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Equilibrium chemical order and segregation at alloy surfaces

Equilibrium chemical order and segregation at alloy surfaces and nanoclusters computed using tight-binding derived coordination-dependent bond energies Micha Polak Department of Chemistry, Ben-Gurion University Beer-Sheva, ISRAEL. ACS meeting San Francisco – September 12, 2006.

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Equilibrium chemical order and segregation at alloy surfaces

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  1. Equilibrium chemical order and segregation at alloy surfaces and nanoclusters computed using tight-binding derived coordination-dependent bond energies Micha Polak Department of Chemistry, Ben-Gurion University Beer-Sheva, ISRAEL ACS meeting San Francisco – September 12, 2006

  2. Motivation • Various applications of alloy nanoclusters in heterogeneous catalysis, magnetic media, etc • Full atomic scale chemical-structural information for alloy nanoclusters is inaccessible by current experimental techniques 

  3. “magic-number” cuboctahedrons (COh) 13 55 147 309 36 concentric shells (sites) around center atom: 0-23 constitute the core 24-28,31 - (100) 29,30,32 - (111) 33-35 - edge 36 - vertex surface 561 13 inequivalent sites (362 atoms) 923

  4. The computational approach: Energetics Surface/subsurface bond energy variation (2-layer) model with data computed by DFT-based Tight-Binding method (NRL-TB) Statistical Mechanics The “Free energy Concentration Expansion Method” (FCEM) adapted to a systemof atom-exchanging equilibratednanoclusters Computational Results: I. Surface segregation profiles for Pt25Rh75(111) – a test case II. Binary & ternary Rh-Pd-Cu 923 atom cuboctahedral clusters 1. Site specific concentrations, surface segregation, core depletion and order-disorder transitions (highlighting bond energy variation effects) Cluster thermodynamic properties: 2. Entropy, Internal-Energy: configurational heat capacity III. Mixing Free-Energy: inter-cluster separation

  5. The alloy systems: basic empirical interatomic energetics (meV) 5750 Rh Cohesive energy (related to ) V<0, endothermic alloying (“demixing” tendency) based on experimental heat of mixing -23 -35 Pd 3890 +33 Cu 3490 V>0 , exothermic alloying (“mixing” tendency)

  6. Energetics Elemental surface-subsurface NN bond energy variation model outmost layer (l=1) subsurface layer (l=2) “bulk” Surface intra-layer and inter-layer elemental bonds are typically stronger than the bulk value

  7. TB computed variations of elemental NN bond energies vs the number of missing bonds after Michael I. Haftel et al, Phys. Rev. B 70, 125419 (2004) Rh Cu Pd (100) (111) surf sub (111) surf sub (100) dw (eV) (110) surf sub (110) DZp # missing bonds of NN pairs, DZp DZp Rh surf Pd surf Parabolical fit to □ Parabolical fit to □ dw (eV) DZp DZp - TB computations for elemental clusters: after C. Barreteau et al. Surf. Sci. 433/435, 751 (1999).

  8. Statistical mechanics FCEM adapted to alloy clusters The FCEM expressions were obtained using NN pair-interaction model Hamiltonian and expanding the free energy in powers of constituent concentrations. The free energy of a system of multi-component alloy clusters capable of atomic exchange: • concentration of constituent I in shell p • number of atoms belonging to shell p • number of nearest-neighbor pairs of atoms belonging to shells p,q (related to coordination numbers) • elemental pair interaction energy for constituent I - heteroatomic interaction and effective interaction energies between constituents I and J advantages: This analytical formula (that takes into account inter-atomic correlations) makes FCEM much more efficient than computer simulations. It can yield large amounts of data: site-specific concentrations and corresponding thermodynamic properties vs. cluster size (up to ~1000 atoms), multi-component composition and temperature

  9. Pt25Rh75(111) as a test case Rh Medium Energy Ion Scattering (MEIS) surf D. Brown et al, Surf. Sci. 497 (2002) 1 dw (eV) sub 1300 K strengthening DZp Pt sub weakening dw (eV) surf Very small V ~ 4 meV, high temperature Are surface-subsurface bond strength variations responsible for the subsurface oscillation? DZp

