC beads in the Newtonian plasticizer at high shear rate. The fifth part deals with discussion and outlook.

We focus on two dispersions of PVC particles. SEM images of the particles are displayed in Fig. 2 while particle size distributions in Dinch are displayed in Fig. 3. In the first dispersion (D1), the mean particle radius, defined as R_{32} = 〈R^{3}〉/〈R^{2}〉, is 1 μm. The size distribution is lognormal and the standard deviation estimated using the volume distribution is 45%. In the second dispersion (D2), the particle size histogram using a volume distribution is trimodal with lognormal peaks around 350 nm (with a standard deviation of 25%), 3.3 μm (with a standard deviation of 55%) and 20 μm (with a standard deviation of 22%). Both dispersions will reach Peclet numbers ( where η_{f} is the suspending fluid viscosity, is the shear rate and R is the particle radius) in the range of 10–10^{6} for which Brownian effects are practically negligible.^{28}

Fig. 2 Scanning Electron Microscopy (SEM) images of (a) PVC used for D1 and (b) PVC used for D2. Scale bar at the bottom right of each image is 1 μm. |

Fig. 3 Particle size distributions of D1 (red) and D2 (blue) obtained with a laser diffraction apparatus (Mastersizer 3000 from Malvern). |

The random close packing fractions ϕ_{RCP} of these dispersions are measured. ϕ_{RCP} corresponds to the value of the solid fraction at which the viscosity diverges at low shear rate under the hypothesis of frictionless particles.^{9} We measure the value of the viscosity at = 10 s^{−1} to get rid of interparticle interactions at low shear rate.^{29} The data are fitted using a Krieger–Dougherty model , where η_{s} is the solvent viscosity. We get ϕ_{RCP} = 69.4% ± 0.25% for D1 suspensions and ϕ_{RCP} = 77.2% ± 0.25% for D2 suspensions. The exponents n of the Krieger–Dougherty models are respectively n = 2.3 for the D1 dispersion and n = 2.9 for the D2 dispersion. η_{s} is equal to 41 mPa s.

This computation resulted from the assumption that (i) the flow is azimuthal (i.e. the flow field is purely tangential), (ii) the fluid is Newtonian and does not slip at the wall. Under these assumptions, and σ are simply proportional to ω and Γ, respectively, with proportionality factors that depend upon the geometrical parameters of the shear cell. For a Couette cell, these links write:

(1) |

(2) |

(3) |

(4) |

As discussed earlier, we use two procedures: an applied shear rate procedure and an applied shear stress procedure. For the applied shear stress procedure, we apply the following instructions. At room temperature (25 °C), we first apply a pre-shear step at = 10 s^{−1} for 60 s that allows us to start the experiment with a well-defined steady state. We then apply a ramp of increasing shear stress. The shear stress is swept logarithmically from 1 to 1300 Pa. When two consecutive measurements are within 5%, an equilibrium is assumed and the shear stress is increased. For the applied shear rate procedure, we also apply a pre-shear step at = 10 s^{−1} for 60 s. We then apply a ramp of increasing shear rate. The shear rate is swept logarithmically from 0.1 to 1000 s^{−1}. When two consecutive measurements are within 5%, equilibrium is assumed and the shear rate is increased. We checked the reproducibility of the flow curve measurements using the two procedures. We anticipate that the shear stress procedure allows a better characterization of the rheological properties in the shear thickening transition.

