Mechanical disorder of sticky-sphere glasses. I. Effect of attractive interactions

Recent literature indicates that attractive interactions between particles of a dense liquid play a secondary role in determining its bulk mechanical properties. Here we show that, in contrast with their apparent unimportance to the bulk mechanics of dense liquids, attractive interactions can have a major effect on macro- and microscopic elastic properties of glassy solids. We study several broadly applicable dimensionless measures of stability and mechanical disorder in simple computer glasses, in which the relative strength of attractive interactions—referred to as “glass stickiness”—can be readily tuned. We show that increasing glass stickiness can result in the decrease of various quantifiers of mechanical disorder, on both macro- and microscopic scales, with a pair of intriguing exceptions to this rule. Interestingly, in some cases strong attractions can lead to a reduction of the number density of soft, quasilocalized modes, by up to an order of magnitude, and to a substantial decrease in their core size, similar to the effects of thermal annealing on elasticity observed in recent works. Contrary to the behavior of canonical glass models, we provide compelling evidence indicating that the stabilization mechanism in our sticky-sphere glasses stems predominantly from the self-organized depletion of interactions featuring large, negative stiffnesses. Finally, we establish a fundamental link between macroscopic and microscopic quantifiers of mechanical disorder, which we motivate via scaling arguments. Future research directions are discussed.

Figure 1

In this work we study the effect of attractive interactions between glasses' constituent particles, on those glasses' elastic properties. (a) Two instances of pairwise potentials studied in what follows, referred to here as “glass 1” and “glass 2.” Despite having similar features, and, in particular, similar attractive parts, the Poisson's ratio $\nu$ (b) and low-frequency vibrational density of states $D\left(\omega \right)$ (c) of the respective glasses made using these two potentials differ dramatically; see text for details. For comparison, the first two columns in panel (b) represent the Poisson's ratio of computer “glass 1” and computer “glass 2,” respectively, while the rest of the columns represent various metallic glasses (data from Ref. [73]).

Figure 2

(a) The pairwise potential ${\phi }_{\text{QSS}}$ of the QSS model [see Eq. (1)], plotted for different powers $q$ as indicated by the legend. (b) The pairwise potential ${\phi }_{\mathrm{PSS}}$ of the PSS model [see Eq. (2)], for the selected dimensionless cutoffs $r_c$ indicated by the legend, expressed in terms of the dimensionless location of the minimum of the potential $x_{\min }=2^{1/6}$.

Figure 3

“Glass stickiness” $s$—defined in Eq. (3)—vs the respective control parameters of the (a) QSS, (b) DSS, and (c) CSS glass ensembles. In what follows, we maintain the same order between figure panels and different glass ensembles as presented here.

Figure 4

Panels (a)–(c) show the sample-to-sample mean athermal shear to bulk modulus ratio $G/K$, and panels (d)–(f) show the Poisson's ratio $\nu \equiv (3-2G/K)/(6+2G/K)$, plotted as a function of the key control parameter pertaining to each glass ensemble.

Figure 5

Relative sample-to-sample fluctuations of shear and bulk moduli as captured by ${\chi }_{{}_G}$ (a)–(c) and ${\chi }_{{}_K}$ (d)–(f), respectively, defined in Eq. (4); see text and Appendix pp1-s1c for more details.

Figure 6

Panels (a), (b), and (c) show the vDOS for the highest and lowest values of the respective key control parameter, in the QSS, DSS, and CSS systems. The dashed lines represent our fits of the low-frequency power-law tails, according to Eq. (5). In panels (d)–(f) we report the fitted prefactors $A_g$ for all systems vs their respective control parameter.

Figure 7

Panels (a)–(c) show the response functions $c\left(r\right)$ to local force perturbation (see precise definition in Appendix pp1-s2b), calculated in glasses of $\approx 250$ K particles. Panels (d)–(f) show the products $r^{6}c\left(r\right)$, that features a crossover to the expected continuum scaling $c\left(r\right)\sim r^{-6}$ [84] beyond a length ${\xi }_g$, defined here as the maximum of the products $r^{6}c\left(r\right)$. ${\xi }_g$ is extracted as explained in Appendix pp1-s2b, and reported in panels (g)–(i).

Figure 8

(a)–(c) The characteristic frequency ${\omega }_g$ of QLMs, obtained via Eq. (6). Panels (d)–(f) show the resulting number density of QLMs $\mathcal{N}=A_g{\omega }_g^5$, estimated for our glass ensembles as a function of their respective key control parameters.

Figure 9

Distributions of relative contributions to QLMs' energies (see text for precise definitions), measured for (a) the QSS system with $q=20$ and (b) for the DSS system with $\rho =0.55$. The vertical dashed lines represent the means of each relative contribution.

