3.1. Bulk rheology

Below the critical volume fraction, i.e. the jamming threshold at which particles can no longer avoid each other and the bulk and shear moduli simultaneously become non-zero (O’Hern et al. Reference Nordstrom, Verneuil, Arratia, Basu, Zhang, Yodh, Gollub and Durian2003), the rheology of granular suspensions is governed by the relative magnitude of the inertial forces of the particles and viscous forces, namely the Stokes number $St=\rho a_1^2\dot {\gamma }/\eta _{f}$ . Considering that in experiments (Chong, Christiansen & Baer Reference Chong, Christiansen and Baer1971; Krishnan & Leighton Reference Krieger and Dougherty1995; Chen et al. Reference Chen, Xue, Li, Wen, Liu, Xue, Shi, Zhou, Xie and Mai2019; Guy et al. Reference Guy, Ness, Hermes, Sawiak, Sun and Poon2020) and simulations (Guy et al. Reference Guy, Ness, Hermes, Sawiak, Sun and Poon2020; Malbranche et al. Reference Luan2023; Singh et al. Reference Singh, Ness, Seto, de Pablo and Jaeger2024), the characteristic stress scales or shear rate scales of bidisperse suspensions adopting different PSDs are generally not distinguished, the radius of the small particles $a_1$ is used to define $St$ . Figures 2(a–c) present rheology data showing the reduced viscosity $\eta _r=\varSigma _{xy}/\eta _{f}\dot {\gamma }$ as a function of $St$ for bidisperse suspensions with different PSDs. Two flow regimes are distinguished for suspensions with $\zeta _1=0.10$ and $0.50$ : a rate-independent viscous regime at low shear rates, followed by an inertial shear thickening regime in which $\eta _r$ scales linearly with $\dot {\gamma }$ . The critical Stokes number for the viscous-to-inertial transition, denoted by $St_{v\rightarrow i}$ , is defined as the intersection of the low shear plateau with this scaling. The values of $St_{v\to i}$ at various volume fractions $\phi$ are listed in table 1. Within the range of $\phi$ studied, $St_{v\rightarrow i}$ decreases with increasing $\phi$ , which implies close proximity to the jamming transition where flow stops. This trend agrees qualitatively with that reported for monodisperse systems when $\phi$ approaches $\phi _J$ (Lerner et al. Reference Lee, Jiang, Wang, Sandy, Narayanan and Lin2012; DeGiuli, McElwaine & Wyart Reference DeGiuli, McElwaine and Wyart2016; Trulsson, DeGiuli & Wyart Reference Trappe, Prasad, Cipelletti, Segre and Weitz2017). The viscous-to-inertial transition for suspensions at $\phi$ exceeding $\phi _J$ occurs at an infinitesimally small shear rate, i.e. $St_{v\rightarrow i}$ approaches zero. The viscosity $\eta _r$ diverges at $St\ll 1.0$ , and suspensions containing jammed particles behave like a solid. Note that ideal hard particles would cease to flow at such a high $\phi$ .

Table 1.

Critical Stokes number $St_{v\to i}$ for viscous-to-inertial transition, shown for two bidisperse suspensions with $\zeta _1=0.10$ and $0.50$ , and for various volume fractions $\phi$ .

Table 1. Long description

The table presents data on the critical Stokes number for the viscous-to-inertial transition in bidisperse suspensions. It includes two sets of data for different particle size distributions, with values of xi1 and xi2 specified. The table has four columns: the first column lists the volume fractions, the second column shows the reduced viscosity values, and the third and fourth columns display the critical Stokes numbers for the two different particle size distributions. The data indicates that as the volume fraction increases, the critical Stokes number decreases, suggesting that the suspensions are approaching the jamming transition where flow stops. This trend is consistent with observations in monodisperse systems. The table provides a clear comparison of how the critical Stokes number varies with volume fraction for different particle size distributions.

