FROM CATALYTICALLY ACTIVE MATERIALS TO ACTIVE CATALYSIS: A MODEL STUDY

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1 KATHOLIEKE UNIVERSITEIT LEUVEN FACULTEIT INGENIEURSWETENSCHAPPEN DEPARTEMENT CHEMISCHE INGENIEURSTECHNIEKEN W. de Croylaan 46 B-31 Leuven, Belgium FROM CATALYTICALLY ACTIVE MATERIALS TO ACTIVE CATALYSIS: A MODEL STUDY Jury: Prof. J. Berlamont, voorzitter Prof. C. Creemers, promotor Prof. J.W. Niemantsverdriet (T.U. Eindhoven) Prof. P. Wollants (K.U.Leuven) Prof. J. Degrève (K.U.Leuven) Prof. B. Blanpain (K.U.Leuven) Dr. S. Cottenier (K.U.Leuven) Proefschrift voorgedragen tot het behalen van het doctoraat in de ingenieurswetenschappen door: Jan LUYTEN UDC September 27

2 Katholieke Universiteit Leuven Faculteit Ingenieurswetenschappen Arenbergkasteel, B-31 Heverlee (Belgium) Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotocopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaandelijke schriftelijke toestemming van de uitgever. All rights reserved. No part of the publication may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher. D/27/7515/9 ISBN

3 Voorwoord Met het beëindigen van dit doctoraat, komt er ook een einde aan een periode van zes jaar onderzoek aan het departement Chemische Ingenieurstechnieken. Tijdens deze jaren heb ik aanmoediging, steun en hulp van heel wat mensen gekregen. Het lijkt mij dan ook gepast om deze mensen hiervoor te bedanken. In de eerste plaats wil ik mij richten tot mijn promotor Prof. Claude Creemers. Ik ben hem zeer dankbaar voor de unieke kans die hij mij heeft geboden om een doctoraat te maken in dit uiterst interessant vakgebied. Hij schonk me hierbij de nodige vrijheid om mijn doctoraatsplan naar eigen wensen uit te stippelen en hij heeft me hierin ook steeds gesteund. Daarnaast wens ik ook de leden van de begeleidingscommissie, Prof. Jan Degrève en Prof. Patrick Wollants, te bedanken voor hun interesse en constructieve opmerkingen tijdens de tussentijdse voortgangsrapporteringen. Verder ben ik ook de overige leden van de examencommissie, Prof. Hans Niemantsverdriet, Prof. Bart Blanpain en Dr. Stefaan Cottenier dank verschuldigd voor hun interessante vragen en suggesties bij het lezen van het manuscript en tijdens de preliminaire verdediging, alsook voor de tijd die ze vrijmaakten om deel uit te maken van de jury. Het is voor mij een hele eer om dit werk voor deze jury te kunnen verdedigen. Tijdens mijn doctoraatsperiode werkte ik nauw samen met Steve Helfensteyn en met Maarten Schurmans. Van beiden heb ik enorm veel kunnen opsteken, waarvoor mijn oprechte dank. Daarnaast zorgden zij samen met de andere collega s Christine, Herman en Marc voor de noodzakelijke afwisseling tijdens de koffiepauzes. Trakteren met koffiekoeken bij speciale gelegenheden of soms ook omdat het al zo lang geleden was is allicht een nieuwe traditie geworden. Voor de DFT berekeningen ben ik in contact gekomen met een groep van de Technische Universiteit Eindhoven. Bouke Bunnik van de groep Moleculaire Heterogene Katalyse en Prof. Gert-Jan Kramer, thans verbonden aan de afdeling Chemische Reactortechnologie hebben mij ingewijd in deze wondere wereld van ab initio modellering en ervoor gezorgd ik zelf de nodige berekeningen kon uitvoeren. Ik wil hen dan ook bedanken voor de mogelijkheden die ze mij geboden hebben.

4 Via Prof. Kramer ben ik in een later stadium van mijn doctoraat in contact gekomen met Prof. Barend Thijsse, auteur van de CAMELION code voor het uitvoeren van Moleculaire Dynamica simulaties gecombineerd met MEAM. Tijdens een driedaags bezoek aan Delft maakte hij mij wegwijs in deze software. Omdat de duur van een doctoraat beperkt is, kon deze samenwerking nog geen concrete resultaten opleveren. Ik ben Prof. Thijsse echter zeer dankbaar voor de tijd die hij heeft vrijgemaakt en ik ben ervan overtuigd dat de geïnitieerde samenwerking in de toekomst zeker zal kunnen leiden tot interessante resultaten i.v.m. het gedrag van nanopartikels. Ik wil ook Maarten, Kim, Jasper, Gert, Ward en Imme bedanken die ervoor kozen om een eindwerk te maken onder mijn begeleiding. Zij getuigden allen van een tomeloze inzet en hun visies hebben zeker geleid tot bijkomende inzichten. Tenslotte wens ik ook mijn familie en vrienden te bedanken. Zij zorgden voor de broodnodige dosis sport en ontspanning zodat ik daarna weer met een heldere geest aan de slag kon. Hierbij zou ik ook mijn ouders speciaal willen bedanken voor hun steun en niet aflatende aanmoediging. Zij stonden steeds achter mijn keuzes en hebben ervoor gezorgd dat ik in alle vrijheid kon doen wat ik graag deed. Zonder hun steun was dit doctoraat allicht niet tot stand gekomen. Een zeer bijzonder woordje van dank wens ik ook te richten aan mijn vrouw Jozefien. Zij heeft eveneens meer dan haar steentje bijgedragen in de realisatie van dit doctoraat. Ook al probeerden we privé en werk zo veel als mogelijk gescheiden te houden, toch was het enthousiasme (en de frustratie als iets niet meteen lukte) soms te groot zodat wetenschappelijke discussies ook af en toe thuis werden verdergezet. Maar vooral wil ik haar bedanken omdat ze steeds zorgt voor een gezellige omgeving en omdat ze mijn rustpunt was in de hectische voorbije maanden. Heverlee, september 27

5 Abstract Transition metal alloys are frequently used as catalyst material in chemical industry. The structure, order and composition at the catalyst surface are crucial for the catalytic properties. The detailed knowledge of the surface behaviour is a necessary condition for understanding the physico-chemical behaviour of the catalytic process and for an optimal scheduling of further experiments. In this dissertation, a modelling platform is presented for a more realistic simulation of a catalyst in action. First, the description of the metallic bond requires a reliable energy model. This is obtained by parameterising the Modified Embedded Atom Method (MEAM) completely based on a consistent set of ab initio data, including bulk as well as surface material properties. Following this novel approach, new MEAM parameters are determined for Cu, Pt, Pd and Rh and all their binary alloys. Subsequently, these parameters are validated by Monte Carlo (MC) simulations of the segregation to low index surfaces of these binary alloys. For disordered solid solutions (Pt-Rh and Pt-Pd), as well as for phase seperating alloys (Pd-Rh and Cu-Rh) and for ordered alloys (Cu 3 Pt and Cu 3 Pd), the results are in good agreement with the experimental evidence and/or theoretical calculations. The impact of a slight deviation from the exact stoichiometry on the surface composition in ordered compounds is illustrated for off-stoichiometric Pt 8 Co 2 (111) alloys. The segregation study is then extended to ternary systems with a case study for Pt- Pd-Rh alloys, showing the typical multi-component effect of co-segregation. Next, segregation to the ideal single crystal surfaces is confronted with the segregation behaviour in nanoparticles. For pure Cu, Pt and Rh the icosahedron is calculated as the more stable particle, while Pd prefers the cubo-octahedral shape. MC simulations for Cu 3 Pt alloy particles elucidate how two distinct segregation mechanisms dominate at low and at high temperatures respectively. Finally, the presence of gaseous adsorbates is taken into account in a case study on the CO adsorption and the CO oxidation reaction on Cu 3 Pt(111) with Kinetic Monte Carlo (KMC) simulations. By comparing the Temperature Programmed Desorption (TPD) spectra for CO on ordered Cu 3 Pt(111) and on disordered Cu 4 Pt 6 (111), a ligand effect could be identified. Equilibrium simulations indicate a strong Pt enrichment of the surface, in contrast with the Cu enrichment in vacuum. KMC simulations at various conditions of temperature and pressure reveal a higher reaction rate on the ordered Cu 3 Pt(111) surface than on the disordered Cu 4 Pt 6 (111) surface.