  10. Computational Results Part I. Surface segregation profiles for Pt25Rh75(111) FCEM (no adjustable parameters): Single layer tension model (SL) Bond energy variations and corresponding layer tension differences Two layer tension model (TL) MEIS: D. Brown et al, Surf. Sci. 497 (2002) 1 (meV) 1300 K Temperature evolution of layer compositions LEED: E. Platzgummer et al, Surf. Sci. 419 (1999) 236 1373 K In the SL model ignoring surface-subsurface bond variations, the subsurface oscillation due to V only is much weaker than in the TL model at all temperatures

  11. Computation procedure for clusters Input: - Cluster geometrical parameters - Energetic parameters Free energy numerical minimization (MATLAB - including Genetic Algorithm confirmation, under the constraint of conservation of the system overall concentration) Output: - set of all site/shell concentrations (e.g., 37 inequivalent sites, 72 independent variables 111 concentrations for ternary COh-923) Cluster thermodynamic functions

  12. Part II. Cluster site specific concentrations, ordering and configurational heat capacity Rh791Cu 132 Rh561Cu 362 1. The case of Rh-Cu (V<0) Surface/core segregation/separation and surface “demixed order” at compositional “magic numbers” 923-COh Rh inclusion (Rh78Cu845) Cu Rh The corresponding heat capacity curves: Cuedge → Cu(100) Cusurf → Cucore Cuedge/vex → Cucore

  13. The surface desegregation process

  14. (Tmax ,Cmax) E vert deseg edge deseg (100) deseg (111) deseg surf segregated T0 Configurational heat-Capacity Schottky anomaly in alloy nanoclusters (111) Rh561Cu362923-COh qCu=1 (100) Desegregation contribution to the cluster heat capacity. The lowest level in the energy scheme corresponds to completely Cu surface segregated cluster. Desegregation excitations of single Cu atom to the Rh core are indicated by vertical arrows. T0 signifies the onset of the desegregation effect involving the lowest (111) excitation.

  15. Cmax & number of deseg. excitations per atom vs. cluster size Rh-CuCOhqCu=1 309 561 923 147 55 Cmax, J/mol/K nsnc Cmax, J/mol/K nsnc ns – fraction of surface sites, nc – fraction of core sites 13 Number of cluster atoms

  16. Order-disorder transitions and desegregation in “magic number” Pd618/923Cu305/923 COh clusters 2. The case of Pd-Cu (V>0) L12-like ordered core (cross-section) Surface “mixed” order Schottky type configurational heat capacity Overall and sublattice concentrations FCEM computations based on NRL-TB energetics ( ) FCEM computations based on simple bond breaking energetics (uniform bond-strength, )

  17. 3. Ternary clusters Surface order-disorder transitions and desegregation “Magic number” Rh561/923Pd150/923Cu212/923vs. Pd618/923Cu305/923 923-COh Pd Cu Core Rh Pd618/923Cu305/923(“substrate” effect)

  18. Part III. Inter-cluster “phase” separation: The case of Rh-Pd (V<0) Mixing free-energies computed for 147-COhclusters Fmix=F-(cRhFRh+cPdFPd) Rh inclusion Rh Pd Convexity between “magic-number” compositional structures (demixed order) inter-cluster separation

  19. Concluding Remarks • The test case for the FCEM/TB approach: good agreement between the two-layer oscillatory profile computed for Pt25Rh75(111) surface and reported experimental data, highlighting the role of subsurface tensions • The relatively high efficiency of FCEM in computing binary & ternary alloy nanocluster compositional structures and related thermodynamic properties enables to predict a variety of phenomena: • Cluster ordering involving “magic-number” low-temperature structures that exhibit - core & segregated surface order-disorder transitions, - enhanced elemental segregation due to preferential surface bond strengthening (Pd-Cu), • Configurational heat capacity Schottky-type anomaly: reflect distinctly thevarious atomic exchange excitation processes: C vs. T experimental measurements are expected to elucidate the energetics of alloy cluster surface segregation (via desegregation peaks) & order- disorder transitions • Surface-Segregation related intra &inter-cluster separation (Rh-Pd) • Ternary alloying effects on surface transitions and segregation