In all cases, the rotational rheometer cannot access normal stress differences N_{1} higher than 1000 Pa due to sample ejection or edge instability. This corresponds to the region of the shear thickening transition close to the maximal viscosity measured. We discarded such data. To circumvent these issues at high shear rate and high normal load, we used two kinds of capillary rheometers: a home-made capillary rheometer to access the intermediate range of shear rates and a commercial capillary rheometer (Gottfert Rheo-Tester) to access larger shear rates. This approach has been used previously for PVC particle suspensions in several works.^{18–20} The Gottfert Rheo-Tester capillary rheometer is equipped with a capillary die of diameter D = 0.5 mm (silicon carbide; 3 length-to-diameter ratios: L/D = 4, 8, and 16) or with a capillary die of D = 0.3 mm (silicon carbide; 2 length-to-diameter ratios: L/D = 4 and 8). L is the length of the capillary. The volumetric flow rate Q is imposed by imposing to the piston successive steps of increasing speed from 0.1 to 1 mm s^{−1}. The drop of pressure is measured thanks to a pressure transducer of 100 bar (10 MPa) full scale.

Home-made capillary rheometers are displayed in Fig. 4. They consist of a manometer (pressure range 0–8 bars) plugged on a compressed air network (0–7 bars), a syringe with a piston and a capillary firmly plugged to the syringe. Two kinds of syringes are used: (i) commercial ones in plastic with a volume of 30 mL (Fig. 4a). The capillary used is made of PEEK (polyether ether ketone) polymer, which ensures good rigidity. The real die diameter was measured with an optical microscope and found to be 1.55 mm. The capillary length used varies between experiments but is always between 12 and 25 mm. (ii) A home-made syringe made up of PMMA with a larger volume of 250 mL (Fig. 4b). The latter one is nearly transparent to X-ray and will be used in the experiments devoted to the measurement of particle volume fractions. The die diameter is 2.0 mm and different lengths are available (L = 6.5, 11.5 or 16.5 mm). In contrast to the commercial capillary rheometer described earlier, these in-house built capillary rheometers work in a stress-controlled mode. Stress is imposed and computed from air pressure. The shear rate is computed from the flow rate, which is measured by weighing the amount of sample going out from a cup after and before a known time of experiments.

Capillary rheometers measure or impose the drop of pressure ΔP required to get a given flow rate Q. From these data, they compute a shear rate at the wall _{w} and a shear stress at the wall σ_{w}. The wall shear stress σ_{w} is given by:

(5) |

The shear rate at the wall is given by the Weissenberg–Rabinowitsch analysis^{31} (Fig. 7):

(6) |

In the low shear rate situation, the exponent does not depend on the dispersion nor on the solid fraction ϕ. It is roughly equal to −0.4. In the high shear rate situation, the exponent of the power law depends upon the volume fraction. The higher the volume fraction, the higher the absolute value of the exponent. Data from D1 and D2 collapse roughly on the same straight line. The point for n close to −0.7 is the only case where the absence of wall slip has not been checked.

Shear thinning with no wall slip is striking for non-Brownian suspensions immersed in a Newtonian fluid. Post-DST shear thinning has been previously associated with shear-induced migration.^{13} In order to probe the homogeneity of the flow, we perform X-ray absorption measurements.

Fig. 9 is a schematic that shows a planar wave of the X-ray penetrating the syringe. In the experiments, the camera's response is first calibrated to provide a homogeneous intensity response I_{0} over all pixels when there are no objects between the X-ray source and the detector. When an object is present, X-rays are absorbed by the various media crossing the beam (the PMMA, the PVC particles, the Dinch, the air) following Beer–Lambert's law. In our situation, the recorded intensity reads:

(7) |

This can be read:

I(y,z,t) = I_{fluid}(y,z)exp(−L_{gap}(y,z)(Φ(y,z,t)μ_{beads}(y,z) − μ_{fluid}(y,z))) | (8) |

This leads to:

(9) |

From the intensity field, eqn (9) allows us to measure the variation (in both time and space) of the averaged particle volume fraction seen by the beam Φ(y,z,t) in the syringe as a function of the applied drop of pressure. Please note that we did not compute directly the local volume fraction ϕ(y,z,t), but only averaged the particle volume fraction seen by the beam along its path through the syringe.

