Microscopic Origin of Frictional Rheology in Dense Suspensions: Correlations in Force Space

We develop a statistical framework for the rheology of dense, non-Brownian suspensions, based on correlations in a space representing forces, which is dual to position space. Working with the ensemble of steady state configurations obtained from simulations of suspensions in two dimensions, we find that the anisotropy of the pair correlation function in force space changes with confining shear stress (${\sigma }_{xy}$) and packing fraction ($\varphi$). Using these microscopic correlations, we build a statistical theory for the macroscopic friction coefficient: the anisotropy of the stress tensor, $\mu ={\sigma }_{xy}/P$. We find that $\mu$ decreases (i) as $\varphi$ is increased and (ii) as ${\sigma }_{xy}$ is increased. Using a new constitutive relation between $\mu$ and viscosity for dense suspensions that generalizes the rate-independent one, we show that our theory predicts a discontinuous shear thickening flow diagram that is in good agreement with numerical simulations, and the qualitative features of $\mu$ that lead to the generic flow diagram of a discontinuous shear thickening fluid observed in experiments.

Figure 1

(a) A snapshot of a sheared suspension of 2000 soft frictional disks. The lines represent the pairwise (lubricated and frictional contact) force vectors between the individual grains. (b) The force tiling associated with this flowing dense suspension. The bonds correspond to the pairwise forces, with larger polygons representing grains with higher stress. The vertices of the tiling represent height vectors $\overrightarrow{h}=(h_x,h_y)$, whose difference provides the pairwise force at each bond. ${\overrightarrow{\mathrm{\Gamma }}}_x=({\mathrm{\Gamma }}_{xx},{\mathrm{\Gamma }}_{xy})$ and ${\overrightarrow{\mathrm{\Gamma }}}_y=({\mathrm{\Gamma }}_{yx},{\mathrm{\Gamma }}_{yy})$ represent the sum of forces in the $x$ and $y$ directions, respectively. The light blue regions represent periodic copies of the system.

Figure 2

(a) Observed pair correlation functions at ${\sigma }_{xy}=2{\sigma }_0$, at packing fractions $\varphi =0.76$, 0.78, and 0.8. $\varphi =0.8$ is above ${\varphi }_{\mathrm{DST}}$: the onset packing fraction for a regime of stress over which the viscosity scales as $\sigma$, which defines DST (see Ref. [13]). The forces (and consequently the heights) have been scaled by the imposed shear stress. The change in symmetry of $g_2(\overrightarrow{h})$ is clearly visible as the packing fraction is increased. (b) Potentials constructed using these pair correlation functions [Eq. (5)]. (c) A comparison with pair correlations obtained from direct Monte Carlo simulations of particles interacting via these potentials.

Figure 3

Sampled values of ${\epsilon }_{\varphi ,\sigma }(\mu ,N_v)$ for $N_v=1024$ and ${\sigma }_{xy}/{\sigma }_0=100$, with $V_2$ derived from simulations at different packing fractions $\varphi$ [Eq. (5)]. Inset: $f(\mu ;\varphi ,\sigma )$ for $N_v=3000$, and $f_p^{*}=6.5\times {10}^{-4}$. The minimum of $f(\mu ;\varphi ,\sigma )$ provides the value of $\mu$ at each $(\varphi ,\sigma )$.

Figure 4

Variation of the macroscopic friction coefficient $\mu$, corresponding to the minimum of the free-energy function in Eq. (7). We find that $\mu$ decreases as packing fraction $\varphi$ and the confining shear stress ${\sigma }_{xy}$ are increased. Inset: Plot of Eq. (8) showing the appearance of two solutions at $\varphi =0.79$, and the second solution moving out to ${\sigma }_{xy}\to \infty$ at $\varphi =0.8$.