Stress-activated friction in sheared suspensions probed with piezoelectric nanoparticles

Shearing a concentrated, or dense, suspension of small particles in a simple liquid drives these particles into contact (1–4). For sufficiently large shear, simulations show that these contacts become frictional and form chains that organize into a network of branching paths along which most stress is transmitted (5, 6). Once this network of frictional contact forces extends throughout the suspension, it changes the overall behavior from easily flowing to nearly rigid, a striking transformation that is fully reversible. Such behaviour has led to these materials being explored in applications that include stress-adaptive materials for impact and penetration mitigation, vibration damping, or flow control (7–9). However, direct experimental evidence for the contact force network is hampered by the difficulty of detecting local particle–particle contacts inside a suspension.

Prior work on dense suspensions has explored particle configurations utilizing confocal microscopy imaging and extracting information from particle center-to-center spacings, either in flowing systems (1) or in suspensions arrested with a photocrosslinker (10). Related approaches have been used in emulsions (11) as well as assemblies of larger hydrogel particles (12). Direct extraction of local contact forces in an assembly’s interior has only been possible in macroscopic dry granular packings of two-dimensional (2D) photoelastic particles (13, 14) and, in dry 3D packings, using magnetic resonance elastography (15) or combinations of x-ray diffraction and tomography together with numerical modeling (16).

Here, we introduce an approach that, simultaneously with measurement of the viscosity, monitors changes in the electrical conductance of a dense suspension of piezoelectric nanoparticles under shear (Fig. 1A). This does not give spatially localized information, but it can signal the emergence of frictional contact force networks with high sensitivity and it allows for in-situ measurements under shear.

When strained, piezoelectric particles develop an electric dipole moment that generates charges at the particle surface. Such particles, either as dense aggregates or embedded at low density in a suitable matrix, have found a range of applications, including as converters of mechanical energy into electric energy for energy harvesting (17, 18). Generally, to obtain a large response to strain from an assembly of spatially fixed, but randomly oriented piezoelectric particles, an initial poling step is applied during which a large electric field aligns the local dipole moments. In a sheared suspension, any effect of poling is negated since particles reconfigure and reorient continually, which keeps randomizing their polarization directions. Furthermore, piezoelectric particles are nonmetallic and their own contribution to the suspension conductivity at rest is typically very small. However, if they are suspended in an aqueous liquid of finite conductivity, the surface charges they produce when strained can transfer into the surrounding liquid through the Faradaic current (19). These charges lead to an increase in conductivity across the suspension. This process is independent of the individual particles’ polarization direction while particles reorient under shear. It can be expected to become particularly efficient when particles are sheared into chain-like arrangements, as happens when contact force networks form during shear thickening. These networks tend to be anisotropic, oriented predominantly along the forward compressive direction, and are highly heterogeneous. As a result, during shear thickening, the uniformly applied stress at the boundaries is focused along a small number of force chains that carry the majority of the load (6). This stress focusing, whereby the local contact stress can largely exceed the average value, serves as an effective mechanism to enhance the piezoelectric response. Similar local stress enhancement has also been found in open-pore, network-type architectures of piezoelectric solids (17). Importantly, to generate an ac conductance signal no system-spanning, fully percolated networks are required. In our suspensions, the ac current is carried by the suspending liquid and not the particles, which are effectively nonconducting. Therefore, an observable increase in the ac conductance can be generated from particle–particle interactions anywhere in the suspension as long as they are strong enough to generate Faradaic current for our bulk conductance measurement to pick up. Thus, while the ac conductance by itself cannot be used to distinguish between different mechanisms producing the shear-induced large local stresses, such differentiation becomes possible in combination with simultaneous rheological measurements.

In the following, we first demonstrate how the emergence, and evolution, of frictional contact force networks during shear thickening in dense suspensions of piezoelectric particles can be tracked by measuring the suspension conductance (Fig. 1A). We show this with ZnO as well as BaTiO${}_3$ nanoparticles for different volume fractions, suspending liquids, and oscillatory shearing conditions. We then show how the frictional, stress-activated piezoelectric response in such suspensions can serve to track structural memory in the contact force network and drive chemical reactions.