The shear thinning behaviours at $\phi =0.65$ ( $\zeta _1=0.10$ ) and $0.69$ ( $\zeta _1=0.50$ ) are essentially soft-particle rheology. At volume fractions above the jamming limit, quasi-static rheology is observed for low and moderate Stokes number, in which the shear stress is almost rate-independent and extrapolates towards a non-zero yield stress (see supplementary material § 4). The shear thinning observed in this quasi-static regime is thus due to the yield stress at volume fractions above $\phi _J$ for particles of finite stiffness. Similar shear thinning phenomena have been reported in experiments on soft colloids above jamming (Nordstrom et al. Reference Ness and Sun2010) and associated theory (Samadani & Kudrolli Reference Sakaï, Moulinet, Lechenault and Adda-Bedia2000). Here, $k_n$ represents the normal contact stiffness between particles, which quantifies the resistance of the particle contact interface to normal deformation under compressive stress. For very hard particles with $k_n \to \infty$ , it would be impossible to assess their soft-particle rheology as $\hat {\dot {\gamma }} \to \infty$ , which would correspond to $St \to \infty$ (Ness & Sun Reference Ness, Seto and Mari2015).

Figure 3.

(a) Plot of $\eta _r$ versus shear strain $\dot {\gamma }t$ for bidisperse suspensions at two critical states before and after the viscosity jump ( $\phi =0.70$ ). Inset: number density curves of small ( $\rho _1$ , solid symbols) and large ( $\rho _2$ , hollow symbols) particles in layers along the gradient direction ( $y$ -axis) at $\dot {\gamma }t=20$ ( $St=1.6$ ), $\dot {\gamma }t=175$ ( $St=1.6$ ) and $\dot {\gamma }t=100$ ( $St=2.0$ ), as marked by arrows with the corresponding symbols. The snapshots on the right-hand sides show the microstructure of the bidisperse suspensions at the steady state. (b) Number density curves of small ( $\rho _1$ ) and large ( $\rho _2$ ) particles in the steady-state suspensions with $\zeta _1=0.10$ ( $\phi =0.64$ ) and $\zeta _1=0.50$ ( $\phi =0.68$ ) at the Stokes numbers marked in figures 2(a) and 2(b).

Figure 3. Long description

The image contains two main sections labeled (a) and (b). Section (a) features a plot of viscosity versus shear strain for bidisperse suspensions at two critical states before and after a viscosity jump. The plot includes data points for two Stokes numbers, 1.6 and 2.0, represented by solid and dashed lines respectively. An inset within this plot shows number density curves of small and large particles in layers along the gradient direction at specific shear strains, marked by arrows. The snapshots on the right-hand side display the microstructure of the bidisperse suspensions at the steady state. Section (b) presents number density curves of small and large particles in steady-state suspensions with different volume fractions of small particles at specified Stokes numbers. The curves are shown for two different volume fractions, 0.10 and 0.50, with corresponding snapshots of the microstructures on the right-hand side. The graphs illustrate the variations in particle distribution and viscosity under different conditions.

Similar features of CST of suspensions with $\zeta _1=0.10$ and $0.50$ suggest a reduction in the flow curves at various $\phi$ values to follow a universal thickening law. Figure 2(d) shows the viscosities normalised by their rate-independent values ( $\eta _r/\eta _{r0}$ ) versus the normalised Stokes number ( $St/St_{v\rightarrow i}$ ) for the suspensions below $\phi _J$ . All data collapse onto one master curve, and the slope of the thickening section is 1. The experimental results (Tapia et al. Reference Sun and Sundaresan2022) of poly(methyl methacrylate) spheres (with mean diameter $d=4.65$ mm) also fall onto this curve. Indeed, this $\tau \propto \dot {\gamma }^2$ Bagnoldian scaling (Bagnold Reference Bagnold1954) has been widely observed in nearly monodisperse suspensions/granular systems ( $\lambda \gtrsim 1$ ) (Fall et al. Reference Fall, Lemaître, Bertrand, Bonn and Ovarlez2010; Fernandez et al. Reference Fernandez, Mani, Rinaldi, Kadau, Mosquet, Lombois-Burger, Cayer-Barrioz, Herrmann, Spencer and Isa2013; Ness & Sun Reference Ness, Seto and Mari2015; Tapia et al. Reference Sun and Sundaresan2022). This universal flow curve indicates that both monodisperse and bidisperse suspensions that do not exhibit significant structural transformation during thickening (as shown in figure 3 b) obey a viscous-to-inertial transition mode of shear thickening.

