The purpose of this project is to calculate the basic structural parameters (including lattice parameter and elastic constants), and generalized stacking fault energies (GSFE) of four equal-molar multi-principal element alloys (MPEAs). The effects of chemical short-range order (CSRO) will be considered.
Please read the following journal articles to understand how the aforementioned material properties can be calculated.
[Elemental materials]:
- Xiaowang Wang, Shuozhi Xu, Wu-Rong Jian, Xiang-Guo Li, Yanqing Su, Irene J. Beyerlein, Generalized stacking fault energies and Peierls stresses in refractory body-centered cubic metals from machine learning-based interatomic potentials, Comput. Mater. Sci. 192 (2021) 110364
- Yanqing Su, Shuozhi Xu, Irene J. Beyerlein, Density functional theory calculations of generalized stacking fault energy surfaces for eight face-centered cubic transition metals, J. Appl. Phys. 126 (2019) 105112
[Random alloys]:
- Abdullah Al Mamun, Shuozhi Xu, Xiang-Guo Li, Yanqing Su, Comparing interatomic potentials in calculating basic structural parameters and Peierls stress in tungsten-based random binary alloys, Phys. Scr. 98 (2023) 105923
- Shuozhi Xu, Arjun S. Kulathuvayal, Liming Xiong, Yanqing Su, Effects of ferromagnetism in ab initio calculations of basic structural parameters of Fe-A (A = Mo, Nb, Ta, V, or W) random binary alloys, Eur. Phys. J. B 95 (2022) 167
- Rebecca A. Romero, Shuozhi Xu, Wu-Rong Jian, Irene J. Beyerlein, C.V. Ramana, Atomistic calculations of the local slip resistances in four refractory multi-principal element alloys, Int. J. Plast. 149 (2022) 103157
- Shuozhi Xu, Saeed Zare Chavoshi, Yanqing Su, On calculations of basic structural parameters in multi-principal element alloys using small atomistic models, Comput. Mater. Sci. 202 (2022) 110942
- Shuozhi Xu, Emily Hwang, Wu-Rong Jian, Yanqing Su, Irene J. Beyerlein, Atomistic calculations of the generalized stacking fault energies in two refractory multi-principal element alloys, Intermetallics 124 (2020) 106844
[Alloys with CSRO]:
- Hui Zheng, Lauren T.W. Fey, Xiang-Guo Li, Yong-Jie Hu, Liang Qi, Chi Chen, Shuozhi Xu, Irene J. Beyerlein, Shyue Ping Ong, Multi-scale investigation of short-range order and dislocation glide in MoNbTi and TaNbTi multi-principal element alloys, npj Comput. Mater. 9 (2023) 89
- Shuozhi Xu, Wu-Rong Jian, Irene J. Beyerlein, Ideal simple shear strengths of two HfNbTaTi-based quinary refractory multi-principal element alloys, APL Mater. 10 (2022) 111107
LAMMPS on OSCER likely does not come with many packages. To build more packages into LAMMPS, please visit this page.
To finish this project, at least three packages are needed.
- MANYBODY package. This is to use the manybody potential such as the embedded-atom method potential.
- EXTRA-COMPUTE package. This is to calculate the elastic constants at finite temperatures using the Born matrix method. To learn more, please visit this page and this page.
- MC package. This is to generate materials with CSRO at a given temperature. This paper and this paper should be cited if one uses this package.
To build LAMMPS with these three packages, use the file lmp_mbecmc.sh. First, cd to any directory on OSCER, e.g., $HOME, then
sh lmp_mbecmc.sh
Note that the second command in lmp_mbecmc.sh will load a module. If one cannot load it, try module purge first.
Once the sh run is finished, we will find a file lmp_mpi in the lammps_mbecmc/src directory on OSCER. And that is the LAMMPS executable with MANYBODY, EXTRA-COMPUTE, and MC packages.