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7 Samenvatting Legeringen van overgangsmetalen worden in de chemische industrie vaak ingezet als katalysor. De structuur, orde en samenstelling van het katalysatoroppervlak spelen daarbij een grote rol. Een goed begrip van het oppervlaktegedrag in een dergelijke legering is dan ook onontbeerlijk om o.a. de fysico-chemie van het katalytisch proces te begrijpen en om gerichter experimenten uit te voeren. In dit doctoraat wordt een modelleerstrategie ontwikkeld voor een meer realistische simulatie van een katalysator in actie. Een betrouwbaar energiemodel voor de beschrijving van de metallische binding wordt bekomen door de parameters van de Modified Embedded Atom Method (MEAM) te bepalen aan de hand van een consistente set van ab initio data, met inbegrip van oppervlakte-eigenschappen. Zo worden nieuwe MEAM parameters voor de elementen Cu, Pt, Pd en Rh en al hun binaire legeringen bepaald en daarna gevalideerd door de segregatie in de binaire deelsystemen te onderzoeken. Zowel voor de ongeordende Pt-Rh en Pt-Pd legeringen als voor de ontmengende Cu-Rh en Pd-Rh legeringen en geordende Cu 3 Pt en Cu 3 Pd verbindingen worden resultaten gevonden in goede overeenstemming met experimentele metingen en/of theoretische berekeningen. Vervolgens wordt de uitbreiding naar ternaire systemen gemaakt in een gevalstudie van segregatie naar Pt-Pd-Rh(111) oppervlakken, waarbij een typisch hoger-orde effect van co-segregatie van Pt en Pd naar boven komt. Daarna wordt de stabiliteit van metallische katalysatorpartikels belicht. Voor zuiver Cu, Pt en Rh wordt de icosaëder als gunstigste partikelvorm voorspeld, terwijl dit voor Pd de cubo-octaëder is. Simulaties voor de segregatie in Cu 3 Pt nanopartikels brengen twee verschillende concurrerende mechanismen aan het licht, die respectievelijk overheersen bij hoge en bij lage temperatuur. Tenslotte wordt de invloed bestudeerd van de aanwezigheid van gassen op de segregatie en de orde aan het Cu 3 Pt(111) oppervlak. Door het gesimuleerde Temperature Programmed Desorption (TPD) spectrum voor CO op geordend Cu 3 Pt(111) te vergelijken met dat op ongeordend Cu 4 Pt 6 (111) wordt een ligand effect gekwantificeerd. Simulaties van de Cu 3 Pt(111) legering in aanwezigheid van CO wijzen, bij volledig thermodynamisch evenwicht, op een sterke Ptaanrijking aan het oppervlak, in tegenstelling met de Cu-aanrijking voor deze legering in vacuüm. Voor ditzelfde systeem wordt ook de kinetica van de oxidatie van CO op beide oppervlakken gesimuleerd. Uit deze simulaties kan afgeleid worden dat de beoogde CO 2 -productie op geordend Cu 3 Pt(111) bij hogere temperatuur aanzienlijk sneller verloopt dan op ongeordend Cu 4 Pt 6 (111).

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9 Contents Abstract Samenvatting Contents List of symbols List of abbreviations i v xi 1. Introduction 1 2. Thermodynamic framework Introduction Thermodynamics of alloy formation Thermodynamics of surface segregation Conclusions Simulation tools Fundamentals of statistical mechanics Monte Carlo simulations Molecular Dynamics Kinetic Monte Carlo simulations Conclusions Energy models Density Functional Theory The Embedded Atom Method The Modified Embedded Atom Method Evaluation New parameterisation method for the MEAM DFT calculations DFT-based MEAM parameterisation Conclusions 69 i

10 ii Contents 6. Surface segregation in disordered alloys Introduction The Pt-Rh system The Pt-Pd system Conclusions Surface segregation in phase separating alloys Introduction The Pd-Rh system The Cu-Rh system Conclusions Surface segregation in ordering alloys Introduction The Cu 3 Pt system The Cu 3 Pd system Conclusions Surface segregation in off-stoichiometric ordered alloys Material properties Experimental and theoretical evidence Simulation results and discussion Conclusions Surface segregation in ternary alloys Introduction The Pt-Pd-Rh system Simulation results Conclusions Catalyst nanoparticles Introduction Computational set-up Pure Cu, Pt, Rh and Pd nanoparticles Nanoparticles of disordered Pt-Pd alloys Nanoparticles of demixing Pd-Rh alloys Nanoparticles of ordering Cu 3 Pt alloys Conclusions Simulation of CO adsorption and CO oxidation on Cu 3 Pt(111) Introduction 18

11 Contents iii 12.2 Computational set-up Overview of experimental and theoretical results Simulation results Conclusions General conclusions and challenges for future research 23 A. Development of the Alloy Simulations Tool 29 A.1 Construction of the simulation slab 29 A.2 Ensembles and sampling 21 A.3 Calculating the energy 211 A.4 Extension to metal-adsorbate interactions 212 B. Numerical aspects of DFT 213 N. Nederlandse samenvatting 217 N.1 Inleiding 217 N.2 Werkwijze 217 N.3 Resultaten 22 N.4 Besluit 245 List of publications 251 Curriculum vitae 257

12 iv Contents

13 List of symbols Symbol Description Unit * a Lattice constant Å a * Dimensionless lattice parameter - a Equilibrium lattice constant Å a atom Surface area covered by one atom at the particular surface concerned m 2 /mol a (1x1) Surface area covered by one atom at the particular surface concerned for an unreconstructed (1 1) slab m 2 /mol A i MEAM parameter - α Regular solution parameter (CBE) J/mol α i Constant for element i (EAM) 1/Å α Variable containing the particle position and spin - B i Bulk modulus of pure i GPa β i Constant for element i (EAM) 1/Å ν β H Heating rate during TPD experiment K/s ( ) β i MEAM parameter - c Shear elastic constant GPa c 11 Elastic constant GPa c 12 Elastic constant GPa c 44 Elastic constant GPa χ Nuclear wave function - d 12 (ijk) First interplanar distance for (ijk) surface Å d 23 (ijk) Second interplanar distance for (ijk) surface Å Extent of reaction - E bcc Energy difference between fcc and bcc structure J E hcp Energy difference between fcc and hcp structure J E sc Energy difference between fcc and sc structure J Standard anti-site defect energy upon thermal disordering J/mol E thermal * The energy units are often expressed in ev rather than in the SI unit J. The conversion can be made according to 1 ev = 1.622e -19 J. v

14 vi List of symbols G segr Standard Gibbs free energy for segregation J/mol H 1 Standard energy contribution associated with the lowering of the surface energy J/mol H 2 Standard energy contribution associated with the lowering of the mixing energy J/mol H 3 Satndard energy contribution associated with the lowering of the relaxation of the elastic strain energy around the solute atoms J/mol H 4 Standard energy contribution associated with the preferential chemisorption on the surface J/mol H mix Enthalpy of formation/mixing J/mol Standard enthalpy of segregation J/mol H segr config S thermal Change in configurational entropy upon thermal disordering J/mol K S segr Standard entropy of segregation J/mol K id S mix Ideal mixing entropy J/mol K e Electronic charge C ε Permittivity of vacuum C²/N m² ε n Energy of n-th electron state J E Potential (internal) energy J/mol E HF Energy corresponding to Hartree-Fock solution J/mol E s Potential (internal) energy of subsystem s J/mol E Activation energy for process i J act i form E ij Formation energy for an intermetallic compound of J/mol elements i and j J/mol E MR {11} (2 1) missing-row reconstruction energy J ads E Unrelaxed adatom adsorption energy on a fcc site of (ijk) ( ijk ) surfaces adatom E Total energy of a slab with two (ijk)-oriented surfaces and two ( ijk ) J/mol adatoms adsorbed J surf E Total energy of a slab with two (ijk)-oriented surfaces J ( ijk ) E reconstruction Energy of formation for a surface reconstruction J E Energy of hcp structure J u hcp u E bcc Energy of bcc structure J u E sc Energy of sc structure J

