First-principles study of the Cu-Pd phase diagram
F. Geng, J. R. Boes and J. R. Kitchin, “First-principles study of the Cu-Pd phase diagram”, Calphad 56, 224-229 (2017). https://doi.org/10.1016/j.calphad.2017.01.009
This summary was written by a language model from the paper and has not been checked line by line. Rely on the published paper for details.
Question
Pd-Cu alloys are candidate hydrogen separation membranes because Cu improves sulfur tolerance over pure Pd. The structure matters: the ordered B2 (bcc) phase has much faster hydrogen diffusion than fcc, while the fcc phase tolerates H2S better, and B2 converts to a disordered fcc solid solution above roughly 833 K. The experimental boundaries between the fcc and bcc regions are not fully established because the transformation is slow, especially above 50 at% Pd. An earlier cluster expansion plus Monte Carlo (CE+MC) study of Cu-Pd that included only configurational free energy did not reproduce the order-disorder temperature or the shape of the fcc/bcc boundary. The paper asks whether a first-principles CE+MC approach, with vibrational free energy added through an approximate model, can reproduce the Cu-rich side of the phase diagram (Geng et al. 2017).
Approach
DFT calculations used VASP with PBE and PAW potentials, a 500 eV cutoff and at least 4000 k-points per reciprocal atom, converging total energies to about 2 meV/atom. Separate cluster expansions were built on the fcc and bcc lattices with ATAT and MAPS. Structures were first relaxed at constant shape; those poorly predicted by the CE were then fully relaxed, and any whose relaxation strain tensor norm exceeded 0.05 were dropped as no longer belonging to the parent lattice. Effective cluster interactions were chosen by minimizing the leave-one-out cross-validation score, then pruned by hand by removing four-body terms until the score or ground-state predictions worsened. Vibrational free energy was estimated with the stiffness versus bond length method: force constants from small-displacement supercell calculations on a few ordered structures per lattice (four fcc, five bcc) were fitted as polynomials in bond length, and thermal expansion was included in a quasi-harmonic way with a 1% volume expansion. The fcc/B2 phase boundary was traced with the ATAT PHB Monte Carlo program, which integrates the chemical potential along temperature starting from a 0 K common tangent (Geng et al. 2017).
Key findings
Computed lattice constants and formation enthalpies agreed closely with earlier PBE values, for example about -119 meV/atom for B2 CuPd and -103 meV/atom for L12 Cu3Pd relative to fcc elements. The final expansions used 69 fcc and 65 bcc configurations (up to eight atoms), with 40 clusters on fcc and 35 on bcc, and cross-validation scores near 5 meV/atom on both lattices.
Without vibrations, MC gave L12 and B2 as the stable ordered phases on the Cu-rich side with a phase diagram shape similar to experiment, but with three discrepancies: an order-disorder temperature near 720 K versus 823 K experimentally, a critical composition near x(Pd) = 0.48 rather than 0.4, and a B2 region that was too narrow. This was still closer to experiment than the 602 K of the earlier CE+MC study. For the force constants, stretching stiffness decreased with bond length for every pair type, while bending terms were small and nearly insensitive, so only stretching was retained.
Adding vibrational free energy widened the B2 region and moved the critical point further toward the Cu-rich side, giving better qualitative agreement, while lowering the critical temperature slightly, to about 700 K (a decrease of roughly 20 K). The B2 widening arises because the ordered bcc structures have lower vibrational free energy than ordered fcc structures between 25 and 50 at% Pd, which the authors tie to different equilibrium bond lengths and hence bond stiffness on the two lattices. They attribute the remaining temperature error to fitting and DFT precision (a 1 to 3 meV/atom change can shift the transition by up to 15 K), the choice of functional, and the neglect of vacancies, and suggest a full phonon treatment might improve agreement further (Geng et al. 2017).
Significance
The work shows that configurational CE+MC alone captures the topology of the Cu-Pd diagram but that vibrational free energy is needed to get the competition between fcc and B2 right, and that a cheap bond-stiffness model can supply it. It extends the group’s Cu-Pd cluster expansions from ground-state searches to finite-temperature phase boundaries relevant to membrane operating conditions.