Unified quantifier of mechanical disorder in solids

Mechanical disorder in solids, which is generated by a broad range of physical processes and controls various material properties, appears in a wide variety of forms. Defining unified and measurable dimensionless quantifiers, allowing quantitative comparison of mechanical disorder across widely different physical systems, is therefore an important goal. Two such coarse-grained dimensionless quantifiers (among others) appear in the literature: one is related to the spectral broadening of discrete phononic bands in finite-size systems (accessible through computer simulations) and the other is related to the spatial fluctuations of the shear modulus in macroscopically large systems. The latter has been recently shown to determine the amplitude of wave attenuation rates in the low-frequency limit (accessible through laboratory experiments). Here, using two alternative and complementary theoretical approaches linked to the vibrational spectra of solids, we derive a basic scaling relation between the two dimensionless quantifiers. This scaling relation, which is supported by simulational data, shows that the two apparently distinct quantifiers are in fact intrinsically related, giving rise to a unified quantifier of mechanical disorder in solids. We further discuss the obtained results in the context of the unjamming transition taking place in soft sphere packings at low confining pressures, in addition to their implications for our understanding of the low-frequency vibrational spectra of disordered solids in general, and in particular those of glassy systems.

Figure 1

(a)–(c) Low-frequency density of states $\mathcal{D}\left(\omega \right)$ averaged over 100 computer glasses of $N=\text{64}\text{000}$ particles in three dimensions, plotted against the rescaled frequency $\omega /{\omega }_0$, where ${\omega }_0\equiv c_s/a_0$ with $c_s$ the (ensemble averaged, $T_{\mathrm{p}}$-dependent) speed of shear waves, and $a_0\equiv {(V/N)}^{1/3}$ is an interparticle distance in a system of volume $V$. (a) Data for glasses of parent temperature $T_{\mathrm{p}}=0.4$ (in simulational units); the disorder-dependent frequency scale ${\omega }_{†}\left(L\right)$ indicated by the dashed vertical line separates the frequency axis into the range in which discrete phonon bands are distinguishable, and the range in which their width surpasses the gaps between them, rendering them indistinguishable (phonon sea). The grey rectangle indicates the zoomed-in window presented in panel (b), where the universal $\sim {\omega }^4$ VDoS of nonphononic modes is observed between and below the discrete phonon bands. (c) Same as (a) but for $T_{\mathrm{p}}=1.3$ glasses of larger mechanical disorder. The dimensionless prefactor $A_{\mathrm{g}}{\omega }_0^5$ is roughly 35 times larger compared to the $T_{\mathrm{p}}=0.4$ glasses. (d) Sample-to-sample probability distribution functions (PDFs) of the shear modulus $\mu$, measured for $T_{\mathrm{p}}=1.3$ and $T_{\mathrm{p}}=0.4$ glasses of $N=$ 16 000 particles.

Figure 2

(a) Phonon-band widths $\mathrm{\Delta }\omega$ plotted against phonon bands frequencies $\omega$ (expressed in simulation units, see Ref. [45]). Data were measured for glasses of various sizes $N$ and parent temperatures $T_{\mathrm{p}}$ as indicated by the legend. (b) Same phonon-band widths $\mathrm{\Delta }\omega$ as reported in (a), this time plotted against the rescaled frequency $\omega \sqrt{n_z}/\sqrt{N}$. The continuous lines are fits to Eq. (3) from which we estimate $\chi \left(T_{\mathrm{p}}\right)$.

Figure 3

Comparison between the direct calculation of $\gamma$ (via sample-to-sample statistics), and the two estimation methods as described in the text for computer glasses of $N=\text{16}\text{000}$ particles. The vertical dashed line marks the crossover temperature $T_{\mathsf{X}}$ [46], which coincides with the onset of the high-$T_{\mathrm{p}}$ plateau of $\gamma$. Also plotted are direct measurements and the outlier exclusion estimations of $\gamma$ for systems of $N=\text{2000}$ particles (using the same ensemble sizes as the $N=\text{16}\text{000}$ systems), which are much noisier [15]. We find that, for $N=\text{16}\text{000}$, the outlier exclusion method estimation follows closely the direct measurement at low $T_{\mathrm{p}}$, and is roughly 55% higher than the median-based estimation throughout the studied $T_{\mathrm{p}}$ range. We therefore adopt the outlier-exclusion estimate of $\gamma$ in what follows.

Figure 4

Parametric plot of $\gamma \left(T_{\mathrm{p}}\right)$ vs ${\chi }^2\left(T_{\mathrm{p}}\right)$ obtained as explained in the text. The main result of this paper is that $\gamma \sim {\chi }^2$, as predicted by our scaling theory in Eq. (10).

Figure 5

Phonon-band widths $\mathrm{\Delta }\omega$ measured in disordered spring networks in two dimensions, see text and Appendix for details. $\mathrm{\Delta }\omega$ is plotted against the frequency $\omega$ in panel (a), and against the rescaled frequency $\omega \sqrt{n_z}/\sqrt{\delta ZN}\sim \chi \omega \sqrt{n_z}/\sqrt{N}$ in panel (b). The data collapse implies that $\chi \sim 1/\sqrt{\delta Z}\sim \sqrt{\gamma }$, strengthening the generality of the scaling relation between the mechanical disorder quantifiers $\gamma$ and $\chi$.

Figure 6

(a) VDoS of our computer glasses of various $T_{\mathrm{p}}$ as indicated by the legend. The dimensionless prefactors $A_{\mathrm{g}}{\omega }_0^5$ are estimated by fitting the low frequency tails to the universal quartic law $\propto {\omega }^4$ of the nonphononic VDoS. The extracted dimensionless prefactors $A_{\mathrm{g}}{\omega }_0^5$ are then plotted in panel (b) vs. the disorder parameter $\gamma$ obtained from the sample-to-sample fluctuations of the shear modulus as explained in the text. In Ref. [13] it is argued that $A_{\mathrm{g}}{\omega }_0^5\sim {\gamma }^{5/3}$, as represented by the continuous line.

Figure 7

(a) Lowest-frequency phonon bands of three realizations of spring networks of $N=\text{1}\text{638}\text{400}$ nodes with coordination $Z=4.1$ generated using the algorithm described in the text. Sound waves are marked by arrows. (b) An example of a spring network of $N=400$ nodes and coordination $Z=4.1$ generated with the algorithm described here.

Figure 8

(a) When an edge $\alpha$ is removed, two remaining angles ${\theta }_i^{\left(\alpha \right)}$ and ${\theta }_j^{\left(\alpha \right)}$ are formed at nodes $i$ and $j$. In each iteration of the algorithm, we aim to remove the edge whose removal forms the smallest remaining angles. (b) Illustration of the key data structure of the algorithm; upon initialization, all edges of the initial network are stored in $b$—an array of linked lists—so $b_i$ contains a linked list of edges with $\frac{2\pi i}{M}<max({\theta }_i^{\left(\alpha \right)},{\theta }_j^{\left(\alpha \right)})<\frac{2\pi (i+1)}{M}$. The edge targeted for removal is shown in pink. (c) An element of a linked list contains an edge $\alpha$ specified by the two nodes $i$ and $j$, the largest bond angle that would open up in case of its removal, and a pointer $p$ to the next element of the linked list.