15 List of symbols vii E r Energy of reconstructed slab J E u Energy of unreconstructed slab J E el Elastic strain energy J/mol LDA E xc Exchange and correlation energy in the local density J approximation J/mol E i Heat of sublimation J/mol E X Energy of configuration X J E i Energy of atom i J E Activation energy barrier for process i J act i ε ii Binding energy of an i-i bond (CBE) J/mol ε ij Binding energy of an i-j bond (CBE) J/mol f Pre-exponential factor (MEAM) - F Kinetic, exchange and correlation energy J F i Embedding energy function of atom i J F s Helmholtz free energy of subsystem s J/mol F i Force acting on atom i N F HK Hohenberg-Kohn density functional J F R Helmhotz free energy of the reservoir J/mol φ Single-particle wave function - φ ij Electrostatic repulsion energy (MEAM) J G i Shear modulus of element i J/Å 3 mol γ i Surface energy of a pure metal i J/m 2 γ i Activity coefficient for species i - ( ) γ Activity coefficient for species i in n-th iteration step (GCE) - i γ (ijk) Surface energy for (ijk) surface J/m² h Planck s constant J s ħ Reduced Planck s constant J s H Hamilton energy J H e Electronic Hamilton energy J H KS Kohn-Sham Hamilton energy J k B Boltzmann s constant J/K J i Rate of event i s -1 J tot Total event rate s -1 ϕ ij Electrostatic repulsion energy between atoms i and j J K thermal Equilibrium constant for the thermal disordering reaction - K segr Equilibrium constant for the segregation reaction - Λ i Thermal De Broglie wavelength of atom i m µ i Chemical potential of element i J/mol µ Standard chemical potential J/mol * i

16 viii List of symbols n Number of sample points - n d,j Number of outer d electrons (EAM) - n s,j Number of outer s electrons (EAM) - N Total number of atoms - N i Number of atoms i - N j Total number of valence electrons of element j (EAM) - ν i Constant for element i (EAM) - ν Attempt frequency for process i s -1 i M i Nuclear mass of atomic species i kg m e Mass of electron kg n Order of desorption process - N s Number of surface sites - Ω i Equilibrium volume per atom i Å 3 p Impuls momentum kg m/s P Ensemble probability of occurrence - P i Ensemble probability of occurrence for state i - P Pressure Pa P i,j Probablity that atom i has an atom j as nearest neighbour - π(i j) Transition probability from state i to state j - Θ Electronic wave function - r i Atomic radius of element i Å r i Rate of event i s -1 r i Position vector of electron i - r Position coordinate - ρ Background electron density in reference structure (MEAM) - R Molar gas constant J/mol K R i Position vector of nucleus i - R ij Distance between atoms i and j Å R Equilibrium distance between atoms i and j (MEAM) Å ij R α ij α-component of the distance between atoms i and j (MEAM) Å ρ Electron density Å -3 ρ b,i Background electron density around atom i Å -3 ρ Host electron density at the site of atom i Å -3 h, i ρ s,i Density associated with the s ground state functions (EAM) Å -3 ρ d,i Density associated with the s ground state functions (EAM) Å -3 ρ Atomic electron density of atom i Å -3 a i

17 List of symbols ix ρ Atomic partial electron density l of atom j (MEAM) - a ( l ) j S Sticking coefficient - S Sticking coefficient prefactor - S ideal Ideal molar entropy of mixing J/mol K S s Configurational entropy on sublattice s J/mol K config S total Total configurational entropy J/mol K s Short-range order - σ Standard deviation - ( ) t i Weighting factor (MEAM) - t Time s T Temperature K T Kinetic energy J T Uncorrelated kinetic energy of electrons J T c Critical temperature K T e Kinetic energy of electrons J T N Kinetic energy of nuclei J T p Temperature corresponding to peak maximum in TPD spectrum K θ Surface coverage - θ ix Fractional covering of i with X (chemisorption) - V Volume of the system m 3 V ext External potential energy J V ee Electrostatic energy between electrons J V H Energy corresponding to Hartree Solution J V Ne Electrostatic energy between electrons and nuclei J V NN Electrostatic energy between nuclei J V c Correlation energy J V x Exchange energy J V xc Exchange and correlation energy J W Number of possible microscopic arrangements - ξ Random number - x i Bulk fraction of element i - ˆX Operator associated with energy contribution X - (X) s Composition of component X on sublattice s - Ψ Many-body wave function - y i Surface fraction of element i - Z Partition function - Z Number of nearest neighbours - z i Surface coordination of i with X (chemisorption) - Z v Number of nearest neighbours in an adjacent atomic plane -

18 x List of symbols Z l Number of in-plane nearest neighbours - Z i Nuclear charge C Z i Effective charge of atom i (EAM) C Z ij Number of nearest neighbours between unlike atoms - Z,i Number of outer electrons of atom i (EAM) -

19 List of abbreviations AES bcc CBE CE CUBO DFT EAM EDX EMT fcc FIM GCE GGA hcp ICO IR ISS KMC LDA LEED LEIS LMTO LRO MC MD MEAM ML MR MTB NMR PAW PW91 QCA RHEED sc Auger Electron Spectroscopy Body Centered Cubic Constant Bond Energy Canonical Ensemble Cubo-octahedron Density Functional Theory Embedded Atom Method Energy Dispersive X-ray Effective Medium Theory Face Centered Cubic Field Ion Microscopy Grand Canonical Ensemble Generalised Gradient Approximation Hexagonal Closed Packed Icosahedron Infrared Ion Scattering Spectroscopy Kinetic Monte Carlo Local Density Approximation Low Energy Electron Diffraction Low Energy Ion Scattering Linear Muffin-Tin Orbitals Long-Range Order Monte Carlo Molecular Dynamics Modified Embedded Atom Method Monolayer Missing-row Modified Tight-Binding Nuclear Magnetic Resonance Projector-Augmented Wave Perdew-Wang exchange correlation functional Quasi-Chemical Approximation Reflection High-Energy Electron Diffraction Simple Cubic xi

20 xii SIMS SRO STM TBIM TEM TOF TPD UHV UPS UV VASP XPS List of abbreviations Secondary Ion Mass Spectroscopy Short-Range Order Scanning Tunneling Microscopy Tight-Binding Ising Model Transmission Electron Microscopy Time Of Flight Temperature Programmed Desorption Ultra High Vacuum Ultraviolet Photoelectron Spectroscopy Ultra Violet Vienna Ab initio Simulation Package X-ray Photoelectron Spectroscopy

21 1 Introduction The tremendous impact of catalysis on our daily life can hardly be overestimated. Actually most reactions in the chemical industry are run with the help of a catalyst. Over 9% of the chemical manufacturing processes throughout the world use a catalyst in one form or another [1]. A catalyst is defined [2] as an entity that accelerates a chemical reaction without itself being consumed in the process. Without catalysts, various chemical reactions of great practical importance would proceed so slowly that their occurrence would not even be detected, even under thermodynamically favourable conditions of temperature and pressure. Most of the catalytic reactions are heterogeneous processes, referring to the fact that the reactants are present in one phase and the catalyst in another, with the catalytic action occurring at the interface or surface between them. Catalysts are used to increase the output of a chemical reaction but also to convert hazardous waste into less harmful products. The most prominent example is the car exhaust catalyst. The enormous technological relevance of catalysts is also reflected by the huge worldwide demand for catalysts. The market for catalysts amounts to ca US dollar per year and more than 8% of the net production in the chemical industry is based on catalytic processes [3]. The large-scale industrial application of heterogeneous catalysis began in the second half of the nineteenth century, although the awareness of the phenomenon of catalysis was already very old at that time [2]. In the 188s and 189s a German chemical company, the Badische Anilin und Soda-Fabrik (BASF), developed a heterogeneous catalytic process (the so-called contact process ) for oxidising sulphur dioxide to sulphur trioxide [4]. In the original BASF process, the reaction was carried out over a platinum catalyst. Not long after the development of the contact process for manufacturing sulphuric acid, two other enormously important processes based on heterogeneous catalysis were 1

22 2 Chapter 1 introduced: (1) the process resulting from the pioneering research of Fritz Haber for the production of ammonia from elemental nitrogen and hydrogen [5], and (2) the Ostwald process for the oxidation of ammonia to nitric oxide in the production of nitric acid [6]. Nowadays, catalysts are indispensable in almost any aspect of petrochemical industry, in the production of fine and bulk chemicals, for pollution prevention and for pollution abatement. Traditionally, catalysts have been designed and improved by a trial and error approach. For example, before the iron-based catalyst for the ammonia synthesis was found, about 1, different catalyst formulations had been tried in the first decade of the twentieth century [7]. The first theoretical breaktrough in the field of catalysis came around the time of World War I, when Langmuir advanced a simple theory of chemisorption and showed how it could be used to formulate rate laws for reactions occurring at surfaces. The connection with the structure and ordering at surfaces was stated and from that time on, surface science has played an important role in heterogeneous catalysis. The role of surface science in catalysis The experimental characterisation of the surface became possible by the late 195s and early 196s. Ideal single crystal surfaces were increasingly being identified under Ultra High Vacuum (UHV) conditions (typically 1-12 bar) in order to accurately probe the surface. Information on the structure, composition, order and energetics of the surface layers is obtained from X-ray Photoelectron Spectroscopy (XPS), Ultraviolet Photoelectron Spectroscopy (UPS), Auger Electron Spectroscopy (AES), Ion Scattering Spectroscopy (ISS), Secondary Ion Mass Spectroscopy (SIMS), Field Ion Microscopy (FIM), Low Energy Electron Diffraction (LEED), Reflection High Energy Electron diffraction (RHEED) and Scanning Tunneling Microscopy (STM). Despite the fact that the surface science and catalysis fields are so complementary, one must be aware of the tremendous gaps between the idealised surface science field and catalysis in practice. Three important gaps are usually mentioned: i) the pressure gap, ii) the material gap and iii) the environment gap. The pressure gap refers to the difference in pressures in surface science experiments, compared to the pressures applied in industrial catalytic reactors. Indeed, most surfacesensitive techniques require a UHV environment, while a realistic catalyst may act under pressures up to a few hundred bar. The effect of pressure on metallic systems is however rather limited. Due to the high bulk modulus of metallic systems, the crystal structure usually does not change much upon a moderate increase in pressure. The second gap (material gap) points to the fact that surface

