3.1.1. Steady viscosity

Figure 6 displays the variation of the measured relative steady viscosity, $\eta _r = \eta /\eta _0$, with applied shear stress, $\varSigma _{12}$, for (a) the suspension $S_{PS40}$ made of spherical PS particles and (b) $C_{PS40}$ made of crushed PS particles. Each coloured point corresponds to an experimental measurement of $\eta _r$ (relative viscosity corrected by (2.1)) at a given $\phi$ and a given $\varSigma _{12}$. The relative uncertainty for each measurement, not represented on the graphs in figure 6 in order to keep them clear, is always smaller than 5 %.

Figure 6. Variation of the measured relative steady viscosity, $\eta _r = \eta /\eta _0$, with applied shear stress $\varSigma _{12}$ for(a) the suspension $S_{PS40}$ made of spherical PS beads and (b) $C_{PS40}$ made of crushed PS particles. Each colour labels a solid volume fraction $\phi$: 0.43 (blue), 0.45 (orange), 0.47 (green), 0.49 (red) and 0.51 (purple). For each given $\phi$, the experimental measurements (coloured dot) are fitted by a power law (coloured straight line): $\varSigma _{12} = K \dot {\gamma }^n$ with $\dot {\gamma } = \varSigma _{12}/\eta$. The resulting parameters of these fits are shown in figure 8.

The values of viscosity measured on $S_{PS40}$ within the explored range of $\varSigma _{12}$ are in quite good agreement with other previous works present in the literature (Blanc et al. Reference Blanc, d'Ambrosio, Lobry, Peters and Lemaire2018; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019; Le et al. Reference Le, Izzet, Ovarlez and Colin2023) and conducted on an identical system (i.e. PS spheres of size close to $40\ \mathrm {\mu } {\rm m}$ dispersed in silicone oil). It appears in figure 6 that $C_{PS40}$ exhibits a rheological behaviour which is broadly similar to that which characterises $S_{PS40}$. In particular, we observe for both suspensions that:

1. (i) as expected, $\eta _r$ increases with $\phi$ for a given $\varSigma _{12}$;

2. (ii) $\eta _r$ decreases with $\varSigma _{12}$ for a given $\phi$, qualifying the non-Newtonian behaviour in the range of applied shear stress ($\varSigma _{12} \in [5\unicode{x2013}100] \ {\rm Pa}$) for both suspensions as shear-thinning;

3. (iii) as expected, the decay of $\eta _r$ with $\varSigma _{12}$ is steeper (meaning the shear-thinning behaviour is more pronounced) at large $\phi$.

On the other hand, the primary distinction between the suspensions is that the shear-thinning behaviour is stronger for $C_{PS40}$ compared with $S_{PS40}$ for a given $\phi$. Figure 7 displays the normalised difference of relative viscosity between the two suspensions, $(\eta _r^{C_{PS40}} - \eta _r^{S_{PS40}}) / \eta _r^{S_{PS40}}$, as function of the applied shear stress $\varSigma _{12}$. One can then easily observe that the suspension made of crushed particles is more viscous than the suspension made of spherical particles at low shear stress, whereas the viscosities of the two suspension are nearly the same at high shear stress.

Figure 7. Variation of the measured relative steady viscosity difference between $C_{PS40}$ and $S_{PS40}$, $\eta _r^{C_{PS40}} - \eta _r^{S_{PS40}}$, normalised by the measured relative steady viscosity for the suspension made of spherical PS particles, $\eta _r^{S_{PS40}}$, as function of the applied shear stress $\varSigma _{12}$. Each colour labels the solid volume fraction $\phi$: 0.43 (blue), 0.45 (orange), 0.47 (green), 0.49 (red) and 0.51 (purple).

According to the literature (Coussot & Piau Reference Coussot and Piau1994; Schatzmann, Fischer & Bezzola Reference Schatzmann, Fischer and Bezzola2003; Sosio & Crosta Reference Sosio and Crosta2009; Mueller, Llewellin & Mader Reference Mueller, Llewellin and Mader2010; Vance, Sant & Neithalath Reference Vance, Sant and Neithalath2015), we can quantify the non-Newtonian behaviour of such suspensions by fitting the experimental measurements by a power law (coloured straight lines in figure 6):

(3.1)\begin{equation} \varSigma_{12} = K \dot{\gamma}^n, \end{equation}

where $K$ and $n$ are the consistency factor and the shear-thinning index, respectively. Their values resulting from the fits of the experimental data in figure 6 are displayed as functions of $\phi$ in figure 8. We observe in figure 8(a) that $K$ increases with $\phi$ as expected. This reflects the increase of the viscosity with volume fraction. On the other hand, we observe in figure 8(b) that $n$ decreases with $\phi$, which accounts for the more pronounced shear-thinning behaviour at large $\phi$. One can also note that $n$ is systematically smaller in the case of $C_{PS40}$ at a given $\phi$, which reflects the more pronounced shear-thinning behaviour for the suspension made of crushed particles. More precisely, we observe that the relative variation of $n$ over the range of studied $\phi$ is roughly twice as large for $C_{PS40}$ than for $S_{PS40}$ (${\Delta n}/{\langle n\rangle } \sim 0.2$ for crushed particles whereas ${\Delta n}/{\langle n\rangle } \sim 0.1$ for spheres). Regarding the consistency factor, $K$, it is interesting to see that apparently $K^{C_{PS40}} \approx K^{S_{PS40}}$ at a given $\phi$. However, any further interpretation of this comparison in $K$ can be difficult since its units are not exactly the same between the two suspensions because $n^{C_{PS40}} \ne n^{S_{PS40}}$ ($[K] = {\rm Pa} \ {\rm s}^{n}$).

Figure 8. Variation of (a) the consistency factor $K$ and (b) the shear-thinning index $n$ with solid volume fraction $\phi$ for the suspensions $S_{PS40}$ made of spherical PS particles (blue discs) and $C_{PS40}$ made of crushed PS particles (orange squares), deduced from (3.1).

From figures 6 and 8, it can be seen that the more pronounced shear-thinning behaviour which characterises the suspension $C_{PS40}$ compared with the same suspension made of spheres ($S_{PS40}$) results from the observations that $\eta _r^{C_{PS40}} \approx \eta _r^{S_{PS40}}$ at large $\varSigma _{12}$ whereas $\eta _r^{C_{PS40}} > \eta _r^{S_{PS40}}$ at small $\varSigma _{12}$.

To conclude this section, we discuss why we have not considered the existence of a yield stress for either suspension. It is true that it is more relevant to characterise the rheological behaviour for some non-Brownian suspensions by using the Herschel–Bulkley (H-B) law:

(3.2)\begin{equation} \varSigma_{12} = \tau_c + K \dot{\gamma}^n \end{equation}

instead of (3.1). According to the literature (Pantina & Furst Reference Pantina and Furst2005; Guy et al. Reference Guy, Richards, Hodgson, Blanco and Poon2018; Richards et al. Reference Richards, Guy, Blanco, Hermes, Poy and Poon2020), it is known that the existence of a yield stress, $\tau _c$, may be caused by the presence of weak adhesive forces between solid particles which would lead to particle aggregation. Thus, the value of $\tau _c$ may be understood as the minimum stress required to break these aggregates. Furthermore, it is expected that the crushed particles, which have some flat faces, favour Van der Waals interactions since they offer a much larger contacting surface between particles compared with spheres, leading to a higher yield stress. In view of this, we have also fitted our experimental measurements in figure 6 by (3.2). The results have shown that the impact of the third fitting parameter $\tau _c$ on $K$ and $n$ is negligible, since we found $\tau _c < 1 \ {\rm Pa}$ for both suspensions and all explored $\phi$. For $S_{PS40}$, one can note that this is in good agreement with the works of Le et al. (Reference Le, Izzet, Ovarlez and Colin2023) who measured $\tau _c=0.3 \ {\rm Pa}$ for a very dense suspension made of PS beads having a size of $40\ \mathrm {\mu } {\rm m}$ and concentration $\phi =0.55$ in a silicone oil (the same system as studied in the present paper). The largest volume fraction studied in the present work being $\phi =0.51$, one can expect that the values of $\tau _c$ for $S_{PS40}$ are even smaller than this value within the range of studied $\phi$. Thus, we can advance with enough confidence that the minimum applied shear stress in our study ($\varSigma _{12} = 5 \ {\rm Pa}$) is at least 10 times larger than $\tau _c$ for $S_{PS40}$ and $C_{PS40}$. We confirm by some measurements from the shear-reversal experiments that adhesive forces do not play a significant role in the rheological behaviour of the studied suspensions within the applied range of shear stress $\varSigma _{12}$.

