Modeling structural transformations in binary alloys with phase field crystals

We develop a phase field crystal model for structural transformations in two-component alloys. In particular, the interactions between components are described by direct correlation functions that are an extension of those introduced by Greenwood et al. [Phys. Rev. Lett. 105, 045702 (2010)] for pure materials. These correlation functions result in broad density modulations that can be treated with high numerical efficiency, hence enabling simulations of phase transformations between a wide range of crystal structures. A simplified binary alloy model is shown to describe the equilibrium properties of eutectic and peritectic binary alloys in two and three dimensions. The robustness and versatility of this method is demonstrated by applying the model to the growth of structurally similar and dissimilar eutectic lamella and to segregation to defects.

Figure 1

Phase diagram for a square-square system with the inset showing the phase diagram of a triangle-triangle mixture. The equilibrium coexistence lines are shown in thick black dots and metastability lines are shown in thin gray dots. The phase diagrams share identical parameters for the noninteracting parts of the energy. Specifically, the parameters for the ideal free-energy contribution were $\eta =1.4$, $\chi =1$, while $\omega =0.02$ and $c_0=0.5$ for the entropy of mixing. For the square-square system, the widths of the correlation peaks (i.e., the setting of the elastic coefficients) correspond to ${\alpha }_{11A}=0.9$, ${\alpha }_{10A}=\sqrt{2}{\alpha }_{11A}$, ${\alpha }_{11B}=0.9$, and ${\alpha }_{10B}=\sqrt{2}{\alpha }_{11B}$. The peak positions for species A correspond to $k_{11A}=2\pi$ and $k_{10A}=\sqrt{2}{k}_{11A}$ and for species B, $k_{11B}=4\pi /\sqrt{3}$ and $k_{10B}=\sqrt{2}{k}_{11B}$. For the triangle-triangle system, peak widths are ${\alpha }_{10A}=0.8$ and ${\alpha }_{10B}=0.8$ and peak positions are $k_{10A}=2\pi$ and $k_{10B}=4\pi /\sqrt{3}$.

Figure 2

Eutectic system for two structurally dissimilar elements. (a) Phase diagram for a mixture of element A (square) and element B (triangle); solid dots are equilibrium points calculated via the single-mode expansion method described in the text, gray dots depict metastable coexistence points, and open circles are dynamic simulations. The parameters for the ideal energy are $\eta =1.4$, $\chi =1$, while $\omega =0.02$ and $c_0=0.5$ for the entropy of mixing. For the square structure, the widths of peaks and peak positions are ${\alpha }_{11A}=1.5$, ${\alpha }_{10A}=\sqrt{2}{\alpha }_{11A}$ and $k_{11A}=2\pi$, $k_{10A}=\sqrt{2}{k}_{11A}$. The triangular structure has peak width and position corresponding to ${\alpha }_{11B}=1.5$ and $k_{11B}=4\pi /\sqrt{3}$. (b) Time slices of an evolving compositional profile through a solid-liquid interface corresponding to a quench to point “a” in the phase diagram. A corresponding density-concentration profile of the advancing 1D solid-liquid interface in a channel is shown on the left. The color represents local composition, where blue is saturated at $c=0.1$ and red is $c=0.6$.

Figure 3

Phase diagram for 3D fcc-bcc mixtures. In both cases the ideal and mixing terms of the free energy are set to $\eta =1.3$, $\chi =1$, and $\omega =0.01$, and a reference composition of $c_o=0.5$. Thick black dots and thin gray dots represent the equilibrium and metastable coexistence lines, respectively. (a) Peritectic phase diagram. The widths of the correlation peaks are set to ${\alpha }_{111A}=0.4$, ${\alpha }_{100A}=0.4$, ${\alpha }_{111B}=0.6$, and ${\alpha }_{100B}=0.6$, and the effective transition temperatures are set to ${\sigma }_{M111A}=1.2$, ${\sigma }_{M100A}=0.8$, ${\sigma }_{M111B}=1.05$, and ${\sigma }_{M100B}=1.05$ for elements A and B, respectively. (b) Isomorphous phase diagram for two elements transforming from liquid to fcc symmetries. The widths of the correlation peaks are set to ${\alpha }_{111A}=0.8$, ${\alpha }_{100A}=0.8$, ${\alpha }_{111B}=0.9$, and ${\alpha }_{100B}=0.9$ and the effective transition temperatures are set to ${\sigma }_{M111A}=1.0$, ${\sigma }_{M100A}=1.0$, ${\sigma }_{M111B}=0.8$, and ${\sigma }_{M100B}=0.8$, for elements A and B, respectively.

Figure 4

Portions of the simulation domain from the eutectic growth channels simulated with the free energy producing the square-square phase diagram of Fig. 1. The temperature is quenched to $\sigma =0.09$ at an average concentration of $\overline{c}=0.56$. Here the color represents concentration, with low concentration being blue (Sq${}_A$) and high being red (Sq${}_B$). Each column of images represents a different misorientation angle of the low-concentration lamella relative to the high-concentration lamella. The misorientations shown are (a) 0, (b) 4, and (c) 20 degrees. The simulation times are increasing from bottom to top.

Figure 5

Portion of the simulation domain from the eutectic growth channel with structurally different phases. Element A prefers to solidify to a square symmetry and element B prefers to solidify to a triangular symmetry. In these structurally different lamellae the surface energy between the triangle-liquid interface (${\gamma }_1$), square-liquid interface (${\gamma }_2$), and the square-triangle interface (${\gamma }_{12}$) are set by the correlation kernel variables ${\alpha }_{Ai}$ and ${\alpha }_{Bi}$.

Figure 6

Solute segregation to an edge dislocation in 2D, forming part of a tilt boundary. The density profile is shown in gray scale. Contour plots show isocontours of $c=0.12$, $0.13$, and $0.14$, from outside to inside, for (a) ${\alpha }_{1A}=0.8\sqrt{2}$ and ${\alpha }_{2A}=0.8$ for the square phase in element A (elastically isotropic case) and (b) ${\alpha }_{1A}=0.8$ and ${\alpha }_{2A}=0.8$ for the square phase in element A (elastically anisotropic case).

Figure 7

Square-square phase diagrams showing both the two- and single-amplitude approximations. Parameters used for the calculation are given in the text.