Testing the Wyart-Cates model for non-Brownian shear thickening using bidisperse suspensions

There is a growing consensus that shear thickening of concentrated dispersions is driven by the formation of stress-induced frictional contacts. The Wyart-Cates (WC) model of this phenomenon, in which the microphysics of the contacts enters solely via the fraction $f$ of contacts that are frictional, can successfully fit flow curves for suspensions of weakly polydisperse spheres. However, its validity for"real-life", polydisperse suspensions has yet to be seriously tested. By performing systematic simulations on bidisperse mixtures of spheres, we show that the WC model applies only in the monodisperse limit and fails when substantial bidispersity is introduced. We trace the failure of the model to its inability to distinguish large-large, large-small and small-small frictional contacts. By fitting our data using a polydisperse analogue of $f$ that depends separately on the fraction of each of these contact types, we show that the WC picture of shear thickening is incomplete. Systematic experiments on model shear-thickening suspensions corroborate our findings, but highlight important challenges in rigorously testing the WC model with real systems. Our results prompt new questions about the microphysics of thickening for both monodisperse and polydisperse systems.

). The grey region is inaccessible due to inertial edge fracture. (b) Symbols, viscosity "branches" for different (fixed) values of σ in (a). Lines correspond to fits by eye to Eq. 1. The jamming volume fraction at which each viscosity branch diverges, φ J , depends on σ . Blue and red lines and symbols correspond respectively to the limiting low-σ and high-σ viscosities, η 0 and η m . (c) φ J (σ ), obtained from the fits shown in (b). φ J (σ ) decreases smoothly from φ 0 , the φ at which η 0 diverges, to φ m , the φ at which η m diverges. In all parts: shear thickening arises at any φ , e.g., φ = 0.51, because increasing σ decreases φ J where conditions exist for which φ > φ J , Fig. 1(c), i.e., the system can exhibit solid-like behaviour.
The WC model, Eq. 1-2, predicts the σ -and φ -dependent viscosity, η WC (σ , φ ), from three inputs: the limiting frictionless and frictional jamming points, φ 0 and φ m , and the σ -dependent fraction of frictional contacts, f . φ m and φ 0 , can be obtained by fitting viscosity branches at different σ , as done in Fig. 1(b). They are not related to shear thickening per se. On the other hand, f , which determines the shape of the flow curve, is currently inaccessible in experiments. Thus, various ansatzs are used to fit the WC model to experiments. For sterically-stabilised PMMA spheres, Guy et al. 3 used a φ -independent sigmoidal form: with β = 0.85. The single stress scale, σ * , scales as the "engineering" onset stress at which η(σ ) visibly begins to increase. Importantly for this study, we note that the particle size does not appear in WC model. On the other hand, and perhaps significantly in light of the findings we will report, Guy et al. found that the onset stress decreases with particle size, with σ * ∝ d −2 , suggesting σ * ∝ F * /d 2 for their PMMA particles, where F * ∼ k B T /nm is a constant force 3,10 . Royer et al. 5 used a similar form to fit data for dispersions of charge-stabilised silica.
In discrete-element (DEM) simulations, f can be calculated directly from particle coordinates. A popular choice is to use the "critical-load model" (CLM), in which µ jumps from 0 to > 0 when the normal contact force between particles, F, exceeds a threshold value, F * (the critical load). This model reproduces 9 the phenomenology of Fig. 1 and unstable flow at high φ . For a bidisperse mixture of spheres with diameter d 1 and d 2 = d 1 /1.4, Mari et al. found 9 a φ -independent f (σ ) of the form Eq. 3, with β = 1.1 and σ * ≈ F * /[6π(d 2 /2) 2 ], and later used it to fit flow curves at a range of φ 18 . Thus, in this one case, the WC model is fully validated: using the measured fraction of frictional contacts in Eq. 2 correctly predicts the viscosity. The similarity between the forms of f (σ ) used to fit experiments and measured in simulations suggests that f in mildly polydisperse experimental systems can indeed be well described by Eq. 3 or some similar form.
In these studies, a weak polydispersity (= standard deviation normalised by the mean of the particle size distribution) of s 20% was used to inhibit shear-induced crystallisation 9 . Industrial dispersions typically have broad, often multimodal, size distributions with s 100%. Nevertheless, the low-s phenomenology in Fig. 1 continues to hold 11,[19][20][21] . However, the validity of the WC model in such higher-s suspensions has not been tested.