Note: if we use the sbatch files from LAMMPSatOU, we may want to change the walltime (default: 12 hours) and/or number of cores (default: 16). For this project, we use
#SBATCH --time=200:00:00
#SBATCH --ntasks-per-node=32
Each time we run a new type of simulation, create a new directory.
Four MPEAs will be considered.
Note: All files for calculations can be found in the CoCrNi directory in this GitHub repository, except the data files which can be found here. The reason is that the data files are too large for GitHub.
All data files are from this paper.
Run the simulation with files lmp_0K.in, min.CoCrNi_27nmx_27nmy_27nmz_random.dat, and CoCrNi.lammps.eam.
Once it is finished, we will find a new file a_E. Run the following script in the terminal
awk NF a_E > new_aE
Then we will find another new file new_aE. The first column is the ratio of the trial lattice parameter to 3.5564, the second column is the trial lattice parameter itself, in units of Angstrom, and the third column is the cohesive energy, in units of eV. If we plot a curve with the second column as the x axis and the third column as the y axis, the curve should look like the ones in Figure 1(a) of this paper.
Then run sh min.sh to find out the trial lattice parameter corresponding to the lowest cohesive energy (i.e., the minimum on that curve), and that would be the actual lattice parameter. Specifically, we will see
1 3.55644549888703 -4.32151351854507
on the screen. Record these three numbers. These are for random CoCrNi.
The simulation requires files
lmp_0K.in, min.CoCrNi_27nmx_27nmy_27nmz_350KMDMC.dat, and CoCrNi.lammps.eam. Make one change in lmp_0K.in:
- Line 10. Change the word
randomto350KMDMC, i.e., to match the new data file's name.
Run the simulation. Once it is finished, run awk NF a_E > new_aE in the terminal to generate a file new_aE. The first column is the ratio of the trial lattice parameter to 3.561; the other two columns have the same meaning as the random case. Repeat the remaining steps in the random case and record the three numbers for CoCrNi with CSRO.
Results, based on the [100]-[010]-[001] system, are in the file CoCrNi/ela_const/0K/data.txt.
Run the simulation with files in.elastic, displace.mod, init.mod, potential.mod, min.CoCrNi_27nmx_27nmy_27nmz_random.dat, and CoCrNi.lammps.eam.
Once it is finished, we will find an output file, *.out, at the end of which we will find values of C11all, C12all etc. Specifically, we will see
Elastic Constant C11all = 309.979695007998 GPa
Elastic Constant C22all = 310.087578803684 GPa
Since the elastic constants are in the [1-10]-[11-2]-[111] system, they should be converted to those in the [100]-[010]-[001] system.
The simulation requires files in.elastic, displace.mod, init.mod, potential.mod, min.CoCrNi_27nmx_27nmy_27nmz_350KMDMC.dat, and CoCrNi.lammps.eam. Make one change in init.mod:
- Change the last number (by default 1.) of line 50 to the correct ratio identified in the prior lattice parameter calculation, i.e., the first of the three numbers we recorded for the CSRO case.
Since the elastic constants are in the [1-10]-[11-2]-[111] system, they should be converted to those in the [100]-[010]-[001] system.
Run the simulation with files lmp_300K.in, min.CoCrNi_27nmx_27nmy_27nmz_random.dat, and CoCrNi.lammps.eam.
Once it is finished, go to the file log.lammps and find the first block of data that starts with a line Step Lx Ly Lz. The first column of the data block starts from 0, increasing in increments of 100, and stops at 10000.
In fact, we will find that the first line of the block is
0 271.59916 274.41197 277.1948
while the last line is
10000 272.5714 275.39428 278.18707
First, calculate the lattice parameter at the last step 10000, using
(Lx'/Lx + Ly'/Ly + Lz'/Lz)/3
where Lx', Ly', and Lz' are taken at step 10000, while Lx, Ly, and Lz are taken at step 0. Record the result.