In our experiments, the piezoelectric particles are suspended in glycerol at different volume fractions $\varphi$. It is worthwhile noting that commercial particles used here have a wide distribution of shapes and sizes. However, while size dispersion tends to broaden the observed shear thickening, it does not eliminate it (20) and increased particle anisotropy tends to enhance shear thickening (21–23). The complex viscosity ${\eta }^{\ast }$ is then measured under oscillatory shear (parallel plate geometry) as a function of dynamic shear rate ${\gamma }_0\omega$, whereby the shear strain amplitude ${\gamma }_0$ is increased while keeping the angular frequency $\omega$ fixed. Simultaneously the electrical ac conductance $\sigma$ is measured by applying an alternating voltage (amplitude 1 V_rms, frequency 100 Hz) across the rheometer plates, which serve as electrodes (Fig. 1A). In order to track changes in conductance with dynamic shear rate, the background conductance ${\sigma }_0$ of the quiescent suspension is subtracted from $\sigma$ (for raw conductance data, see SI Appendix, Figs. S1 and S2). In addition, the slow but steady increase of conductance with time, caused by charge buildup at the electrodes on account of Maxwell–Wagner polarization in the suspending liquid (24), is also subtracted. The remaining, shear-induced enhancement in ac conductance is then calculated as $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0=\left(\sigma -{\sigma }_0-\frac{\partial \sigma }{\partial t}{|}_{t=0}\mathrm{\Delta }t\right)/{\sigma }_0$, which is normalized by the background value ${\sigma }_0$. Here, $\frac{\partial \sigma }{\partial t}{|}_{t=0}$ is the rate of conductance change from Maxwell–Wagner polarization and $\mathrm{\Delta }t$ the measurement time interval. The effect of particle sedimentation in contributing to the measured $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ is negligible given that the stress from particle weight is orders of magnitude lower than the stress where we first pick up a signal in the excess conductance, namely at the onset of strong shear thickening (see Materials and Methods for details).

Fig. 1B shows the complex viscosity ${\eta }^{\ast }$ and excess ac conductance $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ for a suspension of ZnO particles (average diameter $d\simeq 1\mathrm{\mu }\text{m}$) in glycerol at volume fraction $\varphi =0.36$. The rapid rise in ${\eta }^{\ast }$ beyond some characteristic shear rate ${\gamma }_0\omega$ is a signature of pronounced, nearly discontinuous shear thickening associated with the onset of shear-induced, frictional interparticle contacts (5, 6). As the Fig. 1B, Top shows, this rise in ${\eta }^{\ast }$ is mirrored by a corresponding rise in $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$. At the same time, a shift in the onset for shear thickening to larger dynamic shear rates ${\gamma }_0\omega$ is observed in both ${\eta }^{\ast }$ and $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$. This shift shows that the mechanism driving the thickening and the excess conductance is not tied to a specific, characteristic shear rate ${\gamma }_0\omega$, as would have been expected for fluid-mediated particle interactions, where the particles experience rate-dependent lubrication forces but no direct frictional contact (see, e.g., ref. 25). Instead, the shift is a consequence of the condition that the stress $\tau ={\eta }^{\ast }{\gamma }_0\omega$ has to reach a minimum critical magnitude ${\tau }_c$ in order to activate frictional particle–particle contacts, which, for larger $\omega$, moves the onset to smaller viscosity (5, 6). In combination with the viscosity data, the increase in conductance can therefore be associated with a stress-induced transition from frictionless to frictional interactions inside the suspension.

To verify that the measured excess conductance is indeed associated with particle contact formation and not simply with the viscosity magnitude, we can alter the overall rheological behaviour of the suspension by adding salt to the suspending liquid (Fig. 1C). The salt screens the surface charges, lowering the electrostatic repulsion between particles. As a result, a strong shear thinning at low ${\gamma }_0\omega$ is observed along with significant weakening of the degree of thickening under shear at larger ${\gamma }_0\omega$, as seen in the less steep rise of ${\eta }^{\ast }$ beyond the thickening onset. Concomitantly, the magnitude of $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ is reduced by a factor 10. Importantly, $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ in Fig. 1C shows a clear increase only around the point where shear thickening sets in.

At a salt concentration of [NaCl]$=0.1\text{mol}{\text{L}}^{-1}$, as in Fig. 1C, the Debye length ${\kappa }^{-1}$ of a ZnO particle in glycerol becomes sufficiently small (around 1nm) that attractive short-range interactions generate particle agglomerates at rest, which then yield and break up under shear, leading to the observed pronounced shear thinning with increasing ${\gamma }_0\omega$. Importantly, while these attractive forces stabilize the agglomerates, the particles are not sheared into contact, and therefore, no piezoelectric response is observed. Only once the applied stress is increased sufficiently is the piezoelectric response of the particles activated. This behavior differs from high-conductivity carbon black suspensions, where the particles themselves contribute significantly to the overall ac electrical conductivity, which generally drops with increasing shear rate (26). The absence of a piezoelectric response in silane-coated ZnO (27) or of changes in $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ in fumed silica suspensions exhibiting significant shear thickening (SI Appendix, Fig. S3) further supports that the concurrent increase of viscosity and ac conductance originates from stress-induced piezoelectricity between particles.