3.1.2. A stress-dependent jamming volume fraction

We want to recall that the shear-thinning regime observed for a frictional non-Brownian suspension is common and has already been observed extensively in the literature for suspensions made of spheres (Gadala-Maria & Acrivos Reference Gadala-Maria and Acrivos1980; Zarraga et al. Reference Zarraga, Hill and Leighton2000; Dbouk et al. Reference Dbouk, Lobry and Lemaire2013; Vázquez-Quesada et al. Reference Vázquez-Quesada, Tanner and Ellero2016, Reference Vázquez-Quesada, Mahmud, Dai, Ellero and Tanner2017) or even facetted (sugar) particles (Blanc et al. Reference Blanc, d'Ambrosio, Lobry, Peters and Lemaire2018). As explained in the introduction of the present paper, the physical origin of this complex behaviour remains an open question. Some recent works, including an experimental study from Chatté et al. (Reference Chatté, Comtet, Niguès, Bocquet, Siria, Ducouret, Lequeux, Lenoir, Ovarlez and Colin2018) and numerical simulations from Lobry et al. (Reference Lobry, Lemaire, Blanc, Gallier and Peters2019), have demonstrated that the shear-thinning behaviour for frictional spheres could come from a decay of the sliding friction coefficient, $\mu _s$, when the shear stress, $\varSigma _{12}$, increases, which induces an increase of the jamming volume fraction, $\phi _m$ (Wildemuth & Williams Reference Wildemuth and Williams1984; Zhou, Uhlherr & Luo Reference Zhou, Uhlherr and Luo1995; Blanc et al. Reference Blanc, d'Ambrosio, Lobry, Peters and Lemaire2018; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019; Gilbert et al. Reference Gilbert, Valette and Lemaire2022). The introduction of a stress-dependent jamming fraction $\phi _m(\varSigma _{12})$ is thus very useful to describe accurately the complex rheological behaviour of a suspension. Figure 9 displays the evolution of $\eta _r$ with $\phi$ for each applied $\varSigma _{12}$ (see colour code). The coloured points correspond to the experimental data and, for each applied $\varSigma _{12}$, the variation of the reduced viscosity, $\eta _r$, with the volume fraction, $\phi$, is fitted by a Maron–Pierce-type law:

(3.3)\begin{equation} \eta_r = \frac{\alpha_0}{\left( 1 - \dfrac{\phi}{\phi_m(\varSigma_{12})} \right)^2} . \end{equation}

Note that the parameter $\alpha _0$ in (3.3) is used in order to get an accurate fit of our experimental data. If one were to apply (3.3) over the full range of particle volume fractions, $\alpha _0$ would need to be 1 in order that $\eta _r=1$ when $\phi \rightarrow 0$. However, this fit only works in the dense regime, typically for $\phi \gtrsim 0.3$ in the case of frictional spherical particles, and hence $\alpha _0$ can have a value different from 1 in order to describe the variation of $\eta _r$ with $\phi$ accurately within this regime (Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019). In our case, a very good fit for each applied shear stress (plotted as coloured lines in figure 9) is obtained for $\alpha _0 = 0.85$ for both suspensions, a value not too far from the one used in the original equation of Maron & Pierce (Reference Maron and Pierce1956): $\alpha _0 = 1$ when $\phi _m \approx 0.64$. The value chosen here is also in good agreement with the numerical simulations of Lobry et al. (Reference Lobry, Lemaire, Blanc, Gallier and Peters2019) who have found that $0.65 \lesssim \alpha _0 \lesssim 1$ when $0 \lesssim \mu _s \lesssim 2$, which are the typical values of $\mu _s$ for common materials such as PS (Arshad et al. Reference Arshad, Maali, Claudet, Lobry, Peters and Lemaire2021; Le et al. Reference Le, Izzet, Ovarlez and Colin2023), polymethyl methacrylate, glass and rubber. In figure 9(b), we can observe that it satisfactorily fits the experimental data for crushed particles.

Figure 9. Variation of measured relative steady viscosity $\eta _r$ with solid volume fraction $\phi$ for suspensions (a) $S_{PS40}$ made of spherical PS and (b) $C_{PS40}$ made of crushed PS particles. Each colour labels the applied shear stress $\varSigma _{12}$: 5 (blue), 10 (orange), 15 (green), 20 (red), 28 (purple), 36 (brown), 45 (pink), 60 (grey), 80 (yellow) and 100 (cyan). For each given $\varSigma _{12}$, the experimental measurements (coloured dot) are fitted by a Maron–Pierce-type law (see (3.3)). The measurements of $\phi _m(\varSigma _{12})$ resulting from these fits are shown in figure 10.

The good fit obtained with this given value of $\alpha _0$ for both suspensions is not so surprising as it is known that the values of $\eta _r$ when $\phi \rightarrow \phi _m$ are controlled primarily by the value of $\phi _m$ (Blanc et al. Reference Blanc, d'Ambrosio, Lobry, Peters and Lemaire2018; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019). Figure 10 displays the variation of $\phi _m$ with $\varSigma _{12}$, determined by (3.3) with $\alpha _0 = 0.85$ for the suspensions $S_{PS40}$ made of PS beads (blue circle) and $C_{PS40}$ made of crushed particles (orange squares). As expected, we observe that $\phi _m$ increases with $\varSigma _{12}$ for both suspensions. This increase is larger for $C_{PS40}$ when compared with $S_{PS40}$, which illustrates the more pronounced shear-thinning behaviour for the suspension made of crushed particles. To be precise, we observe that $\phi _m^{C_{PS40}} < \phi _m^{S_{PS40}}$ within the smaller end of the $\varSigma _{12}$ range whereas $\phi _m^{C_{PS40}} \approx \phi _m^{S_{PS40}}$ at the largest $\varSigma _{12}$ values. This mirrors the previous observation made from the viscosity comparison between the two suspensions.

Figure 10. Variation of the jamming fraction $\phi _m$ with shear stress $\varSigma _{12}$ for the suspensions $S_{PS40}$ made of spheres PS (blue circle) and $C_{PS40}$ made of crushed PS particles (orange square), deduced from figure 9 and (3.3) with $\alpha _0 = 0.85$. The coloured area for each suspension is related to the possible range of $\alpha _0$: ${0.65 \lesssim \alpha _0 \lesssim 1}$.