Then repeat the calculation above, but using Lx', Ly', and Lz' at steps 9900, 9800, ..., and 9100, respectively. In total, we get ten numbers. Calculate the mean of the ten numbers, and that is the ratio of the lattice parameter for random CoCrNi at 300 K to the initial trial lattice parameter, 3.5564 Angstrom.
We will also find a newly generated file data.relax, which will be used later in elastic constants calculations.
Repeat the steps above, except that
- Use the data file
min.CoCrNi_27nmx_27nmy_27nmz_350KMDMC.datinstead - Change the word
randomto350KMDMCin line 10 of the filelmp_300K.in
Record the lattice parameter, which is for CoCrNi with CSRO. Note that the initial trial lattice parameter is 3.561 Angstrom, different from that for random CoCrNi.
Again, the newly generated file data.relax will be used later in elastic constants calculations.
Results are in the file CoCrNi/ela_const/300K/data.txt.
Run the simulation with files in.elastic, init.in, potential.in, output.in, final_output.in, data.relax, and CoCrNi.lammps.eam. Note that the file data.relax is the one we got from the Lattice parameters at 300 K - Random CoCrNi calculation.
Once it is finished, go to the end of the output file, and we will see
Elastic Constant C11 = 298.291596568703 GPa
Elastic Constant C22 = 297.341418727254 GPa
which are smaller than those calculated at 0 K, as expected.
Since the elastic constants are in the [1-10]-[11-2]-[111] system, they should be converted to those in the [100]-[010]-[001] system.
Repeat the steps above, except that
- Use the
data.relaxfile from theLattice parameters at 300 K - CoCrNi with CSROcalculation instead
Since the elastic constants are in the [1-10]-[11-2]-[111] system, they should be converted to those in the [100]-[010]-[001] system.
Run the simulation with files lmp_gsfe.in, data.CoCrNi_gsfe_random, and CoCrNi.lammps.eam. The first file can be found in the directory CoCrNi/gsfe/ in this GitHub repository. Make sure that line 32 of lmp_gsfe.in contains the correct lattice parameter of random CoCrNi at 0 K, in units of Angstrom.
Once it is finished, we will find a file gsfe_ori, which should contain 101 lines, with the first one being
0 -374943.444563279
Then run sh gsfe_curve.sh in the terminal to generate a new file gsfe.
Note: we have calculated only a single GSFE curve here. A previous paper found that multiple GSFE curves need to be calculated to obtain a good mean GSFE curve.
To obtain other GSFE curves on other shift planes, please change the last integer (by default 1) in line 34 of lmp_gsfe.in to 2, 3, ..., 20, respectively. And run 19 more simulations. Then we will have 20 GSFE curves.
Follow the procedures above, except that
- Use the data file
data.CoCrNi_gsfe_350KMDMCinstead - Change the word
randomto350KMDMCin line 13 oflmp_gsfe.in - Change the number in line 32 of
lmp_gsfe.into the correct lattice parameter of CoCrNi with CSRO at 0 K, in units of Angstrom.
Similarly, we need to obtain 20 GSFE curves.
CoCrNi contains only one pure metal, Ni, that has the same lattice as the alloy. It would be interesting to compare the temperature effect between it and the MPEA.
For CoCrNi, Jian et al. built all atomistc structures, and so we directly used them. For Ni, however, we need to build the atomistic structure ourselves. The first step is to install Atomsk.
Run the atomsk script, atomsk_Ni.sh, which can be found in CoCrNi/ni/ in this GitHub repository, to build a Ni structure named data.Ni, by
sh atomsk_Ni.sh
Then use the data file and the same potential file to calculate its lattice parameters, elastic constants, and GSFE at 0 K, 300 K, 600 K, 900 K, and 1200 K.
Note: The trial lattice parameter is 3.5564 Angstrom. When calculating the lattice parameter at 0 K, in line 44 of lmp_0K.in, change the three numbers 54, 63, and 45, to 30, 30, and 10, respectively. That is because different cell sizes are used here.
