Fig. 1. Radial γ–plot for with various values of a. (a) a=0.2, (b) a=0.23, (c) a=0.3, (d,g) a=0.34, (e,h) a=0.5, (f,i) a=0.9. The marked regions in (a)–(f) indicate the unstable directions where the weighting factor for some angle orthogonal to . The marked regions in (g)–(i) indicate the absolutely unstable directions where the weighting factor for all angles orthogonal to .

Fig. 2. Equilibrium surfaces reached in evolution by solving the surface diffusion equation, Eq. (7), with for various values of a. (a) a=0.2, (b) a=0.3, (c) a=0.5, (d) a=0.9. The (dimensionless) volume of each crystal is 4π/3≈4.2. The initial configuration is the cube with faces in the crystalline orientations. The equilibrium are reached within dimensionless time t=0.00625.

Fig. 3. Equilibrium surfaces reached in evolution by solving the surface diffusion equation, Eq. (7), with for various values of a. (a) a=0.2, (b) a=0.32, (c) a=0.34, (d) a=0.5. The initial block has faces in the directions and the ratios of the three sides as 1:2:3.

Fig. 4. Evolution of the amounts of reduction of the total surface free energy, E(0)–E(t), on logarithmic scales. The symbols represent the reduction with the initial block crystal configuration shown in Fig. 3 and Table 2. The cases are a=0.2 squares, a=0.32 deltas, a=0.34 gradients, and a=0.5 circles. The lines through symbols are the fits by power laws with the exponent shown in Table 2. The long lines represent the reduction with the initial plate crystal configuration shown in Figs. 5–7 and Table 3. The cases are a=0.2 solid line, a=0.3 dashed line, and a=0.35 dotted line.

Fig. 5. Evolution of a crystal to equilibrium using Eq. (7) with the mildly anisotropic . The initial crystal configuration is a square plate with the two faces in the crystalline orientations and the ratio of the thickness to the side of the plate being 1:3. (a) t=0, (b) t=0.00625, (c) t=0.025, (d) t=0.05, (e) t0.4 equilibrium shape.

Fig. 6. Evolution of a crystal to equilibrium using Eq. (7) with the severely anisotropic . The initial crystal configuration and orientation are same as in Fig. 5. (a) t=0.0015625, (b) t=0.0125, (c) t=0.025, (d) t=0.05, (e) t0.1 equilibrium shape.

Fig. 7. Evolution of a crystal to equilibrium using Eq. (7) with the severely anisotropic . The initial crystal configuration and orientation are same as in Fig. 5. (a) t=0.0015625, (b) t=0.0125, (c) t=0.025, (d) t=0.05, (e) t0.2 equilibrium shape.

Total surface free energy for the ECS obtained from Eq. (7) and shown in Fig. 2 with for various values of a

The energy of the (initial) cube of the same volume, , with the same γ is computed and compared to that of the ECS in each case. All energies are computed on the surfaces with 63×64+2 grid points in the spherical coordinate system using the mid-point rule.

Total surface free energy, E(t), for initially block crystals in evolution using Eq. (7) with for values of a shown in Fig. 3

The energy of the ECS obtained from the initial cube with the same γ is also listed. The fits by power laws to the reduction of the energy in the early evolution are listed in the last row. All energies are computed on the surfaces with 63×64+2 grid points in the spherical coordinate system using the mid-point rule.

Total surface free energy for crystals in evolution shown in Figs. 5–7 with for three values of a

The initial crystal configuration is a square plate with two faces in the crystalline orientations as shown in Figs. 5–7. All energies are computed on the surfaces with 63×64+2 grid points in the spherical coordinate system using the mid-point rule.