Amplitude expansion of the binary phase-field-crystal model

Amplitude representations of a binary phase-field-crystal model are developed for a two-dimensional triangular lattice and three-dimensional bcc and fcc crystal structures. The relationship between these amplitude equations and the standard phase-field models for binary-alloy solidification with elasticity are derived, providing an explicit connection between phase-field-crystal and phase-field models. Sample simulations of solute migration at grain boundaries, eutectic solidification, and quantum dot formation on nanomembranes are also presented.

Figure 1

Sample phase diagram for two-dimensional triangular system with parameter set $\left(B_0^x,B_2^{\ell },v,t,\alpha ,u\right)=\left(1,-1.8,1,3/5,0.30,4\right)$. Also $\omega =0.088$ and 0.008 in (a) and (b), respectively. In each panel, the filled regions correspond to regions of phase coexistence.

Figure 2

Solute segregations in symmetric grain boundaries with misorientation $\theta =3.76°$ are shown for $\psi =0$ (a) and $\psi =0.2$ (b). Left and right panels correspond to ${\sum }_j{\left|{\eta }_j\right|}^2$ and $\psi$, respectively. In the corner insets of left panels, the density field $n$ is reconstructed from the amplitudes ${\eta }_j$ for the boxed region. In the right panels, the dark (light) color corresponds to the larger (smaller) of the atomic species. The parameters for (a) and (b) correspond to Figs. 1a, 1b at $\Delta B_0=0.01$, respectively. The dimension of each panel is $55a\times 55a$, where $a$ is the lattice constant.

Figure 3

Eutectic solidification for parameters in Fig. 1b at $\Delta B_0=0.022$ and $\psi =0.0$. Panels (a), (b), and (c) correspond to dimensionless times $30000$, $60000$, and $105000$, respectively. From left to right, the columns correspond to $n$ reconstructed from ${\eta }_j$ (for the boxed region), ${\sum }_j{\left|{\eta }_j\right|}^2$, $\psi$, and the local free-energy density. Dislocations are most easily identified as small black dots in the local free-energy density. The sizes of the panels are $27.5a\times 27.5a$ and $110a\times 110a$ in the first column and last three columns respectively, where $a$ is the lattice constant.

Figure 4

Quantum dot formation on a 5-atomic-layer-thick nanomembrane at times $t=20000$, $60000$, and $100000$ [for (a), (b), and (c), respectively]. The columns from left to right correspond to $\sum {\left|{\eta }_j\right|}^2$, $\psi$, and the local free-energy density. The parameters used for this simulation are from Fig. 1a except $\alpha =0.26$ and $B_0^{\ell }=1.028$. Each $122a\times 53a$ panel shown is a portion of the full $122a\times 211a$ simulation cell, where $a$ is the lattice constant of the unstrained film.

Figure 5

Quantum dot formation on a 12-atomic-layer-thick nanomembrane at times $t=20000$, $40000$, and $60000$ [for (a), (b), and (c), respectively]. The parameters used and system size are identical to those in Fig. 4.

Figure 6

Correlated quantum dot formation on a 3-atomic-layer-thick nanomembrane at times $t=20000$, $40000$, $60000$, $80000$, $100000$, and $120000$ [for (a)–(f), respectively]. The parameters used and system size are identical to those in Fig. 4 except for a higher liquid supersaturation.

Figure 7

Sample phase diagram for three-dimensional bcc system with same parameter set and notation as in Fig. 1 except $\omega =0.05$ in (a).

Figure 8

Sample phase diagram for three-dimensional fcc system with the same parameter set and notation as in Fig. 1 except $\omega =0.02$ in (a).