Hydrodynamic and contact contributions to shear thickening in colloidal suspensions

Shear thickening is a widespread phenomenon in suspension flow that, despite sustained study, is still the subject of much debate. The longstanding view that shear thickening is due to hydrodynamic clusters has been challenged by recent theory and simulations suggesting that contact forces dominate, not only in discontinuous, but also in continuous shear thickening. Here, we settle this dispute using shear reversal experiments on micron-sized silica and latex colloidal particles to measure directly the hydrodynamic and contact force contributions to shear thickening. We find that contact forces dominate even continuous shear thickening. Computer simulations show that these forces most likely arise from frictional interactions.

Shear thickening, the increase of viscosity with shear rate, is ubiquitous in complex fluids [1][2][3][4]. In particular, it plays a controlling role in the flow of concentrated suspensions of micron-sized particles [5]. Such suspensions occur widely in applications, from ceramics and bulletproof armor, through cement and even chocolate. Shear thickening has also become a mainstay of popular science, in the form of running on a pool of corn starch solution.
Despite sustained study, the mechanism of suspension shear thickening is still disputed. A long-standing view is that thickening, especially if it is continuous with shear rate, is predominantly driven by hydrodynamic interactions [2,3,6,7]. In some theories, the increase in viscosity arises directly from enhanced dissipation in the narrow lubrication films between particles [7,8]; in others, it is caused by the formation of transient particle clusters ("hydroclusters") [2,3,6,9,10]. In support, experiments have identified putative hydroclusters [1,11,12] and found an increase in hydrodynamic stresses during thickening [13][14][15].
Recently, contact forces have been shown to mediate a discontinuous jump in viscosity with shear rate in dense suspensions of non-Brownian particles at volume fractions greater than random loose packing φ 0.58 [4,[16][17][18][19][20]. More controversially, simulations [21,22] and theories [23,24] suggest that direct frictional contact can also lead to continuous shear thickening in moderately dense colloids with φ 0.58. This proposal has not yet gained wide acceptance: conceptually, it seems harder to find a role for such forces without the formation of system-spanning contact networks [4].
Experiments that can dissect the relative contributions of hydrodynamics and contact stresses can settle this issue definitively. Here, we demonstrate that shear reversal techniques, pioneered by Gadala-Maria et al. [25,26], can distinguish the relative contributions of these stresses in non-inertial shear thickening systems. The basic idea is simple but powerful: immediately upon shear reversal, instantaneous contact stresses vanish, but hydrodynamic stresses remain because of Stokes flow reversibility. Thus, monitoring time-dependent stresses after reversal will reveal the relative roles of these two interactions in thickening.
Shear reversal measurements were performed in an ARES strain-controlled rheometer (Rheometric Scientific) with roughened cone-plate geometry (25 mm, 0.1 rad) modified with a DAQ directly connected to the analogue output of the stress and strain sensors [28]. Directly measuring the output of these sensors bypasses signal processing that hinders the instantaneous measurement of the system's short-time response.
In our protocol, figure 1(a), a positive shear rateγ is imposed until the accumulated strain γ reaches 10, after which the shear is reversed and a negative shear rate is imposed to accumulate the same amount of strain. Just before shear reversal, the suspension is in a steady state. Immediately upon reversal, the structure is unchanged, so that hydrodynamic forces will remain identical in magnitude but reversed in direction. In contrast, contact forces in hard-sphere systems will drop to zero immediately after reversal, figure 1(b). This qualitative difference between these two forces allows us to disentangle their separate contributions by measuring the transient stress upon reversal. Figure 1(a) illustrates schematically the instantaneous relative viscosity η(t) = σ(t)/(γ(t)η 0 ) as a function of time (or, equivalently, accumulated strain) of a purely Newtonian fluid (green line) and the response where contact forces contribute to the total stress (red line). (See Supplemental Information for calibration.) A rate-sweep measurement of the charge-stabilized sil- Schematic illustrating the difference between hydrodynamic forces, which stay the same in magnitude immediately upon reversal, and contact forces, which drop to zero immediately upon reversal.
