Sharp Phase Field Method

Phase field modeling offers an extremely general framework to predict microstructural evolutions in complex systems. However, its computational implementation requires a discretization scheme with a grid spacing small enough to preserve the continuous character of the theory. We present here a new formulation, which is intrinsically discrete, in which the interfaces are resolved with essentially one grid point with no pinning on the grid and an accurate rotational invariance, improving drastically the numerical capabilities of the method. We show that interfacial kinetic properties are reproduced with a high accuracy. Finally, we apply the model to a situation where conserved and nonconserved fields are coupled.

Figure 1

Left: free energy density $g(\varphi )$ as a function of $\varphi$ for $d/w=2$ [see Eq. (7)]. Right: corresponding 1D phase field profiles for different shifting value $x_0$.

Figure 2

Left: sphericity as function of time for a growing precipitate in a $256^3$ simulation box with gradient terms limited to 1st neighbors (dotted lines) or extended to 3rd neighbors (full lines). Right: phase field profile along a middle [110] line, obtained with the optimization scheme (111).

Figure 3

Comparison between S-PFM (blue) and classical PFM (green). Left: volume fraction of a shrinking precipitate for different values of the ratio $d/w$ (time unit is $t_0=d^2/\mu \sigma$). Right: profiles for a precision on the velocity of 1.7%.

Figure 4

Growing precipitate in a supersaturated matrix. Left, (111) cut through the center of the $256^3$ simulation box; right, concentration and phase field profiles along a middle [110] line.