We focus on the suspension exhibiting DST (D1 at ϕ = 60%). More precisely, we are interested in the DST region and the shear thinning observed beyond. We take advantage of our home-made PMMA syringe that enables viscosity measurements both in the DST and the shear thinning beyond. The syringe was placed in the X-ray tomograph. Viscosity measurements were carried out simultaneously with X-ray absorption recording (cf.Fig. 9).

From the X-ray radiography data, we extract the value of the solid fractions. These 10 measurements have been performed using a high spatial resolution (less than 20 microns). The noise is 1%. This noise is homogeneously distributed in space and time. Averaging these data leads to a variation of 0.3%. We thus conclude that the sample is homogeneous before, during and after the shear thickening transition. Note that this contrasts with the case of the apparent thinning behavior observed in cornstarch suspensions. In this situation, the sample separates into two phases: a dilute phase and a concentrated phase^{13} that clogs the syringe.

To conclude this section, note that this does not preclude anisotropy or local stress heterogeneity.^{22,34}

Fig. 10 Relative variation of the averaged particle volume fraction seen by the beam along (a) the flow direction Z or (b) the radial direction X for D1 60%. 〈ϕ_{i}〉 is the value of ϕ averaged over the x direction. Applied stresses range from 180 (electric blue) to 9.8 × 10^{3} (blue), 1.5 × 10^{4} (cyan blue), 2.3 × 10^{4} (green), 3.0 × 10^{4} (yellow), 3.9 × 10^{4} (orange), 4.6 × 10^{4} (red), and 6.2 × 10^{4} (brown) Pa and correspond to the ones obtained with the large PMMA syringes (from 0 to 8 bars) as shown in Fig. 5a & c (before, in and beyond the DST transition). |

To simultaneously measure the normal and tangential force profiles between the two approaching particles, we simultaneously excite the tuning fork via a piezo-dither at two distinct resonance frequencies f_{N} ≈ 31 kHz and f_{T} ≈ 17 kHz, corresponding to the excitation of both normal (N) and shear (T) modes of the tuning fork (Fig. 11a). Monitoring the changes in the resonance of each mode allows us to measure the conservative force gradient ∇F_{i} [N m^{−1}] and dissipative frictional forces F^{i}_{D} [N] for those two directions (i ∈ {N,T}).^{9} In the following, we show measurements performed with particles of PVC 1. We anticipate similar friction properties for PVC 2 as surface properties are similar.

Fig. 13 Variation of the microscopic interparticle friction coefficient μ = F^{T}_{D}/F_{N} as a function of the normal load. μ decreases with increasing load. Note that the particles involved in this measurement are not the same as that used in Fig. 11. |

To understand this deviation from Coulomb's law, we express the frictional force between the two PVC surfaces as F^{T}_{D} = τ·_{real}, where _{real} is the real area of contact, and the shear strength τ [Pa] characterizes the friction per real contact area between the PVC surfaces. The deviation from the classical Amontons–Coulomb law at large loads, i.e. the non-linear dependence between the tangential friction and the normal load (Fig. 12a), and the decrease in the friction coefficient μ (Fig. 13) could stem from (1) a geometrical origin, i.e. a non-linear variation of the real contact area _{real} with the normal load F_{N} or (2) a physical origin, i.e. a decrease in the shear strength τ [Pa] with the normal load. To disentangle those two effects, we plot in Fig. 12b the tangential stiffness k_{T} [N m^{−1}] as a function of the normal load. k_{T} can be considered to be directly proportional to the real area of contact _{real} (i.e. to the number of contacts). This tangential stiffness is found to vary proportionally to the normal load over the entire range of measurements. We thus deduce that the real area of contact increases proportionally with the normal load, in agreement with classical multi-asperity models. The non-linear variation of tangential dissipation with normal load thus has its origin in a decrease of the shear strength τ with increasing normal loads, and stems from the physical interaction between the two PVC surfaces.

