Unjamming in models with analytic pairwise potentials

Canonical models for studying the unjamming scenario in systems of soft repulsive particles assume pairwise potentials with a sharp cutoff in the interaction range. The sharp cutoff renders the potential nonanalytic but makes it possible to describe many properties of the solid in terms of the coordination number $z$, which has an unambiguous definition in these cases. Pairwise potentials without a sharp cutoff in the interaction range have not been studied in this context, but should in fact be considered to understand the relevance of the unjamming phenomenology in systems where such a cutoff is not present. In this work we explore two systems with such interactions: an inverse power law and an exponentially decaying pairwise potential, with the control parameters being the exponent (of the inverse power law) for the former and the number density for the latter. Both systems are shown to exhibit the characteristic features of the unjamming transition, among which are the vanishing of the shear–to–bulk modulus ratio and the emergence of an excess of low-frequency vibrational modes. We establish a relation between the pressure–to–bulk modulus ratio and the distance to unjamming in each of our model systems. This allows us to predict the dependence of other key observables on the distance to unjamming. Our results provide the means for a quantitative estimation of the proximity of generic glass-forming models to the unjamming transition in the absence of a clear-cut definition of the coordination number and highlight the general irrelevance of nonaffine contributions to the bulk modulus.

Figure 1

Solid realizations of the EXP model. Obtaining a robust disordered solid depends on delicately tuning the model parameters. We show that changing the small-large interaction strength ${\varepsilon }_{ij}$ by a bit more than a percentage, from 1.7 (a) to 1.72 (b), destabilizes the glass and leads to phase separation of the small and large particles, see also Ref. [31].

Figure 2

(a) Shear–to–bulk moduli ratio, measured as a function of density in the EXP system. In addition to the bare data represented by the orange diamonds, we also plot in gray stars the ratio calculated while omitting the terms that contain the interparticle forces, see text for further discussion. (b) We find that the moduli data are consistent with our scaling argument (see Sec. 4) which predicts $\mu /B\sim {\rho }^{1/4}$ (horizontal dash-dotted lines); while this scaling does not yet hold in the density regime accessible by our simulations, it is apparent that the curves are slowly converging to the predicted scaling at lower densities. The inset shows the bare moduli as a function of density.

Figure 3

(a) Shear modulus $\mu$ and bulk modulus $B$ plotted as a function of the exponent $\beta$, measured for the IPL samples, see text for further discussion. (b) Shear–to–bulk moduli ratio for the stressed and unstressed IPL samples. The continuous lines represent the scaling $\mu /B\sim {\beta }^{-1/2}$, derived in Sec. 4.

Figure 4

(a) Density of states of the EXP system, at densities as indicated by the legend. (b) Density of states of the IPL system, calculated in states with exponents $\beta$ as indicated by the legend. We left out of this plot $D\left(\omega \right)$ obtained for the exponents $\beta =8$ and $\beta =512$ for visual clarity; we find that the same trend persists.

Figure 5

Characteristic “unjamming” frequency scale ${\omega }_{*}$ expressed in terms of $\sqrt{B}$ for the EXP system (a) and the IPL system (b). Each data point represents the median calculated over 1600 responses. The insets display the bare medians of ${\omega }_{*}$ vs. the control parameter. The continuous lines correspond to ${\omega }_{*}/\sqrt{B}\sim {\rho }^{0.19}$ and ${\omega }_{*}/\sqrt{B}\sim {\beta }^{-0.40}$ for the EXP and IPL systems, respectively.

Figure 6

Pressure–to–bulk modulus ratio for (a) the EXP and (b) the IPL systems, as a function of the relevant control parameter. The continuous line corresponds to the theoretical predictions of Eqs. (17) and (19) for the EXP and IPL systems, respectively. The prefactor of the scaling $p/B\sim \sqrt{\rho }$ for the EXP system is found to be about 0.58 instead of the predicted 1/2.

Figure 7

Ansatz functions $g\left(\rho \right)$ (a) and $f\left(\beta \right)$ (b), calculated as explained in the text using both the pressure and bulk modulus data. The continuous lines represent Eqs. (20) and (21).