Indeed, it is a surprise for the WC model to work, and work well, even for low-s systems. In suspension rheology, details of the microstructure, fabric of the contact network and distribution of forces matter. Cates pointed out long ago that the relatively small number of nearest neighbours, ∼ O(10), usually precludes any mean-field description 22 . The success of his shear-thickening model with Wyart contradicts this norm. In the WC model, η(σ ) is controlled primarily by a single scalar parameter f that is agnostic to exact microstructural details. For this reason alone, it is important to test the limitations of the WC model.
Here, we do so in suspensions of strongly bidisperse spheres. As before 9,18 , we use DEM simulations to extract f (σ ) for different mixtures and compare the predictions of the WC model to bulk flow curves. We then use the same f (σ ) to test the model against experimental data for bidisperse PMMA spheres. We find that WC works for nearly-monodisperse suspensions [i.e., the simulated f (σ ) correctly predicts η(σ )], but fails in general for bidisperse suspensions. We show that, nevertheless, the model can be extended to fit our data if the fractions of each contact type (large-large, large-small and small-small) are taken into account separately. Our results indicate that, in its original form, the WC model is at least incomplete, and highlight a number of unresolved issues in the current understanding of shear thickening and in the use of the WC framework to make inferences about microphysics. We propose directions for future research to address these issues.

Methods
A binary mixture of spheres is characterised by four parameters: d 1 , d 2 < d 1 , the fraction of small particles ξ = V 2 /(V 1 +V 2 ) (where V 1 and V 2 are the total volumes of large and small particles, respectively) and the total volume fraction φ = (V 1 +V 2 )/V (where V is the total volume). We fix φ and the size ratio α ≡ d 2 /d 1 ≈ 0.25, and vary ξ . At our α, the small spheres are slightly too large to fit inside four touching large spheres (for which α max = 0.225).
We sheared N = 2000 bidisperse, repulsive spheres at fixed φ in a periodic cell with Lees-Edwards boundary conditions. Shortrange lubrication and repulsive contact forces described by linear springs of stiffness k were resolved using a classical DEM code that allows marginal overlaps δ between the surfaces of pairs of particles 23 . We employed a contact model 9 in which Coulomb friction with static friction coefficient µ = 1 appears beyond a critical overlap δ * , corresponding to a critical normal load F * = kδ * . For simplicity, and consistency with experiments for nearly-monodisperse systems 3 , we use a constant F * (and hence δ * ) that is independent of d 1 and d 2 .
Experimental flow curves for binary mixtures of PMMA with α = 0.26 and φ = 0.51, Fig. 3(d), show similar phenomenology, except that the limiting high-σ behaviour is preempted by edge fracture due either to an inertial instability (grey region), or a different fracture mechanism when σ exceeds ≈ 10 3 Pa † . Thus, we cannot access η m (ξ ) directly for all ξ . Shear thickening is preceded by shear thinning, presumably due to residual Brownian motion 3 § ; so, we estimate η 0 by the viscosity minimum before the onset of thickening, Fig. 3 show the same non-monotonicity as the simulated values, but are always too high, by up to a factor of 2 for the two end members (ξ = 0 or 1). Using the experimental values of η 0 (ξ ) in Eq. 1 with φ = 0.53 gives us an experimental estimate of φ 0 (ξ ), Fig. 3(c) (•). Consistent with the experimental viscosities η 0 (ξ ) being higher than simulated values, the experimentally deduced φ 0 (ξ ) are somewhat lower than the simulated values, i.e. the experimental system at φ = 0.53 is closer to jamming than the corresponding simulated system.

Comparing simulations to the WC model
We test the WC model by comparing simulated flow curves, η(σ ), with those calculated using the fraction of frictional contacts, f (σ ), measured from the simulations, η WC (σ ). To explain our procedure, consider data for monodisperse small particles (ξ = 1). First, we calculate f (σ ) directly from inter-particle forces by counting, at each σ , the fraction of contacts with F > F * . The f (σ ) so obtained, Fig. 2(a), is sigmoidal, similar to the f (σ ) in bidisperse mixtures 9 with α = 0.71 and Eq. 3.