It is easy to understand that the values of $\phi _m$ determined by (3.3) might depend slightly on the value of $\alpha _0$, and hence we have also fitted our experimental data of $\eta _r(\phi )$ by (3.3) with the two known extreme values of $\alpha _0$ (Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019): $\alpha _0 = 0.65$ and $\alpha _0 = 1$ (the corresponding curves are not plotted in figure 9 in order to keep the graph clear). The confidence areas plotted in figure 10 for each suspension represent this influence of $\alpha _0$ on $\phi _m$. Thus, one can observe that the values of $\phi _m$ for $S_{PS40}$ are between $0.560 \pm 0.005$ and $0.595 \pm 0.009$ in very good agreement with Lobry et al. (Reference Lobry, Lemaire, Blanc, Gallier and Peters2019), whereas they are between $0.550 \pm 0.004$ and $0.595 \pm 0.009$ for $C_{PS40}$. It is quite satisfying that these values of $\phi _m$ for both types of suspension are globally in very good agreement with the literature when non-Brownian frictional ($\mu _s \ne 0$) suspensions are considered (Zarraga et al. Reference Zarraga, Hill and Leighton2000; Ovarlez et al. Reference Ovarlez, Bertrand and Rodts2006; Boyer et al. Reference Boyer, Guazzelli and Pouliquen2011; Mari et al. Reference Mari, Seto, Morris and Denn2014; Peters et al. Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016; Singh et al. Reference Singh, Mari, Denn and Morris2018; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019; Singh et al. Reference Singh, Ness, Seto, de Pablo and Jaeger2020). Furthermore, it is noteworthy that the primary observations that the jamming volume fraction $\phi _m$ is smaller in the case of non-spherical particles at low shear stress whereas the rheological behaviours of non-spherical and spherical particles are characterised by the same $\phi _m$ at high shear stress is not altered by the value of $\alpha _0$ within its known range.

The rest of the paper focuses on finding a physical mechanism to explain the observed rheological difference between the suspension made of crushed particles and the suspension made of spheres.

3.2.1. Shear reversal experiments and absence of adhesion

As mentioned previously in this paper, shear-thinning behaviour of a suspension is common and can have different possible physical origins depending on the studied system (Gadala-Maria & Acrivos Reference Gadala-Maria and Acrivos1980; Zarraga et al. Reference Zarraga, Hill and Leighton2000; Dbouk et al. Reference Dbouk, Lobry and Lemaire2013; Vázquez-Quesada et al. Reference Vázquez-Quesada, Tanner and Ellero2016, Reference Vázquez-Quesada, Mahmud, Dai, Ellero and Tanner2017; Blanc et al. Reference Blanc, d'Ambrosio, Lobry, Peters and Lemaire2018; Chatté et al. Reference Chatté, Comtet, Niguès, Bocquet, Siria, Ducouret, Lequeux, Lenoir, Ovarlez and Colin2018; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019; Gilbert et al. Reference Gilbert, Valette and Lemaire2022). One of them is adhesion. Weak adhesive forces exist between solid particles that would lead to particle aggregation. In this scenario, two main features would appear. First, a suspension would exhibit a yield stress $\tau _c$ (Brown et al. Reference Brown, Forman, Orellana, Zhang, Maynor, Betts, DeSimone and Jaeger2010), which may be understood as the minimum stress needed to break these aggregates. Second, they would exhibit shear-thinning behaviour, related to the fact that increasing $\varSigma _{12}$ would break more and more aggregates, which would produce as a result a decrease in the viscosity of the suspension. Furthermore, this explanation would be suitable to explain the highest viscosity at low $\varSigma _{12}$ for crushed particles while the viscosity of the two types of suspensions ($S_{PS40}$ and $C_{PS40}$) would tend to be similar at large $\varSigma _{12}$. As already mentioned previously in § 3.1.1, flat surfaces of crushed particles favour particle adhesion. Potential aggregates are then less likely to be destroyed in $C_{PS40}$ than in $S_{PS40}$ when both suspensions are sheared at a given small enough $\varSigma _{12}$. This would make $C_{PS40}$ more viscous than $S_{PS40}$ when $\varSigma _{12}$ is small. In contrast, when $\varSigma _{12} \gg \tau _c$, all the aggregates are destroyed by shear, even in the case of $C_{PS40}$ which then flows similarly to $S_{PS40}$.

The first flaw in this explanation has already been presented in § 3.1.1. Indeed, we have seen that the smallest value of $\varSigma _{12}$ that we apply to shear the suspension is at least 10 times larger than $\tau _c$. At $\varSigma _{12} = 10\ {\rm Pa}$ (second lowest value of applied shear stress), we have $\varSigma _{12} \gtrsim 20 \times \tau _c$. Yet, a significant difference of viscosity between the two suspensions still remains at large $\phi$, which raises doubt that adhesion could be the main physical origin of the stronger shear-thinning for $C_{PS40}$. For instance, at $\phi = 0.51$ and $\varSigma _{12} = 10\ {\rm Pa}$, $\eta _r = (110 \pm 5) \ {\rm Pa} \ {\rm s}$ for $C_{PS40}$ whereas $\eta _r = (80 \pm 5) \ {\rm Pa} \ {\rm s}$ for $S_{PS40}$ (see figure 9), which gives a difference of the order of $30\,\%$. Nevertheless, we understand that this argument about $\varSigma _{12}$ and $\tau _c$ alone is insufficient to support the statement that the adhesion is not mainly responsible for the more pronounced shear-thinning for $C_{PS40}$. It is indeed very difficult to estimate precisely when the adhesive forces can be neglected only from $\tau _c$. To go further, we have conducted a series of shear-reversal experiments on both types of suspension by following the procedure of Blanc et al. (Reference Blanc, d'Ambrosio, Lobry, Peters and Lemaire2018).

A shear-reversal experiment may turn out to be very interesting. It is a very basic experiment (the suspension is simply sheared at a given $\varSigma _{12}$ in a given direction before the flow direction is reversed whereas $\varSigma _{12}$ is kept constant), characterised by a very specific transient response of $\eta$ which has been observed in all shear reversal experiments (Gadala-Maria & Acrivos Reference Gadala-Maria and Acrivos1980; Blanc et al. Reference Blanc, Peters and Lemaire2011a) and in simulations (Ness & Sun Reference Ness and Sun2016; Peters et al. Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016). Figure 11 displays an example of the transient response of $\eta$ for $S_{PS40}$ (in blue) and $C_{PS40}$ (in orange) at $\varSigma _{12}=10 \ {\rm Pa}$ and $\phi = 0.51$. As can be observed, a step-like drop of $\eta$ occurs just after the shear reversal and the viscosity of the suspension reaches a minimum value, $\eta _{min}$, at a strain $\gamma = \gamma _{min}$ ($\gamma =0$ corresponds to the moment of reversal). This drop is then followed by a rebound of the viscosity which reaches the steady value, $\eta _s = \eta _0\eta _r$, it had before the shear reversal, over an accumulated strain, $\gamma$, roughly equal to $\gamma _s \sim 10$. Interestingly, the numerical simulations from Ness & Sun (Reference Ness and Sun2016) and Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) have shown that the hydrodynamic and contact contribution to the viscosity, denoted respectively $\eta ^H$ and $\eta ^C$, are connected directly to the values of $\eta$ and $\eta _{min}$. More precisely, with $\eta _s = \eta ^H + \eta ^C$, Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) have numerically shown in the case of a non-Brownian suspension made of (frictional or frictionless) beads the following relations:

(3.4a,b)\begin{equation} \eta^C = \frac{\eta_s - \eta_{min}}{0.85/\eta_0} \quad {\rm and} \quad \eta^H = \frac{\eta_{min} - 0.15 \eta_s}{0.85/\eta_0} . \end{equation}

Roughly, $\eta ^H \sim \eta _{min}/\eta _0$ and $\eta ^C \sim (\eta _s - \eta _{min})/\eta _0$. We refer readers to the numerical work of Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) to better understand the physical origin of this result. In brief, the particles in contact tend to separate when the shear is reversed. The microstructure of the suspension is thus broken, which induces the drop of the viscosity. Progressively, the microstructure of the suspension is then rebuilt (mirroring the microstructure before the shear reversal since the flow direction has been reversed), which induces the rebound of $\eta$ to its steady value.

Figure 11. Example of transient viscosity response as a function of the accumulated strain $\gamma$ during a shear reversal experiment for $S_{PS40}$ (in blue) and $C_{PS40}$ (in orange) with $\varSigma _{12} = 10 \ {\rm Pa}$ and $\phi =0.51$. $\gamma = 0$ corresponds to the moment when the flow is reversed. Inset: Enlarged view to better visualise the minimum value of viscosity, $\eta _{min}$ reached during the transient when $\gamma = \gamma _{min}$.

