Jamming transition and normal modes of polydispersed soft particle packing

The jamming transition of soft particles characterized by narrow size distributions has been well studied by physicists. However, polydispersed systems are more relevant to engineering, and the influence of polydispersity on jamming phenomena is still unexplored. Here, we numerically investigate jamming transitions of polydispersed soft particles in two dimensions. We find that polydispersity strongly influences contact forces, local coordination, and the jamming transition density. In contrast, the critical scaling of pressure and elastic moduli is not affected by the particle size distribution. Consistent with this observation, we find that the vibrational density of states is also insensitive to the polydispersity. Our results suggest that, regardless of particle size distributions, both mechanical and vibrational properties of soft particle packings near jamming are governed by the distance to jamming.

💡 Research Summary

In this work the authors address a gap in the jamming literature by systematically studying how a broad particle‑size distribution influences the jamming transition and vibrational properties of soft, repulsive particles in two dimensions. Using molecular‑dynamics simulations they generate packings of N = 2048 disks whose radii are drawn from a power‑law distribution P(R) ∝ R⁻³ (ν = 3), a form that mimics the grain‑size statistics observed in seismic faults and Apollonian packings. The degree of polydispersity is quantified by the size ratio λ = R_max/R_min, which is varied from 2 up to 20. Particles interact via a linear spring force f_ij = k δ_ij n_ij, and the total elastic energy is minimized with the FIRE algorithm to obtain mechanically stable, athermal packings at prescribed packing fractions φ.

The first set of results concerns microscopic structural statistics. As λ increases, the distribution of contact forces P(f) broadens dramatically: for λ ≈ 2 the forces are Gaussian about the mean h_fi, whereas for λ ≥ 10 the tail becomes exponential, P(f) ∼ exp(−f/h_fi). The coordination‑number distribution P(z) also widens, developing a power‑law tail P(z) ∝ z⁻⁴·² for the most polydisperse systems, with a cutoff that scales as z* ∝ λ⁰·⁷⁴. These findings indicate that large particles carry many contacts while small particles become under‑coordinated, a direct consequence of the ability of small disks to fill interstices.

A second major observation is that the critical packing fraction φ_c at which the system first develops a finite pressure shifts upward with polydispersity. By fitting the pressure p and excess coordination Δz ≡ ⟨z⟩ − z_c (z_c = 2d − 2d/N) as functions of φ, the authors extract φ_c for each λ. The dependence follows a power law φ_c − φ*_c ∝ (λ − 1)^0·32, where φ*_c ≈ 0.81 corresponds to the monodisperse limit. This monotonic increase reflects the enhanced space‑filling efficiency of broad size distributions.

Despite these pronounced changes in microscopic statistics, the macroscopic scaling relations that define the jamming universality class remain unchanged. The pressure obeys p/k ∝ (φ − φ_c) ∝ Δz², and the excess coordination follows Δz ∝ (φ − φ_c)¹ᐟ², independent of λ. Elastic moduli extracted from the dynamical matrix also collapse onto the same curves: the shear modulus scales as G/k ∝ Δz, the bulk modulus approaches a constant B/k ≈ const as Δz → 0, and the ratio G/B ∝ Δz. Thus, the mean coordination number alone controls linear elasticity, even though the full distribution P(z) is λ‑dependent.

The vibrational analysis further confirms the insensitivity of collective dynamics to polydispersity. Diagonalizing the Hessian yields the vibrational density of states D(ω). For all λ, D(ω) displays the characteristic plateau above a crossover frequency ω* that scales linearly with excess coordination, ω* ∝ Δz, reproducing the well‑known “boson‑peak‑like” plateau observed in monodisperse and bidisperse jammed packings. The shape of D(ω) is essentially identical for λ = 2, 10, 20; only a modest reduction in the participation ratio of intermediate‑frequency modes is observed at the highest λ. Consequently, the vibrational spectrum is governed solely by the distance to the jamming point, not by the underlying size distribution.

In summary, the paper demonstrates that while polydispersity dramatically reshapes local force networks, coordination statistics, and the jamming density, it does not alter the critical scaling of pressure, elastic moduli, or the vibrational density of states. The distance to jamming, measured by the excess coordination Δz, remains the sole control parameter for both mechanical and vibrational properties. This insight is highly relevant for engineering and geophysical applications where particle size distributions are broad: predictions of bulk stiffness, sound propagation, or failure thresholds can safely ignore the details of the size distribution and focus on how close the system is to the jamming transition.