Growth of Dynamic and Static Correlations in the Aging Dynamics of a Glass-Forming Liquid

TL;DR

We address how amorphous order and dynamic heterogeneity grow during aging in a 3D Kob–Andersen glass-former. The study combines extensive molecular dynamics with finite-size scaling of the $ ext{alpha}$-relaxation time $\tau_\alpha(N,t_w)$ and the peak of the four-point susceptibility $\chi_4^p(N,t_w)$, complemented by block analysis and RDF-based estimates of the static length scale. Key findings are that $\tau_\alpha(N,t_w)$ exhibits non-monotonic growth with $N$, whose peak shifts to larger $N$ as $t_w$ increases or $T$ decreases; $\xi_s(t_w)$ grows logarithmically with $t_w$, and $\xi_d(t_w)$ grows as a power law $\xi_d(t_w)\sim t_w^\gamma$ with $\gamma \approx 0.184$–$0.194$. Importantly, the aging-derived static length scale $\xi_s^e(T)$ matches RFOT predictions for the equilibrium static length scale, validating aging data as a proxy for deep-glass behavior and enabling cross-consistency with equilibrium measurements.

Abstract

Using extensive molecular dynamics simulations, we have performed finite-size scaling (FSS) in the aging regime of a model glass-forming liquid to investigate how the length scales associated with amorphous order (static length) and dynamic heterogeneity (dynamic length) evolve with waiting time. The $α$-relaxation time in the aging regime reveals non-monotonic finite-size effects with a peak at an intermediate system size, which, as far as we know, are not found in the equilibrium systems, and the peak position shifts to larger system sizes with decreasing temperature and increasing waiting time, indicating a growth of a characteristic length scale with waiting time. The extracted correlation volume associated with amorphous order increases logarithmically with the waiting time. Detailed analysis of the dependence of the length scale on waiting time allowed us to estimate the static length scale in the deep supercooled liquid regime. The dynamic length scale, obtained from FSS and block analysis of the four-point dynamic susceptibility, follows a power-law growth with waiting time. The values of the length scales obtained agree well with those obtained from different spatial correlation functions.

Figures (6)

• Figure 1: System size dependence of $\tau_\alpha(N,t_w)$ over different waiting times $t_w$ at temperatures (a) $T = 0.395$, (b) $T = 0.385$, and (c) $T = 0.370$. The data corresponding to the maximum $t_w$ are plotted without rescaling, whereas the others are rescaled by an appropriate constant for clarity (see the SM supplement for the raw data). Black circles in (c) guide the eye, showing the apparent peak shift with $t_w$. (d) FSS in aging— Data collapse is obtained by rescaling the x and y-axis using $\xi_s^3(t_w)$ and $\tau_\alpha(\infty, t_w)$, respectively. Collapsed data are fitted with an asymmetric Gaussian function and rescaled by an appropriate constant. (e) Shows growth of the correlation volume $\xi_s^3(t_w, T)$ with $t_w$, calculated from peak positions in (d) for different $T$. The static length scale in equilibrium $\xi_s^{e}(T)$ is estimated by fitting $\xi_s^3(t_w, T)$ with Eq. \ref{['equ_proposed']}. (f) FSS in equilibrium— Similar data collapse as (d) is shown for aging ($T = 0.395$) and equilibrated data ($T \ge T_g$) kallol_2021 for $N \ge 4000$. The inset shows the estimated $\xi_s^e(T)$ from FSS in aging and equilibrium dynamics. The line is a power-law fit according to the RFOT theory.
• Figure 2: (a) System size dependence of $\chi_4^p(N, t_w)$ for different $t_w$ are shown for $T = 0.395$. The dotted line shows the system size at which $\chi_4^p$ becomes constant ($N\ge2000$). (b) A collapse of the data is obtained by rescaling x and y-axis by $\xi_d(t_w)$ and $\chi_4^p(\infty, t_w)$. The inset shows the growth of the normalised $\xi_d(t_w)$ for different temperatures, and saturates at longer $t_w$ for $T = 0.395$. (c) Waiting time dependence of $\chi_4^p$ for different $T$ is shown, and $\chi_4^p$ seems to saturate at longer $t_w$ as it approaches the equilibrium. Inset shows $t_w$ dependence of $\chi_4^p$ for different system sizes at $T=0.395$. (d) Block size dependence of $\chi_4^p$ for different $t_w$ is shown at $T = 0.395$ and $N = 100\,000$. (e) Similar data collapse as (b) is also obtained in block analysis. Lower inset shows growth of $\xi_d(t_w)$ with $t_w$ for different $T$. The upper inset shows a comparison of $\xi_d(t_w)$ with $t_w$ from FSS and block analysis at $T = 0.395$. (f) Waiting time dependency of $\chi_4^p$ for different block sizes at $T = 0.395$ and different $T$ at $L_B = L/3$ is shown in the main and inset, respectively.
• Figure 3: A comparison between (a) the two-point spatial correlation $C_2(r, t_w, st_w)$ and (b) the spatial displacement correlation $\Gamma(r, t_w, st_w)$ is shown for different $t_w$ at $T = 0.370$, $N = 100\,000$, and $s = 1.00$ (see text for details). Lower insets in (a) and (b) show the integrated area (proportional to $\xi_d(t_w)$) of $C_2(r, t_w, st_w)$ and $\Gamma(r, t_w, st_w)$ for different $s$ at $T = 0.370$. Similarly, the upper inset shows the normalised $\xi_d(t_w)$ for different temperatures at $s = 1.00$. (c) A qualitative comparison of the normalised $\xi_d(t_w)$ obtained from different methods is shown. The inset shows the growth of the normalised static and dynamic length scale with waiting time.
• Figure 4: (a) Waiting time dependence of the self-overlap function is shown for $N = 100\,000$ at $T = 0.395$. The black dotted line shows the $1/e$ value, that used to calculate the $\alpha$-relaxation time of the corresponding $t_w$. Inset shows the growth of $\tau_\alpha$ with waiting time, which is fitted with $t_w^{0.82}$ in the aging regime and saturates for $t_w\ge \tau^e_\alpha(T)$ in equilibrium. System size dependence of $\tau_\alpha$ for different waiting times at $T = 0.395$ are shown in (b), (c), and (d). These data are rescaled using the scaled factors of $7.90, 5.70, 2.42, 1.98, 1.16, 1.08, 1.00, 1.00, 1.00$ for $t_w = 5\times10^2, 1\times10^3, 5\times10^3, 1\times10^4, 5\times10^4, 1\times10^5, 2\times10^5, 3\times10^5, 5\times10^5$ respectively, are shown in Fig. \ref{['figure_fss_tau']} of the main text.
• Figure 5: Waiting time variation of $r(g(r,t_w) - 1)$, fitted with Eq. \ref{['equ_gofr_fit']}, is plotted against distance $r$ for $N = 100\,000$ at $T = 0.395$. Inset shows the comparison of normalized static length scales obtained from FSS of $\tau_\alpha$ and RDF.