In the present study, the transient viscosity induced by a shear reversal can be very interesting because, if a stress-dependent particle aggregation occurs, then it should also affect the values of $\eta _{min}$ and the characteristic strains, $\gamma$. Notably, Gilbert (Reference Gilbert2021) has studied the rheology of a non-Brownian frictional suspension composed of homemade soft PDMS particles (Young modulus, $E_{PDMS} = 1.8 \ {\rm MPa} \ll E_{PS} \sim 3 \ {\rm GPa}$) suspended in Span 80 (Newtonian liquid). By using the JKR theory (Johnson, Kendall & Roberts Reference Johnson, Kendall and Roberts1971), the author has observed for this suspension that adhesion plays a role if $\varSigma _{12} \lesssim \tau _a \approx 10 \ {\rm Pa}$. By doing shear reversal experiments, he has then shown (see figures 86-2 of Gilbert Reference Gilbert2021) that $\eta _{min}^{\varSigma _{12} < \tau _a} \gg \eta _{min}^{\varSigma _{12} > \tau _a}$, and that the characteristic deformation of the transient response for a shear reversal, $\gamma _s$, was much larger than 10 ($\gamma _s \sim 50$ for $\phi =0.4$ and $\varSigma _{12} = \tau _a$ in the case of his suspension).

Figure 12 displays the experimental measurements (coloured symbols) of $\eta _{min}/\eta _0$ within the studied range of $\varSigma _{12}$ for (a) $S_{PS40}$ and (b) $C_{PS40}$, and one can see it is not similar to what has been observed by Gilbert (Reference Gilbert2021) for a non-Brownian suspension made of adhesive beads. First, $\eta _{min}^{C_{PS40}} \approx \eta _{min}^{S_{PS40}}$ at a given $\varSigma _{12}$ and a given $\phi$. Second, $\eta _{min}$ is weakly dependent on $\varSigma _{12}$ for a given $\phi$ in the studied range of applied shear stress. Thus, it is likely that $\varSigma _{12} \gg \tau _a$ for both suspensions in the present study and that adhesion can then be neglected. We also want to underline that, based on (3.4a,b), we can determine $\eta ^H \approx 5\unicode{x2013}6$ for $S_{PS40}$ concentrated at $45\,\%$ from the experimental data (roughly independent of $\varSigma _{12}$), which is in very good agreement with numerical simulations from Gallier et al. (Reference Gallier, Lemaire, Peters and Lobry2014) which shows $\eta ^H \approx \eta _\infty \approx 5\unicode{x2013}6$ for a non-Brownian viscous suspensions of (frictionless or frictional) spheres at $\phi = 0.45$. Here $\eta _\infty$ is the high-frequency dynamic viscosity (Van der Werff & De Kruif Reference Van der Werff and De Kruif1989).

Figure 12. Variation of the measured relative minimum value of the viscosity, $\eta _{min}/\eta _0$, with applied shear stress $\varSigma _{12}$ for (a) the suspension made of spherical PS and (b) the suspension made of crushed PS particles. The colour code labels the solid volume fraction $\phi$ of the suspension in the same way as in figure 6. For each given $\phi$, the experimental measurements (coloured symbols) for $\varSigma _{12} \leqslant 45 \ {\rm Pa}$ are fitted by a power law (coloured straight line): $\varSigma _{12} = K_{min} \times \dot {\gamma }^{\nu _{min}}$ with $\dot {\gamma } = \varSigma _{12}/\eta _{min}$. The parameters resulting from these fits are shown in figure 13.

Note that, in figure 12, the experimental data for $\eta _{min}$ of each suspension for $\varSigma _{12} \leqslant 45 \ {\rm Pa}$ have been then fitted by a power law, based on (3.1) where now $K \equiv K_{min}$ and $n \equiv n_{min}$, to quantify these observations of $\eta _{min}$. The fitting parameters $K_{min}$ and $n_{min}$ resulting from this fit are presented in figure 13. The upper limit for the shear stress considered here for the fit ($\varSigma ^{max}_{12} = 45 \ {\rm Pa})$ is imposed due to the poor resolution of the measurement of $\eta _{min}$ when $\varSigma _{12} > \varSigma ^{max}_{12}$. Thus, the apparent plateau of $\eta _{min}$ observed at large $\varSigma _{12}$ has no physical meaning. It is an experimental artifact. Therefore, one can clearly see in figure 13 that $\eta _{min}$ for the suspensions $S_{PS40}$ and $C_{PS40}$ are characterised by the same rheology, as we observe that both $K_{min}$ and $n_{min}$ are independent of the considered suspension. In addition, $S_{PS40}$ and $C_{PS40}$ are both characterised by a Newtonian behaviour when $\eta = \eta _{min}$: $0.94 \lesssim n_{min} \lesssim 1$ for both suspensions, when $0.43 \leqslant \phi \leqslant 0.51$ and $5 \leqslant \varSigma _{12} \leqslant 100 \ {\rm Pa}$. Moreover, the $\eta _{min}$ results indicate that hydrodynamic interactions are not significantly affected by particle shape.

Figure 13. Variation of (a) the consistency factor, denoted $K_{min}$, and (b) the shear-thinning index, denoted $n_{min}$, with the solid volume fraction $\phi$ for the suspension made of spherical PS particles (blue discs) and the suspension made of crushed PS particles (orange squares), when $\eta = \eta _{min}$ and $\varSigma _{12} \leqslant 45\ {\rm Pa}$.

Figure 14 displays the experimental values of characteristic strains, $\gamma _{min}$ (open symbols) and $\gamma _{0.5}$ (closed symbols) for the suspension made of PS spheres (blue) and that made of crushed particles (orange), as a function of $\phi$ when $\varSigma _{12} = 10 \ {\rm Pa}$. Whereas $\gamma _{min}$ corresponds to the accumulated strain from the moment of shear reversal to when ${\eta = \eta _{min}}$, $\gamma _{0.5}$ is defined as the accumulated strain from the minimum state ($\eta = \eta _{min}$) to the moment when the viscosity has recovered 50 % of its reversal-induced deficit:

(3.5)\begin{equation} \eta(\gamma_{0.5}) = \eta_{min} + 0.5 \times \left ( \eta_s - \eta_{min} \right ). \end{equation}

The uncertainties of the experimental measurements for the characteristic strains are estimated at $\pm 5 \times 10^{-2}$. The experimental data are also compared with numerical (Pine et al. Reference Pine, Gollub, Brady and Leshansky2005; Peters et al. Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) and experimental (Pine et al. Reference Pine, Gollub, Brady and Leshansky2005) results from the literature. We recall that Pine et al. (Reference Pine, Gollub, Brady and Leshansky2005) have shown that a particle in a non-Brownian suspension subjected to oscillatory shear flow returns to its initial position at each oscillatory cycle consistent with Stokes flow reversibility as long as the strain amplitude does not exceed a critical value, denoted $\gamma _c$. As explained by Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016), $\gamma _{0.5}$ corresponds to the strain necessary for spherical particles to form a significant amount of solid contacts which would then lead to displacements that violate Stokes flow reversibility.

Figure 14. Characteristic strains $\gamma _{min}$ for the minimum viscosity (open symbols) and $\gamma _{0.5}$ for the partial recovery (filled symbol) as a function of volume fraction $\phi$. The graph on the right is just an enlarged view of the left graph. The experimental measurements ($\varSigma _{12} = 10 \ {\rm Pa}$) from this study are represented for the suspensions $S_{PS40}$ (blue discs) and $C_{PS40}$ (orange squares). From the literature, Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) have performed several numerical simulations to study the influence of the sliding friction coefficient $\mu _s$ and the relative roughness $h_r/d$ on $\gamma _{min}$ and $\gamma _{0.5}$. Some of their results are also plotted (black symbols) on the graph for $h_r/d = 10^{-2}$ (empty symbols for $\gamma _{min}$ and filled symbols for $\gamma _{0.5}$): ($\triangle$) $\mu _s = 0$; ($\circ$) $\mu _s = 0.5$; (${\nabla }$) $\mu _s = 1$. Some experimental measurements ($\star$) from Pine et al. (Reference Pine, Gollub, Brady and Leshansky2005) of the critical strain $\gamma _c$ for which irreversibility occurs are also plotted, as well as the power law ($-$) resulting from their numerical simulations: $\gamma _c = 0.14\phi ^{-1.93}$.

