(A) The simulation code for the first-principles phase field (FPPF) method consists of four Fortran 90 programs,
FPPFModel.f90, Util_mod.f90, Manager.f90, and Variables.f90.
The code can be compiled by using Makefile.
The usage of this code is as follows:
There are three input files, “input”, “coordinate”, and “in_para”.
The file “input” contains the dimension of the simulation space (2 for 2D and 3 for 3D), the average Ti concentration (𝜙), the average V configuration (5-𝜙), the radius of the initial seeds, the simulation mesh spacing (dx), the simulation time step (dt), the gradient surface energy coefficient (ε), the restart flag (0 for new calculation and 1 for continuing calculation), the time steps at which output file is written (for example, 10000), the maximum time step (for example, 100000), the number of meshes in x direction Nx, the number of meshes in y direction (Ny), the number of meshes in z direction (Nz), the amplitude of seed (the c parameter), the amplitude for random force, and the coefficient for the polynomial in the free energy boundary at 𝜙=0. The provided “input” file is for the 3D simulation with Nx=Ny=Nz=40.
For example, “input” file for 6.25 at\% V for FPPF1 (tetrahedron model) is as follows:
3 :: 3D calculation
4.250 :: phi_Ti
0.750 :: phi_V
4.0 :: radius of the seed
0.27d0 :: dx
0.000125d0 :: dt, M = 1.0
0.01 :: the gradient energy coefficient, epsilon_x
0 :: restart
10000 :: time steps interval at which output file is written
100000 :: maximum time step
40 :: Nx
40 :: Ny
40 :: Nz
0.3 :: amplitude of seeds, the c parameter
0.5 :: amplitude for random fluctuation
0.05000 :: coefficient for the polynomial in free energy boundary
Numbers (2nd, 3rd) in the second and third lines should be set as (4.5-n, n+0.5) for Ti4-nVn.
The file “coordinate” contains the positions of the initial fluctuations together with ± 1 for the factor of the c parameter (defined in “input”) of the Ti concentration in the direct (fractional) coordinate. (The minus of these initial fluctuations values are set for the V concentration at the same positions.) The number at the top line denotes the number of the $\pm c$ positions to be given below.
The file “in_para” contains the local free energy values for all configurations of TinVm in a matrix form from n=m=0 to n=m=natom, where natom is written at the top line of this file. The local free energies that are not explicitly defined by the “in_para” file are defined in the quartic function subroutine “F_En” in Util_mod.f90.
In the provided “in_para” file, natom is set at 6, which allows at most two interstitial atoms in the unit cell.
(B) On the other hand, the simulation code for the continuous phase-field (CPF) mode consists of four Fortran 90 programs,
CPFModel.f90, Util_cpf.f90, Manager.f90, and Variables.f90.
The code can be compiled by using Makefile_cpf. The values of A, B, C, and D of TiV at 0 K are set in the function subprogram, F_En(xx,c1), of Util_cpf.f90 as
A = -26.7390818d0
B = -34.16063283d0
C = -27.49d0
D = 0.05d0
The values of A, B, and C should be changed by adding the renormalization correction ΔF for the simulation at 1300 K as
A = -26.7390818d0 + 0.08229089d0
B = -34.16063283d0 + 0.08878314d0
C = -27.49d0 + 0.078d0
D = 0.05d0
The usage of this code is as follows:
There are three input files, “input”, “coordinate”, and “in_para”, although any content of “in_para” is not used in the actual calculation in the CPF method. The files “input” and “coordinate” is the same as that used for FPPF.