Thermal origin of quasilocalized excitations in glasses

Key aspects of glasses are controlled by the presence of excitations in which a group of particles can rearrange. Surprisingly, recent observations indicate that their density is dramatically reduced and their size decreases as the temperature of the supercooled liquid is lowered. Some theories predict these excitations to cause a gap in the spectrum of quasilocalized modes of the Hessian that grows upon cooling, while others predict a pseudogap $D_L\left(\omega \right)\sim {\omega }^{\alpha }$. To unify these views and observations, we generate glassy configurations of controlled gap magnitude ${\omega }_c$ at temperature $T=0$, using so-called breathing particles, and study how such gapped states respond to thermal fluctuations. We find that (i) the gap always fills up at finite $T$ with $D_L\left(\omega \right)\approx A_4\left(T\right){\omega }^4$ and $A_4\sim exp(-E_a/T)$ at low $T$, (ii) $E_a$ rapidly grows with ${\omega }_c$, in reasonable agreement with a simple scaling prediction $E_a\sim {\omega }_c^4$ and (iii) at larger ${\omega }_c$ excitations involve fewer particles, as we rationalize, and eventually become stringlike. We propose an interpretation of mean-field theories of the glass transition, in which the modes beyond the gap act as an excitation reservoir, from which a pseudogap distribution is populated with its magnitude rapidly decreasing at lower $T$. We discuss how this picture unifies the rarefaction as well as the decreasing size of excitations upon cooling, together with a stringlike relaxation occurring near the glass transition.

Figure 1

Schematic density of states for an equilibrated liquid at temperature $T$. When a gapped glass is heated to a temperature $T_a$ for a duration $t_a$, as sketched in the inset, modes beyond the gap act as a reservoir of excitations that can be thermally activated. It fills up the gap, leading, for small $\omega$, to a pseudogap $D\left(\omega \right)\approx D_L\left(\omega \right)\approx A_4{\omega }^4$. This effect is exponentially diminished if ${\omega }_c$ increases (corresponding to a decrease of $T$ as predicted by the infinite-dimensional mean-field description near the glass transition).

Figure 2

(a) Distribution of particle radii, normalized by the number density $\rho =N/\langle V\rangle$, for different values of stiffness $K$ (the particle sizes are narrowly distributed when $K$ is large). (b) Density of quasilocalized modes displaying a finite gap ${\omega }_c$, in contrast to the usual pseudogap scaling $D_L\left(\omega \right)\sim {\omega }^4$ indicated with a dashed line. The gap values ${\omega }_c\approx \{1.64,1.19,0.85,0.65\}$ corresponding, respectively, to $K=\{{10}^2,{10}^3,3\times {10}^3,{10}^4\}$ are indicated using ticks, following the same color code. Physically, decreasing $K$ results to a larger gap and thus a more stable glass, and is associated to a larger polydispersity.

Figure 3

(a) Density of soft modes $D\left(\omega \right)$ after reheating for a fixed duration $t_a=500$ at different temperatures $T_a$ (following the protocol sketched in Fig. 1), at fixed gap ${\omega }_c=1.64$ (the largest we generate, cf. Fig. 2). Note that the order of the legend matches the order of the curves. Furthermore, note that $D_L\left(\omega \right)=D\left(\omega \right)$ for $\omega <{\omega }_e$, with ${\omega }_e$ the frequency of the first plane wave, indicated with a black arrow. For reference, the mode-coupling temperature in this system is $T_c\approx 0.3$, and the gap, ${\omega }_c=1.64$, is indicated by a black tick. We emphasize that, before reheating, we had $D_L(\omega <{\omega }_c)=0$, so that the corresponding modes have been activated by thermal fluctuations. (b) Prefactor $A_4$ as a function of reheating temperature $T_a$ for different durations $t_a$. (c) Collapse of the different curves $A_4(T_a,t_a)$, supporting the functional form $A_4=f(t_a^{\gamma }exp(-E_a/T_a))$ where the function $f$ is linear at small argument, indicating an Arrhenius behavior at small $T_a$. (d) Typical energy scale $E_a$ vs the initial gap ${\omega }_c$, together with the associated scaling prediction $E_a\sim {\omega }_c^4$ (dashed line). Inset: Dynamical exponent $\gamma$ as a function of ${\omega }_c$.

Figure 4

(a) The ensemble averaged shear stress $\langle {\mathrm{\Sigma }}_0\rangle$ as a function of strain $\epsilon$ for our largest gap (${\omega }_c=1.64$). (b) The difference $\langle {\mathrm{\Sigma }}_0-\mathrm{\Sigma }\rangle$ as a function of strain $\epsilon$ for different temperature cycles applied to the gapped glass from (a). As observed, the stress decreases as $A_4$ increases (from bottom to top, $T_a=\{0.15,0.15,0.15,0.2,0.2,0.2,0.23,0.25\}$ corresponding to $t_a=\{5\times {10}^2,2\times {10}^3,{10}^4,5\times {10}^2,2\times {10}^3,{10}^4,5\times {10}^2,5\times {10}^2\}$, respectively). (c) The decrease in the effective shear modulus $\mathrm{\Delta }\mu$ computed at $\epsilon ={10}^{-2}$ is proportional to $A_4$: the dashed line corresponds to $\mathrm{\Delta }\mu \sim A_4$.