One can observe that $\gamma _{min}$ decreases with $\phi$ for both suspensions and that $\gamma _{min}^{C_{PS40}} \sim \gamma _{min}^{S_{PS40}}$ for a given $\phi$. A closer examination shows that $\gamma _{min}^{C_{PS40}}$ is nearly equal to $\gamma _{min}^{S_{PS40}}$ at the highest volume fraction ($\phi = 0.51$) whereas it is smaller than $\gamma _{min}^{S_{PS40}}$ at smaller volume fractions with the largest difference occurring $\phi = 0.43$, the lowest volume fraction studied. Furthermore, one can note that the experimental data are well-predicted by the simulations from Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016), conducted on a non-Brownian suspension of frictional spheres, characterised by a combination of a sliding friction coefficient, $0 \leqslant \mu _s \leqslant 1$, and a relative roughness height, $h_r/d = 10^{-2}$.

Analogous to $\gamma _{min}$, we observe that $\gamma _{0.5}$ decreases when $\phi$ increases, which is in good agreement with the literature. In addition, the experimental values for ${S_{PS40}}$ at ${\varSigma _{12} = 10 \ {\rm Pa}}$ (filled blue discs) are well-captured by the numerical simulations from Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) ($\mu _s=0.5$, $h_r/d=10^{-2}$). One can also note that $\gamma _{0.5}^{C_{PS40}} \gtrsim \gamma _{0.5}^{S_{PS40}}$ for a given $\phi$ even though both are still of the same order and follow the same trend. This slight difference is interesting for two different reasons. First, as we have seen from the work of Gilbert (Reference Gilbert2021), having $\gamma _{0.5}^{C_{PS40}} \sim \gamma _{0.5}^{S_{PS40}}$ (and $\gamma _{s} \sim 10$ as can be observed in figure 11) is consistent with the inference that adhesion forces do not play a predominant role in the rheology of $C_{PS40}$ compared with $S_{PS40}$, within the applied range of $\varSigma _{12}$. Second, Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) have explained that the force network is reestablished over a typical strain equal to $\gamma _{0.5}$ during a shear reversal experiment. According to this assertion, it would be a little harder for the particles in $C_{PS40}$ to rearrange during the transient in order to rebuild the microstructure leading to contact forces (see the works from Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) for details on the physical mechanism). We think this is related to the shape-induced rolling resistance and it could be interesting to study it using numerical simulations, because shear reversal gives access to the separate hydrodynamic and contact contributions to the stress. More generally, Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) have studied the influence of $\mu _s$ and $h_r/a$ on the values of characteristic strains and we think it could be interesting to also quantify the role played by $\mu _r$, if any, in the transient of a shear-reversal experiment.

3.2.2. Variable sliding friction coefficient

In the previous section, we have seen that adhesion cannot account for the shear-thinning behaviour of the two suspensions and that there is no evidence from shear reversal experiments of stronger adhesion in $C_{PS40}$ than $S_{PS40}$. In this section, we show that, unlike adhesion, a variable sliding friction model allows us to explain the shear-thinning behaviour of the suspensions.

From the numerical works of Mari et al. (Reference Mari, Seto, Morris and Denn2014) and Gallier et al. (Reference Gallier, Lemaire, Peters and Lobry2014), it is well known that the jamming volume fraction, $\phi _m$, is strongly dependant on the sliding friction coefficient, $\mu _s$. In addition, as presented in the introduction of the present paper, the recent literature (Chatté et al. Reference Chatté, Comtet, Niguès, Bocquet, Siria, Ducouret, Lequeux, Lenoir, Ovarlez and Colin2018; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019; Arshad et al. Reference Arshad, Maali, Claudet, Lobry, Peters and Lemaire2021; Le et al. Reference Le, Izzet, Ovarlez and Colin2023) relates the shear-thinning behaviour of a non-Brownian frictional suspension to a decay of $\mu _s$ when the normal force $F_N$ between particles (directly proportional to $\varSigma _{12}$) increases:

(3.6)\begin{equation} \mu_s = \mu_s^\infty \times \text{coth} \left[ \mu_s^\infty \left( \frac{\varSigma_{12}}{\varSigma_{c}} \right)^{m}\right] \quad \text{with} \ \mu_s \xrightarrow{\varSigma_{12 \rightarrow \infty}} \mu_s^\infty . \end{equation}

We recall that $\varSigma _c$ is a critical value which characterises the elastoplastic transition of asperities deformation (Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019) and $\mu _s^\infty$ is the constant value reached by $\mu _s$ when $\varSigma _{12} \gg \varSigma _c$. As for the exponent $m$, its value is directly related to the fact that the model (Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019) considers that the contact between two particles occurs at only one or two asperities (mono-asperity contact) and that the particle asperities are supposed to be close to hemispheres (for which $m \sim 1/3$, Brizmer et al. Reference Brizmer, Kligerman and Etsion2007). Recent AFM measurements performed on PS beads ($d \approx 40\ \mathrm {\mu } {\rm m}$) suspended in an aqueous liquid (Arshad et al. Reference Arshad, Maali, Claudet, Lobry, Peters and Lemaire2021) or in silicone oil (Le et al. Reference Le, Izzet, Ovarlez and Colin2023) have given $\mu ^\infty \approx 0.2$, $\varSigma _c \approx 10 \ {\rm Pa}$ and $m \approx 0.5$. Figure 15(a) displays the variation of $\mu _s$ with $\varSigma _{12}$ based on these values ($--$).

Figure 15. (a) Variation of the sliding friction coefficient $\mu _s$ with $\varSigma _{12}$ (see (3.6)) as computed by Brizmer et al. (Reference Brizmer, Kligerman and Etsion2007) and Lobry et al. (Reference Lobry, Lemaire, Blanc, Gallier and Peters2019) ($\cdots\!$) and the variation used to fit the experimental data $\phi _m(\varSigma _{12})$ ($--$). The solid part of the curve corresponds to the experimentally studied range of $\varSigma _{12}$. (b) Jamming volume fraction $\phi _m$ as function of shear stress $\varSigma _{12}$. The experimental measurements for $S_{PS40}$ (blue discs) and $C_{PS40}$ (orange squares) are fitted by the model obtained by combining (3.6) and (3.7), where $\mu _s^\infty = 0.2$, $\varSigma _c = 10 \ {\rm Pa}$, $m = 0.5$ and $\phi _m^0 = 0.65$. Here $\phi _m^{\infty }$ and $X^p$ are free parameters. The best fit gives $\phi _m = 0.555$ and $X^p = 2.3$ in the case of $S_{PS40}$. For $C_{PS40}$, we determined $\phi _m = 0.536$ and $X^p = 1.8$. The two vertical dashed red straight lines on each graph delimit the range of $\varSigma _{12}$ explored experimentally.

