Atomic-scale dynamics of a model glass-forming metallic liquid: Dynamical crossover, dynamical decoupling, and dynamical clustering

The phase behavior of multicomponent metallic liquids is exceedingly complex because of the convoluted many-body and many-elemental interactions. Herein, we present systematic studies of the dynamical aspects of a model ternary metallic liquid ${\mathrm{Cu}}_{40}{\mathrm{Zr}}_{51}{\mathrm{Al}}_9$ using molecular dynamics simulations with embedded atom method. We observed a dynamical crossover from Arrhenius to super-Arrhenius behavior in the transport properties (self diffusion coefficient, self relaxation time, and shear viscosity) bordered at $T_x\sim 1300$ K. Unlike in many molecular and macromolecular liquids, this crossover phenomenon occurs well above the melting point of the system $(T_m\sim 900$ K) in the equilibrium liquid state; and the crossover temperature $T_x$ is roughly twice of the glass-transition temperature of the system $\left(T_g\right)$. Below $T_x$, we found the elemental dynamics decoupled and the Stokes-Einstein relation broke down, indicating the onset of heterogeneous spatially correlated dynamics in the system mediated by dynamic communications among local configurational excitations. To directly characterize and visualize the correlated dynamics, we employed a nonparametric, unsupervised machine learning technique and identified dynamical clusters of atoms with similar atomic mobility. The revealed average dynamical cluster size shows an accelerated increase below $T_x$ and mimics the trend observed in other ensemble averaged quantities that are commonly used to quantify the spatially heterogeneous dynamics such as the non-Gaussian parameter ${\alpha }_2$ and the four-point correlation function ${\chi }_4$.

Figure 1

Partial $g\left(r\right)$ derived using EAM potential for ${\mathrm{Cu}}_{40}{\mathrm{Zr}}_{51}{\mathrm{Al}}_9$ metallic liquids at 1300 K. Al-Al correlation is weak suggesting caging by Cu and Zr atoms.

Figure 2

(a) Temperature evolution of the total neutron structure factor $S\left(Q\right)$ of ${\mathrm{Cu}}_{40}{\mathrm{Zr}}_{51}{\mathrm{Al}}_9$ system. (b) The first peak position of $S\left(Q\right)$ shows a continuous temperature dependence.

Figure 3

$G_s(r,t)$ normalized by the area to measure the probability distribution of finding a particle at a later time. As expected, all three species in the system exhibit a hydrodynamic behavior with a single Gaussian peak that broadens in spatial direction with increasing time at 1450 K. However, at 1100 K, the Gaussian behavior is slightly distorted. At 1450 K, the particle diffuses much faster than at 1100 K evidenced by the broadening of the distribution. Insets show normalized $G_s(r,t)$ in both “$r$” and “$t$” directions.

Figure 4

Temperature dependence of mean-squared displacements (MSD) for constituent elements. Arrow indicates increasing temperature. Al shows largest displacements at short times compared to Zr and Cu.

Figure 5

Self-diffusion coefficients extracted from MSD for each element. Black arrow $(T_x\sim 1300$ K) indicates the onset of correlated dynamics (crossover) marked by a deviation from the respective Arrhenius fits. Red arrow indicates the melting temperature of this system $(T_m\sim 900$ K).

Figure 6

Self-intermediate scattering functions $F_s(Q,t)$ of the constituent elements of a ${\mathrm{Cu}}_{40}{\mathrm{Zr}}_{51}{\mathrm{Al}}_9$ metallic liquid system at $Q=2.6$ Å${}^{-1}$. Cu and Zr atoms show simple two-step relaxations, while Al shows additional peaks around 0.1 ps.

Figure 7

Angell plot of temperature dependence of ${\tau }_{\alpha }$. (a) $\tau$ found using a $1/e$ cut of SISF. (b) $\langle \tau \rangle$ computed by the area under SISF. Al transitions from fast Cu-like dynamics at high T, to much slower Zr-like dynamics at low T.