A similar, direct correspondence between the onset of shear thickening and an increase in $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ is found for barium titanate (${\text{BaTiO}}_3$ or BTO) particles of average diameter $d=500\text{nm}$ suspended in glycerol. Such BTO suspensions exhibit shear thinning at low shear rates even without the addition of salt. Since the piezoelectric constant is almost an order of magnitude larger in BTO compared to ZnO, this enables us to track the piezoelectric response in more detail in BTO suspensions with a smaller solid fraction. Decreasing the BTO particle volume fraction $\varphi$ from 0.49 to 0.25 decreases the degree of both shear thinning and thickening, and with it the magnitude of $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ in the thickening regime (Fig. 2A). However, even at $\varphi =0.25$, where there is only very weak apparent thickening, we can still detect a small increase in $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ (Fig. 2B). Given that system-spanning contact networks are extremely unlikely to form at such low volume fraction, this serves as a reminder that the ac conductance does not rely on a fully percolated network and can detect sufficiently strong stress loading anywhere in a suspension of piezoelectric particles. As shown in Fig. 2 B and C for the $\varphi =0.25$ and $\varphi =0.49$ data (all intermediate concentrations are available in SI Appendix, Fig. S4), increasing $\omega$ from 1 rad/s to 100 rad/s increases the magnitude the $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$-pagination signal.

Since shear thinning and shear thickening arise from different mechanisms and exhibit opposite trends as a function of ${\gamma }_0\omega$, their contributions to the viscosity result in a crossover regime at intermediate shear rate, where thinning can mask the emerging thickening (21, 28). Close inspection of the data in Fig. 2 reveals that the excess conductance $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ rises slightly before the viscosity minimum is reached. This is a further indication that the conductance signal is sensitive to the formation of the frictional force chain network, even in its early stages.

With regard to the apparent peak observed in much of the $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ data in Figs. 1 and 2, it is important to note that oscillatory shear measurements at the highest shear rates in both the ZnO and BTO suspensions are difficult because of interfacial instabilities that can lead to radial ejection of material from the shear cell, and thus, the data beyond the maximum $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ are less reliable. Nevertheless, the clear peak in the data for the largest packing fractions in Fig. 2A could indicate that there are structural changes in the force chain network before the transition into the fully shear-thickened, high-viscosity state has been completed. In particular, the peak could reflect the crossover from a sparser, more anisotropic network to a denser, more homogeneous and isotropic network structure, as found in granular systems (29) and simulations of suspensions (30). As the network becomes denser and the local connectivity increases quicker than the applied stress, forces become more evenly distributed and the amount of stress per contact decreases, resulting in a decrease in ac conductance.

The simultaneous measurement of viscosity and conductance in suspensions of piezoelectric particles provides special access to signatures of particle self-organization and strain-dependent memory associated with the evolution of the frictional force network over time. In studies of jammed and glassy colloidal and granular particle systems under oscillatory shear (31–34), it was found that repeated reversal of the shearing direction at given stress reinforces the most robust network configurations. Effectively, this trains the network to evolve toward the subset of configurations that, for given $\tau$, are most efficient at quick rebuilding after each reversal of the shearing direction.

To observe this structural memory with a BTO suspension, oscillatory shear at constant stress $\tau$ is applied and ${\eta }^{\ast }$ and $\sigma$ are measured as a function of time. This stress is then increased in steps as shown in the Fig. 3A, Inset. Following each increase in $\tau$, the viscosity ${\eta }^{\ast }$ jumps to a new, larger magnitude (Fig. 3A). After each jump, a hint of the network reconfiguration can already be seen in the small, but noticeable, decrease of the viscosity with time, especially at larger stresses (35). Since the applied stress is held constant in these experiments, this gradual reduction of ${\eta }^{\ast }$ indicates a speedup of the dynamic shear rate, consistent with a more effective and thus faster reconfiguration during shear reversal. This small trend in ${\eta }^{\ast }$ is mirrored, in a much more amplified fashion, in the excess conductance $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ in Fig. 3A. Starting near the onset of shear thickening, at stress ${\tau }_1$, the excess conductance is essentially independent of time. As the suspension reaches the middle of the shear thickening regime at ${\tau }_2$, $\mathrm{\Delta }{\sigma }_{\text{net}}/{\sigma }_0$ grows with time at a rate of roughly $8\times {10}^{-5}{\text{s}}^{-1}$. This rate further increases to $3\times {10}^{-4}{\text{s}}^{-1}$ near the peak of thickening at ${\tau }_3$. This growth of conductance over successive shear oscillation cycles would then suggest that such cyclic training sets up particularly robust contact force networks, which in turn generate robust conditions for piezoelectric charge generation that becomes more efficient over time as particles nestle into configurations optimized for shear reversal.