In contrast, the flow curves of bidisperse suspensions with $\zeta _1=0.26$ , i.e. the theoretical optimal PSD, appear somewhat discontinuous. At intermediate volume fractions ( $0.50\leq \phi \leq 0.65$ ), the viscosity $\eta _r$ jumps slightly in the shear thickening regime (figure 2 c). As $\phi$ increases further, the increase in viscosity becomes more pronounced by several orders of magnitude at the critical shear rate $\dot {\gamma }_c$ . To investigate the microstructural picture associated with this viscosity jump, we calculated the relative number densities of small ( $\rho _1$ ) and large ( $\rho _2$ ) particles along the velocity gradient direction ( $y$ -axis) at two critical states before ( $St=1.6$ ) and after ( $St=2.0$ ) the viscosity jump at $\phi =0.70$ (see inset of figure 3 a). The oscillating number density curves represent a layered particle structure with alternating aggregated large and small particles, whereas the almost flat number density curves indicate uniformly distributed disordered particles. At $St=1.6$ , the viscosity decreases by an order of magnitude upon reaching a steady state. Correspondingly, the initially disordered particles are rearranged into an ordered layered structure, as illustrated by the change in the pattern of the particle number density curves. Early studies on monodisperse suspensions also attributed shear thinning to the formation of ordered structures (Ackerson & Pusey Reference Ackerson and Pusey1988; Hoffman Reference Hoffman1998; Lee et al. Reference Kwan, Wong and Fung2018). At a slightly higher shear rate ( $St=2.0$ ), the viscosity remains essentially unchanged around a considerably high value, with no significant structural transformation occurring during the flow. Therefore, the particle phase change results in a discontinuous viscosity jump in inertial flows of bidisperse granular suspensions. Similar OD transitions are also observed at the remaining volume fractions at $\phi \geq 0.50$ ; see supplementary material § 6. As $\zeta _1$ increases from 0.10 to 0.26, the flows exhibit a gradual transition from CST to DST (see supplementary material § 2). The abruptness of the viscosity jump is positively correlated with $\zeta _1$ . For the range $0.26\leq \zeta _1\leq 0.50$ , the phenomenon of viscosity jump is pronounced at $\zeta _1=0.26$ and $\zeta _1=0.34$ ; with a further increase in $\zeta _1$ , the DST behaviour weakens.

Figure 3(b) shows the evolution of the particle number densities of suspensions with $\zeta _1=0.10$ and $0.50$ (the corresponding models are marked in figures 2 a,b). Neither suspension exhibits significant structural changes during shear thickening. At $\zeta _1=0.50$ ( $\phi =0.68$ ), the relative content of large particles ( $\zeta _2=0.50$ ) is not sufficient to form a complete layered structure. Large particles form shear-induced strips or a partially layered structure, as shown by oscillating density curves milder than those in figure 1(a). These partially layered structures are more stable than the layered structure at the theoretical optimal PSD, with no significant change in the particle arrangement during flow regime transition. Compared with the suspensions with $\zeta _1=0.26$ and $0.50$ , the $\zeta _1=0.10$ suspension maintains a disordered structure throughout the entire flow regime transitions, as shown by the nearly flat density curves in figure 3(b). Overall, the $\zeta _1=0.26$ suspensions that form layered structures under low stresses have the lowest structural stability.

The black dashed line in figure 2(c) shows the critical points of the OD transition. Evidently, this transition depends on both $\phi$ and $\dot {\gamma }$ . At low shear rates, the particles are lubricated apart by a strong network of lubrication films, which retards inertial collisions. Under the constant energy input provided by simple shear, large particles self-organise into ordered layers. As the volume fraction increases, the distance between large particles within one layer is reduced, leading to an increase in the overlapping depletion zone. This leaves relatively more space for small particles to move. Consequently, the ordering becomes more pronounced at higher shear rates as the volume fraction increases, as shown by the positive slope of the OD transition line. As $\phi$ approaches $\phi _J$ , the flow is primarily dominated by the collective rearrangements of particles owing to sufficient constraints to lock the structure (Seto et al. Reference Samadani and Kudrolli2013). This results in rapid shear jamming, i.e. the suspensions transition from layered to a solid-like disordered state at a diminishing critical shear rate $\dot {\gamma }_c$ , as indicated by the almost flat OD transition line at $\phi \geq 0.67$ in figure 2(c).