23 Introduction 3 science experiments deal with idealised, well-defined surfaces with a very low defect concentration. On the other hand, in the practice of catalysis, small catalyst nanoparticles (1-1 nm) are deposited on a porous substrate in order to maximise the surface area exposed to the gaseous reacting species. Reducing the size of the material already has a major impact on the thermodynamic properties of metallic systems. Furthermore, the interaction with the substrate may influence the particle shape and the reaction kinetics. Finally, the environment gap refers to the difference in chemical environment between quasi-ideal, vacuum-housed surfaces in surface science and the reactive gas atmosphere for an actual catalyst in an industrial process. The gaseous reactants can have a large influence on the catalyst s surface properties. A Low Energy Ion Scattering (LEIS) study indeed revealed a strong Pt segregation on the UHV-clean Pt-Rh surface in contrast to a weak Rh segregation on an oxygen covered surface [8]. Apart from these gaps, it must also be pointed out that true thermodynamic equilibrium is not always reached in a catalytic process. Given all incoming flow rates, the system may evolve to a steady-state regime that does not necessarily correspond to thermodynamic equilibrium. In terms of the study of chemical reactions at the atomic level, significant and exciting advances have recently come from the development of STM and atomic force microscopy (AFM). These scanning microscopes provide the unique ability to image individual surface atoms and to follow their fate in real time as they react. STM studies have already begun to provide new information on the role of steps, kinks and other defects. Recently, attempts have been undertaken successfully to study the real time chemical dynamics at surfaces [9]. The main role of surface science in catalysis is to expand the basic knowledge about the surface reactions, relevant to catalysis. The better the microscopic details of surface reactions are understood, the less guess-work will be required for the design of new processes. Computational materials science Meanwhile, in the last two decades there has been a tremendous progress as far as the accurate and reliable theoretical description of the geometric and electronic structure of surfaces is concerned and their interaction with adsorbed atoms and molecules [1]. This progress was made possible partially by the increasing computer power, but also to a large extent by the improvement of programs for ab initio electronic structure calculations [11-15] and of other methods that use information from electronic total-energy calculations as input. The great advantage of such calculations compared to experiments is that the calculations can shed light on the underlying electronic factors that determine

24 4 Chapter 1 bond making and breaking at surfaces. In addition, one can follow the microscopic steps of a reaction while in an experiment usually only the initial reactants and final products can be determined. Ultimately, the holy grail in catalysis is to be able to design catalysts from ab initio data [16]. However, it is believed that the prospect of designing a catalyst from ab initio principles is still far away [1,17]. Therefore, the present goal of computations is not to create a new product theoretically from scratch but, much more humbly, to bring about more relevant information [7] for the research and development process. The aim of this work is to contribute to this relevant information by creating a computational strategy for the simulation of catalysts in action. The demands for a suitable modelling strategy are very high. It must be able to deal with the earlier mentioned gaps as well as with the totally different nature of the various chemical bonds. Furthermore, simulations are at best performed with a technique at atomic resolution and should yield results within a reasonable time scale. First, a reliable potential for metallic systems must be constructed. For this work, the Modified Embedded Atom Method (MEAM) has been selected as a good candidate. However, the success of the MEAM in quantitative predictions of surface properties strongly depends on the parameterisation. The original parameterisation by Baskes [18] was entirely based on a few experimentally determined material properties of the bulk only. In this work, the MEAM is upgraded by determining its parameters from a large amount of accurate ab initio data, including various surface properties. Following this novel parameterisation approach, new MEAM parameters for Cu, Pt, Pd and Rh and their binary alloys are determined. These potentials are incorporated in Monte Carlo (MC) simulations in order to study the equilibrium properties of metallic systems as a function of temperature. The bulk behaviour is tested by calculating typical quantities for the particular alloy. Next, the surface segregation equilibrium is studied in all these binary systems and in one ternary system. Complementary to classical equilibrium MC simulations, Kinetic (or Dynamical) MC simulations are able to simulate kinetic processes that are important in catalysis. With a hybrid model for all interatomic interactions, the kinetics and thermodynamics of CO adsorption and of the CO oxidation reaction on Cu 3 Pt(111) are simulated.

25 Introduction 5 Guide for further reading The remainder of this dissertation is built up as follows Chapter 2 gives an introduction to the thermodynamic framework for the surface segregation phenomenon. The driving forces for surface segregation emerge from a simple Quasi-Chemical Approximation (QCA) with constant bond energies. Chapter 3 describes the physical background of the most relevant simulation tools used in this work. Chapter 4 presents the Density Functional Theory (DFT) framework, which made feasible calculations for metallic systems with (periodic) unit cells, containing some tens of atoms. A short introduction into this, most widely used, ab initio approach to the solution of the Schrödinger equation is presented as well as the relation with the family of EAM methods. The chapter concludes with a critical evaluation of the energy models. In Chapter 5, the novel MEAM parameterisation for the pure transition metals Cu, Pt, Rh and Pd is described, as well as for all (binary) cross-potentials containing these 4 elements. The newly parameterised MEAM is then tested by recalculating the input data and validated with the calculation of material properties that were not included in the input data set. Chapter 6 reports on the simulation of surface segregation in disordered alloy systems. To this end, surface segregation is studied in Pt-Rh and Pt-Pd alloys. Chapter 7 deals with phase separating alloys Pd-Rh and Cu-Rh. The phase diagram for these systems is first simulated and compared to the experimental evidence. Below the critical temperature for demixing, a core-shell configuration emanates from the simulations, with the phase of lower surface energy located in the (sub)surface layers of the simulation cell. Chapter 8 focuses on segregation in yet another class of alloys, such as Cu 3 Pt and Cu 3 Pd, in which ordered compounds are formed. First, the critical temperature for the order-disorder transition is determined and compared to experimental values. Next, surface segregation to the low index single crystal surfaces of these alloys is simulated and compared to experimental and theoretical results. Chapter 9 extends the study of ordered alloys to (slightly) off-stoichiometric alloys. The large impact of this deviation from the ideal stoichiometry is clearly illustrated in the case study of Pt 8 Co 2 (111).

26 6 Chapter 1 Chapter 1 investigates the typical effects occurring in multi-component systems by studying the ternary Pt-Pd-Rh alloy. First, the phase diagram at 6 K is simulated. Next, surface segregation is studied for the whole ternary diagram. Three particular compositions were selected for a more detailed study. Chapter 11 partially narrows the material gap by comparing the segregation and ordering behaviour at the extended flat low-index surfaces with the corresponding behaviour in nanoparticles. This is done for Pt-Pd, Pd-Rh and Cu 3 Pt particles. In Chapter 12, the environment gap is tackled by the simulation of the presence of adsorbates CO and O 2 on Cu 3 Pt(111). For this system, the desorption kinetics are simulated for the Temperature Programmed Desorption (TPD) spectrum of CO on ordered and disordered Cu 3 Pt(111). Finally, the CO oxidation reaction on Cu 3 Pt(111) is simulated at different pressures and temperatures. All simulations were performed with a C ++ program that was developed as a part of this work. More details about the implementation of the code are described in Appendix A. Finally, the details of the parameters controlling the numerical evaluation of the DFT equations are presented in Appendix B. References [1] J.M. Thomas and W.J. Thomas, Principles and Practice of Heterogeneous Catalysis, VCH, Weinheim, [2] J.H. Sinfelt, Surf. Sci. 5 (22) 923. [3] W.A. Herrmann, in: U.-H. Felcht, Chemie - Eine reife Industrie oder weiterhin Innovationsmotor?, Verlag der Universitätsbuchhandlung Blazek und Bergmann seit 1891 GmbH, Frankfurt, 2. [4] E.B. Maxted, Catalysis and its Industrial Applications, J.&A. Churchill, London, [5] W.G. Frankenburg, The catalytic synthesis of ammonia from nitrogen and hydrogen, in: P.H. Emmett (Ed.), Catalysis, vol. III, Reinhold, New York, [6] G. Chinchen, P. Davies and R.J. Sampson in: J.R. Anderson, M. Boudart (Eds.), Catalysis - Science and Technology, vol. 8, Springer, Berlin, [7] E. Wimmer, J.-R. Hill, P. Gravil and W. Wolf, Industrial Use of Electronic Structure Methods, Newsletter 34 of the Psi-k Network, 1999,