We compare this flow curve calculated using the measured f (σ ) with the simulated η(σ ), Fig. 2(d) (symbols). The two calculated plateaux agree with the simulated values by construction. The WC model is judged instead by how well it captures the shear ¶ We expect this to be a reasonable approximation, since Eq. 1 has previously been used to fit η(φ ) for various frictional, bidisperse sphere mixtures 26 . thickening process. It does this well for monodisperse spheres. Both model-predicted and simulation viscosities start to increase at σ /σ 0 ≈ 1, reaching saturation at σ /σ 0 30.
We repeat this procedure for our bidisperse suspensions with Note, first, that f (σ ) for the monodisperse end members, ξ = 0 and 1, are identical in shape, Fig. 4(a), but with the former shifted to the left by a factor of (d 2 /d 1 ) 2 = α 2 = 0.0625 due to a decrease in σ * by the same factor for the larger particles 3 . Addition of just 20% of small spheres to a suspension of large spheres (ξ = 0.2) produces a dramatic effect, Fig. 4(a). While frictional contacts still start to form at σ * 1 ≈ 0.06σ 0 , the onset is now much more gradual, until σ ≈ σ * 2 ≈ σ 0 , whereupon d f /dσ abruptly becomes as large as the monodisperse case (either ξ = 0 or 1), before f saturates at a σ that is comparable to (but slightly later than) that of monodisperse small spheres, even though only 20% of these are present. By ξ = 0.5, f (σ ) becomes very similar to that of that of monodisperse small spheres (ξ = 1); although, the onset is still clearly somewhat earlier and the saturation somewhat lower. These features become progressively less obvious at ξ = 0.65 and 0.80. The effect of bidispersity is therefore asymmetrical: the presence of 20% of large spheres in 80% of small spheres has far smaller an effect on f (σ ) than the reverse situation.
Turning to the shear-thickening flow curves, Fig. 4(d), we see that, as expected, the WC model reproduces the simulated data for the two monodisperse end members. It gives a tolerable representation of the data at ξ = 0.8, i.e. when there are 20% of large spheres present in a predominantly small-sphere system; but, it fails badly in the reverse situation, when there are 20% of small spheres in a mainly large-sphere system (ξ = 0.2). The disagreement between the WC prediction and simulation data is still substantial at ξ = 0.5, and remains perceptible at ξ = 0.65.

Comparing experiments to the WC model
Testing the WC model against experimental data is more involved. Figure 3(a) shows clearly that introducing bidispersity alters shear thickening; however, simulations 9 and recent experiments 6,27 have shown that, even for nearly monodisperse suspensions, thickening is also sensitive to the relationship between the particle static friction coefficient, µ, and the normal contact force, F. The function µ(F) is fully prescribed in our simulations: µ = 0 below a threshold force F * , and µ = µ m > 0 above F * , allowing us to isolate the effect of bidispersity on the shearthickening phenomenology. However, for our sterically stabilised PMMA particles the load-dependent friction µ(F) has not been measured; hence, we do not know a priori the role of the specific tribology of our particles.
For simplicity, we assume that the experimental µ(F) obeys the same contact model (CLM) as in simulations and treat the critical load, F * , as an unknown parameter; as in simulations, we take F * to be independent of the size of the contacting particles. F * defines a contact stress scale σ 0 ∼ F * /d 2 2 , the unit of stress in our simulations. For each σ and ξ in experiments we calculate σ /σ 0 and read off the corresponding fraction of frictional contacts, f (σ /σ 0 ), from simulations, Fig. 4(a).
The all-large (ξ = 0) flow curve is well fit by the WC model, in agreement with our simulations. This justifies a posteriori our use of the CLM for µ(F) for this sample. The model should equally well describe the all-small (ξ = 1) flow curve; however, this is not the case. Although the onset of shear thickening is correctly predicted, the rise in η(σ ) is overestimated by the model, implying a different µ(F) for the small particles, e.g., CLM with a lower µ m than for the large particles 9 . As a consequence, the present map between σ and f is not reliable for our bidisperse suspensions; to calculate f (σ ) properly, one would have to independently characterise µ(F) experimentally for the different contact types (largelarge, small-small and large-small) and compare with simulations employing a representative DEM contact model. We do not do so here; however, we point out that all the same trends noted when we compared simulations with the WC model are clearly reproduced in our bidisperse experiments ([0 < ξ < 1 in Fig. 5]). In particular, there is a striking disagreement between model and experiment at ξ = 0.2, for which, as in Fig. 4(d), the onset of shear thickening is grossly underestimated. Based on this similarity, we can infer already that the same shortcomings of the WC model as applied to bidisperse simulations should also apply to bidisperse experiments.