Our main assumption is that (3.6) can describe the variation of $\mu _s$ with $\varSigma _{12}$ in both suspensions. In addition to the form of the function, we assume that the values of $\mu _s^\infty$, $\varSigma _c$ and $m$ are also identical for both types of particles: spheres and crushed. We understand that this statement is critical but several arguments tend to support it. We recall that sliding friction should depend on the local interaction of two surfaces. In the present study, the same PS particles (in size and material) constitute the two studied suspensions and, even though the crushing process does change the radii of curvature of particles in some places, we assume that it does not significantly affect the topology of the asperities. Thus, as considered from Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) for spheres, we assume that the solid contact between particles occurs through only a few asperities even to the crushed particles in $C_{PS40}$. In this scenario, the value of $m$ determined by Arshad et al. (Reference Arshad, Maali, Claudet, Lobry, Peters and Lemaire2021) and Le et al. (Reference Le, Izzet, Ovarlez and Colin2023) for PS spherical particles ($m \approx 0.5$) can be applied for the crushed ones. Moreover, the values of $\mu _s^\infty$ and $\varSigma _c$ determined for PS beads ($\mu _{\infty } \approx 0.2$ and $\varSigma _c \approx 10 \ {\rm Pa}$) by Arshad et al. (Reference Arshad, Maali, Claudet, Lobry, Peters and Lemaire2021) and Le et al. (Reference Le, Izzet, Ovarlez and Colin2023), depending on the properties of the solid particle material (Young's modulus $E$, Poisson's ratio $\nu$, yield strength $Y_0$) and asperity height $h_r$, can also be kept the same for the crushed PS particles. We want to underline that the assumption that $\mu _s^{C_{PS40}}(\varSigma _{12}) \approx \mu _s^{S_{PS40}}(\varSigma _{12})$ is also consistent with the results displayed in figure 14 for the characteristic strain, $\gamma _{min}$. Indeed, Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) have shown the role played by $\mu _s$ on $\gamma _{min}$, and the experimental data from the present study tend to show that the values of $\mu _s$ are between 0 and 1 for the studied suspensions and are very similar between the two.

Lobry et al. (Reference Lobry, Lemaire, Blanc, Gallier and Peters2019) have proposed the following phenomenological function $\phi _m(\mu _s)$ relating the jamming volume fraction to the sliding friction coefficient:

(3.7)\begin{equation} \phi_m = \phi_m^{\infty} + \big(\phi_m^{0} - \phi_m^{\infty}\big) \left[ \frac{\exp({-X^p \,\text{atan}(\mu_s)}) -\exp({-{\rm \pi} X^p/2})}{1 -\exp({-{\rm \pi} X^p/2})} \right] ,\end{equation}

where $\phi _m^{\infty }$ and $\phi _m^{0}$ are specific values of $\phi _m$ when the particles cannot slide ($\mu _s \xrightarrow {\varSigma _{12} \rightarrow 0} \infty$) and when the suspension is frictionless ($\mu _s \rightarrow 0$), respectively. The expression contains a fitting parameter $X^p$. Figure 15(b) redisplays the variation of the jamming volume fraction, $\phi _m$, with shear stress, $\varSigma _{12}$ (already shown in figure 10). In this new figure, the experimental data (represented as blue discs for $S_{PS40}$ and orange squares for $C_{PS40}$) are fitted by the model described by (3.6) and (3.7). Additionally, $\phi _m^{\infty }$ and $X^p$ are left as free parameters while $\phi _m^{0}$ is set equal to 0.65, in good agreement with the literature when the frictionless ($\mu _s = 0$) regime is considered (Gallier et al. Reference Gallier, Lemaire, Peters and Lobry2014; Mari et al. Reference Mari, Seto, Morris and Denn2014; Gallier, Peters & Lobry Reference Gallier, Peters and Lobry2018; Singh et al. Reference Singh, Mari, Denn and Morris2018; Le et al. Reference Le, Izzet, Ovarlez and Colin2023). We show later in the present paper (in § 3.2.4) that it is also in very good agreement with the rheology of $S_{PS40}$ and $C_{PS40}$ sheared in the frictionless regime.

We observe that the experimental data are well-predicted by the model within the experimentally explored range of shear stress ($\varSigma _{12} \in [5\unicode{x2013}100] \ {\rm Pa}$, coloured solid lines in figure 15b). By coupling figures 15(a) and 15(b), one can note that $\phi _m^{S_{PS40}} \approx (0.585 \pm 0.008)$ when $\mu _s \approx 0.5$ ($\varSigma _{12} \sim 45 \ {\rm Pa}$) and $\phi _m^{S_{PS40}} = (0.568 \pm 0.006)$ when $\mu _s = 1$ ($\varSigma _{12} \sim 10 \ {\rm Pa}$) for the suspension made of spheres, which is in quite good agreement with numerical simulations from the literature. For instance, Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) and Gallier et al. (Reference Gallier, Peters and Lobry2018) found $\phi _m \approx 0.59$ and $\phi _m \approx 0.58$, respectively, when $\mu _s = 0.5$. The numerical simulations of Mari et al. (Reference Mari, Seto, Morris and Denn2014) and the ones from Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) predict $\phi _m \approx 0.58$ and $\phi _m \approx 0.56$ for $\mu _s = 1$, respectively.

Then, one can observe in figure 15(b) that the variation of $\phi _m$ with $\varSigma _{12}$ deduced from the fit exhibits two plateaus ($--$), each located at extreme values of shear stress: the first when $\varSigma _{12} \lesssim 10^{-1} \ {\rm Pa}$ and the second when $\varSigma _{12} \gtrsim 10^{3} \ {\rm Pa}$. According to (3.6) (Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019), the plateau when $\varSigma _{12} \rightarrow +\infty$ is due to the saturation of $\mu _s$ (plastic regime) when $\varSigma _{12}/\varSigma _c \gg 1$ (see figure 15a). The other plateau predicted by the fit when $\varSigma _{12} \rightarrow 0$ is explained by the weak influence of $\mu _s$ on the values of $\phi _m$ when $\mu _s$ is larger than 1 or 2, as demonstrated by the numerical works of Mari et al. (Reference Mari, Seto, Morris and Denn2014), Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) and Lobry et al. (Reference Lobry, Lemaire, Blanc, Gallier and Peters2019).

In figure 15(b), the function $\phi _m(\varSigma _{12})$ deduced from the fit is then characterised by:

1. (i) $\phi _m \xrightarrow [\mu _s \rightarrow \infty ]{\varSigma _{12} \rightarrow 0} \phi _m^{\infty } = (0.555 \pm 0.005)$ for $S_{PS40}$ and $\phi _m^{\infty } = (0.536 \pm 0.004)$ for $C_{PS40}$;

2. (ii) $\phi _m \xrightarrow [\mu _s \rightarrow 0.2]{\varSigma _{12} \rightarrow \infty } \phi _m^{0.2} = (0.61 \pm 0.01)$ for $S_{PS40}$ and $C_{PS40}$.

Note that the estimated values of $\phi _m^{\infty }$ and $\phi _m^{0.2}$ for $S_{PS40}$ are in very good agreement with the literature (Fernandez et al. Reference Fernandez, Mani, Rinaldi, Kadau, Mosquet, Lombois-Burger, Cayer-Barrioz, Herrmann, Spencer and Isa2013; Gallier et al. Reference Gallier, Lemaire, Peters and Lobry2014; Mari et al. Reference Mari, Seto, Morris and Denn2014; Peters et al. Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019; Le et al. Reference Le, Izzet, Ovarlez and Colin2023). Mari et al. (Reference Mari, Seto, Morris and Denn2014) and Lobry et al. (Reference Lobry, Lemaire, Blanc, Gallier and Peters2019) determined $\phi _m \approx 0.56$ and $\phi _m \approx 0.546$ when $\mu _s \rightarrow +\infty$, respectively, whereas Le et al. (Reference Le, Izzet, Ovarlez and Colin2023) obtained $\phi _m^{\infty } \approx 0.55$ by studying experimentally the same suspension as $S_{PS40}$. Peters et al. (Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016) found $\phi _m \approx 0.61$ when $\mu _s = 0.3$ and Lobry et al. (Reference Lobry, Lemaire, Blanc, Gallier and Peters2019) determined $\phi _m \xrightarrow [\mu _s \rightarrow 0.27]{\varSigma _{12} \rightarrow \infty } \phi _m^{0.27} = 0.625$. Regarding $C_{PS40}$, one can observe that

$\phi _m^{\infty }|_{C_{PS40}} < \phi _m^{\infty }|_{S_{PS40}}$ and $\phi _m^{0.2}|_{C_{PS40}} \approx \phi _m^{0.2}|_{S_{PS40}}$, as expected.