  20. Relevant publications: M. Polak and L. Rubinovich, Surface Science Reports 38, 127 (2000) L. Rubinovich and M. Polak, Phys. Rev. B 69, 155405 (2004) M. Polak and L. Rubinovich, Surf. Sci. 584, 41 (2005) M. Polak and L. Rubinovich, Phys. Rev. B 71, 125426 (2005) L. Rubinovich, M.I. Haftel, N. Bernstein, and M. Polak, Phys. Rev. B 74, 035405 (2006) NRL M. Polak and L. Rubinovich, (submitted to Phys. Rev. B, 2006) Future Plans: 1. Refinement of FCEM energetics: TB-computed bonding in clusters, including also: Hetero-atomic interactions; NNN pairs; deeper subsurface layers; Higher accuracy by inclusion of on-site contributions 2. Comparative computations for icosahedrons 3. Effects of chemisorption (O,S) 923 atom icosahedron This research is supported by THE ISRAEL SCIENCE FOUNDATION

  21. Thank you!

  22. Bond energy between atoms i and j from states a and b is estimated as the corresponding contribution to ( - the density-operator) (the summation is over atom labels i , j and state labels a, b, and implicitly includes an integral over k) denote orbitals and angular momenta, s,p,d : pair contributions to the total bond energy ss, sp, pp, sd, pd, dd (M. I. Haftel, N. Bernstein, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B 70, 125419 (2004).) The system is described by an ensemble of quantum states with probabilities , The density-operator

  23. Effects of elemental bond-energy variations on Pd-Cu cluster surface segregation Schematics of two models ((100) face) Simple bond breaking ( ) Two layer tension model ( ) Esite Pd surf subsurf bulk Cu segregation Extra Cu segregation Cu depletion Cu Extra Cu enrichment in surface layer & subsurface oscillation (core depletion) surf subsurf bulk depth

  24. Effects of cluster size on Cmax Estimation of the number of surface-core desegregation excitations Rh Cu Initial fully segregated state Final randomized state The number of excitations (core-surface atomic exchanges): The number of excitations per atom: ncore – fraction of core sites, nsurf – fraction of surface sites

  25. Introduction Typical shapes of free clusters numbers and colors mark distinct “surface” shells (sites)

  26. DFT formalism DFT key variable is the electron density, The Kohn-Sham equation is solved in a self-consistent (iterative) way: - An initial guess for - Calculation of the corresponding Kohn-Sham potential - Solution of the Kohn-Sham equation - The eigenvalue spectrum and the orbitals (of an auxiliary non-interacting system (1-electron Hamiltonian), which reproduce the density of the original many-body system) - Calculation of a new density The procedure is repeated until convergence is reached

  27. DFT formalism (continue) Solution of Kohn-Sham equations by augmented plane-wave method (APW) Plane waves (PW’s) - inefficient basis set for describing the rapidly varying wave function (around the nuclei) In the APW scheme the unit cell is divided into two regions (mixed basis set): (i) The muffin-tin (MT) region which consists of spheres centered at the nuclear position, inside which the APW’s satisfy the atomic Schrodinger equation (ii) The interstitial region I, where the APW’s consist of PW’s, Notes: - Only the density has strict physical meaning in the Kohn-Sham equations. - Eigenvalues of an auxiliary single-body Schrodinger equation are artificial objects - Total energy is not simply the sum of all : the integral is over the first Brillouin zone, the first sum is over occupied states • a functional of the density (includes the repulsion of the ionic cores, correlation effects, and part of the Coulomb interaction)

