Rigidity transition in polydisperse shear-thickening suspensions

TL;DR

This work investigates how polydispersity influences the rigidity transition and shear jamming in dense, non-Brownian suspensions. Using 2D LF-DEM simulations of polydisperse size distributions, it analyzes rigidity development via the pebble game and compares with statistically equivalent bidisperse systems to validate microstructural mappings. The study finds a rigidity transition at $φ_c$ below the shear-jamming packing fraction $φ_J^{μ}$, with Ising-like exponents $β=1/8$ and $γ=7/4$ and a finite-size scaling exponent $ν=1.64$ consistent with 2D continuum percolation; these results underscore a percolation/isings universality in shear jamming. Furthermore, both $φ_c$ and $φ_J^{μ}$ depend non-monotonically on the polydispersity index $α$ and on particle stiffness $k_n$, with softer contacts pushing the thresholds higher. Overall, the bulk rheology and microstructure of polydisperse suspensions can be effectively predicted from their statistically equivalent bidisperse counterparts, linking shear jamming to classical critical phenomena.

Abstract

Shear thickening suspensions of non-Brownian polydisperse particles are simulated in 2D using a discrete element method based algorithm (LF-DEM) at high packing fractions ($φ$) and large non-dimensional stresses ($σ$). Rigidity analysis of the stress induced particle clusters is carried out using \textit{pebble game} algorithm for polydisperse suspensions and compared with the statistically equivalent bidisperse systems. A critical value of the packing fraction, $φ_c$, close to the shear jamming transition, $φ_J^μ$, ($φ_c<φ_J^μ$) is obtained where rigid particle clusters begin to grow sharply. The growth is found to be characterized by a critical transition of an order parameter ($f_\text{rig}$), defined by the fraction of rigid particles which scales as, $f_\text{rig}\sim (φ-φ_c)^β$ for $φ>φ_c$, and by the susceptibility scaling, $χ_\text{rig}\sim|φ-φ_c |^{-γ}$, with exponents $β=1/8$ and $γ=7/4$, which are consistent with the critical exponents in 2D Ising model. The variations of $f_\text{rig}$ and $χ_\text{rig}$ in polydisperse suspensions are found to be identical to that of the statistically equivalent bidisperse suspensions. Finite size scaling analysis shows a divergence of correlation length near $φ_c$ following critical exponent $ν=1.64$, consistent with the 2D continuum percolation transition. Furthermore, $φ_c$ and $φ_J^μ$ are found to vary non-monotonically with polydispersity index and depend on the particle stiffness.

Figures (9)

• Figure 1: Histograms of polydisperse suspensions, following (a) normal distribution and (b) log-normal distribution, compared to the statistically equivalent bidisperse systems.
• Figure 2: Divergence of relative viscosity $\eta_r$ as a function of packing fraction $\phi$ for (a) normal and its statistically equivalent bidisperse distributions with polydispersity index $\alpha=0.133$, at stress $\sigma=25$ and for (b) log-normal and its statistically equivalent bidisperse distributions, with $\alpha=0.374$, at $\sigma=100$. The lines represent M-P fit: $\eta_r \sim \left(1-\frac{\phi}{\phi_J(\sigma)}\right)^{-2}$, where $\phi_J(\sigma)$ is the packing fraction in the shear jamming limit.
• Figure 3: Variation of the order-parameter $f_\text{rig}$ with solid packing fraction $\phi$ for normal and its statistically equivalent bidisperse distributions with $\alpha=0.133$, at stress (a) $\sigma=25$ and (b) $\sigma=100$. (c) Variation of $f_\text{rig}$ with $\phi$ for log-normal and its statistically equivalent bidisperse distributions with $\alpha=0.374$, at $\sigma=100$. $\phi_c$ denotes the corresponding critical packing fraction. The solid lines indicate the fit analogous to the 2D-Ising model with scaling exponent $\beta=1/8$.
• Figure 4: Variation of the susceptibility $\chi_\text{rig}$ with solid packing fraction $\phi$ for normal and its statistically equivalent bidisperse distributions with $\alpha=0.133$, at stress (a) $\sigma=25$ and (b) $\sigma=100$. (c) Variation of $\chi_\text{rig}$ with $\phi$ for log-normal and its statistically equivalent bidisperse distributions with $\alpha=0.374$, at $\sigma=100$. The vertical line at $\phi=\phi_c$ indicates the critical point at which $\chi_\text{rig}$ diverges.
• Figure 5: Variation of (a) the order-parameter $f_\text{rig}$ and (b) susceptibility $\chi_\text{rig}$ as a function of the relative distance from the critical point ($\phi-\phi_c$) for different conditions of polydispersity at stress values of 100 and 25.