27 Introduction 7 [8] B. Moest, S. Helfensteyn, P. Deurinck, M. Nelis, A.W. Denier van der Gon, H.H. Brongersma, C. Creemers and B.E. Nieuwenhuys, Surf. Sci. 536 (23) 177. [9] M. Bonn, A.W. Kleyn and G.J. Kroes, Surf. Sci. 5 (22) 475. [1] A. Gross, Surf. Sci. 5 (22) 347. [11] M. Head-Gordon, J. Phys. Chem. 1 (1996) [12] P. Blaha, K. Schwarz, P. Sorantin and S.B. Trickey, Comput. Phys. Commun. 59 (199) 399. [13] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 145. [14] G. Kresse, J. Furthmüller, Comp. Mat. Sci. 6 (1996) 15. [15] M. Bockstedte, A. Kley, J. Neugebauer and M. Scheffler, Comput. Phys. Commun. 17 (1997) 187. [16] F. Zaera, Surf. Sci. 5 (22) 947. [17] I. Chorkendorff and J.W. Niemantsverdriet, Concepts of Modern Catalysis and Kinetics, Wiley-VCH, 23. [18] M.I. Baskes, Phys. Rev. B 46 (1992) 2727.

28 8 Chapter 1

29 2 Thermodynamic framework * In our universe, any system strives for the equilibrium situation of minimum Gibbs free energy and unless kinetic limitations prevent it, this situation will be the endpoint of the spontaneous evolution. In a piece of material, this equilibrium configuration is accomplished by the subtle interplay of interatomic interactions. The very same effects are thus governing alloy formation and bulk phase behaviour of alloys and surface segregation and other effects at alloy surfaces. The common thermodynamic basis of these processes is therefore briefly outlined Introduction For the sake of simplicity, only binary alloys in the Constant Bond Energy (CBE) approximation [1-3] are considered. This simple pair-potential model assumes that the configurational energy of an alloy is determined solely by the pair-wise interactions between its constituent atoms. The constant bond energy between two atoms is further taken as being independent on the distance between them. This model thus assumes that the atoms reside on fixed lattice sites and interact only with their nearest neighbours. The bond strength (ε ii ) between two identical atoms sub can directly be derived from the heat of sublimation for the pure metal ( ) H i * The content of this chapter is primarily based on a text by C. Creemers, S. Helfensteyn, J. Luyten and M. Schurmans, that appeared as Chapter 5 in: G. Bozzolo, P. Abel, R.D. Noebe (Eds.), Applied Computational Materials Modeling: Theory, Experiment, and Simulations, Kluwer Academic Publishers, New York, 27. 9

30 1 Chapter 2 sub 2 Hi εii = (2.1) Z with Z the number of nearest neighbours of an atom in the pure i lattice. The energetic interactions between unlike atoms (ε ij ) are crudely characterised by one single parameter only, the regular solution parameter α, defined as εii + ε jj α = εij (2.2) 2 and can be calculated from the molar heat of mixing of an alloy A x B y, that supposedly corresponds to a regular solution, as H mix α = (2.3) Z x (1 x) The CBE model only differs from regular solution theory in that the latter also assumes ideal entropy of mixing id S = R x ln x (2.4) mix i i i while CBE combined with Monte Carlo simulations can handle various degrees of short- and long-range order as deviations from perfectly random solutions Thermodynamics of alloy formation The cohesion of a solid is the result of the overall attractive interactions between the constituting particles. These interactions determine the properties of a pure type of solid: the stronger these interactions, the higher the melting and boiling points. The particular crystal lattice is thereby determined by the chemical identity of the atoms and by a.o. the balance in bond strength between first and second nearest neighbours. When different metals are alloyed, interatomic interactions between unlike particles arise. In ideal solutions and alloys the energy of these new AB interactions is simply the average of AA- and BB-bonds, causing a zero enthalpy of mixing, when preparing the alloy from the juxtaposition of the pure elements.

31 Thermodynamic framework 11 As a consequence, the atomic arrangement will be ideally random and the molar entropy of mixing will be given by the well-known expression for the ideal entropy of mixing (Eq. 2.4). In non-ideal alloys preferential interactions exist, either between unlike neighbours AB or between like atom pairs AA and BB. Inevitably, the atomic stacking will then no longer be random and the assumption of ideal mixing entropy will fail. A preference for AB bonds leads to varying degrees of shortrange order and eventually long-range order while repulsive AB interactions will favour AA and BB bonds and will eventually induce phase separation. As being responsible for these preferential interactions, firstly chemical attractions or repulsions must be recognised, caused by the degree of chemical likeness of the constituting elements. These effects are caused by the behaviour of the valence electrons, that to a large extent also determine the crystallography / lattice of the pure elements. In addition to this chemical resemblance, the size of the atoms is a very important factor. Unlike in liquid mixtures, where size differences between the molecules are easily accommodated by a change in the local composition or number of neigbours, in a solid with a more or less fixed lattice, size differences create strain fields in the lattice and only limited size differences can be accommodated before phase separation occurs. If the overall interactions in the alloy are less attractive than in the juxtaposition of the pure elements, AB interactions will be more repulsive (due to chemical repulsions or to largely different atomic sizes) than the average of AA and BB bonds and alloying will be endothermic, e.g. the systems Pd-Rh and Cu-Rh. There will be a preference for like neighbours causing a form of short-range order or deviation from ideal randomness. These repulsions will somewhat destabilise the solid phase, causing a lower melting point and a downward curvature of the melting region. Sufficiently strong repulsions will cause a miscibility gap and a two-phase region in the existence domain of the otherwise single phase solid solution. At temperatures higher than the critical temperature for demixing, the repulsions are overcome and the alloy becomes a single solid solution again, in which however the preference for like neighbours still subsists. As the repulsions become stronger, the melting region becomes azeotropic with a common minimum for the solidus and the liquidus, as is shown in Fig At the same time the two-phase region of immiscibility becomes broader and higher and the critical temperature for overruling the repulsions by thermal motions becomes higher. In the case of even stronger repulsions, the region of phase separation and the melting region coalesce into a eutectic phase diagram. Extreme repulsions will ultimately result in complete immiscibility and the disappearance of the side - phases with solid solutions α and α.

32 12 Chapter 2 Figure 2.1. Evolution of the phase diagram of an endothermic alloy for increasing repulsive interactions between A and B [4]. If, on the other hand, the overall interactions are more attractive than in the juxtaposition of the pure elements, alloying will be exothermic. A limited degree of exothermicity will cause some degree of short-range order. By this favourable energy effect, the solid phase will be stabilised and typically, the two-phase melting region has a tendency to be curved upwardly (see Fig. 2.2). Increasing energetic preference for AB bonds will cause the formation of (long-range) ordered compounds. At first this long-range order only holds up to a critical orderdisorder temperature that is much lower than the melting point, and above which the thermal energy of the atoms will again randomise the alloy. The new ordered phase α will also be able to accommodate significant amounts of excess A or B, allowing large deviations from the ideal stoichiometry. The domain for this ordered compound is separated from the domain of the random solid solution α by rather narrow two-phase domains in which the ordered and the disordered phases coexist. As this intermetallic compound formation becomes more and more exothermic, the order-disorder critical temperature will increase, making the ordered compound stable up to the point of (distectic or peritectic) melting. The phase boundaries for the ordered phase will merge with the melting region. At the same time, the domain of existence of the ordered compound in the phase diagram will become (much) narrower and the correct stoichiometry will be better respected. As a consequence, the two-phase regions (ordered compound β coexisting with the random solid solution) become broader. Eventually more and more ordered compounds with different well-defined stoichiometries will emerge from the phase diagram, separated by two-phase regions in which two immiscible compounds coexist (e.g. the Pt-Sn system). This evolution with increasing magnitude of α is shown in Fig. 2.2.