How the WC model fails
Previous experiments and simulations find, and we confirm, that the WC model works well in the quasi-monodisperse limit (s 0.2) 18 . This phenomenological model is designed to reveal the consequences of a simplified set of assumptions in the most perspicuous way, and (according to its authors 10 ) not meant for the fitting of data. Thus, that it works quantitatively in the small-s limit is itself non-trivial, especially given its mean-field nature 22 .
It is perhaps unsurprising that we find the WC model fails to account for a binary mixture with α = 0.25 (size ratio 1:4), which translates, using a previously-proposed definition 28 , to an effective polydispersity s eff = (d 1 −d 2 )/(d 1 +d 2 ) ≈ 60% . The pertinent question is: precisely where is the WC model failing in this case? In a monodisperse system, there is a single kind of frictional contact. In a bidisperse system, such contacts are of three kinds: large-large ('11'), large-small ('12'), and small-small ('22').  contribution is from 'mixed' contacts, f ∞ 12 ≈ 0.5 5 f ∞ 11 , which start to form at σ * 12 ≈ 0.2σ 0 (perhaps fortuitously close to σ * 1 σ * 2 ). In its original form, the WC model is agnostic to particle size. Consistency with this feature requires that, when applied to our biphasic system, we take f = f 11 + f 12 + f 22 , so that any new frictional contact formed as stress builds up contributes equally to the lowering of φ WC J , and therefore to the viscosity increment via Eqs. 1 and 2. Thus, because f 12 ≈ f 22 f 11 at ξ = 0.2, the WC flow curve at this composition is much more similar in shape to that for the all-small (ξ = 1) system than that for the ξ = 0 system. In reality, the simulated flow curves start to shear thicken at σ * 1 , which is where f 11 starts to increase; i.e., large-large contacts dominate despite the smallness of f 11 , and many small-large and small-small contacts seem to contribute little to the shift in φ J .
This suggests that we should write φ J a function of f 11 , f 12 and f 22 , separately. A simple ansatz is to retain the functional form of φ J in Eq. 2 and replace f with a polydisperse crossover function, f poly = f denotes the corresponding large-σ limit of f i j in Fig 6(a). We choose κ 11 + κ 12 + κ 22 = 1 to ensure f poly (σ /σ 0 1) = 0 and f poly (σ /σ 0 1) = 1. So, our extended WC model reads We continue to use Eq. 1 for the relative viscosity.
The weighted fraction of frictional contacts of type (i j), f poly i j (σ ) needed to fit our data, shown in Fig. 6(b) for ξ = 0.2, has the same shape as f i j (σ ), Fig. 6(a), but scaled up by a factor of κ i j , which sets the limiting value of f poly i j as σ /σ 0 −→ ∞. [Note that the slight non-monotonicity of f 11 (σ ) in Fig. 6(a) (red) means that f poly exceeds 1 using our normalisation scheme.] By varying the free parameters κ 11 and κ 12 (with κ 22 = 1 − κ 11 − κ 12 ), we can readily fit all of our bidisperse simulation flow curves; Fig. 6(c) (solid line) shows the best-fit curve, obtained by eye, for ξ = 0.2. In Fig. 7(a), we plot the best-fit coefficients for ξ = 0.2, along with the coefficients for other ξ (for the full fits, see the ESI † ).
