Fig. 1. Critical nucleus size calculated by CNT at different temperatures is compared with the results of MD simulation for (a) Al, (b) Fe and (c) Mg.

Fig. 2. Nucleation for various solidification temperatures considering (a) Al, (b) Fe and (c) Mg.

Fig. 3. Visualization of heterogeneity in homogenous nucleation. (a) The complete simulation box at an intermediate stage of nuclei formation from Al melt. The solid circles show nuclei with TBs, and the dotted circles show nuclei where embryos form as precursors for TBs/GBs. (b) Two time steps showing embryo formation on top and side of an initially formed nucleus. (c) Formation of TBs and GBs ii the initial nucleus in (b); the orientations are shown in the top image by orientation-coloring, and CNA method shows two crystal types in the below figure. All the red atoms with hcp stacking are TBs or GBs, or stacking faults that later transform to TB or GBs. Some amorphous GBs are not shown and they can be observed from the orientation coloring.

Fig. 4. Twin boundaries between nuclei. Snapshots showing the atomic configuration at the initial stages of nuclei formation. (a) Al nuclei forms with a TB, and there is 122° angle between atoms on the opposite sides of the TB. The coloring method is CNA, where red atoms are hcp, and green atoms are fcc; (b) Formation of a GB with -109° misorientation in Fe; colors show different orientations and (c) Mg nuclei formation with stacking faults and a GB with misorientation angle of -112°. CAN shows that red atoms are hcp, and green atoms are fcc.

Fig. 5. Density and probability density versus bond orientational order (Q6). Density($ρ,\dot{A}^{-3}$) versus the bond order parameter for (a) Al at 500 K, (c) Fe 1100 K and (e) Mg 550 K considering the same probability distribution function. The probability density function for bond orientation order parameter Q6 for (b) Al at 500 K, (d) Fe at 1100 K and (f) Mg at 550 K; the dotted red rectangles show the SRO atoms, the green rectangles indicate the mix or SRO and crystalline atoms, and the blue rectangles indicate crystalline atoms.

Fig. 6. Densification and crystallization steps in solidification. (a) A nucleus is shown during its growth in isothermal solidification of liquid Al at 500 K at 125 ps by CNA. Red atoms are hcp, green atoms are fcc, and grey atoms are liquid. (b) At the same timestep density is shown for the Al matrix. The area around the nucleus shows higher density. The formation of the first critical nucleus by CNA (c-f), density (g-j), and interatomic distances (k-n) are shown at 10 ps, 25 ps, 50 ps and 75 ps, from top to bottom. The colorbar on the left is for density, and the colorbar on the right is for interatomic distances in Å. The dotted circles show the common neighbor analysis and density of the nucleation sites.

Fig. 7. Time evolution of ico-bcc atoms during solidification. (a) Time evolution of bcc and ico atoms in nucleation of Fe at 1100 K isothermal solidification simulation. (b) Bcc and ico Fe atoms before the formation of critical nuclei. (c)-(e) The CNA and the bond orientational order parameters at 180 ps, 204 ps and 240 ps. In CNA, the white atoms are liquid/amorphous solid, blue atoms are bcc and yellow atoms are ico. Bond order coloring is shown in the color bar.

Fig. 8. Landau free energy. The joint probability P(Q6,ρ) of the density and the bond orientational order parameter is plotted for Al at 500 K at (a) 0 ps, (b) 90 ps, (c) 135 ps and (d) 180 ps.