Universal stress correlations in crystalline and amorphous packings

We present a universal characterization of stress correlations in athermal systems, across crystalline to amorphous packings. Via numerical analysis of static configurations of particles interacting through harmonic as well as Lennard-Jones potentials, for a variety of preparation protocols and ranges of microscopic disorder, we show that the properties of the stress correlations at large lengthscales are surprisingly universal across all situations, independent of structural correlations, or the correlations in orientational order. In the near-crystalline limit, we present exact results for the stress correlations for both models, which work surprisingly well at large lengthscales, even in the amorphous phase. Finally, we study the differences in stress fluctuations across the amorphization transition, where stress correlations reveal the loss of periodicity in the structure at short lengthscales with increasing disorder.

Figure 1

(a) Spatial distribution of the orientational order parameter (${\psi }_6$) in the system of harmonic disks for different polydispersities and initial conditions. (b) Spatial distribution of pressure for different initial conditions, with $\eta =0.5$; see text for the definition of labels. (c) $\left(a1\right)–\left(f1\right)$ Exact predictions for stress correlations in Fourier space using Eq. (9). $\left(a2\right)$–$\left(f2\right)$ Radially integrated stress correlations in Fourier space $C_{ijkl}\left(\theta \right)/\langle \delta \tilde{P}\delta \tilde{P}\rangle ={\int }_{k_{\text{min}}}^{k_{\text{max}}}dk\langle \delta {\tilde{\sigma }}_{ij}\left(\vec{k}\right)\delta {\tilde{\sigma }}_{kl}(-\vec{k})\rangle /{\int }_{k_{\text{min}}}^{k_{\text{max}}}dk\langle \delta \tilde{P}\left(\vec{k}\right)\delta \tilde{P}(-\vec{k})\rangle$, with $k_{\text{min}}=0.5$ and $k_{\text{max}}=1.5$; comparison between theory and numerical data. The system size is $N=6400$, and the data has been averaged over 400 configurations in order to obtain the ensemble averaged stress correlations.

Figure 2

Bond orientational order parameter (${\psi }_6$) and its fluctuation (${\chi }_6$) with variation in the disorder parameter $\eta$ for (a) short ranged Harmonic model and (b) long-ranged LJ model. For each value of $\eta , {\psi }_6,{\chi }_6$ are obtained by averaging over $50000$ disordered configurations of system size $N=256$.

Figure 3

Stress correlations $[C_{xxxx}\left(\vec{k}\right)=\langle \delta {\tilde{\sigma }}_{xx}\left(\vec{k}\right)\delta {\tilde{\sigma }}_{xx}(-\vec{k})\rangle ]$ in the harmonic model. (a) Theoretical prediction in the $\eta \to 0$ limit, (b)–(d) numerical results with polydispersities varying across the amorphization transition, at fixed initial packing fraction $\varphi =0.92$. These stress correlations display the same large lengthscale behavior, with observable changes at larger values of $|\vec{k}|$, near the edges of the Brillouin zone.

Figure 4

Stress correlations $[C_{xxxx}\left(\vec{k}\right)=\langle \delta {\tilde{\sigma }}_{xx}\left(\vec{k}\right)\delta {\tilde{\sigma }}_{xx}(-\vec{k})\rangle ]$ in the harmonic model plotted for $|\vec{k}|\le 11.5$. (a)–(e) Numerical results with polydispersities varying across the amorphization transition, at fixed initial packing fraction $\varphi =0.92$, (f) stress correlations for jammed packings quenched from random initialization with $\eta =0.5$. These stress correlations display the same large lengthscale behavior, with observable changes at larger values of $|\vec{k}|$, near the edges of the Brillouin zone.