Critical fluctuations of elastic moduli in jammed solids

TL;DR

This work demonstrates that near the jamming transition, sample-to-sample fluctuations of the shear modulus diverge with a critical exponent that is independent of the interparticle potential, a result confirmed in both 3D and 2D packings. The authors quantify fluctuations using χ_G ~ δz^{-1/2} for the shear modulus and χ_{δz} ~ δz^{-2/5} for the excess contact number, showing potential-independent behavior for G fluctuations across harmonic and Hertzian interactions. They dissect stressed and unstressed moduli, relate the findings to heterogeneous-elasticity theory (HET) and effective-medium theory (EMT), and discuss implications for Rayleigh-type acoustic attenuation in amorphous solids. The results suggest a common elastic-length scale l_c governing fluctuations and vibrational properties, while indicating that the physical interpretation of δz fluctuations remains dimension-dependent and warrants further investigation with larger systems and lower pressures.

Abstract

We investigate sample-to-sample fluctuations of the shear modulus in ensembles of particle packings near the jamming transition. Unlike the average modulus, which exhibits distinct scaling behaviours depending on the interparticle potential, the fluctuations obey a critical exponent that is independent of the potential. Furthermore, this scaling behaviour has been confirmed in two-dimensional packings, indicating that it holds regardless of spatial dimension. Using this scaling law, we discuss the relationship predicted by heterogeneous-elasticity theory between elastic-modulus fluctuations and the Rayleigh scattering of sound waves across different pressures. Our numerical results provide a useful foundation for developing a unified theoretical description of the jamming critical phenomenon.

Figures (5)

• Figure 1: Critical scaling of the average shear moduli $G_1$ and $G_0/2$ for harmonic and Hertzian packings with $N = 32000.0$. The solid and dashed lines represent $G \sim \delta z$ and $G \sim \delta z^2$, respectively.
• Figure 2: Probability distributions of (a) the excess contact number $\delta z$ and (b) the shear modulus $G$ for the ensemble of Hertzian packings ($N = 32000.0$) at various pressures. To enable comparison between different pressures, each quantity $X$ is rescaled as $\hat{X} = X/\Braket{X} - 1$. In panel (b), the distributions of the stressed modulus $G_1$ are shown as solid curves, whereas those of the unstressed modulus $G_0$ appear as dashed curves.
• Figure 3: Critical fluctuations of the excess contact number $\delta z$. The dashed line represents the scaling $\chi_{\delta z} \sim \delta z^{-2/5}$. The shaded region marks the glassy regime, $\delta z \gtrsim 7.0e-1$, where this scaling behaviour is no longer expected to hold Giannini2024.
• Figure 4: Critical fluctuations of the shear moduli. The unstressed modulus $G_0$ and stressed modulus $G_1$ are shown in panels (a) and (b), respectively. The dashed lines indicate the scaling $\chi_G \sim \delta z^{-1/2}$. The legend is identical to that in Fig. \ref{['fig:chi_dz']}. The shaded region denotes the glassy regime Giannini2024.
• Figure 5: Critical fluctuations in two-dimensional harmonic disk packings. Panels (a), (b), and (c) display data of the excess contact number, the unstressed modulus, and the stressed modulus, respectively. In panel (a), the dashed line in black and the solid line in grey represent $\chi_{\delta z} \sim \delta z^{-0.55}$ and $\chi_{\delta z} \sim \delta z^{-0.6}$, respectively. Dashed lines in panels (b) and (c) represent $\chi_G \sim \delta z^{-1/2}$. The pressure range spans to $p = e-7$ for $N = 4000.0$ and $p = e-6$ for $N = 32000.0$.