Fig.1 Δfsqmin and Δfstrmin as function of ψ for ω4=λ/2 Square+ and square-correspond to the positive and negative sign adopted in Eq. (18), respectively.

Fig.2 (a) Δfrect and Δfstrmin as function of ψ; (b) the optimal solution of (ηⅠ, ηⅡ, ηⅢ) versus ψ

Fig.3 A circular grain embedding in a otherwise perfect rectangular matrix simulated by the fast APFC model with (a) ψ=0.1; (b) ψ=0.2; (c) ψ=0.3; (d) ψ=0.4, respectively. The field plotted is A2. The figure (e)-(f) is the corresponding reconstructed atom number density field of (a)-(d) from the same region which is specially illustrated in (a) by the white solid square (Other model parameters are ε=0.3, g=0, λ=3.0, $\stackrel{◆}{\omega}_4 $=2.1λ, A0=0.2, b=1.25, χani=1.0 and κ=0.0. The structure in (e) and (f) is apparently different from that of (g) and (h), which is a new phase, referred to as the orthogonal layered (OL) phase in this work. In the figure (d) or (h), the liquid phase begins to grow at the grain boundary.)

Fig.4 Area of the circular grain versus time calculated by the fast APFC model for (a) hexagonal lattice; (b) square lattice and (c) rectangular lattice (For the rectangular lattice, the grain area is estimated by ra× rb, ra (rb) is the long (short) axis of the elliptical grain. For the other two lattices, the grain area is estimated by the square of the radius of the circular grain. In (c), the three knees corresponding to the change of the shrinking rate are marked by A, B and C.)

Fig.5 (a) The trajectory of the dislocations of the circular grain in the hexagonal matrix, where the color is encoded by A2; (b) atom number density evolution reconstructed from the position as pointed by the arrow in the figure (a) ((b) 1-4 correspond to the moment of 600, 800, 1 000 and 1 200, where the mixed dislocation cores with different burgers vectors are depicted by the yellow and cyan lines mark. Specially, the [${\rm{\bar 1}}$2${\rm{\bar 1}}$0] dislocation components depicted by the yellow lines in the figure 1 or 2 of (b) cancel out after the dislocation reaction.)

Fig.6 (a) The trajectory of the dislocations of the circular grain in the square matrix, where the color is encoded by A2; (b) Atom number density evolution reconstructed from the position as pointed by the arrow in the figure (a)((b) 1-4 correspond to the moment of 260 000, 268 000, 272 000 and 276 000, where the dislocation cores with different burgers vectors are depicted by the yellow and cyan lines mark. Specially, the <01> dislocation components with the opposite sign depicted by the yellow lines in 1 and 2 of (b) cancel out after the dislocation reaction.)

Fig.7 The trajectory and shape of the dislocations of the circular grain in the rectangular matrix (Where the color is encoded by A2.Dislocation cores on the same dashed circle corresponds to the same moment.)

Fig.8 (a) A2 field of dislocation reaction Ⅱ, figure (a)-(d) correspond to the moment of 22 400, 22 800, 2 3000 and 23 200, respectively (The inset shows the reconstructed atom number density from the region marked by the white solid square. The yellow and cyan lines depict the <10> and <01> dislocation, respectively.)

Fig.9 A2 field of dislocation reaction Ⅲ, figure (a)-(d) correspond to the moment of 8 000, 8 300, 8 500 and 8 700, respectively (Other conventions are the same as that of Fig. 8.)