33 Thermodynamic framework 13 Figure 2.2. Evolution of the phase diagram of an exothermic alloy with increasing ordering tendency due to preferential attractions between A and B [4]. Effect of temperature on the order in stoichiometric ordered compounds The configurational entropy can be evaluated using Boltzmann s well-known expression S = k B lnw, with k B Boltzmann s constant and W the number of possible microscopic arrangements yielding the same macroscopic situation. When applied to a randomly disordered mixture, Boltzmann s formula directly yields the expression for the ideal mixing entropy (Eq. 2.4). When calculating the change in mixing entropy generated by a chemical reaction, the classical equilibrium constants directly emanate from the minimisation of the Gibbs free energy. For ordered alloys, the entropy can be calculated elegantly by using the concept of sublattices. The changes in configurational entropy can be evaluated by artificially splitting up the entire lattice of the ordered alloy in a number of sublattices [5-7]. Each of these sublattices is mainly filled with one species, with the minority species randomly distributed over the sites of the sublattice, hence again forming an ideal mixture. The total configurational entropy is then calculated as the sum of the entropy contributions in each sublattice. This approach will be demonstrated next on the basis of a simple example. An ideally ordered alloy A 75 B 25 is assumed, with a face centered cubic (fcc) lattice (L1 2 structure) in which two sublattices can be discerned. Sublattice α comprises the midplane sites, i.e. 75% of the total number of sites and is exclusively occupied by A atoms. Sublattice β groups the corner sites, represents 25% of the sites and is exclusively occupied by B atoms. This ideal situation of zero configurational entropy can be disturbed or by a deviation from the exact stoichiometry, or by the thermal randomising tendency at higher temperatures, or else by segregation.

34 14 Chapter 2 Next, assume that, as a consequence of thermal randomising at higher temperatures, a fraction of A atoms from the α sublattice has exchanged places with an equal number of B atoms from the β sublattice A + B A + B (2.5) α β β α The entropy of mixing for both sublattices can then be written as Sα ( ) = R.75 ln + ln S β ( ) = R.25 ln + ln (2.6) The total configurational entropy is calculated as the sum of the entropy contributions in each sublattice Rearrangement gives S ( ) = S ( ) + S ( ) (2.7) config total α β config Stotal (.75 ) ln(.75 ) + ln( ) + ( ) = R (.25 ) ln(.25 ) ( ) ln( ) ln(.75).25ln(.25) (2.8) In this expression, the first two lines represent the usual expressions for the ideal molar entropy for each of the two sublattices forming ideal mixtures. The two terms on the third line bring into account the entropy decrease due to the order in the alloy: one mole of the highly ordered alloy consists of.75 mole of a sublattice α and.25 mole of sublattice β. The change in configurational entropy upon the atomic rearrangement at higher temperature is given by config config config S ( ) = S ( ) S ( = ) (2.9) thermal total total The second term equals zero as it corresponds to the perfectly ordered lattice. In the procedure of minimising G by putting its derivative with respect to equal to zero, the two last constant terms in Eq. 2.8 will vanish.

35 Thermodynamic framework 15 In order to estimate the equilibrium situation corresponding to minimum Gibbs free energy, this entropy balance must be combined with an energy balance. Straightforward summing of the detailed changes in bond energy leads to a standard anti-site defect energy of 16 Ethermal ( ) = 8α 1 3 (2.1) Continuing the quasi-chemical treatment, the equilibrium of the thermal rearrangement is described by RT K + E = (2.11) ln thermal thermal ( ) with K thermal 2 [( A) β ] [( B) α ] = = [( A) ] [( B) ] α β ( ) ( ) (2.12) It appears that the order in the bulk lattice is not appreciably degraded as long as the temperature does not approach too closely the critical order-disorder temperature. As an example, Pt 8 Fe 2 can be considered (although this composition is off-stoichiometric). CBE predicts 175 K as the order-disorder temperature for this alloy [8]. Even at 9 K, amounts to only 2%. In the simplified quasi-chemical treatment, one can then safely assume perfect bulk order as the starting situation for segregation. In summary, Table 2.1. shows the four different possible combinations, depending on whether alloy formation is endothermic or exothermic and depending on the temperature being higher or lower than the critical temperature. T > T c T < T c Exothermic mixing Endothermic mixing (α < ) (α > ) Disordered solid solution e.g. Pt-Rh, Pt-Pd, Stoichiometric ordered Demixing into two phases intermetallic compounds e.g. Cu-Pt, Cu-Pd, e.g. Cu-Rh, Pd-Rh, Table 2.1. Possible bulk configurations for non-ideal alloys.

36 16 Chapter Thermodynamics of surface segregation Apart from bulk effects, also surface processes can contribute significantly in achieving the minimal Gibbs free energy for a piece of material, in concreto, by attenuating its surface energy. Surface segregation, surface relaxations in bond energy and in interlayer spacing and possibly surface reconstructions play a role in this respect. In a first instance, it is the altered surface composition after segregation that is explored in view of improved catalytic properties of an alloy. Surface segregation can be defined as the spontaneous enrichment of the surface, at equilibrium, in one of the components, caused by the mere presence of the surface. Because of the altogether different nature in origin and extent, a clear distinction will be made between segregation in disordered alloys on the one hand, and in ordered intermetallic compounds on the other hand. Surface segregation in disordered alloys In randomly disordered alloys, surface segregation is essentially a partial demixing of the surface layers. This demixing causes a lowering of the configurational entropy. For segregation to be spontaneous and assuming reasonable values for the excess entropy, it must therefore necessarily be exothermic. As a consequence, segregation in disordered alloys is always less pronounced at higher temperatures. In the concept of the quasi-chemical approximation (QCA), atomic exchanges in segregation can be approximated as a quasi-chemical reaction [1-3]: surface segregation is considered to be a process where atoms B from the bulk of an AB alloy are swapped with atoms A from the surface. Formally written as a chemical reaction, this becomes A + B A + B (segregation of B) (2.13) surface bulk bulk surface The equilibrium constant for this reaction can be written in the usual way K segr [ Abulk ][ Bsurface ] y(1 x) = = [ A ][ B ] (1 y) x surface bulk (2.14) where y and x are the surface and bulk concentrations respectively, expressed in at.% B. This traditional way of writing the equilibrium constant assumes ideal

37 Thermodynamic framework 17 entropy of mixing in disordered alloys. The classical thermodynamics of the chemical equilibrium leads to the corresponding segregation equation RT y(1 x) (1 y) x G ln + segr = (2.15) The standard Gibbs free energy change for segregation can be written as G = H T S (2.16) segr segr segr with H segr the standard enthalpy change and S segr the standard entropy change upon segregation. The condition for disordered alloys is clearly incorporated in the formalism since Eq is the pure translation of the ideal mixing entropy in a stochastic disordered system. Eq allows the calculation of the surface concentration y at temperature T for an alloy with bulk composition x, if the change in Gibbs free energy upon the exchange of atoms between bulk and surface can be evaluated. Most often, the standard entropy change is negligible =, and thus S segr G = H T S H (2.17) segr segr segr segr and RT y(1 x) (1 y) x ln + H segr = (2.18)

38 18 Chapter 2 Figure 2.3. Graphical illustration of the segregation in a binary disordered alloy for constant values of the reduced, dimensionless segregation energy (Eq. 2.15). If the driving force for segregation can be assumed to be constant, independent of the bulk composition, Fig. 2.3 gives the solutions to this equation for different values of the (dimensionless) segregation enthalpy. Three different situations can be observed. First, when the heat of segregation equals zero (K segr = 1), no segregation occurs and the surface composition remains identical to the bulk composition. Next, when the segregation enthalpy is negative, B atoms segregate spontaneously by an exothermic process. At higher temperatures, segregation becomes less pronounced. Finally, when the heat of segregation is positive, segregation of B atoms is endothermic and not spontaneous. The reaction will proceed in the opposite direction and atoms A will segregate instead. Looking at the nature of the segregation enthalpy, up to four different contributions can be distinguished, each representing a possible driving force for segregation H = H + H + H + H (2.19) segr where H 1 corresponds to the lowering of the surface energy, of the mixing energy, H 2 the lowering H 3 comes from the partial relaxation of elastic strain

