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Ordering Tendencies in Pd-Pt, Rh-Pt, and Ag-Au Alloys Z.W

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Ordering Tendencies in Pd-Pt, Rh-Pt, and Ag-Au Alloys Z.W Section I: Basic and Applied Research Ordering Tendencies in Pd-Pt, Rh-Pt, and Ag-Au Alloys Z.W. Lu and B.M. Klein Department of Physics University of California Davis, CA 95616 and A. Zunger National Renewable Energy Laboratory Golden, CO 80401 (Submitted October 7, 1994; in revised form November 9, 1994) First-principles quantum-mechanical calculations indicate that the mixing enthalpies for Pd-Pt and Rh-Pt solid solutions are negative, in agreement with experiment. Calculations of the diffuse-scatter- ing intensity due to short-range order also exhibits ordering tendencies. Further, the directly calcu- lated enthalpies of formation of ordered intermetallic compounds are negative. These ordering tendencies are in direct conflict with a 1959 prediction of Raub that Pd-Pt and Rh-Pt will phase-sepa- rate below ~760 ~ (hence their mixing energy will be positive), a position that has been adopted by all binary alloy phase diagram compilations. The present authors predict that Pdl_xPtx will order in the Llz,Llo, and L12 structures ([001] superstructures)at compositionsx = 4' 2' and 4' respectively, I I while the ordered structures of Rhx_/Ptx are predicted to be superlattices stacked along the [012] di- rections. While the calculated ordering temperatures for these intermetallic compounds are too low to enable direct growth into the ordered phase, diffuse-scattering experiments at higher tempera- tures should reveal ordering rather than phase-separation characteristics (i.e., off-F peaks). The situation is very similar to the case of Ag-Au, where an ordering tendency is manifested both by a dif- fuse scattering intensity and by a negative enthalpy of mixing. An experimental reexamination of Pd- Pt and Rh-Pt is needed. 1. Introduction ing Pd-Pt and Rh-Pt as phase-separating (miscibility gap) sys- tems. In 1959, Raub 1 suggested the existence of miscibility gaps in First-principles quantum mechanical stability calculations in the Pd-Pt and Rh-Pt systems with fairly high maximum misci- conjunction with cluster expansions ~6 offer an alternative bility gap temperatures of 770 and 760 ~ respectively. He in- method to the phenomenological approaches.8, 9 Here, note ferred these results from the assumed correlation between the that such calculations 7 predict Pd-Pt and Rh-Pt to exhibit or- maximum miscibility gap temperature and the difference in dering rather than phase separating at low temperatures. This melting points of the constituents and from the experimentally conclusion comes from the first-principles quantum mechani- observed miscibility gaps in similar alloys, such as Pd-Rh, Ir- cal calculations 7 backed up by experimental evidence, 17-19 Pt, and Ir-Pd. 1Attempts to observe these miscibility gaps have, which seems to have been largely overlooked by the alloy however, consistently failed. 2-5 In agreement with the conjec- phase diagram community. The situation in Pd-Pt and Rh-Pt ture of Raub, tight-binding d-band only alloy theories 6 pre- systems is very similar to that in Ag-Au where an ordering ten- dicted phase-separation for all d-electron-rich alloys, e.g. the dency 20 was recognized by ample experiments 21-24 and theo- late transition metal alloys (because the anti-bonding states are retical calculations 25-30 despite the fact that the ordering partially occupied). That this is not a compelling argument is temperatures24. 30 are too low to actually grow ordered com- clear from the fact that Ni-Pt orders strongly despite its high d- pounds. An experimental reexamination of the phase stability electron count. The reason for the failure of such arguments of Pd-Pt and Rh-Pt is thus needed. was recently clarified. 7 Phenomenological theoriesS, 9 are not clear on whether Pd-Pt and Rh-Pt should phase-separate or or- The following gives a brief account of the theoretical strategy der: Miedema's models predicts AH = +(1 to 3) kJ/mol for Pd- and then presents the results and discussion. For comparison Pt and AH --- -(2 to 3) kJ/mol for Rh-Pt, both values being close and as a "sanity test," parallel calculations are also conducted to the limit of accuracy of the model. Thus, neither experi- for Pd-Rh, which is known experimentallyl, 31,32 and theoreti- ment 2-5 nor simple phenomenological theories s shed light on cally 7,33-35 tO phase separate. In the Rh-Pd-PI triangle, Pd-Rh whether these systems order or not. Yet, all phase-diagram should phase separate (in agreement with experiment), but Pd- compilations, 10-15 including the latest assessments by Massal- ~t and Rh-Pt should order (in distinction to current phase-dia- ski et al. 13 and Okamoto, 14A5 have adopted Raub's view, label- gram assessments). 