# Characterizing nonaffinity upon decompression of soft-sphere packings

## TL;DR

This study investigates how nonaffine deformation contributions to the bulk modulus vary near the unjamming transition in soft sphere packings, revealing model-dependent behaviors and providing a theoretical scaling framework.

## Contribution

It demonstrates that the relative nonaffine contribution to the bulk modulus can differ between models, and offers a scaling analysis to predict this behavior based on interaction potentials.

## Key findings

- In canonical models, nonaffine bulk modulus ratio approaches a constant upon decompression.
- In some models, the nonaffine contribution vanishes as unjamming is approached.
- A theoretical scaling analysis explains the observed behaviors and predicts scaling laws.

## Abstract

Athermal elastic moduli of soft sphere packings are known to exhibit universal scaling properties near the unjamming point, most notably the vanishing of the shear-to-bulk moduli ratio $G/B$ upon decompression. Interestingly, the smallness of $G/B$ stems from the large nonaffinity of deformation-induced displacements under shear strains, compared to insignificant nonaffinity of displacements under compressive strains. In this work we show using numerical simulations that the relative weights of the affine and nonaffine contributions to the bulk modulus, and their dependence on the proximity to the unjamming point, can qualitatively differ between different models that feature the same generic unjamming phenomenology. In canonical models of unjamming we observe that the ratio of the nonaffine to total bulk moduli $B_{na}/B$ approaches a constant upon decompression, while in other, less well-studied models, it vanishes. We show that the vanishing of $B_{na}/B$ in non-canonical models stems from the emergence of an invariance of net (zero) forces on the constituent particles to compressive strains at the onset of unjamming. We provide a theoretical scaling analysis that fully explains our numerical observations, and allows to predict the scaling behavior of $B_{na}/B$ upon unjamming, given the functional form of the pairwise interaction potential.