Figure 8

Inverse of structural relaxation time ${\tau }_{\alpha }$ plotted against $Q^2$ shows a linear dependence for small $Q$ values. The slope of the curve, when fitted with a linear curve, can be used to extract the self-diffusion coefficient $D$.

Figure 9

(a) Normalized stress autocorrelation function of melted ${\mathrm{Cu}}_{40}{\mathrm{Zr}}_{51}{\mathrm{Al}}_9$ at various temperatures. (b) The computed shear viscosity of liquid ${\mathrm{Cu}}_{40}{\mathrm{Zr}}_{51}{\mathrm{Al}}_9$ using the Green-Kubo relation and the temperature dependence is fitted using a VFT type equation. (c) Testing linearity between Maxwell relation ${\tau }_M$ and alpha relaxation time ${\tau }_{\alpha }$. ${\tau }_{\alpha }$ is roughly three times of ${\tau }_M$.

Figure 10

Decoupling of elemental dynamics revealed by taking the (a) ratio of self-diffusion coefficients and (b) ratio of $\alpha$-relaxation time of all three elements. Al and Zr dynamics is highly coupled while Cu shows decoupling in the entire temperature range but accelerated below $T_x\sim 1300$ K.

Figure 11

Breakdown of Stokes-Einstein relation is observed below $T_x\sim 1300$ K, which is well above the glass transition temperature of the system, and above the melting temperature of the system. The arrow indicates the onset temperature $T_x$ where the dynamics crossovers from the uncorrelated to the correlated state.

Figure 12

Effective hydrodynamic/Stokes-Einstein diameter of Cu, Zr, and Al using (a) viscosity $\eta$ and respective self-diffusion coefficients $D$ (b) using VFT fitted $\eta$. Clearly, at higher temperatures, the effective diameter is constant, while it is decreasing sharply below the dynamical crossover temperature $T_x$.

Figure 13

(a)–(f) Visualization of dynamical clusters with more than two atoms in each group, which are classified based on displacement of atoms over a time period of $\alpha$-relaxation time at 1000 and 2000 K. In this analysis, distribution of displacements is divided into 25 bins and average cluster size in each mobility group is shown in (g) and (h). Clusters are colored by the number of atoms in groups (3–43 atoms). Larger clusters are formed at lower temperatures in the slow and intermediate mobility groups. Faster mobility groups do not show appreciable changes over the temperature range.

Figure 14

Mean number of atoms in dynamical clusters $\langle n_{\mathrm{dc}}\rangle$ classified into five major groups. Clearly, cluster size increases with decreasing temperature, but it increases sharply below the dynamical crossover temperature $T_x\sim 1300$ K. This feature is observed across all five groups as well as the mean value for each temperature. Average cluster size computation includes single particles. Displacement distribution is divided into 250 bins and average cluster size for each bin (or mobility group) is computed. Next, the slowest 20% of atoms are classified as Grp 1 (slow) particles, while the most mobile 20% are in Grp 5.

Figure 15

(a)–(c) Non-Gaussian parameter ${\alpha }_2\left(t\right)$ for constituent elements spanning a temperature range of 1100–2300 K. Al shows a distinct short time peak below the main structural relaxation time. This behavior is consistent with medium time scale oscillations in $F_s(Q,t)$ and MSD of Al. (d)–(f) Temperature dependence of four point correlation function ${\chi }_4\left(t\right)$ of constituent elements [27]. As $T$ decreases, both ${\alpha }_2\left(t\right)$ and ${\chi }_4\left(t\right)$ increase in peak height and shift to a larger time value.

Figure 16

(a) The peak value of ${\alpha }_2\left(t\right)$, denoted as ${\alpha }_2^{*}$, shows a distinct increase below the dynamical crossover temperature. (b) ${\chi }_4^{*}$ is defined as the maximum of ${\chi }_4\left(t\right)$. The temperature dependence of ${\chi }_4^{*}$ shows similar behavior as that of ${\alpha }_2^{*}$, the mean cluster size $\langle n_{\mathrm{dc}}\rangle$, and $D{\tau }_{\alpha }/T$. Arrows indicate the dynamical crossover temperature $T_x\sim 1300$ K.