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

B. M. Guy, M. Hermes, and W. C. K. Poon

Phys. Rev. Lett. 115, 088304 – Published 20 August 2015; Erratum Phys. Rev. Lett. 116, 059901 (2016)

Abstract

The rheology of suspensions of Brownian, or colloidal, particles (diameter $d ≲ 1 μ m$ ) differs markedly from that of larger grains ( $d ≳ 50 μ m$ ). Each of these two regimes has been separately studied, but the flow of suspensions with intermediate particle sizes ( $1 μ m ≲ d ≲ 50 μ m$ ), 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 (noninertial) 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.

Received 17 March 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.088304

Erratum

Erratum: Towards a Unified Description of the Rheology of Hard-Particle Suspensions [Phys. Rev. Lett. 115, 088304 (2015)]

B. M. Guy, M. Hermes, and W. C. K. Poon

Phys. Rev. Lett. 116, 059901 (2016)

Authors & Affiliations

B. M. Guy, M. Hermes, and W. C. K. Poon

SUPA, School of Physics and Astronomy, The University of Edinburgh, King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom

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Vol. 115, Iss. 8 — 21 August 2015

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Images

Figure 1
Rheology in the colloidal, intermediate, and granular size regimes. (a) Relative viscosity $η$ versus shear stress $σ$ in units of Pa and $k_{B} T / d^{3}$ for $d = 404 nm$ spheres at different volume fractions (%), as labeled. Solid lines, fits based on [12]; finely dotted lines, schematics based on literature data (with $sparsely dotted = unstable states$ ). Samples shear thicken above a $ϕ$ -independent onset stress $σ^{*}$ (vertical dashed line). Color scheme: blue, frictionless interactions ( $σ < σ^{*}$ ); black, shear thickening; red, frictional interactions ( $σ ≫ σ^{*}$ ). The unshaded region is accessible using our rheometer, which reaches maximum and minimum shear rates of 8000 and $10^{- 3} s^{- 1}$ , respectively, and a minimum stress of $10^{- 2} Pa$ ; the maximum accessible stress is set by a $d$ -dependent fracture stress $σ^{†}$ . (b) Main: $η (ϕ)$ for the limiting high-shear viscosity, $η_{1}$ , in (a) [blue (dark gray) filled square]; the lower, $η_{1} (ϕ)$ [red (light gray) filled square], and upper, $η_{2} (ϕ)$ (red square), branches in (c); the upper branch (red triangle) in (d). Solid red line, least squares fit to $η_{2} (ϕ) = A (1 - ϕ / ϕ_{m})^{- n}$ with $A = 0.20 (9)$ , $ϕ_{m} = 0.558 (5)$ and $n = 2.2 (2)$ . $η_{1} (ϕ)$ data for other sizes of PMMA spheres in this work, $d = 912 nm$ (triangle down) and 1800 nm (filled triangle down). Other symbols, literature high shear viscosities (with $ϕ$ shifted by up to 5%) for sterically stabilized PMMA [5, 6, 14, 15], sterically stabilized silica [3], and glass beads [11]. (filled circle) and (circle), lower and upper branches from [15]. Inset, $η_{2} (ϕ)$ versus ( $ϕ_{m} - ϕ$ ) including the upper branch from [15]. (c) $η (σ)$ for $d = 3770 nm$ spheres. The flow in both (a) and (c) was unsteady for $ϕ \geq 0.56$ , and points represent temporal averages. (d) $η (σ)$ for $d = 45 μ m$ spheres, with $ϕ$ shifted up so that the data agree with $η_{2} (ϕ)$ in (b).
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Figure 2
Onset stress $σ^{*}$ versus particle diameter $d$ for PHSA-stabilized PMMA spheres in this and previous [9, 14, 16, 17, 18, 19] work. The solid line is a least-squares fit to a power law $σ^{*} = B d^{α}$ , with $B = 7 \times 10^{- 11} J {nm}^{- 1.1}$ and $α = - 1.9 (1)$ .
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Figure 3
Relative viscosity $η$ versus dimensionless stress $\hat{σ}$ . The observable window (shaded) for colloids shows shear thinning, which begins at $\hat{σ} ≃ 1$ ; shear thickening, beginning at a dimensionless onset stress of ${\hat{σ}}^{*} = σ^{*} d^{3} / k_{B} T = f^{*} d / k_{B} T$ , occurs towards the high end of accessible stresses. As the particle diameter $d$ increases, ${\hat{σ}}^{*}$ shifts right as $d$ , and the second Newtonian regime in the $η (\hat{σ})$ curve is stretched out. However, the (shaded) observable window shifts right faster, as $d^{3}$ . Thus, shear thinning becomes unobservable for intermediate particle sizes, and the shear-thickened state fills the observable window of the largest particles (grains). Figures 1, 1, and 1 show three snapshots of this scenario (also, see Supplemental Material Fig. S1 [20]).
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