Mechanical disorder of sticky-sphere glasses. II. Thermomechanical inannealability

Many structural glasses feature static and dynamic mechanical properties that can depend strongly on glass formation history. The degree of universality of this history dependence and what it is possibly affected by are largely unexplored. Here we show that the variability of elastic properties of simple computer glasses under thermal annealing depends strongly on the strength of attractive interactions between the glasses' constituent particles—referred to here as glass “stickiness.” We find that in stickier glasses the stiffening of the shear modulus with thermal annealing is strongly suppressed, while the thermal-annealing-induced softening of the bulk modulus is enhanced. Our key finding is that the characteristic frequency and density per frequency of soft quasilocalized modes becomes effectively invariant to annealing in very sticky glasses; the latter are therefore deemed “thermomechanically inannealable.” The implications of our findings and future research directions are discussed.

Figure 1

Illustration of the concept of thermomechanical inannealability. The top panel depicts the potential energy landscape (PEL) of a generic glass. In generic glasses the characteristic frequencies associated with local minima increase with decreasing energy. As we increase the relative strength of attractive forces and decrease their range (see insets)—namely, we make a “stickier” glass—the PEL is altered (see bottom panel): characteristic frequencies associated with local minima become largely independent of the energy.

Figure 2

The Piecewise Sticky Spheres (PSS) pairwise interaction potential employed in this work. The interaction cutoffs $r_c$—marked by the color-coded vertical dotted lines—are expressed in terms of the dimensionless distance $x_{\min }=2^{1/6}$ at which the canonical Lennard-Jones potential attains a minimum.

Figure 3

Glass potential energy per particle $u\left(T_p\right)$, shifted by the interpolated $u\left(T_{\mathrm{co}}\right)$, and rescaled by the crossover temperature $T_{\mathrm{co}}$. The inset shows the extracted crossover temperatures $T_{\mathrm{co}}$ for the computer glasses studied in this work.

Figure 4

(a) Sample-to-sample average shear modulus $G$, rescaled by its high-temperature plateau value $G_{\infty }$, and plotted against $T_p/T_{\mathrm{co}}$. (b) same as (a) for the sample-to-sample mean bulk modulus $K$.

Figure 5

The shear modulus $G$ can be decomposed into an affine part $G-G_{\mathrm{na}}$, and a nonaffine part $G_{\mathrm{na}}$ that features an explicit dependence on the vibrational spectrum of a glass; see precise definitions in Appendix pp1. Here we show the fraction $G_{\mathrm{na}}/G$ vs parent temperature $T_{\mathrm{p}}$ for all $r_c$-ensembles.

Figure 6

(a) Poisson's ratio $\nu$ is plotted against the parent temperature $T_p$, for all $r_c$-ensembles. (b) The circled data points represent ${\nu }_{\infty }$, used for rescaling $\nu$.

Figure 7

The vDOS $\mathcal{D}\left(\omega \right)$, calculated for low and high parent temperatures $T_p$, in our $r_c=1.2$ systems (left) and for our $r_c=1.5$ systems (right). We find that the prefactor $A_g$ of the ${\omega }^4$ scaling depends very weakly on $T_p$ for moderate particle stickiness ($r_c=1.2$) and shows a much more pronounced dependence on thermal annealing for weak particle stickiness ($r_c=1.5$).

Figure 8

Prefactors $A_g$ of the universal nonphononic vDOS $\mathcal{D}\left(\omega \right)=A_g{\omega }^4$, made dimensionless by scaling by ${\omega }_0^5$, measured across and plotted against parent temperatures $T_p$, for our different $r_c$-ensembles.

Figure 9

Distributions $p\left({\omega }_g^{\left(ij\right)}\right)$ of the dipole-response frequencies ${\omega }_g^{\left(ij\right)}$ [cf. Eq. (3)] vs the dimensionless ${\omega }_g^{\left(ij\right)}/{\omega }_0$, for (a) our stickiest glasses ($r_c=1.1$) and (b) for the least sticky glasses ($r_c=1.5$), measured for their respective highest and lowest parent temperature $T_p$ as indicated by the legends.

Figure 10

(a) Characteristic frequency ${\omega }_g$ of QLMs, made dimensionless by rescaling by ${\omega }_0\equiv c_s/a_0$, and plotted against the parent temperature $T_p$. (b) The same as (a), but rescaled by the high-$T_p$ plateau ${\omega }_{\infty }$.

Figure 11

Scaled mean low-frequency plateau of the participation ratio, $N{e}_0$, plotted against the parent temperature $T_p$; here again we observe that increasing glass stickiness leads to thermomechanical inannealability. For a visual example of how $N{e}_0$ is estimated; see Fig. 14 in Appendix pp1-s2.

Figure 12

(a) Scaled participation ratio $N{e}_0$ plotted against the rescaled characteristic frequency ${\omega }_g/{\omega }_0$. Each point represents a different parent temperature $T_p$. The dashed line represents the relation $N{e}_0\sim {({\omega }_g/{\omega }_0)}^{-3}$. (b) Dimensionless vDOS prefactor $A_g{\omega }_0^5$ plotted against the dimensionless characteristic frequency ${\omega }_g/{\omega }_0$ of QLMs.

Figure 13

We consider two dimensionless forms for our glasses' pressure: $p/K$ and $p/p_0$ (see text for the definition of $p_0$); both indicate only minor variations between the our different $r_c$ ensembles. See further details in Appendix pp1-s1.

Figure 14

Scatter plot of the scaled participation ratio $Ne$ vs frequency $\omega$, for our sticky-sphere glasses with $r_c=1.5$ and $T_p/T_{\mathrm{co}}\approx 3.0$.

Figure 15

Potential energy per particle, expressed in terms of simulational units, for all studied parent temperatures $T_p$, and different glass stickiness as obtained by tuning the interaction cutoff $r_c$. The empty stars represent the interpolated energies $u\left(T_{\mathrm{co}}\right)$; see discussion in Sec. 3.

Figure 16

Shear-to-Bulk moduli ratio $G/K$, rescaled by their high-$T_p$ limit $G_{\infty }/K_{\infty }$, and plotted against the rescaled parent temperature $T_p/T_{\mathrm{co}}$. Here $T_{\mathrm{co}}$ is defined as the intersection of the linearly extrapolated $G/K$—marked, for example, by the nearly vertical dashed line—with the high-$T_p$ limit. The inset compares between the crossover temperatures extracted as shown here and those extracted by the $u\left(T_p\right)$ collapse shown in Fig. 3.

Figure 17

Comparison of the estimation of ${\omega }_g$ obtained as explained in Sec. 5b, to that obtained in Ref. [39] by following the scheme explained in this Appendix, for all $r_c$-glass ensembles (we consider the highest $T_p$'s).

Figure 18

Stress autocorrelation functions for our model glass formers with different pair interaction cutoffs $r_c$.