Simulations of the steady flow of bidisperse suspensions with larger particle size ratios $\lambda =7$ and $\lambda =9$ show a similar phenomenon of viscosity jump resulting from the OD transition (see supplementary material § 3). Although the magnitude of viscosity jump and critical shear rate vary systematically with $\lambda$ , the qualitative trend of DST behaviour (dependence on $St$ and $\phi$ ) is consistent across models with different size ratios. The layered ordered states of suspensions with different size ratios, volume fractions and shear rates primarily fall within the same region on the reduced Stokes number $St\times (a_2/a_2^0)^2$ reduced volume fraction $\phi / \phi _{rcp}$ plane (where $a_2^0$ represents the large particle radius in the model with $\lambda =5$ ). For suspensions with a wider particle size range, the $\phi$ window of the OD transition is larger due to higher $\phi _{rcp}$ , while the $St$ window of the OD transition is smaller. When the volume fraction $\phi$ falls within the range $0.80\phi _{rcp}{-}0.83\phi _{rcp}$ , the range of $St$ over which layered structures can form is the largest, indicating the greatest possibility of forming such ordered structures. When $\phi$ exceeds this range, however, the range of $St$ forming a layered structure decreases rapidly, making it difficult for the system to order.

In brief, the competition between the depletion effect, tending towards layering when particles have sufficient room for movement, and the jamming effect, which tends to form a solid-like disordered structure at $\phi$ near $\phi _J$ , results in the curved OD transition line. Although being more localised, the shear-induced order in monodisperse colloidal suspensions also becomes more pronounced as $\phi$ and $\dot {\gamma }$ increase, but it may not be present at small $\phi$ or $\dot {\gamma }$ values (Ackerson Reference Ackerson1990).

3.2. Microstructure

We analysed the contact networks of the suspension with $\zeta _1=0.26$ at $\phi =0.70$ to understand the changes in stress transmission resulting from the OD transition. In the layered structure ( $St=1.6$ ), the contact force chains are sparse and weak (figure 4 a). More than $90\,\%$ of the bond angles between the large–large particle contacts and the flow-vorticity plane ( $xz$ plane) fall within the range $0^\circ$ – $12^\circ$ (see supplementary material § 7), indicating that most large particles are aligned into layered ‘channels’ along the flow direction ( $x$ -axis). Small particles can move through these channels. The momentum is transferred mainly through hydrodynamic interactions, rather than collisions between particles. Occasional intra-layer large–large particle collisions and inter-layer large–small particle collisions lead to two principal stress orientations in the radial force distribution (in the $xy$ plane, as shown in figure 4 c for $St=1.0$ and $1.6$ ). The layered particles cannot elastically support infinitesimal loads above the critical shear rate. This entails shear-induced dissipative particle reorganisation from the ordered state to the disordered state. The contact network shifts from strongly aligned to almost percolated following the structural transformation, as evidenced by the more isotropic radial force distribution.

Figure 4.

(a) Particle contacts are visualised as bonds before and after the viscosity jump; $\phi =0.70$ , $\zeta _1=0.26$ , $N=9306$ . Red bonds indicate large–large particle contact; blue bonds indicate large–small particle contact; grey bonds indicate small–small particle contact. The width of the line segments represents the contact force magnitude. The forces are scaled in magnitude by the respective maximum values. For clarity, the width of the line segments in the right-hand image ( $St=2.0$ ) is reduced by half. (b) Shear component of the fabric tensor $A_{12}$ plotted against $St$ for $\zeta _1=0.26$ suspensions at different volume fractions $\phi$ , and $\zeta_1 =0.50$ suspensions at $\phi =0.67$ . (c) Radial force distributions of contact forces in $\zeta _1=0.26$ suspensions ( $\phi =0.70$ ) at various $St$ marked in (b).

Figure 4. Long description

The image contains three graphs. The first graph (a) visualizes particle contacts as bonds before and after a viscosity jump in granular suspensions. Red bonds indicate large particle contact, blue bonds indicate large-small particle contact, and grey bonds indicate small-small particle contact. The width of the line segments represents the contact force magnitude, scaled by the respective maximum values. The second graph (b) plots the shear component of the fabric tensor against St for suspensions at different volume fractions and suspensions at various values. The third graph (c) shows radial force distributions of contact forces in suspensions at different values marked in graph (b).

