图1 ω4=λ/2，Δfsqmin和Δfstrmin与ψ的函数关系 square+和square-分别对应式(18) 中采用的正负号。

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.

图2 (a) Δfrect和Δfstrmin与ψ的函数关系；(b) (ηⅠ, ηⅡ, ηⅢ)的最优值与ψ的函数关系

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

图3 快速APFC模型模拟的完美矩形矩阵中的圆形晶粒嵌入情况(a) ψ=0.1；(b) ψ=0.2；(c) ψ=0.3；(d) ψ=0.4，绘制的场为A2; (e)~(f)是(a)~(d)中同一区域的相应重建原子数密度场，图(a)中的白色实心方框对其进行了特别说明 ε=0.3，g=0，λ=3.0，$\stackrel{◆}{\omega}_4 $=2.1λ，A0=0.2，b=1.25，χani=1.0，κ=0.0。(e)和(f)中的结构显然不同于(g)和(h)中的结构，这是一种新的相，在本文中称为正交分层(OL)相, 在(d)或(h)中，液相开始在晶界处生长。

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.)

图4 快速APFC模型计算的(a)六角晶格、(b)方形晶格、(c)矩形晶格的圆形晶粒面积与时间的关系(对于矩形晶格，晶粒面积由ra× rb估算，ra (rb)是椭圆晶粒的长(短)轴。对于其他两个晶格，晶粒面积按圆形晶粒半径的平方估算。图(c)中用A、B和C标出了与收缩率变化相对应的三个拐点。)

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.)

图5 (a) 六角晶中圆形晶粒的位错轨迹，颜色由A2编码；(b)图(a)中箭头所指位置重建的原子数密度演化(图(b)中1~4分别对应于600，800，1 000，1 200时刻，图中黄色和青色线条分别表示具有不同柏氏矢量的混合位错芯。特别地，[${\rm{\bar 1}}$2${\rm{\bar 1}}$0] 位错分量在位错反应后被抵消。)

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.)

图6 (a) 方形晶中圆形晶粒的位错运动轨迹，其中颜色由A2编码；(b)图(a)中箭头所指位置重建的原子数密度演化(图(b)中1~4分别对应于260 000，268 000，272 000，276 000时刻，黄色和青色线条分别表示不同柏氏矢量的位错芯。特别地，图(b)中1和2中黄线所示的符号相反的<01>位错分量在位错反应后抵消。)

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.)

图7 矩形晶中圆形晶粒的位错轨迹和形状(颜色由A2编码同一虚线圆上的位错芯对应同一时刻。)

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.)

图8 位错反应Ⅱ的A2场，图(a)~(d)分别对应无量纲时间22 400，22 800，23 000，23 200 插图表示从白色实线方框标记的区域重建的原子数密度。黄色和青色线分别表示<10>和<01>位错。

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.)

图9 位错反应Ⅲ的A2场，图(a)~(d)分别对应无量纲时间8 000，8 300，8 500，8 700(图中标记的含义与图 8相同。)

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.)