Figure 5

(a) Mean (dashed black) and median (green) of the number of particles $N{P}_r$ involved in an individual excitation $\mathit{vs} {\omega }_c$ at $t_a=500$ (at the lowest temperature we probe for each gap). The error bars stand for the standard deviation of $N{P}_r$ which indicates that the distribution of $N{P}_r$ is broader for smaller ${\omega }_c$. (b) Mean (dashed black) and median (green) of the displacement norm of the most mobile particle in an excitation, as a function of ${\omega }_c$. $d_0^{*}$ is the most frequent diameter of the smaller particles ($d_0^{*}=\{0.738,0.918,0.963,0.988\}$ for the different ${\omega }_c$). (c),(d) Thermally induced rearrangement, respectively for our largest (${\omega }_c=1.64$) and smallest (${\omega }_c=0.65$) gaps, projected on the $xy$ plane. (e) Ensemble and radially averaged (on all observed excitations) Van Hove correlation function $G_d(r,t_a)$ computed for all (in black and having largest $G_d$ at large $r$), only small (red dotted) or only large (blue) particles for ${\omega }_c=1.64$. The peak around $r=0$ corresponds to permutations of particles. The black and red curves overlap, indicating that permutations only occur on small particles. (f) Average number of permutations $\langle n_p\rangle \mathit{vs} {\omega }_c$, for two different cutoff distances $r_c/d_0^{*}=\{0.025,0.05\}$ (dashed and solid, respectively).

Figure 6

Mean interaction potential energy $\langle u\rangle$ after sample preparation with breathing dynamics for varying parent temperature $T_p$ and two different waiting times $t_p$. The different panels correspond to different $K$ as indicated. The selected temperature $T_p$ for which the potential energy is lowest for the largest practically reachable $t_p={10}^4$ is indicated using vertical lines (see Appendix pp1 for numeric values).

Figure 7

Mean interaction potential energy as obtained by sample preparation using breathing dynamics (at $T=0$ for $K={10}^2, t_p={10}^4$, and $T_p=0.2$, in black) and using normal dynamics at different cooling rates (cooling from $T=0.4$) as indicated in the legend. Note that in both cases the ensemble comprises $n=10$ samples, but that normal dynamics are run using smaller than usual samples comprising $N=2000$ particles ($N=8000$ is used throughout). We verify the representativeness of these smaller samples using $N=8000$ for $\dot{T}={10}^{-3}$, shown using a dashed red line (that indeed coincides with the solid red line for $N=2000$). The required CPU time to run the entire simulation with $N=2000$ particles is indicated. From bottom to top, $\dot{T}$ increases. Note that for our breathing dynamics the time has been divided by four to correct for the difference in system size.

Figure 8

Fit of ${\omega }_c$ by asserting the power law scaling in Eq. (D6) and check by extreme value statistics as in Eq. (D7) (insets), for all considered $K$.

Figure 9

(a) $D\left(\omega \right)$ for different system sizes $N$ and different activation temperatures $T_a$ (as indicated in the legend $T_a$ increases from bottom to top). (b) $A_4$ fitted on (a) as a function of $1/T_a$. Both plots are for $K={10}^2$ and $t_a=500$.

Figure 10

Fitting of $E_a$ and $\gamma$ for all $K$ not shown in the main text. The fitted values are reported in Appendix pp1.

Figure 11

$E_a$ as a function of ${\omega }_c$ (open markers) or as a function of ${\omega }_{min}$ (solid markers).

Figure 12

Double-well potential.

Figure 13

Scatter plot ${\chi }_1$ as function of ${\lambda }_1$ for our largest gap (${\omega }_c=1.64$). ${\lambda }_1$ follows as ${\lambda }_1={\sum }_{\alpha ,\beta =1}^3{\sum }_{k,l=1}^N{s}_l^{\alpha }{H}_{lk}^{\alpha \beta }{s}_k^{\beta }$.

Figure 14

Individual local rearrangements projected on the $xy$ plane, shown using different colors, in five randomly chosen examples (from those samples that show more than one local rearrangement) for: (top) $K={10}^2, T_a=0.15$, and $t_a=500$, (bottom) $K={10}^4, T_a=0.03$, and $t_a=500$.

Figure 15

Probability distribution of (a) the participation ratio $N{P}_r$ and (b) the maximal particle displacement ${max}_i\left\{\right||\delta {\vec{r}}_i\left|\right|\}$ at different ${\omega }_c$. (c) The distinct part of the Van Hove correlation normalised by the number density $\langle G_d\rangle /\rho$ at different ${\omega }_c$. (d) Histogram of the number of particles that permute per realisation, $\#n_p$ (for largest ${\omega }_c$) at two different cutoff distances $r_c/d_0^{*}=\{0.025,0.05\}$.