Figure 4(b) shows the variation in the $xy$ component of the fabric tensor $A_{12}$ at various $\phi$ . A negative $A_{12}$ represents the alignment of contacts along the compression axis ( $y=-x$ ), and $A_{12}=0$ indicates perfect isotropy. As large particles transition from ordered to disordered state, $A_{12}$ moves closer to zero, which is consistent with the characteristic of a more isotropic contact network. In contrast, the change in $A_{12}$ in the suspension that does not exhibit obvious structural transformation ( $\zeta _1=0.50$ ) is more subtle (see the dashed line in figure 4 b), suggesting that the orientation of the contact network aligns consistently along the compression direction. Following the OD transition, $A_{12}$ is almost rate-independent, especially at $\phi$ approaching $\phi _J$ . The average number of contacts with non-rattlers, $Z^{\prime }$ , reflects the degree of dense packing in a system. In the $d$ dimensions, a particle can be a rattler if it has fewer than $d+1$ neighbours (Kumar et al. Reference Kulkarni and Morris2021). In the shear jammed state ( $St=50.0$ , $\phi =0.70$ ), the $Z^{\prime }$ values for large and small particles reach $\approx 23$ and $\approx 4$ , respectively, indicating that small particles are near the jamming criteria above the isostatic condition $Z^{\prime } \gt Z_{iso}^{\mu =\infty }=4$ for near-monodisperse systems (van Hecke Reference Tsai, Voth and Gollub2010; Singh et al. Reference Sierou and Brady2020). Such almost percolated particles result in system-spanning contact networks.

For polydisperse suspensions/granular matter, different types of particle contacts generally do not contribute equally to the increase in viscosity (Voivret et al. Reference Voivret, Radjai, Delenne and El Youssoufi2009; Guy et al. Reference Guy, Ness, Hermes, Sawiak, Sun and Poon2020). To understand their role in phase transition, we counted the average numbers of small-to-small ( $C_{11}$ ), small-to-large ( $C_{12}$ ), large-to-small ( $C_{21}$ ) and large-to-large ( $C_{22}$ ) contacts for the entire flow regimes, comparing the results of suspensions with and without phase transition (figure 5). Following the OD transition in the suspension with $\zeta _1=0.26$ , $Z$ increases suddenly by $30.9\,\%$ , primarily owing to large-to-large and large-to-small contacts (see the inset of figure 5 a). The value of $C_{22}$ remains approximately constant ( $C_{22}\approx 1.7$ ) in the disordered state, indicating that large particles form a relatively stable contact network. For suspensions in which no apparent phase transition occurs during flow ( $\zeta _1=0.50$ ), both $Z$ and $C_{ij}$ increase continuously in the thickening regime (figure 5 b), as expected. Note that in both $\zeta _1=0.26$ and $\zeta _1=0.50$ suspensions, the variation of $C_{21}$ with increasing $St$ is much more pronounced than that of $C_{12}$ , where $C_{21}$ and $C_{12}$ represent the contacts between small and large particles normalised by different particle size classes. This is mainly due to the fact that the number of small particles is more than 40 times that of the large particles in both suspensions.

Figure 5.

Simulated rate-dependent average number of particle–particle contacts $Z$ (black solid lines with square symbols) and the partial mean coordination number of small-to-small ( $C_{11}$ ), small-to-large ( $C_{12}$ ), large-to-small ( $C_{21}$ ) and large-to-large ( $C_{22}$ ) contacts for (a) $\zeta _1=0.26$ suspension ( $\phi =0.67$ ) and (b) $\zeta _1=0.50$ suspension ( $\phi =0.67$ ).

