Characterizing nonaffinity upon decompression of soft-sphere packings

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 differ qualitatively 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 noncanonical 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 us to predict the scaling behavior of $B_{na}/B$ upon unjamming, given the functional form of the pairwise interaction potential.

Figure 1

Plotted are the harmonic, cubic, and bump pairwise potentials as a function of the overlap $(R_i+R_j)-r_{ij}$ of particles $i$ and $j$. Potentials have been factored for visualization purposes.

Figure 2

(a) The shear-to-bulk modulus ratio, $G/B$, vs $p/B$ for all investigated models of soft repulsive spheres. The scaling $G/B\sim \sqrt{p/B}$ appears to universally hold across all studied models. (b) The relative nonaffine contribution to the bulk modulus, $B_{na}/B$, plotted against the key dimensionless control parameter $p/B$. The harmonic and cubic systems show a saturation as $p/B\to 0$, while in stark contrast with these canonical models, the EXP and bump models feature a vanishing $B_{na}/B$ as $p/B\to 0$, with different scaling laws.

Figure 3

$B_{na}/B$ vs ${\mathrm{\Xi }}^2/\left(V{B}^2\right)$ for all investigated models. The dashed line represents the fit $B_{na}/B=5{\mathrm{\Xi }}^2/\left(2V{B}^2\right)$, validating Eq. (19).

Figure 4

Probability distribution functions of the proportionality coefficients $c\equiv {\phi }^{\prime }/{\phi }^{\prime \prime }r$ for the EXP [panel (a)] and cubic [panel (b)] models, plotted against the rescaled variables $c/(p/B)$. We see a clear narrowing of the distributions in the EXP model, whereas they approach a limit form in the case of the cubic model. We note that for the EXP model, only interactions within the first coordination shell were considered in this analysis.

Figure 5

Plots of $B_{na}/B$ vs $p/B$ (closed symbols) and ${\mathrm{\Xi }}^2/\left(V{B}^2\right)$ vs $p/B$ (open symbols), plus the squared full width at half maximum (${\mathrm{\Lambda }}^2$) of the distributions of $c/(p/B)$ (stars); see Fig. 4 for examples of these distributions. Panels (a) and (b) show the harmonic and cubic interactions, with a putative line of constant $B_{na}/B$. Panels (c) and (d) show the EXP and bump models, respectively. The data make clear that all three quantities—$B_{na}/B, {\mathrm{\Xi }}^2/\left(V{B}^2\right)$, and ${\mathrm{\Lambda }}^2$—all share the same scaling (per model) with respect to $p/B$.

Figure 6

Rescaled force distributions for the different potentials. Clearly the distributions obtain a constant shape as the unjamming transition is approached. This can be used to determine the scaling of $\mathrm{\Delta }r$ as a function of the control parameter.

Figure 7

$B_{na}/B$ at low pressures for the canonical models with different exponents $n$. The continuous line represents a fit to the form $1/{(n-1)}^2$ as predicted by Eq. (28).