For comparison, we consider the special case in which κ 11 = f ∞ 11 , κ 12 = f ∞ 12 and κ 22 = f ∞ 22 , so that f poly reduces to f , the unweighted fraction of frictional contacts, Fig. 4(a), and the original WC model is recovered, Fig. 6(c) (dashed line). We plot the coefficients for this case, f ∞ 11 (ξ ), f ∞ 12 (ξ ) and f ∞ 22 (ξ ), in Fig. 7 For ξ = 0.2, our set of fitted {κ i j }, Fig. 7(a), reaffirms quantitatively what we proposed qualitatively earlier. The largest contribution to changes in f poly (and hence φ J ) is from large-large contacts, for which κ 11 = 0.7, while there is a negligible contribution from small-small contacts, κ 22 ≈ 0. In contrast, in the original WC model κ 11 = f ∞ 11 ≈ 0.1 and κ 22 ≈ 0.4 , Fig 7(b). Increasing ξ to 0.5 sees the increasing importance of large-small contacts and decreasing importance of large-large contacts, while small-small contacts remain irrelevant. At ξ = 0.65, f poly is determined almost entirely by large-small contact formation. Only for ξ = 0.8 do small particles have a measurable contribution, where they enter on equal footing with large-small contacts; large-large contacts are, by this point, irrelevant.

Discussion
The WC coefficients, f ∞ i j , in Fig. 7(b) correspond to the relative numbers of each kind of contact ('11', '12' or '22') in the high-σ /σ 0 limit. Thus, the ratio ∆ i j ≡ κ i j / f ∞ i j measures the relative contribution to φ WC J (and hence to η) due to the formation of a single frictional contact of type i j. In the WC model, all contact types give rise to the same increment in φ J and ∆ 11 = ∆ 12 = ∆ 22 = 1. In Fig. 8, we plot the ∆s as a function of ξ (colours) and overlay the WC prediction (horizontal dashed line). Strikingly, ∆ 11 ∆ 12 ∆ 22 for all our bidisperse mixtures. Thus, at ξ = 0.2, for example, a largelarge contact contributes over an order of magnitude more than a large-small one; while, the effect of forming a small-small contact is negligible (∆ 22 ≈ 0); i.e., small particles are largely redundant for stress tranmission. Only at ξ = 0.8, where the fraction of small particles is largest and there are no large-large contacts (∆ 11 is undefined here; so, we do not plot it), do small-small contacts have an appreciable contribution (∆ 22 becomes non-zero). Even then, a single large-small contact contributes the same as ∼ O(10) small-small contacts.
Importantly, none of the bidisperse ∆s follow the WC prediction (dashed line). Since different contact types do not contribute equally to changes in φ WC J , one needs to know not only the total number of frictional contacts, but the sizes of the particles participating in all those contacts to predict η. Thus, f , which assumes that all frictional contacts contribute the same, is inherently unsuitable for modelling bidisperse systems.
The particle-size-dependence in Fig. 8 is reminiscent of sheared polydisperse dry granular packings, in which stress transmission is strongly spatially heterogeneous with contacts between larger particles carrying a higher load on average than those between smaller particles 28 . In bidisperse dry granular systems, the detailed partition of stress is sensitive to both ξ and the size ratio α; e.g., as the size disparity grows (α decreases), the contribution of contacts involving small particles progressively decreases 29 . While this problem has been studied at length in dry systems under imposed particle pressure ( =⇒ varying φ ) [30][31][32][33] , it has received relatively little attention for fixed-φ , granular suspensions, in which particles interact not only through contact-, but also hydrodynamic forces. Presumably, the trend with α is similar to the dry-grain case, so that, as α → 1, the disparity between different contact types vanishes, i.e., ∆ 11 ≈ ∆ 12 ≈ ∆ 22 = 1, and hence f eventually becomes a reasonable approximation for f poly , which would explain the success of the WC model for weakly bidisperse mixtures 18 . Clearly, focussed work is needed to understand the details and origins of stress partitioning, and its relation to shear thickening, before a realistic model can be constructed. Minimally, the the relative weights of different contact types should be allowed to vary, e.g., like our Eq. 5.
Alternative existing models of frictional shear thickening could prove more successful in capturing bidisperse flowcurves. A re-  Fig. 7(a) and (b). The horizontal dashed line is the WC prediction, cent idea is that thickening is not driven by the formation of frictional contacts per se, but by the changes in anisotropy of stress tranmission that this induces 34,35 * * . In particular, Thomas et. al. proposed an ab initio model for two-dimensional systems based on the ratio of the shear stress and particle pressure, σ xz /P, which they relate to the anistropy of contact forces. Interestingly, for simulations of dry grains with a uniform particle size distribution, σ xz /P is found to be independent of polydispersity (controlled via the span of the distribution) 28,33 , which suggests that their approach may account for polydispersity effects naturally in a way that the WC model, which is agnostic to the spatial distribution of contact forces, does not. This merits a thorough study of the role of stress anisotropy in bidisperse systems, and the extension of Thomas et. al.'s theory to 3-d and polydisperse systems.