  28. The Naval Research Laboratory tight-binding (NRL-TB) method R. E.Cohen, M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 50 14694 (1994) M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 54 4519 (1996) NRL-TB maps results of a limited set of first-principles calculations to a two-center non-orthogonal Slater-Koster TB Hamiltonian: 1) Construction of a first-principles database of eigenvalues εn(k) and total energies E. 2) Shift of the Kohn-Sham potential by ( - the number of electrons in the unit cell) 3) Definition of shifted the eigenvalues, corresponds to the self-consistent charge (SCC-TB) correction The total energy is simply the sum, 4) In the two-center TB approximation, are dependent on the terms, ( - the two-center part of the Hamiltonian) The integral depends on quantum numbers denoting orbitals and angular momenta, s,p,d, and on the component of the angular momentum relative to the direction (specified by ) and, for non-orthogonal orbitals, are dependent on the terms, and are parameterized in order to reproduce the eigenvalues (for ss, sp, pp, sd, pd, dd at a large number of k-points for fcc and bcc structures for several volumes each).

  29. 3D representation of thermodynamic functions of ternary clusters elucidating composition-dependent properties (Rh-Pd-Cu 147-COh) Mixing free-energy and configurational entropy plotted with respect to the concentration Gibbs triangle 1000 K 10 K 1000 K 500 K - Convexity in Fmix indicates inter-cluster separation. Minima in S indicate intra-cluster separation or ordering - Note: hundreds computed data points constitute each plot

  30. Pt& Rh (111) surface & subsurface layer tensions oscillatory profile (meV) Schematics of two models Two layer tension model (TL) Single layer tension model (SL) El Rh Eb Pt Pt enrichment Eb Pt depletion Layer 1 2 3

  31. C vs. T curves for different overall compositions Rh-Cu 923-COh Surface-core processes with increasing desegregation excitation energies and Tmax Cu (111)→Cu core Intrasurface exchange processes E Rh561Cu 362 (qCu=1) Cu (100)→Cu core (111) (100) Rh641Cu 282 edge Cu edge→Cu(100) Cu(100)→Cu(111) Cu edge/vex→Cu core Rh791Cu 132

  32. The Naval Research Laboratory tight-binding (NRL-TB) method R. E.Cohen, M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 50 14694 (1994) M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 54 4519 (1996) NRL-TB maps results of a limited set of first-principles calculations to a two-center non-orthogonal Slater-Koster TB Hamiltonian: - Construction of a first-principles database of eigenvalues εn(k) and total energies E. • Finding “shift potential” V0 and shift the eigenvalues in order to get total energy: • Finding a set of parameters which generate non-orthogonal, two-centre Slater-Koster Hamiltonians H which will reproduce the energies and eigenvalues in the database. Estimation of bond energy: Effective bond energy between nearest neighbor (NN) atoms i and j from states a and b is defined as the corresponding contribution to E: - the Hamiltonian operator - the density-operator, (M. I. Haftel, N. Bernstein, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B 70, 125419 (2004).)

  33. The total energy of the system in the TB method: (the summation is over atom labels i , j and state labels a, b, and implicitly includes an integral over k as well) - the density-operator - the Hamiltonian operator, - wave-functions associated with atomic orbitals - lattice vector - atom position - Bloch vector

  34. SRO in an alloy with LRO: - 100% probability - smallest probability

  35. Relevant bulk phase diagrams Rh-Pd Rh-Cu Pd-Cu

  36. The Statistical-Mechanical Theory

  37. Background: The segregation process

  38. “Magic number” Rh561Pd302Cu60 923-COh clusters Surface order-disorder transition and desegregation Pd-Cu edge disordering Edge-vex order (40 K) Pd Cu Core Rh Pd-Cu desegregation Pd-Cu site competition and co-desegregation at vertexes Pd-Cu edge disordering Pd863 Cu60(“substrate” effect)

  39. Rh inclusion in Rh78Cu845 923-COh Rh

  40. Rh core inclusion in Rh19Pd128 147-COh

  41. The attraction of a solute atom to local compositional fluctuations (SRO) in a binary alloy segregation suppression due to higher atomic bulk coordination

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