Figure 5. Long description

The image contains two line graphs labeled (a) and (b), each showing the simulated rate-dependent average number of particle-particle contacts (black solid lines) and the partial mean coordination number of small-to-small, small-to-large, large-to-small, and large-to-large contacts. Graph (a) represents a suspension with a volume fraction of 0.26, while graph (b) represents a suspension with a volume fraction of 0.50. The x-axis on both graphs represents the Stokes number (St) on a logarithmic scale, ranging from 0.1 to 100. The left y-axis represents the average number of particle-particle contacts (Z), and the right y-axis represents the partial mean coordination number (C_ij). In graph (a), the black squares indicate the average number of particle-particle contacts, which increases with the Stokes number. The blue diamonds, red circles, purple triangles, and gray inverted triangles represent the partial mean coordination numbers for small-to-small, small-to-large, large-to-small, and large-to-large contacts, respectively. Similar trends are observed in graph (b), with the black squares showing an increase in the average number of particle-particle contacts with the Stokes number. The blue diamonds, red circles, purple triangles, and gray inverted triangles again represent the partial mean coordination numbers for the respective contacts. The insets in both graphs provide a zoomed-in view of the data for lower Stokes numbers. All values are approximated.

Figure 6.

(a) Average forces of small–small (S–S), small–large (S–L) and large–large (L–L) contacts in $\zeta _1=0.26$ and $0.50$ suspensions ( $\phi =0.67$ ) at $St=0.2$ and $St=50$ , as marked by the red circles in figure 5. (b) Contact networks corresponding to the models in (a). Red bonds indicate large–large particle contact; blue bonds indicate large–small particle contact; grey bonds indicate small–small particle contact. The width of the line segments represents the magnitude of the contact force. The forces are scaled in magnitude by the respective maximum values.

Figure 6. Long description

The image consists of two main parts. Part (a) features a bar graph that compares the average forces of small-small (S-S), small-large (S-L), and large-large (L-L) particle contacts in suspensions with different solid volume fractions (ζ1 = 0.26 and ζ1 = 0.50) and Stokes numbers (St = 0.2 and St = 50). The bars are color-coded, with yellow representing ζ1 = 0.26 and green representing ζ1 = 0.50. The graph shows significant differences in forces, with notable increases in the L-L contacts for higher Stokes numbers and solid volume fractions. Part (b) displays 3D visualizations of the contact networks corresponding to the conditions in part (a). The networks are color-coded: red bonds indicate large-large particle contacts, blue bonds indicate large-small particle contacts, and grey bonds indicate small-small particle contacts. The width of the line segments represents the magnitude of the contact force, scaled by the respective maximum values. The visualizations show how the contact networks change under different conditions, highlighting the complexity of particle interactions in dense granular suspensions.

Different types of particle contacts differ in their carrying forces, which constitute the heterogeneity of the contact network along with the varying number of contacts in polydisperse systems (Voivret et al. Reference Voivret, Radjai, Delenne and El Youssoufi2009; Desu & Annabattula Reference Desu and Annabattula2019). Figure 6(a) compares the average forces of different types of contact in $\zeta _1=0.26$ and $0.50$ suspensions before and after phase transition. The average forces, which are equal to the total force of a specific contact type divided by the number of occurrences of that contact type, are normalised by the maximum force for a $2\,\%$ overlap of the small particle radius ( $0.02k_na_1$ ). This value corresponds to the penalty parameters $k_n$ and $k_t$ in the monodisperse hard-sphere algorithm (Seto et al. Reference Seto, Mari, Morris and Denn2019). In the Newtonian regime ( $St=0.2$ ), most particles are lubricated apart, as reflected in the extremely small magnitudes of the contact forces in both suspensions. The viscosity of the $\zeta _1=0.26$ suspension, which has fewer particle contacts owing to the formation of a layered structure, is relatively lower.

The particles in both suspensions are close to being percolated at a high shear rate following phase transition ( $St=50$ ). The contact force in the $\zeta _1=0.26$ suspension is almost six times greater than that in the $\zeta _1=0.50$ suspension. Consequently, the viscosity is approximately 140 % higher, although the average contact number $Z$ is still lower. Fewer contacts do not naturally lead to higher contact forces; the elevated inter-particle contact forces in the layered structure stem from stress concentration on the remaining contacts. Under external shear, the applied stress in a disordered system is dispersed across a large number of random contacts, and transmitted through the particles along a principal axis opposing the shear flow. In the layered structure, however, the applied stress is redistributed and concentrated onto the limited intra-layer large–large particle contacts and inter-layer large–small particle contacts. Thus the contact forces are higher compared to those in a disordered structure. In summary, owing to the distinct thickening mechanisms, the suspensions with $\zeta _1=0.26$ and $0.50$ exhibit striking differences in their flow characteristics in terms of both the evolution of contact networks (figure 6 b) and the onset of shear thickening.