36 Journal of Phase Equilibria Vol. 16 No. 1 1995 Basic and Applied Research: Section I 2. Method of Calculation search by expanding Eq 1 in a finite Ising-like cluster expan- sion (CE): 2.1 Cluster-Expanding Lattice Quantities 2; (zq 3) Short- and long-range order (SRO and LRO) in binary AI~,:B x 1 f systems can be interpreted in terms of spin--~ lattice models in where Jj(V) are volume-dependent interaction energies of which each lattice site i (i = 1..... N) is labeled by a spin vari- some basic lattice figures f (e.g., nearest-neighbor pairs, next able, ~i, taking on the value -1 (+ 1) if site i is occupied by atom neighbor pairs, triangles, etc.) as shown in Fig. I. Their coordi- A(B). The excess energy AE(c~,V) of any of the 2 N configura- n__ates are defined in Table 1. The lattice averaged spin products tions (y obtained by occupying N sites by A or B is: [If (13) are the product over the figurefof the variables ~i, av- eraged over all symmetry equivalent figures of the lattice. All AE(~,V) = E(G,V) - [(1 -X)EA(VA) +xEB(VB)] (Eq I) of the terms on the right-hand side of Eq 3 are trivially deter- The excess energy in Eq 1 depends on the volume, V, and is de- mined geometrical quantities, with theexception of Jf, the ef- fined with respect to the energies of equivalent amounts of the fective interaction energies. Since {Hf(~)} is a complete, pure solids A and B at their respective equilibrium volumes VA orthonormal set of polynomials, the expansion Eq 3 is exact, 41 and liB. The low-temperature ground state structures of a given if not truncated. In practice, one hopes that this series con- lattice type are then interpreted as the configurations 6 that verges reasonably rapidly so only -O(10) interaction are re- give the lowest AE(G,Vo) at the equilibrium volume Vo. For tained (the first few pair interactions, as well as many-body phase-separating systems (miscibility gap) AE(cy,V) are posi- tive for all structures, while for an ordering system, AE(cy,V) are negative for at least the ground-state structures. Cluster Figures in FCC ] Advances in the electronic structure theory36 now enable prac- tical and accurate quantum-mechanical calculations of the to- tal energy of simple configurations G: A <tlJIHIW> Ed,rect(G,V) = <tls[ tlJ> (Eq 2) where H and te are the Hamiltonian and wave function of the electronic system. In practice, most calculations are performed using the density functional formalism,37.38 as implemented numerically by, for instance, the highly precise, full-potential, linearized augmented plane wave (LAPW) method. 39.4~Such calculations include electrostatic, ionic, and exchange-corre- lation effects. They avoid tight-binding or spherical-potential approximations and account for metallic, ionic, and covalent Fig. 1 Atomic figures used in the cluster expansion, which in- bonding. However, since quantum-mechanical calculation of clude pair figures, three-body figures, and four-body figures. Ta- an astronomic number of configurational energies AE(o,V) is ble 1 defines the figure coordinates. prohibitive, one approach is to perform such a ground state Table 1 Definition of the Figures fUsed in Cluster Expansion in Terms of the Vertices of the fcc Structure Figure type Designation Vertices Pd-Pt Rh-Pt Pd-Rh Ag-Au Empty ........................................ J0 -28.2 -11.6 65 0 .-43 41 Point ....................................... Jt (000) -3. l 4.2 1.9 1.95 Pairs ...................................... J, (000),(110) 22 6 43.2 49.2 41 l0 K 2 (000),(200) 4).7 0.1 0.4 0.7 l L 2 (000),(211 ) 10.9 -2 1 -9.9 3 65 M~ (000),(220) -1 0 8 3 10.4 -0.48 N 2 (000),(310) ......... -0.03 O 2 (000),(222) ........ -0.89 Triplets ..................... "].3 (000),(1101,(10l) 3.5 -6 6 0.7 -0 56 K 3 (000),(110),(200) ........ -2.22 L 3 (000),(l 10),(211 ) ... 0.89 Quadruplets ........... Ja (000),( 110),(101 ),(011) 0'3 0.7 -39 -031 , a 1 Note: Verhces are in umts of~, where a is the lattice parameter See Fig 1 The effective T = 0 cluster lnteracnons Df.lf(m meV/atom) at x = 97 [or corresponding to V = (Va + V,~)/2I, are hsted The dominant negative (positive) J2 denote phase-separating (compound-fonmng) mteracttons. One meV/atom equals 9.649 x 10 2 Id/mol. Journal of Phase Equilibria Vol. 16 No. 1 1995 37 Section I: Basic and Applied Research terms). If the series Eq 3 converges after M terms, one can cal- nuity in the internal energy as a function of temperature and the culate an equivalent number of interaction energies {Jr(V) } by ground state detennined by the state of the simulation at a tem- equating Eq 2 with Eq 3.
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