Such a decrease of the friction coefficient with load has already been reported in the literature for strongly compressed polymer brushes in good solvents.^{35,36} It is important to underline that the friction coefficient is a well-defined property between two particles. However, it displays a rather large distribution. Over 30 different pairs of beads, we find a mean μ coefficient under a small load equal to 0.45 ± 0.35.^{9}

To go further and quantitatively analyse our results, we extract the repulsive force profiles from the rheological measurements and we compare them to the data obtained using our atomic force microscope. We start by building upon the analyses of Wyart and Cates for dense suspensions.^{7} For the sake of simplicity, we assume as in ref. 7 that the pressure P and the shear stress σ are proportional, and can be expressed as

σ/ = BP/ = η | (10) |

(11) |

(12) |

(13) |

(14) |

W(h) = PR = σR/B = W_{0}(σ/σ*) | (15) |

To find n and ϕ_{RCP}, we measure the value of the viscosity η* at * where the viscosity vs. shear rate curve goes through a minimum as a function of the solid fraction, and fit the curve using a Krieger–Dougherty model. We get ϕ_{RCP} = 69.4% ± 0.25% for D1 suspensions and ϕ_{RCP} = 77.2% ± 0.25% for D2. The exponents n of the Krieger–Dougherty models are respectively n = 2.3 for the D1 dispersion and n = 2.9 for the D2 dispersion.

Fig. 15 displays the values W(h) from eqn (15), which lead to the four rheological curves displayed in Fig. 5. All the data collapse on a single curve, which is very close to the repulsive profile measured by the tuning fork, shown as the black line in Fig. 15.^{9} To compute this curve we use R equal to 1 μm for D1 and R equal to 1.5 μm for D2. The slight differences between the measurements and the theoretical model may be related to the polydispersity of the samples.

This quantitative analysis shows that the existence of short-ranged repulsive forces along with lubricated contacts at low particular pressure is responsible for the shear thinning behavior observed at low shear rate.

6.2.1 Analysis in the framework of the models existing in the literature.
It has been proposed^{11,25} that flow may deform the particles at high shear rate. The deformation δ varies as σ/G where G is the elastic modulus of the beads. The elastic modulus of PVC is equal to 3 × 10^{9} Pa. In the shear thinning regime, the shear stress is less than 10^{6} Pa. The deformation is less than 3 × 10^{−4} and cannot be the origin of shear thinning. The created film h = Rδ due to particle deformation is thus much less than 3 × 10^{−10} m. Such a film is not physical since it does not contain a single molecule of Dinch.

Shear thickening relies on the transition between the lubricated contact and frictional ones. The solvent film has to be drained during the collision between the particles. At low shear rates, this process is likely to happen. However at high shear rates, one may suggest that the time required to go to the contact and drain the liquid is less than the time involved in the collision 1/. This would lead to fewer contacts when is increased and thus to a decrease of η with . Let us estimate the time required for the particles to form the contact. By neglecting the inertial forces, the time required for the particles to go into contact may be estimated by balancing the normal dissipative forces F_{d} with the forces involved during the contact PR^{2} − F_{repulsive}. F_{repulsive} is the force that has to be overcome to enter into contact. Then,^{9} in the shear thinning region at high shear rates, F_{repulsive} is much less than PR^{2}. We assume P close to σ and write . τ is the characteristic time to reach contact. We get . In all the situations involved in the shear thinning regime, τ is much less than 1/, which suggests that the particles have time to come close to each other during the collision. At this stage, Dinch has to be drained from the brushes of polymers in order to obtain a frictional contact.

Mass conservation of the solvent writes , where c is the solvent fraction. During the collision, i.e. for a time equal to 1/, this leads to . In the frictional situation, the contact has become fully plastic and deforms so that the normal stress remains quasi-constant P = H, where H is the hardness of the material. At room temperature, for polymer glasses, H/E = 10^{−1}–10^{−2}. a is the size of the micro-contact and is roughly given by , where s is the roughness of the material and R_{asperity} is the radius of curvature of the asperity.