ica suspension gives its flow curve, η(γ), figure 2(a) (thick grey line), which shows shear thickening atγ 0.3 s −1 . Next, we monitor the viscosity after shear reversal, η rev , as a function of accumulated strain after reversal, γ, at four representative shear ratesγ = 0.50, 1.00, 2.00 and 5.00 s −1 , in the shear thickening regime, figure 2(b). Immediately before reversal, η has a value (not shown) equal to that measured in the steady state (γ = 10). The viscosity immediately after reversal drops to a value, η 0 rev , that remains constant, to within experimental error, until γ 0.1. We take η 0 rev to be the hydrodynamic contribution to the total steady-state viscosity. From γ ≈ 0.3, η rev rises, reaching a steady-state value, η ∞ rev , that is the same as the steady-state viscosity before reversal. We take η ∞ rev − η 0 rev to be the contact contribution. The hydrodynamic and contact contributions to the total viscosity so obtained are plotted as a function ofγ rev , is taken to be the average of the relative viscosity over the range 0.01 < γ < 0.2. The contact component is taken to be the difference between the steady state value at γ = 10, η ∞ rev , and η 0 rev .
in figure 2(a). Strikingly, while the contact contribution increases withγ, the hydrodynamic contribution remains constant as the suspension shear thickens. This demonstrates the essential role played by contact forces in the continuous shear thickening of this silica suspension. Contact forces depend sensitively on the nature of particle surfaces. We therefore repeated our experiments using a PMMA suspension. In contrast to our silica particles, which are charge stabilized, these PMMA particles are stabilized by (neutral) surface 'hairs'. Our φ = 0.51 PMMA suspension shear thickens more readily, atγ ≈ 0.01 s −1 , figure 2(c). Again using shear-reversal to measure η rev , figure 2(d), and analyzing the data in the same way to obtain the hydrodynamic and contact contributions, figure 2(c), we find essentially the same pattern of behavior already uncovered for the φ = 0.49 silica suspension. Thus, contact forces also dominate the continuous shear thickening in this sterically-stabilized system.
To validate this interpretation, and to probe the nature of these contact forces, we conducted computer simulations of repulsive spheres (radius a) in which the shortrange lubrication and repulsive contact forces were mim- The total viscosity as a function of shear rate, and its decomposition into hydrodynamic and contact contributions done in two ways: directly calculated from simulations, and 'read off' the data shown in part (a) in the same way as the experimental data is analyzed into these contributions, for which see figure 2 icked using a previously-established critical load model [21], implemented in a classical discrete element method code (details of which are identical to those in [29]; also, see Supplemental Information). In this model, frictional interactions appear beyond a critical normal force between particles, F CL , which also sets a shear rate scale, γ 0 = F CL /(6πη 0 a 2 ). We explored shear reversal at φ = 0.51.
The observed evolution of the viscosity with accumulated strain after reversal, figure 3(a), is qualitatively similar to experiments. In the simulations, however, we can follow the hydrodynamic and contact contributions as a function of accumulated reversed strain by direct evaluation; figure 3(b) shows the results forγ = 0.74γ 0 . Consistent with the interpretation we have offered for our experimental data, the contact contribution drops essentially to zero immediately after reversal, and only increases back to its steady-state magnitude after an accumulated stain of order unity. By contrast, the hydrodynamic contribution remains more or less constant.
We cannot access directly the hydrodynamic and contact stresses in our experiments. Instead, we have to 'read off' these contributions to the reversed viscosity from the data as η 0 rev and η ∞ rev − η 0 rev , respectively, figure 2(b). We validate this procedure using simulations. First, we 'read off' the hydrodynamic and contact contributions to the reversed viscosity data, figure 3(a), in exactly the same way as in experiments, giving results, figure 3(c), analogous to the experimental data shown in figure 2(a) and (c). Secondly, we evaluated these contributions directly from the raw simulation data, and overlaid the results in the same graph, figure 3(c). The near identity of the two sets of results validates our identification of η 0 rev and η ∞ rev − η 0 rev with the hydrodynamic and contact contributions, respectively. Thus, simulations confirm the key role played by contact forces in continuous shear thickening.