For GSFE at 0 K, make the following two changes in the lmp_gsfe.in file
- line 17. Delete
CoandCr - line 32. Use the correct lattice parameter for Ni at 0 K
For GSFE at finite temperatures, make the following three changes to the lmp_gsfe.in file:
-
line 17. Delete
CoandCr -
line 32. Use the correct lattice parameter for Ni at the specified temperature
-
add the following lines immediately before the first
displace_atomscommand (note: there are two of them):variable myTemp equal 300 neighbor 0.3 bin neigh_modify delay 10 thermo 1 velocity all create ${myTemp} 1917 thermo_style custom step lx ly lz fix 1 all npt temp ${myTemp} ${myTemp} 0.1 x 0. 0. 1. y 0. 0. 1. run 10000 unfix 1In the first line above, the default temperature is 300. Change it to 600, 900, and 1200, respectively, in the corresponding calculation.
Similar to Ni, we need to build the atomistic structures for MoNbTa ourselves.
Run the atomsk script, atomsk_MoNbTa.sh, which can be found in MoNbTa/random/ in this GitHub repository, to build a random MoNbTa structure named data.MoNbTa_random.
In the data file, change the masses section to
1 95.96000000 # Mo
2 92.90638000 # Nb
3 180.94788000 # Ta
The lattice parameter, elastic constants, and GSFE of random MoNbTa have been calculated. They are summarized in the file MoNbTa/random/data_random.txt in this GitHub repository. Most results were based on the EAM potential while those at 0 K were also based on the MTP.
First, calculate the radial distribution functions (RDF) for random MoNbTa. To do that, run a LAMMPS simulation with three files data.MoNbTa_random, CrMoNbTaVW_Xu2022.eam.alloy, and lmp_vcsgc.in. The second file can be found in another GitHub repository. The last file is from the MoNbTa/csro/ directory in this GitHub repository and should be modified as follows:
- Lines 3 and 4. Change the two large numbers to zero
- Linne 25. Change the data file name to
data.MoNbTa_random
Once the simulation is finished, we will find a file cn.out, which contains RDF information. Based on the information, one can calculate the WC parameters. The codes used are not shared here.
Density functional theory (DFT) calculations will be conducted using VASP to calculate the lattice parameter, elastic constants, and GSFE of random MoNbTa.
All files can be found in MoNbTa/random/dft/a0/ in this GitHub repository. Create a new directory named a0 on OSCER and move files there. Enter that directory.
First, create POTCAR by
cat POTCAR_Mo POTCAR_Nb POTCAR_Ta > POTCAR
Second, submit the job by
sbatch vasp.batch
Once the calculation is finished, open the file CONTCAR and record lx, ly, and lz which appear in line 3, line 4, and line 5, respectively. The lattice parameter can be calculated by
(lx/sqrt(6)+ly/(sqrt(2)*6)+lz/sqrt(3))/3
All files can be found in MoNbTa/random/dft/ela_const/ in this GitHub repository. Create a new directory named ela_const on OSCER and move files there. Enter that directory.
First, copy POTCAR, which we just created, into that directory.
Second, copy CONTCAR, which was just generated by the lattice parameter calculation, into that directory. Then rename the file to POSCAR.
Third, submit the job by
sbatch vasp.batch
Once the calculation is finished, open the file OUTCAR. The elastic constants are below the line TOTAL ELASTIC MODULI (kBar) in that file. The constants can converted from kBar to GPa following
1 kBar = 0.1 GPa
Also, since the calculated elastic constants are in the [11-2]-[111]-[1-10] system they should be converted to those in the [100]-[010]-[001] system.
All files can be found in MoNbTa/random/dft/gsfe/ in this GitHub repository. Create a new directory named gsfe-5 on OSCER and move files there. Enter that directory.
First, copy POTCAR, which we just created, into that directory.