39 Thermodynamic framework 19 energy around the solute atoms upon segregation and due to preferential chemisorption on the surface. H 4 is the energy change The first two energy contributions can be straightforwardly evaluated in the CBE. For the calculation of the first energy contribution, the assumption that the surface energy equals half the energy of the missing bonds at the surface leads to Zv H1 = aatom ( γ B γ A) = ( ε BB ε AA) (2.2) 2 with a atom the surface area covered by one atom at the surface in question, γ i the surface energy of pure metal i, Z v the number of bonds with an adjacent layer and ε ii the bond energy of an i-i bond. The element with the lower surface energy or heat of sublimation will normally segregate to the surface. The segregation is face-dependent because the surface energy γ i and the area of an atom a atom or alternatively the number of neighbours with an adjacent layer Z v depend on the orientation of the surface. As a consequence, segregation in a fcc lattice is most pronounced at the (11) face, followed by the (1) and finally is less pronounced at the (111) surface. In addition to the previous effects, it has to be mentioned that all kinds of relaxation and reconstruction processes diminish the surface energy as compared to the first estimate on the basis of bond breaking only. If only nearest neighbours are accounted for, the influence of the segregation behaviour is restricted to one layer. From first principles calculations, it is shown [9] that the surface energy of transition elements behaves more or less parabolically with the number of d-electrons (N d ), with a maximum at N d = 5. Concerning the second energy term, the solution of the segregation equation is presented for regular solutions, in terms of the regular solution parameter α H 2 = 2 α[ Zl ( x y) + Zv( x.5)] (2.21) with Z l the number of in-plane neighbours (Z = Z l + 2 Z v ). Fig. 2.4 illustrates the effect of the second energy term for different values of α/rt. For x =.5 the solution is y = x, independent of the value of the regular solution parameter α. Positive values of the regular solution parameter and the corresponding AB repulsions enhance the segregation of the minority component and give rise to a monotonous depth profile. In case of a negative value for the regular solution parameter, segregation of the minority component is reduced and an oscillating depth profile appears. The effect of this energy term must be

40 2 Chapter 2 Figure 2.4. Graphical illustration of the effect of the non-zero bulk mixing energy on the segregation to the (111) surface (Z l = 6, Z v = 3) of a binary disordered alloy (Eqs and 2.21). evaluated together with the other driving forces and can extend up to four layers in depth. A third energy contribution pertains to the partial relaxation of elastic strain energy due to the difference in atomic size between A and B. The solution of atoms with different sizes in an alloy creates local elastic deformations in the crystal lattice around these atoms. Upon segregation, the accompanying elastic strain energy can be (partially) released and this contributes a third driving force for segregation. This effect favours the segregation of the minority component when the difference in size exceeds 1% and is commonly believed to be of importance only for larger solute atoms. Moreover, this is always an exothermic effect. A coarse approximation of this elastic strain energy is obtained by extrapolating macroscopic elasticity theory to the atomic scale [1]. The elastic strain energy associated with a sphere of radius r A of solute A incorporated into solvent B with a spherical cut-away of radius r B is classically calculated as

41 Thermodynamic framework 21 E 24 πb r ( r r ) G r 2 A A A B B B el = = H3 3BArA + 4GBrB (2.22) with B A the bulk modulus of solute A and G B the shear modulus of solvent B. The extrapolation of the elasticity theory for macroscopic continua down to the discrete atomic level is very questionable. Furthermore, this estimation is only valid for very dilute solutions, since it is assumed that the regions of elastic deformation do not influence each other. Next, it is not entirely correct to treat this effect separately from the second driving force H 2, since the elastic strain is generated during the alloying process and the associated energy is an essential part of the mixing energy for the solid alloy. It is as such already taken into account by the regular solution parameter. Finally, it is possible that the elastic strain around larger solute atoms is not entirely relaxed upon segregation. Up to now, segregation has been considered for a surface in vacuum. In reality, surfaces (e.g. of a catalyst) are often in contact with a (reacting) gas phase. Chemisorption of gaseous components can take place to a different extent on different atomic species at the surface. When a gaseous component X adsorbs preferably on one of the constituent atoms A or B (because of a different heat of adsorption), this preferential chemisorption of adsorbate X may enhance or reduce the surface enrichment caused by the other driving forces. In some cases the segregation can even be reversed. Continuing the parallel with classical chemical equilibrium, this effect can formally be accounted for by a set of consecutive reactions A + B A + B H + H + H surface bulk bulk surface ( - ) B + X B X H surface gas surface (2.23) The enthalpy effect due to the preferential chemisorption is given by H = z ε θ z ε θ (2.24) 4 B BX BX A AX AX with z i the surface coordination of i with X, ε ix the bond energy between i and X and θ i the fractional coverage of i by X. An exothermic fourth driving force for segregation causes a preference for B-X bonds and a further enrichment in B atoms. In the opposite case, segregation of B is diminished and eventually reversed to segregation of A. In this case, the segregation of A that is not spontaneous by itself ( > ) is entrained by the subsequent adsorption that G segr causes an even larger decrease of the Gibbs energy. The species X adsorbing on

42 22 Chapter 2 the surface acts as a chemical pump to make the more reactive component A or B migrate to the surface. An example of this behaviour is observed in the Pt-Rh system. Under Ultra High Vacuum (UHV) Pt is the segregating component. Traces of carbon at the surface or oxygen in the surrounding gas phase make Rh segregate instead [11]. Surface segregation in ordered alloys In ordered alloys, the situation is altogether different and somewhat more complex. Ordered alloys are formed in an exothermic process and exist because of the energetic preference for bonds between unlike atoms. In an ideally ordered intermetallic compound the maximum number of such AB bonds is realised, therefore yielding a situation of minimum energy. Both the long- and short-range order (see Eqs and 3.25) are maximum and equal to one, corresponding to zero configurational entropy. Higher temperatures tend to first disturb and later destroy this order. Also segregation disturbs this ideal order. In the next paragraphs, segregation in stoichiometric and slightly off-stoichiometric ordered alloys is described. a) Segregation in stoichiometric ordered compounds As already mentioned, a stoichiometric ordered compound is a system at minimum energy and with perfect order. Segregation disturbs this ideal situation, hence is normally endothermic and causes partial atomic mixing at the surface with an increase in the configurational entropy. Whereas segregation in disordered alloys is always energy-driven, segregation in stoichiometric ordered alloys is most often (only) entropy-driven. It takes an unusually large difference in surface energy between the components A and B to turn this type of segregation into an exothermic process. For the estimation of the segregation energy, in the QCA/CBE approach, four possibilities have to be considered. Considering segregation to the (111) surface of a fcc A 75 B 25 compound, both atomic species (A and B) can segregate to the surface. Moreover, this segregation can occur by a nearest neighbour exchange between the outermost layer and the second atomic layer or an exchange of atoms between the surface layer and a deeper bulk layer:

43 Thermodynamic framework 23 o α, bulk β, surface β, surface α, bulk 1 atom A B A + B A + B E ( ) = a ( γ γ ) 5α o α,2 nd layer β, surface β, surface α,2nd layer 2 atom A B A + B A + B E ( ) = a ( γ γ ) 3α o β, bulk α, surface α, surface β, bulk 3 atom B A B + A B + A E ( ) = a ( γ γ ) 9α o B ( ) ( ) 7 β,2 nd layer + Aα, surface Bα, surface + Aβ,2nd layer E4 = aatom γ B γ A α (2.25) The first term in the four energy balances corresponds to the change in surface energy. This change in surface energy is realised at the cost of respectively 3, 5, 7 or even 9 times ( α). In ordering alloys α <, and, unless the difference in surface energy is unusually large, all four energy balances represent endothermic processes. These processes can then only occur because of the disorder (entropy) they entail. Van Santen et al. [2,3] argued that at low to moderate temperatures, the least endothermic of these four possible processes would most likely occur and dominate the segregation. This would result in segregation of the component with the lower surface energy by nearest neighbour exchange from the second to the first atomic layer. In this simple approach, which is valid for moderate temperatures, the two outermost atomic layers form a closed system. The second atomic layer is depleted in the segregating element to the same extent as the enrichment in the first monolayer, leading to an oscillating concentration profile with the bulk concentration present from the third layer on. This is well illustrated by the Pt 75 Sn 25 (111) alloy, where the minority component Sn segregates [2,3]. The oscillatory composition variation over the first few atomic layers is often observed for exothermic alloys in quantitative Low Energy Electron Diffraction (LEED) experiments. However, this reasoning is primarily based on energy considerations and overlooks one particular entropy aspect: the number of possibilities for swapping atoms between the surface layer and the bulk of the alloy is so very much larger than the number of possible nearest neighbour exchanges between the first and the second layer. As a consequence, the first reaction features a significantly more favourable reaction entropy. While it remains valid that the component with the lower surface energy will segregate, the segregation will be dominated by the first process at all realistic temperatures. This is confirmed by more elaborate multilayer segregation models. Continuing the above example and supposing that A features the lower surface energy, the first reaction will determine the surface composition, which can be calculated from the equilibrium equation

