Large temperature-up-jump simulations of a binary Lennard-Jones system

TL;DR

The paper probes the limits of the Tool-Narayanaswamy material-time approach in describing aging after large temperature up-jumps in a modified Kob–Andersen binary Lennard-Jones liquid. By defining a material time $ξ(t)$ from the potential-energy time-autocorrelation $C_{uu}$ and testing several two-time observables, it finds that the triangular relation holds for $C_{uu}$ and that $ξ(t)$ becomes proportional to real time at long times, but a single global material time does not collapse all observables for the largest jump. For a smaller jump, partial collapse is observed for some observables (notably $C_{uu}$ and $F_s$), while others show limited or no collapse, suggesting that the TN framework best describes aging near equilibrium. The results imply that dynamic heterogeneity or observable-specific clocks may be needed to fully capture far-from-equilibrium aging, guiding future work on local material times or multi-clock TN formalisms with potential practical impact on predicting long-timescale aging behavior in glasses.

Abstract

This paper presents simulations of the physical aging of a binary Kob-Andersen-type Lennard-Jones liquid following large temperature up-jumps from equilibrated states of high relaxation time. The purpose is to investigate how well the Tool-Narayanaswamy (TN) material-time concept works for this rather extreme case of aging. First the triangular relation of the potential energy is studied. This is found to be well obeyed, making it possible to define a potential-energy-based material time $ξ$. We proceed to study aging toward equilibrium at the final temperature 0.48 for jumps from the temperatures 0.43 and 0.37, monitoring the following five quantities: the potential energy, the self-intermediate scattering function, the mean-square displacement, the dynamic susceptibility $χ_4$, and the non-Gaussian parameter $α_2$. The TN material-time prediction is that all time-autocorrelation functions should collapse to only depend on the material-time difference $ξ_2-ξ_1$. This is found to work much better for the $0.43\to 0.48$ temperature jump than for the $0.37\to 0.48$ jump. Our findings thus confirm the general understanding that the TN aging formalism works best for systems that are never very far from equilibrium. This raises two questions for future work: Is the collapse significantly improved if each aging quantity is allowed its own material time? Can better collapse be obtained if the material-time is generalized to be defined locally in order to reflect dynamic heterogeneity?

Figures (4)

• Figure 1: Physical aging. (a) Schematic drawing of a temperature up jump starting and ending in thermal equilibrium (with time on the x-axis). Even though not all quantities relax in identical manner, the standard TN aging formalism operates with a single "global" clock controlling all relaxations toward equilibrium. (b) Results for the potential energy per particle following one down jump and several up jumps to the same final temperature $0.48$, illustrating the noted "asymmetry of approach" seen in all aging simulations and experiments. The black filled rectangles mark the potential energy at $T=0.48$.
• Figure 2: Parametric representation of the triangular relation, Eq. (\ref{['eq:triang']}). (a) Illustration of the sampling of time triplets used to calculate correlation "triangles". The values on the time axis illustrate the logarithmic increase in time intervals within a single simulation block. (b) Illustration of the triangle binning algorithm. Each pixel contains a distribution of triangles $(C_{12}, C_{23}, C_{13})$. (c) The average value $\left\langle C_{13}\right\rangle$ as a function of $C_{12}$ for given values of $C_{23}$. This is the mean value used to determine the function $F_A$ in Eq. (\ref{['eq:triang']}). The form of $F_A$ averaged over each pixel remains unchanged as the amplitude of the jumps changes, as predicted by the TN formalism in which $F_A$ is determined by the equilibrium dynamics. The panel shows that $F_A$ is well defined. The standard error $\sigma_{C_{13}}/\sqrt{n}$ is indicated by the width of the colored lines in which $n$ is the number of triangles and $\sigma_{C_{13}}$ the absolute standard deviation of $C_{13}$ in the pixel. (d) shows a heat map of the average $\left\langle C_{13}\right\rangle$ as a function of $C_{12}$ and $C_{23}$.
• Figure 3: Numerical test of the triangular relation Eq. (\ref{['eq:triang']}). $\sigma_{C_{13}}$ is the standard deviation of the single-particle potential energy's time-autocorrelation function $C_{13}\equiv C_{uu}(t_1,t_3)$ for fixed values of $C_{12}$ and $C_{23}$. The lower the value of $\sigma_{C_{13}}$ , the better is Eq. (\ref{['eq:triang']}) satisfied. (a) The absolute standard deviation $\sigma_{C_{13}}$ is close to 0 if Eq. (\ref{['eq:triang']}) applies. The statistical noise is illustrated by the equilibrium heat map at $T=0.48$ (left panel). In the middle and right heat maps, the deviation is shown for the two temperature up jumps. A small increase in the deviation is observed for the large up jump compared to the smaller one. (b) Number of pixels whose deviation from the triangular relation exceeds a given threshold, plotted as a function of the threshold for a number of different up jumps. A consistent increase of the deviation is observed as the size of the jump is increases, but the largest values of $\sigma_{C_{13}}$ are only roughly twice as large as at equilibrium. The initial number of pixels vary for different jumps because we disregard pixels with too little data. (c) The material time $\xi$ is a non-linear function of time during aging, as shown here for two temperature up jumps. At long times equilibrium at $T=0.48$ is approached and $\xi$ becomes proportional to time.
• Figure 4: Left Two-time functions plotted during aging as functions of the time difference $t_2-t_1$. The left column gives data for the small up jump, the right for the large one. The color of the curves vary from blue for small values of $t_1$ to red as this value increases. The following two-time observables are considered as functions of $t_2-t_1$: (a-b) Per-particle potential energy autocorrelation function; (c-d) Self part of the intermediate scattering function; (e-f) Mean-square displacement; (g-h) Dynamical susceptibility (Eq. (\ref{['eq:def_chi4']})); (i-j) Non-Gaussian parameter (Eq. (\ref{['eq:def_alpha2']})). The curves are identical at large $t_1$ for both jumps because they share the same final temperature while they differ at small $t_1$s. Right Same data plotted as functions of the material-time difference $\xi_2-\xi_1$. There is a significantly better collapse for the small up jump than for the large one.