Towards a Unified Description of the Rheology of Hard-Particle Suspensions

Abstract

The rheology of suspensions of Brownian, or colloidal, particles (diameter d ≲ 1 µm) differs markedly from that of larger grains (d ≳ 50 µm). Each of these two regimes has been separately studied, but the flow of suspensions with intermediate particle sizes (1 µm ≲ d ≲ 50 µm), which occur ubiquitously in applications, remains poorly understood. By measuring the rheology of suspensions of hard spheres with a wide range of sizes, we show experimentally that shear thickening drives the transition from colloidal to granular flow across the intermediate size regime. This insight makes possible a unified description of the (non-inertial) rheology of hard spheres over the full size spectrum. Moreover, we are able to test a new theory of friction-induced shear thickening, showing that our data can be well fitted using expressions derived from it.Complex fluids, polymers, colloids and surfactant solutions find wide applications, partly because of their highly tuneable behavior under deformation and in flow. The success of the mean-field 'tube' model for polymers [1], which describes how each chain is constrained by thousands of neighbours, means it has long been possible to predict ab initio their linear and non-linear rheology from the molecular topology with very few free parameters. In particular, a scaling description is available of the dependence of rheology on molecular weight.However, progress in suspension rheology has been more difficult [2]. The small number of nearest neighbours (order 10) rules our any mean-field description: local details matter. It is now possible to predict the lowshear viscosity of a suspension of Brownian hard spheres (HS, diameter d ≲ 1 µm) up to volume fractions of φ ≲ 0.6, and the rheology of granular HS (d ≳ 50 µm) is increasingly being studied. Surprisingly, however, how the rheology of HS changes over the whole size spectrum remains unknown, because the behavior in the industriallyubiquitous intermediate size regime, 1 ≲ d ≲ 50 µm, has not been systematically explored. We offer such an exploration in this Letter, and show that the physics bridging the colloidal and the granular regimes is shear thickening.The rheology of colloidal HS is well known [3][4][5]: the viscosity is determined by the particle volume fraction, φ, and the dimensionless shear rate, or Péclet number, Pe (= τ Bγ , the shear rateγ non-dimensionalised by the Brownian time, τ B , needed for a free particle to diffuse its own radius). At Pe ≪ 1 the flow is Newtonian; the viscosity becomes immeasurably large at φ g ≈ 0.58 [5,6]. Shear thinning starts at Pe ≲ 1, reaching a second Newtonian regime at Pe ≫ 1 with a viscosity that diverges at random close packing [2], φ RCP ≈ 0.64, the densest amorphous packing for lubricated (frictionless) HS.Since τ B scales as d 3 , granular HS inhabit the Pe ≫ 1 regime at all practical shear rates. Extrapolating naïvely from the above description of colloidal flow, one expects Newtonian behaviour with a viscosity diverging at φ RCP . Experiments do find a Newtonian viscosity, but i...

Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions

“…3), both PVC and cornstarch suspensions exhibit discontinuous shear thickening where the gradient d(log η )/d(log σ ) reaches 1 (vertical flow curve when plotting ). Within experimental uncertainty, shear thickening begins at a fixed onset stress, which depends only on the studied system and not on the volume fraction ϕ 2 (see dotted lines on Supplementary Fig. 3).…”

“…3 for details). All these suspensions exhibit a discontinuous shear thickening transition above a critical shear stress σ C (defined as the critical shear stress above which the viscosity starts to increase as a function of the shear rate2). As found previously, σ C (contrary to the corresponding shear rate ) does not depend upon the solid fraction of particles ϕ 2 for high solid fractions (Fig.…”

“…Shear thickening, where the resistance to flow increases when the dispersion is stirred is one of the most intriguing behaviour12. In the extreme situation of discontinuous shear thickening, shear viscosity increases by orders of magnitude at a given shear rate.…”

“…At this stage, this picture and the role played by frictional forces have been validated through numerical simulations but only indirectly through experiments at the level of the suspension2616.…”

Revealing the frictional transition in shear-thickening suspensions

“…Supporting results are also provided by recent experimental investigations. For instance, standard rheological measurements were performed on suspensions of small poly(methyl methacrylate) (PMMA) particles sterically stabilized by a coating of poly-12-hydroxystearic acid (14). The suspension was indeed found to follow two separate viscosity curves with distinct critical volume fractions, depending on what shear rate was applied.…”