Universal scaling law in frictional non-Brownian suspensions

We compare the rheology of two kinds of non-Brownian suspensions. One is made of spherical monodisperse polystyrene particles $(80\mu \text{m}$ in diameter), and the other is made of faceted sugar particles (average size, $100\mu \text{m})$, both suspended in a Newtonian silicon oil. We perform shear reversal experiments on both suspensions for several particle volume fractions, $\varphi$, and several shear stresses, $\tau$. The two suspensions behave in a quite different fashion. For the faceted particle suspensions (FPSs), a large shear thinning is observed, while it is much more moderate for the spherical polystyrene particle suspensions (SPSs). Another striking difference lies in the value of the jamming packing fraction, ${\varphi }_m$, which is much lower for FPSs than for SPSs. Despite these differences, we will show that shear reversal experiments make it possible to obtain a universal scaling that holds for both FPSs and SPSs. In this scaling, the difference between the steady viscosity and the viscosity at the minimum that follows the shear reversal, normalized by the steady viscosity, is shown to depend only on the ratio $\varphi /{\varphi }_m\left(\tau \right)$. The collapse of all the data suggests that concentrated non-Brownian suspensions behave in a universal way regardless of the mechanisms responsible for flow hindering (rotation frustration or sliding friction).

Figure 1

(a) Sugar particles. (b) Polystyrene particles. Scale bar $=200\mu \text{m}$.

Figure 2

Transient response of the viscosity under shear reversal. The reduced viscosity is plotted against the accumulated strain after shear reversal. Orange squares: Sugar particle suspension (FPS). Blue disks: polystyrene particle suspension (SPS). $\varphi =0.43$ and $\tau =28\text{Pa}$.

Figure 3

The reduced viscosity curves as a function of shear stress for both FPSs (squares) and SPSs (disks). Blue: $\varphi =0.3$. Orange: $\varphi =0.35$. Green: $\varphi =0.40$. Red: $\varphi =0.43$. Purple: $\varphi =0.45$. Brown: $\varphi =0.47$. Pink: $\varphi =0.49$. Gray: $\varphi =0.51$. Steady viscosity of the SPSs (a) and FPSs (b). Viscosity at minimum of the SPSs (c) and FPSs (d). The sugar particle suspension exhibits a highly shear-thinning behavior, while the shear-thinning behavior demonstrated by the spherical particle suspension is much weaker. The viscosity at the minimum only slightly decreases with increasing shear stress for both FPSs and SPSs.

Figure 4

The inverse of the square root of the reduced steady viscosity as a function of particle volume fraction for all the tested stresses. Each color corresponds to a given stress $(\tau \in [5\text{Pa}-100\text{Pa}]$ for SPSs and $\tau \in [10\text{Pa}-100\text{Pa}]$ for FPSs). (a) SPSs, (b) FPSs. Symbols: experimental data. Solid lines: fits.

Figure 5

Variation of the jamming packing fraction with the shear stress for FPSs (orange squares) and SPSs (blue disks). Insert: Variation of ${\alpha }_0$ with $\tau$.

Figure 6

Steady viscosity as a function of $\varphi /{\varphi }_m$. Squares: Sugar particles. Disks spherical polystyrene particles. Blue: $\varphi =0.3$. Orange: $\varphi =0.35$. Green: $\varphi =0.40$. Red: $\varphi =0.43$. Purple: $\varphi =0.45$. Brown: $\varphi =0.47$. Pink: $\varphi =0.49$. Gray: $\varphi =0.51$.

Figure 7

Ratio of the difference between the steady viscosity and the viscosity at minimum normalized by the steady viscosity as a function of the ratio $\varphi /{\varphi }_m$. Squares: FPSs, disks: SPSs. Blue: $\varphi =0.3$. Orange: $\varphi =0.35$. Green: $\varphi =0.40$. Red: $\varphi =0.43$. Purple: $\varphi =0.45$. Brown : $\varphi =0.47$. Pink: $\varphi =0.49$. Gray: $\varphi =0.51$. Black disks: numerical simulations from Ref. [11] obtained for different sliding friction coefficients between 0 and 1 (no rolling friction). Empty triangles: PMMA particles $31\mu \text{m}$ in diameter (Microbeads, CA30), dispersed in a Newtonian liquid from [13] $\left({\varphi }_m=0.534\right)$. Solid triangles: polyamideparticles (see Supplemental Materials [22])

Figure 8

(a) Aspect ratio distribution. (b) Equivalent size distribution.

Figure 9

Polyamide particles. Scale bar $=100\mu \text{m}$.

Figure 10

(a) Aspect ratio distribution. (b) Equivalent size distribution.