In Fig. 3B, we start from stress level ${\tau }_3$ and after 3,600 s abruptly cut the oscillation strain amplitude by five orders of magnitude, thereby switching from a stress ${\tau }_3=1\times {10}^4\text{Pa}$ to ${\tau }_0=10\text{Pa}$. Since this BTO suspension is strongly shear thinning, its viscosity ${\eta }^{\ast }$ jumps up by two orders of magnitude (Fig. 3A, Inset). In this state, the system is far below the onset of shear thickening so that a stress-activated frictional force chain network can no longer be maintained and the particles reconfigure into a frictionless configuration held together by attractive forces. As Fig. 3B shows, this disassembly of the trained-in network is very slow and likely occurs by particle diffusion, given that the Péclet number dropped from $\text{Pe}\approx 5,000$ to $\text{Pe}=0.01$ during shear cessation ($\text{Pe}=6\pi {\eta }_{\text{f}}{a}^3{\gamma }_0\omega /k_{\text{B}}T$ characterizes the ratio of advective to diffusive transport rates for a particle of diameter $a$ in a liquid of viscosity ${\eta }_{\text{f}}$). Yet even in a partially disintegrated frictional network, oscillatory excitation with very small strain evidently still generates a piezoelectric response that shows up as clear signal in the excess conductance.

To further confirm that the measured excess ac conductance is due to a change in the stress state of individual particles during shear-induced frictional network formation, a piezochemical reaction was examined. Such reactions require that particles experience significant stresses, typically produced by ball milling (36, 37), ultrasound (38, 39), or electrodynamic agitation (27). By using a polymerization reaction known to be driven by mechanically activated piezochemistry, the detection of crosslinking under shear can be used as a readout to signal that sufficiently large stress on the particles has changed the local charge environment and enabled a chemical reaction in the surrounding polymeric fluid. To this end, ZnO particles are suspended in a monomer mixture of dipentaerythritol hexakis (3-mercaptopropionate) (DiPETMP) and tri(ethylene glycol) divinyl ether (TEGDE) (Fig. 4A) and the kinetics of the thiol-ene crosslinking reaction are observed over time. This reaction has been shown to be mediated by electrodynamic agitation of a solution containing a low concentration of ZnO particles (27). Fig. 4B demonstrates that oscillatory shear ($\omega =50\text{rad}{\text{s}}^{-1}$ and ${\gamma }_0=80\%$) can indeed transform the initially viscous fluid into a polymerized solid. In Fig. 4C, the polymerization reaction kinetics are tracked by measuring the gelation time $t_{\text{gel}}$ for a 70 wt.% ZnO suspension in a DiPETMP and TEGDE mixture solution (1:3 molar ratio), using oscillatory strain of different amplitude with a fixed frequency $\omega =50\text{rad}{\text{s}}^{-1}$. As ${\gamma }_0\omega$ is increased and the onset of shear thickening is reached, $t_{\text{gel}}$ starts to become reduced, eventually dropping by as much as a factor 4 at large dynamic shear rate ${\gamma }_0\omega$ (see SI Appendix, Fig. S5 for data with $\omega =10\text{rad}{\text{s}}^{-1}$). The result is a striking correlation between the onset of piezochemical reaction kinetics and the onset of shear-thickening.

In conclusion, our simultaneous measurements of viscosity and excess ac conductance, together with the observed enhancement of the reaction kinetics, provide strong experimental evidence for the emergence of networks of frictional particle–particle contacts in the shear-thickening regime of piezoelectric dense suspensions. Further evidence for frictional stress in these suspensions comes from our finding that oscillatory shear trains the particle configurations in these networks and, as highlighted by the evolution of the excess conductance, leads to structural memory. Overall, our observations demonstrate that the excess conductance in piezoelectric suspensions acts as a macroscale reporter and amplifier of very subtle, local and much more microscopic changes in the nature of the particle interactions as increased stress drives these interactions from lubricated to frictional. These findings also open up opportunities for further exploring how frictional contacts aid in the memory formation process and in the relaxation of the memory-formed state. Finally, while the piezoelectric particle suspensions studied here exhibit strong friction-induced shear thickening, by inserting a low concentration of piezoelectric particles as probe particles, it may be possible to track the onset of large local stresses also in suspensions with other particle types or where shear thickening is lubrication-dominated (40).