Second, copy CONTCAR, which was just generated by the lattice parameter calculation, into that directory. Then rename the file to POSCAR_0.
Third, edit line 14 of the file gsfe_curve.sh; by default, c equals 1, change the value of c to
16021.8/(lx*lz)
where 1/(lx*lz) is to divide the energy by area, and 16021.8 is to convert the unit from eV/angstrom2 to mJ/m2.
Fourth, build 41 subdirectories by
sh build.sh
Fifth, submit 41 jobs by
sh run.sh
Once all calculations are finished, generate the GSFE curve file gsfe by
sh gsfe_curve.sh
The USFE is the maximum GSFE value.
So far, we have calculated only a single GSFE curve. To obtain other GSFE curves on other shift planes, please change the last number (by default 5) at the end of line 12 of build.sh to 3, 4, 6, and 7, respectively. Then copy all necessary files into four directories: gsfe-3, gsfe-4, gsfe-6, and gsfe-7.
In each directory, run sh build.sh and sh run.sh; then when all calculations are finished, run sh gsfe_curve.sh. That way, we will get four more GSFE curves, and hence four more USFE values. Calculate the mean USFF value based on the five USFE values and report the mean value in the paper.
The first step is to determine the chemical potential difference between Mo and Nb, and that between Mo and Ta, respectively. To this end, run a hybrid molecular dynamics (MD) / Monte Carlo (MC) simulation in a semi-grand canonical (SGC) ensemble using lmp_sgc.in and CrMoNbTaVW_Xu2022.eam.alloy.
Once the simulation is finished, we will find a file statistics.dat, which should contain one line:
-0.021 0.32 0 0.0005 0.9995
The first two numbers are the two chemical potential differences we provided in lines 10 and 11 of lmp_sgc.in, while the last three numbers are the concentrations of Mo, Nb, and Ta, respectively. Since they are not close to equal-molar, modify the two numbers in lines 10 and 11 of lmp_sgc.in, and run the simulation again. We can make the modification in the same folder and a new line will be appended to statistics.dat once the new simulation is finished. Therefore, no need to delete the file statistics.dat each time we submit a new job. Iteratively adjust the two numbers until the material is almost equal-molar, meaning that the three concentrations should be between 3.18 and 3.48 simultaneously. The procedure is similar to what is described in Section B.2 of this paper.
Note: either chemical potential difference should not have a too small absolute value; otherwise the next step will not work properly.
Once the two chemical potential differences are identified, change the two chemical potential differences in lines 10 and 11 in file lmp_vcsgc.in to the opposite of the values identified earlier. Then run the atomsk script, atomsk_Mo.sh to build a Mo structure named data.Mo.
Next, make two changes to data.Mo:
-
Line 4. Change the first number
1to3 -
Line 12 contains the atomic mass of Mo. Add two lines after it, i.e.,
Masses 1 95.96000000 # Mo 2 92.90638000 # Nb 3 180.94788000 # Ta Atoms # atomic
Next, run a hybrid MD/MC simulation in variance constrained semi-grand canonical (VC-SGC) ensemble using lmp_vcsgc.in, data.Mo, and CrMoNbTaVW_Xu2022.eam.alloy.
Once the simulation is finished, we will find a file data.MoNbTa_CSRO, which is the CSRO structure annealed at 300 K, and a file cn.out.
We can also check whether the potential energy converges to a constant. For that, plot a curve with pe as the y axis and step as the x axis. We can find pe and step in the log file; only use the data in the first run. The curve may look like Figure 1(a) of this paper, which is for CoCrNi.
Use the data file data.MoNbTa_CSRO to calculate its lattice parameters and elastic constants at 0 K, 300 K, 600 K, 900 K, and 1200 K. Also calculate its GSFE at 0 K.
Note that the calculated elastic constants are in the [11-2]-[111]-[1-10] system, and so they should be converted to those in the [100]-[010]-[001] system.
In all calculations, use the same method for CoCrNi. Remember to modify the input files accordingly and use the appropriate potential.