Perhaps as important as the contact-type-dependent contributions to φ J is our observation that the original WC model fits our monodisperse simulation data. This result is non-trivial: it implies that the microphysics of shear thickening can be captured by a single scalar parameter ( f ) that is agnostic to the spatial distribution of contacts. Remarkably, there is evidence in the literature that this scenario may be true, at least in the quasimonodisperse limit. By simulating a weakly bidisperse mixture of particles interacting via the critical-load model and the same mixture containing particles with different but load-independent µ, Dong and Trulsson 37 showed that η is uniquely defined by f , even though the microstructure for both setups is very different. For strongly bidisperse suspensions, the roles of microstructure and stress paritioning in shear thickening remain to be disentangled. If they bear similarity to polydisperse dry grains, then the two should be strongly correlated. Specifically, we would expect big particles, which carry the largest loads, to align with the compressive axis; whereas small particles, which carry a negligible load, would form an almost isotropic background of "specta- * * This is reminiscent of the "hydrocluster"-driven thickening of lubricated spheres observed in Stokesian dynamics simulations 36 , which, however, is distinct from the contact-driven thickening we observe 4 tor" particles 28,38 . Studying the spatial distribution of contacts systematically in these systems, e.g., in the vein of Dong and Trulsson 37 , would shed light on this issue and help to establish whether the independence to microstructure in the monodisperse case has deeper physical meaning, or if it is entirely fortuitous.
Before concluding, we comment on the utility of experimental data in testing the WC model. Although our experiments qualitatively support the notion that the WC model fails for bidisperse suspensions, the inability of the model to decribe the all-small η(σ ) based on simulation f (σ ) highlights an important obstacle to rigorous testing: the F-dependent tribology of interparticle contacts is a priori unknown. Our work with PHSA-stabilised PMMA dispersions suggests that, even for particles with ostensibly well-controlled surface properties, µ(F) may vary considerably from batch to batch. For example, in the ESI † we show flow curves for quasi-monodisperse PMMA spheres showing "twostage" shear-thickening, with two distinct onset stresses. Such behaviour clearly cannot arise from the single-stress-scale CLM used here. Thus, microtribology experiments 6,39,40 must play a central role in future model testing. Indeed, the scarce tribology measurements that already exist for sterically-stabilised particles indicate a rich behaviour, particularly at large normal loads 27 . Finally, we note that even in experimental systems where µ(F) is described by the CLM, "fitting" the WC model to experimental data with a presumed form for f (σ ) will result in a f (σ ) that is not the fraction of frictional contacts except in the monodisperse limit 3,5 .

Summary and conclusions
We have simulated and experimentally measured the rheology of a bidisperse suspension of repulsive spheres to test the validity of the WC model of shear thickening. By using the fraction of frictional contacts f extracted directly from simulations, we showed that the WC model works in the special case of monodisperse particles, but is incomplete when applied to bidisperse mixtures. While our study focusses on continuous shear thickening, we expect all the same conclusions to apply at higher volume fractions, where discontinuous shear thickening is observed.
In practical terms, our results suggest caution when using the WC model as anything other than an empirical fitting tool. Specifically, little, if any, meaning can be ascribed to f extracted from fits to flow curves. On a fundamental level, our work highlights the need for a focussed effort to understand the link between σdependent frictional contact formation and dissipation. Existing studies of shear thickening consider either bulk rheology 3,5 or exsitu, two-particle properties 6 , with little or no concerted effort to bridge the two regimes. We believe that any serious effort to make this link should consider polydispersity from the outset in its own right, rather than merely as a means of mitigating crystallisation. Indeed, our work hints that the monodisperse limit is a singular one, and so cannot be used as a guide to developing models for the flow of polydisperse systems.

Conflicts of interest
There are no conflicts to declare.