For this problem we want to find the formation energy of bcc CuPd alloy from fcc Cu and fcc Pd. Then use that to justify why CuPd is the favored low temperature crystal structure of Pd and Cu when they are mixed with this stoichiometry.
Methods
The optimal lattice parameter for fcc Cu, fcc Pd and bcc CuPd was be found, then using the optimal lattice parameter the cohesive energy of each crystal was calculated.
The exchange-correlation functional used was PBE, the pseudopotential used was OTFG ultrasoft and the relativistic treatment was that that of Koelling-Harmon. The pseudo atomic configuration of the pseudo potential was 3d10 4s1 for the copper atom and 4s2 4p6 4d10 5s0.5 for the palladium atom.
Convergence tests with respect to the number of k-points and the energy cutoff energy were preformed for each of the three crystals and the results of which can be found in the following section. To get a convergence of 0.01eV in the cohesive energy a energy cutoff of 4000eV and a k-point grid of 10x10x10 were used. Also, the SCF tolerance was set to 1.0e-6eV/atom with a convergence window of three steps, all other parameters where kept to the quality fine preset for CASTEP.
For the calculation the space group for Cu and Pd was Fm-3m and Pm-3m for the CuPd alloy.
Convergence
k-points
For each crystal the k-point grid was varied and the energy calculated, the results shown in Figures 1,2 and 3.
Figure 1: fcc Cu Cohesive Energy vs. Number of k-points, a=3.615Å and Ecut=408.2eV
Figure 2: fcc Pd Cohesive Energy vs. Number of k-points, a=3.891Å and Ecut=560.6eV
Figure 3: bcc CuPc Cohesive Energy vs. Number of k-points, a=3.012Å and Ecut=560.6eV
From these three graphs one can see that a k-point grid of 10x10x10 will converge the cohesive energy of all the crystals to at least 0.01ev
Energy Cutoff
For each crystal the energy cutoff was varied and the energy calculated, the results shown in Figures 4, 5 and 6. The k-point grid was selected to be 11x11x11 for each of the crystals to ensure that the convergence of the k-point grid did not effect the convergence of the energy cutoff.
Figure 4: fcc Cu Cohesive Energy vs. Energy Cutoff, a=3.615Å and k-points 11x11x11
Figure 5: fcc Pd Cohesive Energy vs. Energy Cutoff, a=3.891Å and k-points 11x11x11
Figure 6: bcc CuPd Cohesive Energy vs. Energy Cutoff, a=3.012Å and k-points 11x11x11
From these three graphs we can see that an energy cutoff of 5000eV converged the cohesive energies to at least 0.02eV, the limiting case being the CuPd alloy in Figure 6.
Optimization
For each of the three crystals, the lattice parameter was optimized in CASTEP, using the exchange correlation functional PBE, a k-point grid of 10x10x10, a cutoff energy of 5000eV, the BFGS algorithm for geometry optimization with full cell optimization. For the convergence tolerance for the energy was 1e-5 eV per atom, for the max force was 0.03 eV per Å, for the max stress was 0.05 GPa, and for the max displacement was 0.001 Å.
For both the fcc Cu and fcc Pd crystal CASTEP changed the unit cell to reduce the computational cost, in the new cell a, b and c are still equal, but the angles α, β, γ, changed from 90° to 60°.
The results of the optimizations are listed in Table 1.
Table 1: Lattice parameters after optimization of the crystals unit cell| Crystal | Lattice Parameter |
|---|---|
| fcc Cu | 2.566 |
| fcc Pd | 2.786 |
| bcc CuPd | 3.013 |
Results
For each of the three crystals the energy of the crystal was taken from the last cycle of the geometry optimization and can be found in Table 2.
Table 2: Cohesive Energies of optimized crystal structures, the Final Free Energy is per unit cell| Final free energy (E-TS) (eV) | Energy of Free Cu atom (eV) | Energy of Free Pd atom (eV) | Cohesive Energy (eV) | |
|---|---|---|---|---|
| Cu | -1680.93 | -1677.14 | -3.79 | |
| Pd | -3493.26 | -3490.46 | -2.79 | |
| CuPd | -5174.43 | -1677.14 | -3490.46 | -6.83 |
From the cohesive energies in Table 2, the formation energy of the bcc CuPc alloy was found to be -0.24eV.
From the literature we know that the bcc CuPd alloy is the favored low temperature crystal structure of Pd and Cu when they are mixed with this stoichiometry, this would mean that the CuPd alloy is more thermodynamically stable than the two separate mono-atomic crystals, leading to a formation energy that is negative. This matches with the results of the calculations.