Figure 10

The pairwise potentials ${\phi }_{\text{QSS}}$ (a) and ${\phi }_{\mathrm{PSS}}$ (b) are superimposed with their resulting glasses' radial distribution functions $g\left(r\right)$, calculated for different pair types as detailed in the legends and plotted against the dimensionless distance $r_{ij}/{\lambda }_{ij}$. The key observation here is that large, negative stiffnesses in the pairwise potential lead to a depletion of the population of interactions featuring those stiffnesses, which can affect, in turn, QLMs' stiffnesses and thus glass stability. The vertical dashed line in panel (b) marks the cutoff $r_c=1.2$ used for the DSS system. Panels (c) and (d) are zoomed-out representations of panels (a) and (b), respectively.

Figure 11

Parametric plots of the radial distribution $g\left(r\right)$ vs the pairwise stiffness ${\phi }^{\prime \prime }\left(r\right)$ for (a) the QSS system with $q=20$ and (b) the DSS system with $\rho =0.55$. Here we consider only distances $r$ for which ${\phi }^{\prime \prime }\left(r\right)<0$. We clearly see that the population of interactions featuring large negative stiffnesses becomes smaller at more negative stiffnesses.

Figure 12

The bars cover the second and third quartiles of the ratio of shear to dilational strain coupling of QLMs, and the middle horizontal line represents these ratios' averages; see Appendix pp1-s2c for precise definitions.

Figure 13

Comparison between simulational and experimental [73] size of STZs, and their dependence on the Poisson's ratio. We estimate the STZ size for our computer glasses to be proportional to QLMs' core size $c_{\mathrm{STZ}}{\xi }_g^3$, and plot it against those glasses' Poisson's ratio (we used $c_{\mathrm{STZ}}=2$). STZ sizes were estimated experimentally in various laboratory metallic glasses in Pan et al. [73].

Figure 14

Dimensionless prefactors $A_g{\omega }_0^5$ of the nonphononic vDOS [see Eq. (5) and Fig. 6], plotted against the Poisson's ratio $\nu$.

Figure 15

Dimensionless prefactors $A_g{\omega }_0^5$ of the nonphononic vDOS [see Eq. (5) and Fig. 6], plotted against the dimensionless, $N$-independent quantifier ${\chi }_{{}_G}$ of sample-to-sample shear modulus fluctuations, for the sticky-sphere glass ensembles, and for five additional glass models: polydisperse power-law (POLY), Stillinger-Weber (SW), Hertzian soft spheres (HRZ), orthophenyl (OTP), and polymeric (PG). See text for details.

Figure 16

The crossover length ${\xi }_g$ (extracted as shown in Fig. 19), plotted against the sample-to-sample shear modulus fluctuations as quantified by ${\chi }_{{}_G}$. We find ${\xi }_g\sim {\chi }_{{}_G}^{2/3}$, for which scaling arguments are provided in the text.

Figure 17

(a) and (b) The dimensionless pressure $p/p_0$ for the QSS and CSS ensembles, respectively. We have tuned the density carefully in the QSS system such that $p/p_0\approx 0.05$, in order to achieve maximum glass stability while maintaining a positive pressure. In the CSS system we kept the density fixed. (c) and (d) The pressure made dimensionless via rescaling by the bulk modulus $K$, for comparison.

Figure 18

Robustness of the scheme employed for estimating the disorder quantifier ${\chi }_{{}_G}$. The scheme is tested first on data obtained for a computer glass former of soft repulsive spheres and various glass sizes $N$; model details, glass preparation protocol and glass-ensemble sizes can be found in Ref. [90]. Panel (a) compares between the value of ${\chi }_{{}_G}$ obtained by a direct calculation, and that obtained by fitting the probability distribution function $p\left(G\right)$ to a Gaussian, a “3/4 Gaussian,” and a skewed Gaussian. The inset of panel (a) demonstrates that the generic form of $p\left(G\right)$ features a non-Gaussian tail towards negative $G$'s [118]. Panels (b), (c), and (d) test the $\delta$-dependence of our scheme applied to (b) the same system shown in (a), to the CSS model with $r_c=1.1$ (c), to the CSS model with $r_c=1.2$ (d). The insets of panels (b), (c), and (d) report the percentage of outliers removed from the original data set by our scheme, for each $\delta$ (color coded as the legend). We conclude that the most effect percentage $\delta =1\%$.

Figure 19

Decay functions $c\left(r\right)$ [defined in Eq. (A13)] measured in the CSS system with different cutoffs $r_c$, factored by $r^6$ and plotted vs $r/a_0$, where $r$ is the distance from the applied local force dipole, and $a_0$ is a characteristic interparticle distance. The big arrow indicates the direction of increasing interaction cutoff $r_c$. With this figure we illustrate how the length at which the crossover to the continuum scaling, ${\xi }_g$, is extracted. We fit a third-order degree polynomial to the (logarithm base 10 of the) signal in the bump range (highlighted with yellow markers). The fit is represented by the black curves on top of the yellow markers, and the full black squares mark the extracted values for ${\xi }_g$.