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Register a new userRheological instability in a simple shear thickening model
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We study the strain response to steady imposed stress in a spatially homogeneous, scalar model for shear thickening, in which the local rate of yielding Gamma(l) of mesoscopic `elastic elements is not monotonic in the local strain l. Despite this, the macroscopic, steady-state flow curve (stress vs. strain rate) is monotonic. However, for a broad class of Gamma(l), the response to steady stress is not in fact steady flow, but spontaneous oscillation. We discuss this finding in relation to other theoretical and experimental flow instabilities. Within the parameter ranges we studied, the model does not exhibit rheo-chaos.
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We study a simple scalar constitutive equation for a shear-thickening material at zero Reynolds number, in which the shear stress sigma is driven at a constant shear rate dotgamma and relaxes by two parallel decay processes: a nonlinear decay at a nonmonotonic rate R(sigma_1) and a linear decay at rate lambdasigma_2. Here sigma_{1,2}(t) = tau_{1,2}^{-1}int_0^tsigma(t)exp[-(t-t)/tau_{1,2}] {rm d}t are two retarded stresses. For suitable parameters, the steady state flow curve is monotonic but unstable; this arises when tau_2>tau_1 and 0>R(sigma)>-lambda so that monotonicity is restored only through the strongly retarded term (which might model a slow evolution of material structure under stress). Within the unstable region we find a period-doubling sequence leading to chaos. Instability, but not chaos, persists even for the case tau_1to 0. A similar generic mechanism might also arise in shear thinning systems and in some banded flows.
Colloidal shear thickening presents a significant challenge because the macroscopic rheology becomes increasingly controlled by the microscopic details of short ranged particle interactions in the shear thickening regime. Our measurements here of the first normal stress difference over a wide range of particle volume fraction elucidate the relative contributions from hydrodynamic lubrication and frictional contact forces, which have been debated. At moderate volume fractions we find $N_1<0$, consistent with hydrodynamic models, however at higher volume fractions and shear stresses these models break down and we instead observe dilation ($N_1>0$), indicating frictional contact networks. Remarkably, there is no signature of this transition in the viscosity, instead this change in the sign of $N_1$ occurs while the shear thickening remains continuous. These results suggest a scenario where shear thickening is driven primarily by the formation of frictional contacts, with hydrodynamic forces playing a supporting role at lower concentrations. Motivated by this picture, we introduce a simple model which combines these frictional and hydrodynamic contributions and accurately fits the measured viscosity over a wide range of particle volume fraction and shear stress.
Nearly all dense suspensions undergo dramatic and abrupt thickening transitions in their flow behavior when sheared at high stresses. Such transitions occur when the dominant interactions between the suspended particles shift from hydrodynamic to frictional. Here, we interpret abrupt shear thickening as a precursor to a rigidity transition, and give a complete theory of the viscosity in terms of a universal crossover scaling function from the frictionless jamming point to a rigidity transition associated with friction, anisotropy, and shear. Strikingly, we find experimentally that for two different systems -- cornstarch in glycerol and silica spheres in glycerol -- the viscosity can be collapsed onto a single universal curve over a wide range of stresses and volume fractions. The collapse reveals two separate scaling regimes, due to a crossover between frictionless isotropic jamming and a frictional shear jamming point with different critical exponents. The material-specific behavior due to the microscale particle interactions is incorporated into an additive analytic background (given by the viscosity at low shear rates) and a scaling variable governing the proximity to shear jamming that depends on both stress and volume fraction. This reformulation opens the door to importing the vast theoretical machinery developed to understand equilibrium critical phenomena to elucidate fundamental physical aspects of the shear thickening transition.
We investigate shear thickening and jamming within the framework of a family of spatially homogeneous, scalar rheological models. These are based on the `soft glassy rheology model of Sollich et al. [Phys. Rev. Lett. 78, 2020 (1997)], but with an effective temperature x that is a decreasing function of either the global stress sigma or the local strain l. For appropiate x=x(sigma), it is shown that the flow curves include a region of negative slope, around which the stress exhibits hysteresis under a cyclically varying imposed strain rate gd. A subclass of these x(sigma) have flow curves that touch the gd=0 axis for a finite range of stresses; imposing a stress from this range {em jams} the system, in the sense that the strain gamma creeps only logarithmically with time t, gamma(t)simln t. These same systems may produce a finite asymptotic yield stress under an imposed strain, in a manner that depends on the entire stress history of the sample, a phenomenon we refer to as history--dependent jamming. In contrast, when x=x(l) the flow curves are always monotonic, but we show that some x(l) generate an oscillatory strain response for a range of steady imposed stresses. Similar spontaneous oscillations are observed in a simplified model with fewer degrees of freedom. We discuss this result in relation to the temporal instabilities observed in rheological experiments and stick--slip behaviour found in other contexts, and comment on the possible relationship with `delay differential equations that are known to produce oscillations and chaos.
We analyse the flow curves of a two-dimensional assembly of granular particles which are interacting via frictional contact forces. For packing fractions slightly below jamming, the fluid undergoes a large scale instability, implying a range of stress and strainrates where no stationary flow can exist. Whereas small systems were shown previously to exhibit hysteretic jumps between the low and high stress branches, large systems exhibit continuous shear thickening arising from averaging unsteady, spatially heterogeneous flows. The observed large scale patterns as well as their dynamics are found to depend on strainrate: At the lower end of the unstable region, force chains merge to form giant bands that span the system in compressional direction and propagate in dilational direction. At the upper end, we observe large scale clusters which extend along the dilational direction and propagate along the compressional direction. Both patterns, bands and clusters, come in with infinite correlation length similar to the sudden onset of system-spanning plugs in impact experiments.
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