Figure 7.

(a) Friction coefficient $\mu$ as a function of the viscous number $I_v$ for monodisperse and bidisperse suspensions at $\phi \geq 0.55$ . Filled symbols indicate $\zeta _1=0.26$ suspensions; semi-filled symbols indicate $\zeta _1=0.10$ and $0.50$ suspensions (the same symbols as in figure 2 d). Inset: shear stress $\tau$ versus shear rate $\dot {\gamma }$ for $\zeta _1=0.26$ suspensions. (b) Variation in the contribution of the fluid to the total stress $\sigma _{xy}^{F}/\sigma _{xy}$ . Inset: $\sigma _{xy}^{F}/\sigma _{xy}$ versus Stokes number $St$ for $\zeta _1=0.26$ suspensions with different volume fractions $\phi$ . The colour legend in (a) refers also to (b).

Figure 7. Long description

The image contains two graphs. The first graph (a) shows the friction coefficient as a function of the viscous number for monodisperse and bidisperse suspensions. Filled symbols represent suspensions, while semi-filled symbols indicate other types of suspensions. The inset in graph (a) displays shear stress versus shear rate for suspensions. The second graph (b) illustrates the variation in the contribution of the fluid to the total stress. The inset in graph (b) shows the ratio of fluid stress to total stress versus Stokes number for suspensions with different volume fractions. The color legend in graph (a) applies to both graphs.

3.3. Friction coefficient

The rheology of monodisperse suspensions and dry granular systems can be unified by the evolution of the macroscopic friction coefficient $\mu$ (in simple shear, $\mu \equiv \sigma /P_p$ , where $P_p$ is a measure of the particle pressure) with the viscous number $I_v \equiv \eta _f\dot {\gamma }/P_p$ (Boyer et al. Reference Boyer, Guazzelli and Pouliquen2011). We plotted $\mu$ against $I_v$ for all suspensions at $\phi \geq 0.55$ (figure 7 a) to explore whether this governing law also applies to bidisperse suspensions with different PSDs, especially those that may undergo an OD transition. All data for suspensions in which no significant structural transformations occur during shear thickening ( $\zeta _1=0.10, 0.50$ , semi-solid symbols) collapse onto a single curve. The simulated data for monodisperse systems at different $\phi$ and $\dot {\gamma }$ also fall onto this curve. The friction coefficient is $\mu \approx 0.3$ when $I_v \to 0$ , and $\mu$ increases with increasing $I_v$ . This trend is qualitatively similar to that reported for dense suspensions containing monodisperse poly(methyl methacrylate) and polystyrene spheres (Boyer et al. Reference Boyer, Guazzelli and Pouliquen2011). The simulated $\mu (I_v)$ data for suspensions with the theoretical optimal PSD ( $\zeta _1=0.26$ , solid symbols) at various $\phi$ also collapse onto a single curve. However, the friction coefficient $\mu$ is lower than that of the other suspensions, especially in the viscous regime in which a layered structure is formed. As discussed below, this can be attributed to the additional viscous contribution to the total shear stress.

At the onset of the viscous-to-inertial transition, the lubrication films between the particles rupture, initiating mechanical contact. As the particle inertia increases with increasing $\dot {\gamma }$ , the contribution of the fluid to the total stress diminishes, as shown by the reduction in $\sigma _{xy}^F/\sigma _{xy}$ in the inset of figure 7(b). Here, $\sigma _{xy}^F/\sigma _{xy}$ can be generalised using $I_v$ for both monodisperse and bidisperse suspensions without obvious phase transition during shear thickening. Consistent with a lower friction coefficient, the values of $\sigma _{xy}^F/\sigma _{xy}$ for the suspensions with $\zeta _1=0.26$ are slightly higher in the viscous regime. The layered structure that is formed in this regime reduces particle collisions as well as contact dissipation, in part because the particles maintain a relatively large average separation (Ackerson & Pusey Reference Ackerson and Pusey1988). Interestingly, the layered structure achieves an effect similar to that of a reduced $\phi$ owing to the greater contribution of fluid to the stress (Huang et al. Reference Hsu, Ramakrishna, Zanini, Spencer and Isa2005).