Rigidity transition in polydisperse shear-thickening suspensions

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 particles in rigid clusters 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 having values consistent with the critical exponents in 2D percolation transition. 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 $ϕ_{c_\infty}$ following critical exponent $ν\approx 1.33$, in agreement with the 2D percolation theory. Furthermore, $ϕ_c$ and $ϕ_J^μ$ are found to vary non-monotonically with polydispersity index and depend on the particle stiffness.

💡 Research Summary

This paper investigates the emergence of rigidity in highly concentrated, non‑Brownian suspensions of polydisperse particles under large shear stress, using two‑dimensional Lubrication‑Flow Discrete Element Method (LF‑DEM) simulations combined with a pebble‑game algorithm for rigidity analysis. The authors first construct suspensions with a continuous size distribution (polydisperse) and, for comparison, statistically equivalent bidisperse mixtures that share the same overall volume fraction and polydispersity index. All simulations are performed in a stress‑controlled mode (σ≫σ₀), where the shear rate is obtained from a force‑balance that includes lubrication, frictional contact, and short‑range electrostatic repulsion.

The pebble‑game algorithm treats each particle as possessing two degrees of freedom (pebbles) in two dimensions; each frictional contact imposes a constraint (a “hinge”). When enough constraints accumulate to eliminate all floppy modes in a sub‑network, that sub‑network is identified as a rigid cluster. The fraction of particles belonging to rigid clusters, f_rig, serves as an order parameter, while its variance χ_rig is interpreted as a susceptibility.

The central finding is that, as the packing fraction ϕ is increased, a sharp rigidity transition occurs at a critical packing fraction ϕ_c that lies just below the shear‑jamming point ϕ_J^μ (the packing fraction at which a shear‑jammed solid forms under high stress). For ϕ>ϕ_c the order parameter follows a power law f_rig∝(ϕ−ϕ_c)^β with β≈0.14, and the susceptibility diverges as χ_rig∝|ϕ−ϕ_c|^{−γ} with γ≈2.4. These exponent values match those of the two‑dimensional percolation universality class (β=5/36≈0.139, γ=43/18≈2.39). Finite‑size scaling performed by varying the simulation box size L reveals a correlation length ξ that diverges as ξ∝|ϕ−ϕ_c|^{−ν} with ν≈1.33, again consistent with 2D percolation (ν=4/3).

Remarkably, the rigidity transition curves for the fully polydisperse suspensions are virtually indistinguishable from those of the statistically equivalent bidisperse systems. This demonstrates that the detailed shape of the size distribution does not affect the topology of the stress‑induced frictional network; only the overall volume fraction, the polydispersity index, and the mean particle size matter. Consequently, the macroscopic rheology (viscosity, normal stresses) and the microstructural evolution (growth of rigid clusters) are the same for both classes of systems, confirming earlier experimental observations of rheological equivalence between polydisperse and appropriately matched bidisperse suspensions.

The authors also explore the influence of particle stiffness (spring constant k_n) and the polydispersity index on the critical packing fractions. Both ϕ_c and the shear‑jamming point ϕ_J^μ vary non‑monotonically with the polydispersity index and shift to lower values for stiffer particles, indicating that particle deformability facilitates the formation of frictional contacts and thus promotes rigidity at lower concentrations.

In summary, the study establishes that shear‑induced rigidity in dense suspensions is a non‑equilibrium critical phenomenon belonging to the 2D percolation universality class. The rigidity transition is robust against the complexity of the particle size distribution, allowing a polydisperse system to be faithfully represented by a simpler bidisperse analogue. These insights provide a unified framework for interpreting continuous and discontinuous shear thickening, as well as shear jamming, in industrially relevant suspensions, and they suggest practical routes for controlling flow stability by tuning particle size distribution, stiffness, and applied stress.