Traditionally, shear thickening is supposed to occur beyond a critical onset strain rate [30]. Recent theories, simulations and experiments [5,8,21,22,24] suggest that, instead, the phenomenon appears at a critical onset shear stress. Indeed, this assumption has been built into the critical load model used in our simulations. To explore the role of stress in the onset of shear thickening, we performed rate sweep measurements in our two systems at different temperatures, T , to change the solvent viscosity, η 0 (T ), over an order of magnitude.
We find that as the solvent viscosity varies, the onset of thickening (dashed lines in figure 4(a) and (c)) occurs at different strain rates for both of our suspensions. Thus, in the silica suspension, figure 4(a), the highest solvent viscosity (lowest temperature) data set shows shear thickening atγ 200 s −1 , but this onset progressively moves to higher rates as we lower the solvent viscosity, until in the lowest viscosity data set, we barely see thickening over our range of shear rates, but observe a small degree of shear thinning instead. Similarly, in the PMMA suspension, figure 4(c), we see the onset of thickening at around 0.05 s −1 in the highest viscosity data set; this onset also moves to higher shear rates as the solvent viscosity decreases.
Each of these two data sets, however, can be scaled onto a master curve if we plot the relative viscosity η(γ) = σ/(γη 0 (T )) against the shear stress σ [31]. The master curve for the silica particles, figure 4(b), indeed shows a single onset stress at ≈ 10 Pa, while that for the PMMA system, figure 4(d), shows an onset stress at ≈ 1 Pa. Thus, in both systems, our experiments show that shear thickening occurs above a critical stress. The difference in the magnitude of the onset stress of the two suspensions is consistent with the expectation that the load needed to press particles into contact should be sen-sitive to details of the stabilization mechanism [32].
Our observation of contact-driven shear thickening contradicts the conclusions drawn from rheo-SANS [14] and scattering dichroism measurements on Brownian hard spheres [13]. The rheo-SANS data provides an approximation of the hydrodynamic stresses through a nontrivial analysis of the structural anisotropy in the suspension. However, even the latest rheo-SANS measurements show thickening that is only a small fraction of that observed in rheometry [14]. In the scattering dichroism measurements, the total suspension viscosity, η, and the viscosity contribution from Brownian stresses, η B , can be measured independently. The authors found that the contribution to the viscosity from other stresses, η − η B , increased upon shear thickening, and attributed this to an increase in hydrodynamic stresses [13]. However, the remaining viscosity may in fact be dominated by contact contributions, an interpretation that is in alignment with results from dense non-Brownian suspensions [33] as well as the data presented in the current work.
Scattering dichroism measurements [34] also found evidence of particle clusters in the shear-thickened state (later confirmed by real-space imaging [1]), which were supposed to originate from large lubrication stresses between particles. The origin of these clusters remains unclear, however -the present work suggests that any clustering is likely to involve frictional contacts.
To conclude, we have used shear reversal to show that, for two canonical model colloidal systems, continuous shear thickening does not originate from hydrodynamic interactions but from the formation of particle contacts. The onset of contact formation, and thus shear thickening, is found to occur above a critical stress whose value is sensitive to whether the particles are charge-or sterically-stabilized. Further work is needed to establish whether shear thickening of nanoparticles, for which Brownian stresses are not negligible, is also driven by contact formation; this should be possible with shear reversal.
While shear reversal alone provides no a priori information about the nature of the interactions between contacting particles, the quantitative agreement between the experimental and simulation viscosity response after reversal, figure 2(b), (d) and 3(a), strongly suggests that static friction is present. Indeed, evidence for frictional contacts was recently found in a similar model system [5]. A careful study of the effect of particle surface properties -specifically surface roughness -on suspension rheology would be required to validate this claim, which in turn motivates the need for robust measurements of particle friction. We