  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 $R_p
\rightarrow R_c^*$ 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.
