Asymmetric crystallization during cooling and heating in model glass-forming systems

We perform molecular dynamics (MD) simulations of the crystallization process in binary Lennard-Jones systems during heating and cooling to investigate atomic-scale crystallization kinetics in glass-forming materials. For the cooling protocol, we prepared equilibrated liquids above the liquidus temperature $T_l$ and cooled each sample to zero temperature at rate $R_c$. For the heating protocol, we first cooled equilibrated liquids to zero temperature at rate $R_p$ and then heated the samples to temperature $T>T_l$ at rate $R_h$. We measured the critical heating and cooling rates $R_h^{*}$ and $R_c^{*}$, below which the systems begin to form a substantial fraction of crystalline clusters during the heating and cooling protocols. We show that $R_h^{*}>R_c^{*}$ and that the asymmetry ratio $R_h^{*}/R_c^{*}$ includes an intrinsic contribution that increases with the glass-forming ability (GFA) of the system and a preparation-rate dependent contribution that increases strongly as $R_p\to R_c^{*}$ from above. We also show that the predictions from classical nucleation theory (CNT) can qualitatively describe the dependence of the asymmetry ratio on the GFA and preparation rate $R_p$ from the MD simulations and results for the asymmetry ratio measured in Zr- and Au-based bulk metallic glasses (BMG). This work emphasizes the need for and benefits of an improved understanding of crystallization processes in BMGs and other glass-forming systems.

Figure 1

Snapshots of the nucleation and growth of crystal clusters at several temperatures $T$ as a monodisperse Lennard-Jones system is heated from zero temperature to $T_f=2.0$ at a rate $R_h<R_h^{*}$ (top row) and cooled from initial temperature $T_i=2.0$ to zero temperature at a rate $R_c<R_c^{*}$ (bottom row). Distinct, disconnected crystal nuclei are identified using the technique in Ref. [32] and are shaded different colors. The far-right panel indicates the number of clusters $N_c$ normalized by $L^3/{\sigma }_A^3$ as a function of temperature during a typical heating (top) and cooling (bottom) trajectory. The maximum number of clusters $N_c^{\max }$ is indicated by the horizontal dashed line.

Figure 2

Shifted and normalized probability for crystallization $[P\left(R_{h,c}\right)-P_{h,c}^{\infty }]/(P_{h,c}^0-P_{h,c}^{\infty })$ versus the scaled heating or cooling rate ${log}_{10}{(R_{h,c}/R_{h,c}^M)}^{1/{\kappa }_{h,c}}$. Circles (squares) indicate data for cooling (heating) for diameter ratios $\alpha =1.0$ (filled symbols) and $0.97$ (open symbols). The insets show the fraction of crystal-like particles $N_{\text{cr}}/N$ as a function of temperature $T$ during cooling (lower left) and heating (upper right) for 12 configurations with $\alpha =1.0$. The four solid, dashed, and dot-dashed curves in each inset correspond to cooling and heating trajectories with rates slower than $R_{h,c}^{*}$, near $R_{h,c}^{*}$, and faster than $R_{h,c}^{*}$, respectively. Trajectories for which $N_{\text{cr}}/N$ exceeds $0.5$ (above the horizontal dashed line) are considered to have crystallized during the heating or cooling protocol.

Figure 3

Maximum value $N_c^{\max }$ of the number of crystal clusters $N_c\left(T\right)$ normalized by $L^3/{\sigma }_A^3$ (averaged over $1000$ trajectories) during the cooling (squares) and heating (circles) protocols at rates $R_c\approx 0.5R_c^{*}$ and $R_h\approx 0.5R_h^{*}$ for LJ mixtures with diameter ratios $\alpha =1.0$, $0.97$, and $0.95$. For all systems, the maximum number of crystal clusters is larger for the heating protocol compared to that for the cooling protocol and $N_c^{\max }$ decreases with increasing glass-forming ability (decreasing $\alpha$).

Figure 4

Intrinsic asymmetry ratio $R_h^{*}\left(\infty \right)/R_c^{*}$ versus the critical cooling rate $R_c^{*}$ (for diameter ratios $\alpha =1.0$, $0.97$, $0.96$, $0.95$, and $0.93$) normalized by $R_0=1\mathrm{K}/\mathrm{s}$ on a logarithmic scale. The inset shows the intrinsic asymmetry ratio versus ${log}_{10}{R}_c^{*}/R_0$ on an expanded scale. The filled circles indicate data from the MD simulations and filled squares indicate data from experiments on Zr- and Au-based BMGs [15, 16]. The prediction [Eq. (12)] from classical nucleation theory (solid line) with $A^{\prime }=\left(8\pi A{D}_0^4\right)/3a^3=0.5$ [in units of ${\epsilon }^2/\left(m^2{\sigma }_A^4\right)]$, $\Sigma =0.26$, and $Q_{\text{eff}}=2.6$ interpolates between the MD simulation data at high $R_c^{*}$ and experimental data from BMGs at low $R_c^{*}$.

Figure 5

The nucleation $I/A{D}_0$ (solid lines; left axis) and growth $Ua/D_0$ (dashed lines; right axis) rates as a function of temperature $T$ for increasing values of the glass-forming ability (GFA) $c=1.2$, $1.1$, $1.0$, $0.9$, $0.8$, $0.7$, $0.6$, and $0.5$ (from top to bottom) that span the range of diameter ratios from $\alpha =1.0$ to $0.93$. The filled circles indicate the maximum rates ($I^{*}$ and $U^{*}$) for each GFA. As the GFA increases, $I^{*}$ and $U^{*}$ decrease and the difference $T_U-T_I$ between the temperatures at which the maxima in $U\left(T\right)$ and $I\left(T\right)$ occur increases (inset).

Figure 6

The time-temperature-transformation (TTT) diagram during cooling is visualized by plotting the probability to crystallize $P$ (increasing from light to dark) from 96 samples as a function of temperature $T$ and waiting time $t$ for LJ systems with diameter ratio $\alpha =1.0$. A sample is considered crystalline if the number of crystal-like particles satisfies $N_{\text{cr}}/N>0.5$. The initial states are dense liquids equilibrated at $T=2.0$. Each initial state is cooled (at rate $R_c\gg R_c^{*}$) to temperature $T<T_l$, where $T_l\approx 1.4$ is the liquidus temperature, and then run at fixed $T$ for a time $t$.

Figure 7

Asymmetry ratio $R_h^{*}\left(R_p\right)/R_c^{*}$ plotted versus the preparation cooling rate $R_p$ normalized by the critical cooling rate $R_c^{*}$ from MD simulations with $\alpha =1.0$ (filled circles) and the prediction from CNT (solid line) with the same parameters used for the fit described in the legend of Fig. 4 and the GFA parameter set to $c=1.2$. The vertical dashed line indicates $R_p=R_c^{*}$. The horizontal dashed lines $R_h^{*}\left(R_p\right)/R_c^{*}=1.18$ and 1 indicate the plateau value in the $R_p\gg R_c^{*}$ limit and $R_h^{*}=R_c^{*}$, respectively. The gap between the horizontal dashed and dotted lines give the magnitude of the intrinsic asymmetry ratio for this particular GFA (cf. Fig. 4).