Interaction network analysis in shear thickening suspensions

Dense frictional particulate suspensions in a viscous liquid undergo increasingly strong continuous shear thickening as the solid packing fraction, $\varphi$, increases above a critical volume fraction, and discontinuous shear thickening is observed for even higher packing fractions. Recent studies have related shear thickening to a transition from mostly lubricated to predominantly frictional contacts with the increase in stress, with the transition determined by overcoming a repulsive force. The rheology and networks of frictional forces from two- and three-dimensional simulations of shear-thickening suspensions are studied. These are analyzed using measures of the topology of the network, including tools of persistent homology. We observe that at low stress, the frictional interaction networks are predominantly quasilinear along the compression axis. With an increase in stress, the force networks become more isotropic, forming loops in addition to chainlike structures. The topological measures of Betti numbers and total persistence provide a compact means of describing the mean properties of the frictional force networks, and provide a link between macroscopic rheology and the microscopic interactions. A total persistence measure describing the significance of loops in the force network structure, as a function of stress and packing fraction, shows behavior similar to that of relative viscosity, and displays a scaling law near the jamming fraction for both two- and three-dimensional systems considered. The total persistence measures for both dimensions are found to be very similar.

Figure 1

(a),(b) Snapshots of the force network at $\tilde{\sigma }=1$ for (a) ${\varphi }_{\mathrm{A}}=0.78$ in 2D and (b) $\varphi =0.56$ in 3D. Line thickness is proportional to the normalized force $F$ with red, green, and blue lines showing frictional contact, frictionless contact, and lubrication interactions, respectively. (c),(d) Relative viscosity, ${\eta }_{\mathrm{r}}$, as a function of dimensionless shear rate $\dot{\gamma }/{\dot{\gamma }}_0$ in stress-controlled simulations in (c) 2D and (d) 3D for several packing fractions.

Figure 2

Flow-state diagram in the shear stress vs 2D packing fraction $(\tilde{\sigma },{\varphi }_{\mathrm{A}})$ plane for $\mu =1$; for the 3D version, see [8]. The red dashed line shows the locus of points for DST (where $\partial \dot{\gamma }/\partial \sigma =0$) and the leftmost point on this curve is the minimum packing fraction, ${\varphi }_{\mathrm{A},\mathrm{C}}$, at which DST is observed. The blue solid curve shows packing-fraction-dependent maximum flowable stress above which the suspension is shear-jammed. Dashed and dotted-dashed black lines represent frictional $({\varphi }_{\mathrm{A},\mathrm{J}}^{\mu }=0.795)$ and frictionless $({\varphi }_{\mathrm{A},\mathrm{J}}^0=0.855)$ jamming points in 2D, respectively.

Figure 3

Interaction network of all frictional contacts (normal force above the critical value, $F_0$) corresponding to a snapshot in steady state obtained in simulations with $\tilde{\sigma }=$ (a) 0.1, (b) 0.5, (c) 1.0, (d) 2.0, (e) 5.0, (f) 10, (g) 50, and (h) 100 in 2D. Line thickness is proportional to the normalized force, $F$.

Figure 4

Filtration of the interaction network of all frictional contacts for $\varphi ={\varphi }_{\mathrm{A}}=0.76$ and $\tilde{\sigma }=10$ corresponding to a snapshot in steady state for several values of the threshold, in order from left to right and top to bottom: $F=$ (a) 4, (b) 3, (c) 2, (d) 1.6, (e) 1.3, (f) 1, (g) 0.5, and (h) 0. Here, black dots show particle centers, and the thickness of the red line corresponds to the strength of the normal force at the contact.

Figure 5

Betti numbers corresponding to the filtration in Fig. 4 for components (a) and loops (b). The (red) dots are at the force threshold levels corresponding to the values illustrated in Fig. 4. Note the difference in the axes range for (a) and (b).

Figure 6

Persistence diagrams corresponding to the filtration in Fig. 4 for components (a) and loops (b).

Figure 7

Total persistence for 2D data as a function of $\tilde{\sigma }$ for components (a) and loops (b), for several values of packing fraction ${\varphi }_{\mathrm{A}}$.

Figure 8

Interaction networks for the 3D data for $\varphi =0.56$, for $\tilde{\sigma }=0.5$ (a), $\tilde{\sigma }=1$ (b), $\tilde{\sigma }=2$ (c), and $\tilde{\sigma }=10$ (d), each showing a single instant from the simulation once the structure has developed.

Figure 9

Total persistence for 3D data as a function of $\tilde{\sigma }$ for components (a) and loops (b), for several values of packing fraction $\varphi$.

Figure 10

Scaled total persistence for 3D data for $\varphi =0.56$ as a function of $\tilde{\sigma }$: (a) ${\tau }_0$ and (b) ${\tau }_1$.

Figure 11

(a) Relative viscosity ${\eta }_{\mathrm{r}}$ vs scaled total persistence (dimension 1) ${\tau }_1/N$ for different packing fractions for two- and three-dimensional data. Solid and empty symbols are used for 2D and 3D, respectively. The dashed lines represent ${\eta }_{\mathrm{r}}\propto {[({\tau }_1^{\mathrm{J}}-{\tau }_1)/N]}^{-\alpha }$ with ${\tau }_1^{\mathrm{J}}/N=0.34$ and 0.42 for 2D and 3D, respectively, and $\alpha =-1.8$. Red and blue symbols represent two- and three-dimensional data. (b) The same data on a log-log scale showing ${\eta }_{\mathrm{r}}$ vs $({\tau }_1^{\mathrm{J}}-{\tau }_1)/N$. The dashed line shows the power law of $-1.8$.

Figure 12

(a) Relative viscosity ${\eta }_{\mathrm{r}}$ plotted as a function of stress $\tilde{\sigma }$ for area fraction ${\varphi }_{\mathrm{A}}=0.79$. Density plots of all the ${\beta }_0$ persistence diagrams superimposed for the given parameter values for the 2D simulations colored according to the density of points. The color bar indicates the density normalized across all the plots. The plots correspond to ${\varphi }_{\mathrm{A}}=0.79$ and for (b) $\tilde{\sigma }=0.5$, (c) $\tilde{\sigma }=1$, (d) $\tilde{\sigma }=2$, (e) $\tilde{\sigma }=5$, (f) $\tilde{\sigma }=10$, (g) $\tilde{\sigma }=50$, and (h) $\tilde{\sigma }=100$.

Figure 13

Betti numbers for several values of $\tilde{\sigma }$, scaled by the number of particles $N$ for 2D simulations (a),(b) for ${\varphi }_{\mathrm{A}}=0.79$, and for 3D simulations (c),(d) for $\varphi =0.56$. Parts (a),(c) show ${\beta }_0$, and (b),(d) show ${\beta }_1$. Here we have analyzed only the frictional contacts, i.e., contacts for which force is larger than the critical normal force $F_0$ and is normalized with the applied stress. Since there are no frictional contacts for $F<F_0$, the Betti number plateaus for small forces, and due to the normalization with applied stress, the onset of the plateau decreases with an increase in stress.