In D1 and D2 suspensions, R_{asperity} is roughly equal to 100 nm and s is 2 nm, which gives a = 14 nm. k is the ratio of the permeability of the PVC brushes to Dinch over the viscosity of Dinch, k = 10^{−20} kg^{−1} m^{3} s. We get or 10^{3}/, which proves that all the Dinch has time to drain out of the contact during the collision at least for a shear rate of less than 10^{3} s^{−1}.

6.2.2 Variation of the microscopic friction coefficient.
The second shear thinning behavior at large shear rates occurs after the shear thickening transition and thus happens in the context of a frictional rheology (see Fig. 16a). To explain this shear thinning behavior, we thus turn to our experimental measurements of the friction coefficient between two beads, as shown in Fig. 12 and 13. We pointed out that μ decreases as a function of the normal load, going down from 0.12 to as low as 0.03 for normal forces up to 1 μN. In our experiments, the shear stress varies between 10^{4} Pa and 10^{5} Pa in the shear thinning region. Normal forces are of the order of 10^{5}–5 × 10^{5} Pa with B = 0.016 or B = 0.025 (eqn (10)). Even though local heterogeneities might be present at the level of the stress field,^{34} we use a simple argument to estimate the normal load. Assuming that the stress is homogeneous in the sample, the normal force applied on a single bead of radius 1 μm comprises between 0.3 and 1.5 μN, which corresponds to the range of normal loads for which μ decreases.

_{s} is the solvent viscosity. Here, we assume that the exponent n of the Krieger–Dougherty model does not depend upon the nature of the contact, and thus can be considered equal to the exponent found for the lubricated rheology at low shear rates (eqn (11)). This leads to . The shear rate is given by . Knowing experimentally the link between μ and F_{N}, we display in Fig. 17 the evolution of ϕ_{m}() as a function of μ. The data obtained by using the rheological measurements displayed in Fig. 17 collapse on a single curve for each of the dispersions revealing the validity of our analysis. The variations of ϕ_{m}() required to explain the shear thinning behavior are of the order of 10% when μ varies between 0.08 (close to the entry of the shear thickening zone) and 0.04–0.01 in the shear thinning zone. Such behavior is in agreement with simulations.^{38}

To go further in our analysis, we follow the model of Wyart and Cates.^{7} As shown in Fig. 16b, a decrease of μ will increase the value of the friction-dependent jamming density ϕ_{m} and thus will increase the distance between the volume fraction ϕ and the critical volume fraction ϕ_{m}. It is worth noting that ϕ_{m}(μ) is very sensitive to μ in the range 0–0.2, which corresponds to our microscopic friction coefficient variations. At the scale of the suspension, an increase in the shear rate will lead to an increase in the particular pressure and thus a decrease of the value of the inter-particle friction coefficient μ. To validate this picture, we correlate the critical volume fraction ϕ_{m} with the microscopic friction coefficient μ in the following.

For a frictional rheology, the viscosity of the suspension can be expressed as:

The second shear thinning regime at high shear rate occurs after the shear thickening transition and thus for a frictional rheology (Fig. 16). At the scale of two particles, we measure a decrease of the friction coefficient at large normal loads (Fig. 13). In this second frictional regime, an increase in the shear rate leads to an increase in the particular pressure, a decrease of the inter-particle friction coefficient, and thus an increase of the critical volume fraction ϕ_{m} at which viscosity diverges for frictional spheres. This increase in the critical volume fraction ϕ_{m} with shear rate explains the observed shear thinning behavior.

In conclusion, let us evaluate the generality of our results. As pointed out in the introduction, the second shear thinning regime observed after the shear thickening transition has also been observed in cornflour suspensions. In this situation, the second shear thinning regime has been attributed to phase separation.^{13} The behavior reported in this paper for PVC suspensions is thus not universal. Our ongoing work deals with experiments on silica particles stabilized by electrostatic forces. Both shear thinning regimes will be analyzed.

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