Constraint ratio controls viscosity in shear thickening suspensions

Abstract

The dramatic viscosity increase observed in dense suspensions under shear poses a major challenge in our understanding of how microscopic contact mechanics translate into macroscopic flow resistance. Here, we introduce a constraint-counting model that incorporates friction and dimensionality naturally without additional assumptions and allows for collapsing of rheological data onto a universal master curve. In this model, we borrow ideas from dry granular jamming physics and classify contacts as either locked or non-locked to define a single state variable, the constraint ratio, which measures the average strength of mechanical constraint per particle. By identifying the constraint ratio as the key control parameter, our framework provides a unifying route toward predictive modeling and rational design of shear-thickening materials.

Figures (9)

• Figure 1: Contact classification and rheological flow curves. ( A) Three types of interparticle contacts include (i)frictionless interactions mediated by lubrication forces, (ii) frictional contacts where the Coulomb threshold has been exceeded and sliding occurs, and (iii) frictional contacts below the Coulomb threshold. These types are grouped into two sets: locked and non-locked. Snapshots of the contact networks obtained from simulations: ( B) 2$D$, ( C) 3$D$. Rheological flow curves for different particle packing fractions $\phi$: ( D) 2$D$, $\mu=0.1$, ( E) 2$D$, $\mu=1$, ( F) 3$D$, $\mu=0.1$, ( G) 3$D$, $\mu=1$. The packing fraction, indicated by the color of the curves, is specified by the scales in F for 2$D$ and in G for 3$D$ systems. Similar rheological behavior, including strong or even discontinuous shear thickening, is obtained if the packing fraction is adjusted appropriately.
• Figure 2: Viscosity master curves in 2$D$. ( A) Collapsing of viscosity curves for different $\phi$ and $\mu$ using best $\alpha$ and $\beta$ values. ( B) The loss function $L$ as a function of $\alpha$ and $\beta$ for different $\mu$s. Dashed white line: theoretical value. Errorbars: 1 standard error of the best fitting parameters.
• Figure 3: Viscosity master curves in both 2$D$ and 3$D$. The collapsing of viscosity curves for different $\mu$ in ( A) 2$D$ and ( C) 3$D$. Comparison of the best fit values and the theoretical ones in ( B) 2$D$ and ( D) 3$D$.
• Figure 4: Constraint ratio $\chi$ predicts viscosity for widely differing contact networks. All systems are chosen with $\chi=0.72\pm0.01 (2$D$), 0.66\pm0.01 (3$D$)$ and $\eta_r=100\pm10$. The locked contacts are shown as thick red bonds and the unlocked ones as thin blue.
• Figure 5: Connect constraint ratio $\chi$ with rigid percolation and jamming. ( A) Power-law-like relationship between $\eta_r$ and $1-\chi$, as the system approaching the jamming transition. ( B) The median size of rigid clusters as a function of $\chi$, suggesting a percolation transition happening at $\chi_p$. Inset: log-log plot of $S_{max}/N$ with $\chi_p-\chi$.