To sum up, we have observed by fitting the experimental data $\phi _m(\varSigma _{12})$ by (3.6) and (3.7) that the shear thinning behaviour of the two studied suspensions ($S_{PS40}$ and $C_{PS40}$) is induced by the same variable friction law, $\mu _s(\varSigma _{12})$. The main difference between the two is in $\phi _m^\infty$, whose value is smaller in the case of globular/crushed PS particles compared with the PS spheres. One can note that $X^p|_{C_{PS40}} \sim X^p|_{S_{PS40}} \sim 2$, which supports the statement about the sliding friction being the same for the two types of particles. Moreover, $X^p|_{S_{PS40}} \approx 2.3$ is a value which is in good agreement with the literature (Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019; Arshad et al. Reference Arshad, Maali, Claudet, Lobry, Peters and Lemaire2021; Le et al. Reference Le, Izzet, Ovarlez and Colin2023).

3.2.3. Geometry-related rolling resistance

A decade ago, Estrada et al. (Reference Estrada, Taboada and Radjai2008, Reference Estrada, Azéma, Radjai and Taboada2011) have simulated rolling regular polygons and shown that the stress was the same as discs (with the same $\phi$) equipped with a rolling friction coefficient, $\mu _r$ (see the schema in figure 2). This would then mean that the geometric effect is a rolling resistance which, in the case of equivalent discs, can be obtained with a $\mu _r$.

More recently, in the frame of a study characterising the shear-thickening behaviour of suspensions made of hard spheres (for which $\mu _s$ is kept constant), Singh et al. (Reference Singh, Ness, Seto, de Pablo and Jaeger2020) have numerically studied the role of torque-activated (or stress-activated) rolling resistance, which can be simply induced by the ‘rough’ particle shape of particles in real-life suspensions. Note that adhesive surfaces can also induce a resistance to rolling motion but we eliminated this physical origin in § 3.2.1. To this aim, the authors have simulated spherical particles with a rolling resistance characterised by a rolling friction coefficient, $\mu _r$. Singh et al. (Reference Singh, Ness, Seto, de Pablo and Jaeger2020) have then studied the role played by different combinations of $\mu _r$ and $\mu _s$ in determining the value of the jamming volume fraction, $\phi _m$. As shown in figure 16, which displays their result, Singh et al. (Reference Singh, Ness, Seto, de Pablo and Jaeger2020) demonstrated interestingly on the one hand that $\phi _m$ depends weakly on $\mu _r$ when $\mu _s$ is small enough (typically, $\mu _s \lesssim 0.35$). For instance, their results show that $\phi _m$ decreases from 0.62 to 0.60 when $\mu _r$ increases from $10^{-3}$ (vanishing rolling resistance) to $10$ (extremely strong rolling resistance), and $\mu _s = 0.2$ (see the blue curve in figure 16). Note that we determined in the present work: $\phi _m = 0.61 \pm 0.01$ when $\mu _s = 0.2$ (see figure 15), which is in very good agreement with this observation. On the other hand, Singh et al. (Reference Singh, Ness, Seto, de Pablo and Jaeger2020) have predicted that $\phi _m$ is strongly dependent of $\mu _r$ when $\mu _s \gtrsim 0.5$. For instance, within the same range of rolling friction coefficient ($\mu _r \in [10^{-3} - 10]$), the authors showed that $\phi _m$ decreases from $0.57$ to $0.36$ when $\mu _s = 10$ (see the purple curve in figure 16). Typically, this corresponds to the case where sliding is prevented and only rotation can occur ($\mu _r \ll \mu _s \rightarrow \infty$). In the frame of the present study, this latter result from the literature (Singh et al. Reference Singh, Ness, Seto, de Pablo and Jaeger2020) is very interesting since, based on the assumption that $\mu _r|_{C_{PS40}} > \mu _r|_{S_{PS40}}$, it can explain the main observation obtained in the previous section: $\phi _m^{\infty }|_{C_{PS40}} < \phi _m^{\infty }|_{S_{PS40}}$ and $\phi _m^{0.2}|_{C_{PS40}} \approx \phi _m^{0.2}|_{S_{PS40}}$. To go further, we plotted in figure 17 the variation of $\phi _m$ with $\mu _s$, determined by our experimental data in the case of the suspensions $S_{PS40}$ (blue solid line) made of spherical particles and $C_{PS40}$ (orange dashed line) made of crushed particles. We compared these two different variations with the numerical simulations from Singh et al. (Reference Singh, Ness, Seto, de Pablo and Jaeger2020) for suspensions of spheres with two values of the rolling friction coefficient $\mu _r = 0.03$, and $\mu _r = 0.10$ (i.e. the two values of $\mu _r$ that we determined in figure 16 with the corresponding value of $\phi _m^\infty$ for the two suspensions). One can observe very good agreement between our experimental results and the numerical simulations from Singh et al. (Reference Singh, Ness, Seto, de Pablo and Jaeger2020), which tends to confirm the assessment already formulated by Estrada et al. (Reference Estrada, Taboada and Radjai2008, Reference Estrada, Azéma, Radjai and Taboada2011) that the non-spherical globular particles can be approximated as spheres as long as the effect of their shape is reflected by a rolling resistance, characterised by $\mu _r$. In the case of the rheological measurements, it is then captured by the value of $\phi _m^\infty$.

Figure 16. The jamming volume fraction $\phi _m$ as a function of the sliding friction coefficient, $\mu _s$, and rolling friction coefficient, $\mu _r$, as computed by Singh et al. (Reference Singh, Ness, Seto, de Pablo and Jaeger2020). Their data can be obtained at: https://acdc.alcf.anl.gov/mdf/detail/singh_rolling_friction_prl_2020_v1.4/. Bottom right: Variation of $\phi _m$ with $\mu _r$ for $\mu _s = 0.2$ (blue) and $\mu _s = 10$ (purple). The latter allows one to predict the values of $\mu _r$ for $S_{PS40}$ and $C_{PS40}$ from the experimental values of $\phi _m^{\infty }|_{S_{PS40}}$ (light blue) and $\phi _m^{\infty }|_{C_{PS40}}$ (orange), respectively: $\mu _r^{\phi _m^\infty }|_{S_{PS40}} = 0.03 \pm 0.02$ and $\mu _r^{\phi _m^\infty }|_{C_{PS40}} = 0.10 \pm 0.01$.

Figure 17. Variation of the jamming volume fraction, $\phi _m$, as a function of the sliding friction coefficient, $\mu _s$. The coloured symbols are deduced from the experimental determination of $\phi _m$ for $S_{PS40}$ (blue circles) and $C_{PS40}$ (orange squares), and $\mu _s$ is determined from (3.6) with $\mu _s^\infty = 0.2$, $\varSigma _c = 10 \ {\rm Pa}$ and $m = 0.5$. The coloured lines are displayed from the variable sliding friction model (see (3.6) and (3.7)) used to fit the experimental data in figure 15 for the suspensions $S_{PS40}$ (blue solid line) and $C_{PS40}$ (orange dashed line). The variation of $\phi _m(\mu _s)$ is compared for each suspension with the numerical results from Singh et al. (Reference Singh, Ness, Seto, de Pablo and Jaeger2020) (open black symbols) for $\mu _r = 3 \times 10^{-2}$ (circles) and $\mu _r = 1 \times 10^{-1}$ (squares).

