Description of order-disorder transitions based on the phase-field-crystal model

Order-disorder transition is an attractive topic in the research field of phase transformation. However, how to describe order-disorder transitions on atomic length scales and diffusional time scales is still challenging. Inspired from high-resolution transmission electron microscopy, we proposed an approach to describe ordered structures by introducing an order parameter into the original phase-field-crystal model to reflect the atomic potential distribution. This new order parameter contains information about kinds of atoms, showing that different kinds of sublattices in ordered structures can be distinguished by the amplitude of the order parameter. Two case studies, growth of ordered precipitations and evolution of antiphase domains, are also presented to demonstrate the capabilities of this approach.

Figure 1

(a) The ordered ground state of an $AB$ alloy on a square lattice, where blue and red circles represent the two kinds of atoms. Density fields $n_A\left(r\right), n_B\left(r\right), n\left(r\right)$ and the atomic potential distribution fields $\eta \left(r\right)$ in an ordered structure (b) and a disordered structure(c).

Figure 2

(a) Direct pair correlation function for the order square phase with different temperature parameters $\sigma$ when $c=c_{\mathrm{order}}=0.5$, where the first peak in the direct pair correlation shows the long-periodic arrangement and the latter two peaks show the basis crystal structure. (b) A comparison of η between the ordered and disordered structures. When $\sigma =0.7$, the stable structure is the disordered phase, while the ordered structure is stable for the other two cases. (c) and (d) show the spots of η in Fourier space for the ordered and disordered structures, respectively. The spots of η in (d) show the fundamental lattices [$(1,0)$ and $(1,1)$], while those in (c) show a superlattice [$(\frac{1}{2},\frac{1}{2})$].

Figure 3

Calculated phase diagram showing the coexistence between the ordered and disordered structures.

Figure 4

Simulated results of the η field for the ordered structures listed in Table 1. (a) The slice of (110) face for $B2$ structures and (b) the slice of (111) for $L{1}_2$ structures. Different colors of atoms reflect different amplitudes of η field. The rectangles in (a) and the triangles in (b) show the two kinds of superlattice in $B2$ and $L{1}_2$ structures, respectively.

Figure 5

Simulated results of the ordered precipitate growth process at $c_0=0.47,\sigma =0.05$. (a)–(c) show the evolution of the $\eta$ field at $t=10000$, 100 000, and 200 000, respectively; (d) shows the variation of precipitate volume fraction with time; (e) shows the concentration profiles across the interface.

Figure 6

(a) The interface width $W$ as a function of the peak width $\alpha$; (b) the variation of APB energy (a) and order-disorder interface energy (b) with the peak width α.

Figure 7

Evolution of APD when $c_0=0.47,\sigma =0.05$. (a)–(c) show the atomic image at $t=10000$, 1 000 000, and 2 000 000, respectively; (d) shows the concentration profiles at different times.

Figure 8

Effect of temperature σ on the interface energy of APBs at $\alpha =0.8$ (the black line); the red line shows the ${\gamma }_{\mathrm{APB}}$ of ${\gamma }^{\prime }$ vs temperature in the Ni-Al system by using the CVM [34].