Rigid clusters in shear-thickening suspensions: a nonequilibrium critical transition

TL;DR

This study shows that dense 2D shear-thickening suspensions develop system-spanning rigid clusters above a stress-dependent critical packing $φ_c(σ,μ)$, with a clear critical scaling characterized by $β=1/8$ and $γ=7/4$, consistent with 2D Ising universality for the order parameter and susceptibility. Using the 3,3 pebble-game, the authors define a rigidity order parameter $f_{\rm rig}$ and its fluctuations $χ_{\rm rig}$, observing data collapse when incorporating an effective field $h(μ)$ tied to friction. The cluster-size distribution follows a power-law with exponent $τ\approx1.8$, and a scaling collapse reveals a line of critical points $φ_c(σ,μ)$ that shifts as stress decreases toward the jamming fraction $φ_J^{μ}$. These findings link rigidity percolation to jamming in high-stress suspensions and suggest Ising-like criticality governs the onset of large-scale rigidification. The work provides a framework to understand how microstructural rigidity emerges under shear and its relation to flow transitions in DST.

Abstract

The onset and growth of rigid clusters in a two-dimensional (2D) suspension in shear flow are studied by numerical simulations. The suspension exhibits the lubricated-to-frictional rheology transition, but the key results here are for stresses above the levels that cause extreme shear-thickening. At large solid fraction, $φ$, but below the stress-dependent jamming fraction, we find a critical $φ_{c}(σ,μ)$ where $σ$ is a dimensionless shear stress and $μ$ is the interparticle friction coefficient. For $φ>φ_c$, the proportion of particles in rigid clusters grows sharply, as $f_{\rm rig} \sim |φ-φ_{c}|^β$ with $β=1/8$. The fluctuations in the fraction of particles in rigid clusters yield a susceptibility measure $χ_{\rm rig} \sim |φ-φ_{c}|^{-γ}$ with $γ= 7/4$. The system is thus found to exhibit criticality. The results are shown to depend on an effective field $h(μ)$, which provides data collapse near $φ_c$ for both $f_{\rm rig}$ and $χ_{\rm rig}$. This behavior occurs over a range of stresses, with $φ_c(σ,μ)$ increasing as the stress decreases.

Figures (6)

• Figure 1: (a) Flow curves for 2D suspensions with particle size ratio $a_{\ell}/a_s=1.4$, with 50% of the area occupied by each size, and friction coefficient $\mu=100$, at packing fraction $\phi$ ranging from 0.74 to 0.78. (b) Flow-state diagram with critical transition concentration $\phi_{c}(\mu,\sigma)$. The solid black line passing through the $\phi_{c}(\mu,\sigma)$ data points represents the critical transition. The solid red line indicates the CST-DST transition and the solid blue line represents the shear jamming fraction, $\phi_{\rm J}^\mu(\sigma)$, with $\phi_{\rm J}^\mu (\infty) \doteq 0.784$. Regions shaded in gray indicate the conditions studied in the present work.
• Figure 2: Particles in rigid clusters contributing to $f_{\rm rig}$ (solid red shading), surface particles of rigid clusters (line shaded gray), and non-rigid (unfilled), for $\phi = 0.756$ and $N=2000$ particles; periodic in both horizontal and vertical directions.
• Figure 3: Order parameter $f_\text{rig}$ as a function of distance from the critical packing fraction $\phi_c$ at $N=2000$ for different stresses $\sigma$ and friction coefficients $\mu$. The solid black line represents $1.53\,[\phi-\phi_{c}(\sigma,\mu)]^\beta$ with $\beta = 1/8$, as in the Ising 2D model. The inset presents the variation of $\phi_c(\sigma,\mu)$ with $\mu$. For $\sigma=1$, $f_{\rm rig}\approx 0$ and we have taken $\phi_c=\phi_\text{max}\doteq 0.83$, which is seen to approach $\phi_{\rm J}^0$.
• Figure 4: Susceptibility $\chi_{\rm rig}$ as a function of distance from the critical packing fraction $\phi_c$ at $N=2000$ for different stresses $\sigma$ and friction coefficients $\mu$. For $\sigma=1$, see comment in caption of Fig. 3.
• Figure 5: Scaling collapse of rigid cluster size distribution for a range of packing fraction at stresses $\sigma =$ 10, 20 and 100. The data collapse is shown here for $N=2000$ and friction coefficient $\mu=100$. The inset shows the average cluster size $\langle n\rangle$ as a function of $\phi-\phi_c$ for different stress values.