To sum up, the rheology of the suspensions ($S_{PS40}$ and/or $C_{PS40}$) is solely determined by $\mu _r$ (induced by the non-spherical particle shape) when $\phi _m \rightarrow \phi _m^\infty$ ($\mu _s \xrightarrow {\varSigma _{12} \rightarrow 0}\infty$), whereas it is nearly independent of shape when $\phi _m \rightarrow \phi _m^{0.2}$ ($\mu _s \xrightarrow {\varSigma _{12} \rightarrow \infty } \mu _s^\infty = 0.2$) or $\phi _m \rightarrow \phi _m^0$ ($\mu _s \rightarrow 0$). Estrada et al. (Reference Estrada, Taboada and Radjai2008) have indeed demonstrated in the frame of a numerical study on granular material that the dominant mode of relative motion at the contacts (sliding or rolling) is that which minimises the coefficient of internal friction. This simply means that the particles prefer rolling if $\mu _r\ll \mu _s$ or sliding if $\mu _r\gg \mu _s$. The case where $\mu _r \sim \mu _s$ is obviously more complex since it involves rolling and sliding motion at the same time. Thus, by considering the most extreme case where $\mu _s = 10$ in figure 16 (rolling mode) and having deduced the values of $\phi _m^{\infty }$ for each type of suspension (see § 3.2.2), a value of the rolling friction coefficient $\mu _r^{\phi _m^\infty }$ for each suspension can be predicted from the rheological measurements: $\mu _r^{\phi _m^\infty }|_{S_{PS40}} = 0.03 \pm 0.02$ and $\mu _r^{\phi _m^\infty }|_{C_{PS40}} = 0.10 \pm 0.01$. Note that the uncertainty in $\mu _r^{\phi _m^\infty }$ for each suspension is due to the uncertainty in the value of $\phi _m^\infty$ related to the possible range of $\alpha _0$ (see (3.3)).

3.2.4. Frictionless suspensions made with the same particles

We have briefly studied the rheology of the frictionless case ($\mu _s = 0$) of the two suspensions studied in the present paper, by dispersing the same PS particles present in $S_{PS40}$ and $C_{PS40}$ in an aqueous solution, labelled $AQ0$, and shearing the suspensions in a vane tool geometry. The aqueous solution is a mixture of deionised water with a small amount (less than $3 {\rm wt}\%$) of Triton-X-100 (surfactant, Sigma Aldrich) and sodium iodide. We encourage the reader to see the supplementary material of Madraki et al. (Reference Madraki, Oakley, Nguyen Le, Colin, Ovarlez and Hormozi2020) for more details about this experimental procedure. Furthermore, the critical normal load $f_N^C$ (occurrence of the frictionless–frictional transition) has been measured by AFM measurements by Madraki et al. (Reference Madraki, Oakley, Nguyen Le, Colin, Ovarlez and Hormozi2020) for PS beads ($d \approx 140 \ \mathrm {\mu } {\rm m}$) in this aqueous solution $AQ0$. The authors found $f_N^C = (12 \pm 4) \ \mathrm {\mu } {\rm N}$, which gives $\sigma _{in}^{fft} \approx 0.3 \times f_N^C/(6 {\rm \pi}a^2) \sim 40 \ {\rm Pa}$ (Mari et al. Reference Mari, Seto, Morris and Denn2014) for this type of suspension (PS beads in $AQ0$).

Figure 18 displays the experimental measurements of $\eta _r$ (coloured symbol) for spherical PS particles (blue discs) and crushed PS particles (orange squares) in aqueous solution $AQ0$, when $\varSigma _{12} \approx 10^{-2} \ {\rm Pa}$ (frictionless case: $\varSigma _{12} \ll \sigma _{in}^{fft} \leftrightarrow \mu _s = 0$). As expected, the variation of the reduced viscosity $\eta _r$ with the volume fraction $\phi$ follows a Maron–Pierce law (coloured solid straight lines in figure 18, see (3.3)) (Peters et al. Reference Peters, Ghigliotti, Gallier, Blanc, Lemaire and Lobry2016; Lobry et al. Reference Lobry, Lemaire, Blanc, Gallier and Peters2019). Analogous to § 3.1.2, the fit using (3.3) has been done for $\alpha = 0.85$ and a confidence area is displayed according to the fits of the experimental data when $\alpha = 0.65$ and $\alpha = 1$. The result of the fit of $\eta _s$ as function of $\phi$ gives $\phi _m \approx 0.66 \pm 0.01$ for the suspension made of spherical particles (blue solid line) whereas $\phi _m \approx 0.66 \pm 0.02$ for the suspension made of the non-spherical particles (orange dashed line). Several observations can be underlined from this result. First, in the case of frictionless spherical particles, the value of the jamming fraction is in good agreement with the literature (Mari et al. Reference Mari, Seto, Morris and Denn2014; Gallier et al. Reference Gallier, Peters and Lobry2018; Singh et al. Reference Singh, Mari, Denn and Morris2018) and this confirms that the suspension is frictionless. Second, it confirms our previous choice to have assumed $\phi _m^{0} \approx 0.65$ in order to fit the experimental data for $\phi _m(\varSigma _{12})$ of the suspensions $C_{PS40}$ and $S_{PS40}$ by (3.6) and (3.7). Finally, $\phi _m^0 \approx \phi _m|_{\text {crushed PS in AQ0}} \approx \phi _m|_{\text {spheres PS in AQ0}}$ is consistent with the numerical results of Singh et al. (Reference Singh, Ness, Seto, de Pablo and Jaeger2020) who found that $\phi _m$ is independent of $\mu _r$ when $\mu _s \rightarrow 0$.

Figure 18. Variation of the shear viscosity, $\eta _r$, against volume fraction, $\phi$, for two different suspensions: the first made of the same spheres as in $S_{PS40}$ (blue circles) and the second made of the same crushed particles as in $C_{PS40}$ (orange squares). Here, the PS particles are dispersed in an aqueous solution ($\eta _0 = 10^{-3} \ {\rm Pa} \ {\rm s}$). The two suspensions sheared at $\varSigma _{12} = 10^{-2} \ {\rm Pa}$ in a vane tool geometry are frictionless ($\mu _s = 0$). A fit of the experimental data by (3.3) ($0.65 \leqslant \alpha _0 \leqslant 1$) gives $\phi _m \approx 0.66 \pm 0.01$ for the suspension made of spheres (blue solid line) whereas $\phi _m \approx 0.66 \pm 0.02$ for the suspension made of the non-spherical particles (orange dashed line).

We mention that some literature (Donev et al. Reference Donev, Cisse, Sachs, Variano, Stillinger, Connelly, Torquato and Chaikin2004; Baule & Makse Reference Baule and Makse2014; Kallus Reference Kallus2016) shows that particles having a shape which deviates slightly from spheres are characterised by a random close packing concentration, $\phi _m^{RCP}$, which is slightly larger than the known value for spheres, $\phi _m^{RCP}|_{spheres} = 0.64$. To the best of the authors’ knowledge, there is no strong evidence in the literature showing that the jamming volume fraction $\phi _m^0$ (frictionless case: $\mu _s = 0$) and $\phi _m^{RCP}$ have to be equal, even though it is known that both have the same value in the case of spherical particles. In the present paper, we find that $\phi _m^0|_{\text {crushed PS}} \approx \phi _m^0|_{\text {spheres PS}}$, but it is possible that a slight difference is hidden by the uncertainties. Nevertheless, we want to emphasise that, although the value of $\phi _m^0$ in (3.7) plays a significant role in the high-shear-stress regime ($\varSigma _{12} \gtrsim 10^2 \ {\rm Pa}$), the shape-induced rolling resistance, related most strongly to $\phi _m^\infty$ rather than $\phi _m^0$, dominates the low-shear-stress regime ($\varSigma _{12} \lesssim 10^1 \ {\rm Pa}$).

In the second part of the present paper, we describe how we can determine a value of $\mu _r$ for each type of particles (spheres and crushed), based on image analysis. The goal is to compare these new values with the ones predicted by the combination of the numerical works of Singh et al. (Reference Singh, Ness, Seto, de Pablo and Jaeger2020) based on shear rheology measurements coupled with the experimental data $\phi _m(\varSigma _{12})$ (see in §§ 3.2.2 and 3.2.3) that we recall here: $\mu _r^{\phi _m^\infty }|_{S_{PS40}} = 0.03 \pm 0.02$ and $\mu _r^{\phi _m^\infty }|_{C_{PS40}} = 0.10 \pm 0.01$.