The initial trial lattice parameter is 3.135 Angstrom, but after running LAMMPS simulations, it might change. The new trial lattice parameter can be calculated by
(lx/(10*sqrt(6.))+ly/(46*sqrt(3.)/2.)+lz/(14*sqrt(2.)))/3.
where lx, ly, and lz can be found in the data file data.MoNbTa_CSRO, i.e.,
lx = xhi - xlo
ly = yhi - ylo
lz = zhi - zlo
In particular, when calculating the lattice parameter at 0 K, additionally change line 44 of lmp_0K.in to
variable lat_para equal (lx/(10*sqrt(6.))+ly/(46*sqrt(3.)/2.)+lz/(14*sqrt(2.)))/3.
Based on the file lmp_gsfe.in for CoCrNi, make the following changes:
- line 13. Use the data file for MoNbTa
- line 17. Use the potential file for MoNbTa
- line 32. Use the lattice parameter for MoNbTa
- line 35. Change the last part to
${tmp2}-1.+${latparam}/sqrt(2.)*(${d}-10) - line 57. Change the last part to
(${latparam}*sqrt(3)/2)/${stepn}
Then iteratively change the value of d in line 34 to obtain 20 GSFE curves.
Follow the steps in the random MoNbTa case to calculate the WC parameters in the CSRO MoNbTa structure.
Eventually, use all WC parameters to make a plot similar to Figure 2(d) of this paper.
MoNbTa contains three pure metals having the same lattice as the alloy. It would be interesting to compare the temperature effect between them and the MPEA.
Lattice parameters, elastic constants, and USFEs of the three pure metals are in Mo.txt, Nb.txt, and Ta.txt. They were studied at 0 K, 300 K, 600 K, 900 K, and 1200 K, respectively. The only exception is that Nb becomes unstable at 1200 K so there is no data for that case.
Calculations at 0 K were performed when preparing this paper, although USFE data were not reported there. All data at finite temperatures are newly calculated for the current project. In all cases, simulation cells with size D (see Table II of the paper) were used.
Lattice parameters, elastic constants, and USFEs are summarized in the directory HfMoNbTaTi in this GitHub repository. USFEs, taken on the {110} plane, are in units of mJ/m2.
The random material was studied at 0 K, 300 K, 600 K, 900 K, and 1200 K, respectively. Results are in data_random.txt.
Following this paper, four levels of CSRO were considered, with the material annealed at 300 K, 600 K, and 900 K, respectively. In what follows, let's call them 300KMDMC, 600KMDMC, 900KMDMC, respectively.
All materials were studied at 0 K, 300 K, 600 K, 900 K, and 1200 K, respectively. Results are in data_300KMDMC.txt, data_600KMDMC.txt, and data_900KMDMC.txt.
Calculations at 0 K were performed when preparing this paper, although USFE data were not reported there. All data at finite temperatures are newly calculated for the current project. In all cases, simulation cells with size D (see Table II of the paper) were used.
Lattice parameters, elastic constants, and USFEs are summarized in the directory HfNbTaTiZr in this GitHub repository. USFEs, taken on the {110} plane, are in units of mJ/m2.
The random material was studied at 0 K, 300 K, 600 K, 900 K, and 1200 K, respectively. Results are in data_random.txt.
Following this paper, four levels of CSRO were considered, with the material annealed at 300 K, 600 K, and 900 K, respectively. In what follows, let's call them 300KMDMC, 600KMDMC, 900KMDMC, respectively.
All materials were studied at 0 K only. Results are in data_300KMDMC.txt, data_600KMDMC.txt, and data_900KMDMC.txt.
If you use any files from this GitHub repository, please cite
- Subah Mubassira, Wu-Rong Jian, Shuozhi Xu, Effects of chemical short‑range order and temperature on basic structure parameters and stacking fault energies in multi‑principal element alloys, Modelling 5 (2024) 352--366