Thinning or thickening? Multiple rheological regimes in dense suspensions of soft particles

The shear rheology of dense colloidal and granular suspensions is strongly nonlinear, as these materials exhibit shear-thinning and shear-thickening, depending on multiple physical parameters. We numerically study the rheology of a simple model of soft repulsive particles at large densities, and show that nonlinear flow curves reminiscent of experiments on real suspensions can be obtained. By using dimensional analysis and basic elements of kinetic theory, we rationalize these multiple rheological regimes and disentangle the relative impact of thermal fluctuations, glass and jamming transitions, inertia and particle softness on the flow curves. We characterize more specifically the shear-thickening regime and show that both particle softness and the emergence of a yield stress at the jamming transition compete with the inertial effects responsible for the observed thickening behaviour. This allows us to construct a dynamic state diagram, which can be used to analyze experiments. Copyright c © EPLA, 2014 Introduction. – Understanding the shear rheology of dense colloidal and granular suspensions remains a central challenge at the crossroad between nonequilibrium statistical mechanics and soft matter physics, with a clear technological relevance [1–3]. Simple liquids display simple rheological properties characterized by linear Newtonian behaviour [4]. In a simple shear flow, for instance, the rate of deformation, γ̇, is proportional to the applied shear stress, σ, such that the viscosity η = σ/γ̇ uniquely characterizes the rheological response. In dense particle suspensions such as emulsions, colloidal assemblies, or granular materials, the viscosity is usually a nonlinear function of the applied flow rate. To characterise these materials, an entire flow curve η = η(γ̇) is thus needed [1–3]. Because the applied deformation now determines the response of the system, understanding nonlinear flow curves obviously requires a more detailed analysis, which must deal with both nonlinear and nonequilibrium effects. When the viscosity varies with the applied shear rate, the system can either flow more easily as γ̇ increases (shear-thinning), or offer increasing resistance to flow (shear-thickening). We are familiar with both these effects, as most of the complex fluids used for cosmetics or in food products display these amusing nonlinearities, which can be technologically both useful and annoying [3,5]. In practice, most of the experimental flow curves measured even in model suspensions display a complex mixture of both these nonlinear effects [2,3,5]. As two typical examples, we show flow curves measured in a colloidal dispersion of latex particles [5,6] (diameter a = 250 nm, fig. 1(a)), and in an oil-in-water emulsion [7] (diameter a = 20μm, fig. 1(b)). For a given volume fraction φ, the flow curves may display an initial Newtonian regime at low enough γ̇ and φ, or a strong shear-thinning regime when φ is larger. This thinning regime is followed, for intermediate φ and larger γ̇, by a Newtonian plateau regime. At larger γ̇, shear-thickening sets in, and the magnitude of the viscosity increase clearly depends on the density regime. In some cases, shear-thickening is interrupted and flow curves display a viscosity maximum. Finally shear-thickening is not observed when density is too large, see, for instance, the large density data in fig. 1(b). Books and reviews, of course, offer an even broader range of possible behaviours [1–3,5,8,9], but the data in figs. 1(a), (b) are representative of the typical behaviour of dense suspensions. The primary purpose of this work is to show that a simple model of soft repulsive particles can exhibit a similarly complex rheology, despite the fact that it does not incorporate several of the physical ingredients usually put forward to account for these nonlinearities. We argue that understanding first such a simple model is useful