44 24 Chapter 2 RT ln K + H = (2.26) segr o 1 with K segr [ A ][ B ] 2 [ A ][ B ] (.75 )(.25 ) β, surface α, bulk = = α, bulk β, surface (2.27) The release of elastic strain energy upon segregation is not accounted for by CBE. In ordered compounds, however, elastic strain energy is normally not a major driving force, as the lattice parameter automatically adjusts to the ordered alloy stacking. It then follows that segregation in stoichiometric ordered alloys is in general less pronounced than in disordered alloys: if the phase diagram of a particular alloy system shows one or more true intermetallic compounds, which follow from strong AB interactions, the surface situation often corresponds to a truncated bulk without appreciable segregation enrichment. b) Segregation in off-stoichiometric ordered compounds In stoichiometric compounds, segregation in the perfectly ordered lattice is necessarily accompanied by the creation of anti-site defects. This is not the case for off-stoichiometric alloys: new channels for segregation become possible in which the excess amount of one component, accommodated on the wrong sublattice, segregates to the surface from either the second layer or from the bulk. These processes are exothermic, since they somewhat restore the ideal situation of a maximum number of AB bonds in the bulk where the atoms are fully coordinated. In this case, segregation of the excess component from the bulk is even more exothermic than segregation from the second layer. At first sight, it would seem that these processes would then occur more likely and more frequently than the (almost always) endothermic segregation in the stoichiometric compound. However, by segregation according to this mechanism, the disorder caused by the excess amount on the wrong sublattice is partially eliminated and the associated entropy decreases. This would again lead to typical equilibrium processes. Nevertheless, more abundant segregation is commonly observed in LEED or LEIS analyses of these alloys, often up to 1% A for e.g. A 8 B 2 alloys [8,12]. Summarising the analysis of all the possible segregation paths in an L1 2 ordered fcc alloy A75 ± δ B25 δ leads to the following conclusion: purely on the basis of the alloying parameter α and thus making abstraction of the difference in surface

45 Thermodynamic framework 25 energy, the majority component A will always segregate. However, this surface enrichment in A occurs to a different extent for under- or overstoichiometric compositions. In A75 δ B25+ δ, A segregates to the surface up to the stoichiometric composition of 75%; in A75 + δ B25- δ, the same component A tends to enrich the surface up to 1%, irrespective of the magnitude of δ. At K, this leads to a discontinuity: the surface concentration changes from 75% to 1% for an infinitesimal change in the bulk composition. This is confirmed by calculations performed by Ruban [13]. At higher temperatures the transition is still discontinuous, but the surface enrichment is less extreme as becomes evident from our simulations. In addition to these effects induced by the alloying parameter α, the change in surface energy, that amounts to a atom (γ A -γ B ) must also be taken into account. Finally, it should be noted that in off-stoichiometric alloys, the relaxation of strain energy can again become an appreciable issue and this effect is always exothermic. Suppose that in A75 + δ B25- δ the A atoms are the larger ones. The excess A atoms are accommodated on the wrong β lattice and indeed experience strain that can be released upon segregation to the surface. These A atoms replace the smaller B ones on β lattice sites where B atoms normally fit without appreciable strain. In the L1 2 structure these β lattice sites are completely surrounded by the larger A atoms on α sites. The approximation of the strain energy based on continuum-mechanical theory (Eq. 2.22) can then again cautiously be applied, but both elastic constants now pertain to material A E el 2 24 π BArA ( ra rb ) GArB = = H 3B r + 4G r A A A B 3 (2.28) More sophisticated energy models will however automatically and much more correctly account for this effect Conclusions In this chapter, the thermodynamic framework for alloy formation has been reviewed. Depending on the alloy parameter, the alloy components have a tendency to demix or to form ordered compounds instead. Next, surface segregation is described for two limiting states of order in the bulk, e.g. perfect disorder and (nearly) perfect order. First estimates for the degree of surface enrichment can be obtained from a quasi-chemical approach. The real state of order in an alloy at a given temperature, however, usually deviates from these two limiting cases, due to a non-negligible degree of short-range order. The equations

46 26 Chapter 2 are then no longer valid and the use of more sophisticated techniques is necessary to account for these short-range order effects. Monte Carlo (MC) simulations are ideally suited to overcome this deficiency and will be described in the next chapter. Furthermore, the assumed constant bond energies are only a first estimate for the bond strength in metals. More sophisticated energy models, that can include the typical many-body character of the metallic bond, must be used for more accurate estimates of the surface energies. Suitable energy models, together with their anchoring basis in quantum mechanics will be described in Chapter 4. References [1] F.L. Williams and D. Nason, Surf. Sci. 45 (1974) 377. [2] R.A. Van Santen and W.M.H. Sachtler, J. Catal. 33 (1974) 22. [3] W.M.H. Sachtler and R.A. Van Santen, Appl. Surf. Sci. 3 (1979) 121. [4] R.A. Swalin, Thermodynamics of solids, 2nd edition, Wiley (1972). [5] B. Sundman and J. Ågren, J. Phys. Chem. Solids 42 (1981) 297. [6] B. Sundman and J. Ågren, Mat. Res. Soc. Symp. Proc. 19 (1983) 115. [7] B. Sundman, Anal. Fis. B 86 (199) 69. [8] C. Creemers and P. Deurinck, Surf. Interface Anal. 25 (1997) 177. [9] M. Methfessel, D. Hennig, M. Scheffler, Phys. Rev. B 46 (1992) [1] D. McLean, Grain boundaries in metals, Clarendon, Oxford, [11] N. Sano, T. Sakurai, J. Vac. Sci. Technol. A 8 (199) [12] M.A. Vasylyev, V.A. Tinkov, A.G. Blaschuk, J. Luyten and C. Creemers, Appl. Surf. Sci. 253 (26) 181. [13] A.V. Ruban, Phys. Rev. B 65 (22)

47 3 Simulation Tools A more detailed description of typical equilibrium processes in solid state physics (surface segregation, order-disorder transformations,...) requires the treatment of a large collection of atoms. An interesting computational approach is provided by statistical mechanics. Equilibrium properties are derived by calculating statistical averages for the desired quantities. In practice, the MC simulation technique naturally emerges when calculating these average properties in a computationally economical way. Another option is to solve the equations of motion for each individual atom, according to Newton s law of motion. This is exactly what is done in Molecular Dynamics simulations (MD). Detailed information about the kinetics can be obtained. However, the time-scale on which processes can be studied is typically limited to a few nanoseconds, which is usually not sufficient to reach equilibrium. Kinetic information and the computational ease of MC simulations are unified in Kinetic Monte Carlo simulations (KMC). These three computational methods are discussed more in detail in this chapter. 3.1 Fundamentals of Statistical Mechanics Although developed historically after the introduction of thermodynamics, statistical mechanics provides a microscopic justification for various laws of thermodynamics and relates thermodynamic quantities to more fundamental physical properties [1]. In order to develop statistical mechanics in a concrete way, an ensemble is defined as a collection of particles, subject to certain constraints. With respect to the problems relevant for this work, the Canonical Ensemble (CE) and the Grand Canonical Ensemble (GCE) are introduced. In the CE, the number of atoms for each species N i, the total volume V and the 27

48 28 Chapter 3 temperature T are kept fixed, while in the GCE, only the total number of atoms N tot, the volume V and temperature T are constant. In the CE with N particles, the probability of occurrence P ( r, p) for a given state, characterised by 3N positional coordinates ( r ) and 3N momenta ( p ) is given by the Boltzmann distribution function P r p (, ) c( N) exp H ( r, p) / kbt = Z (3.1) with c(n) a prefactor, assuring equivalence between quantum-mechanical and classical systems, c(n) = 1/(N! h 3N ), H ( r, p ) the Hamilton function for the given H r, p is defined as system and Z the partition function. The Hamilton function ( ) 2 p H ( r, p) = + E r 2m ( ) (3.2) The first term expresses the kinetic energy and the second one the potential (internal) energy. The partition function Z normalises the probality between and 1 and is calculated as a 6N dimensional integral 1 3N 3N Z = exp H 3 ( r, p) / k N BT d rd p N! h (3.3) The partition function is related to the thermodynamic properties of the system through the Helmholtz free energy. The Helmholtz free energy F can indeed be written as follows F = k T ln( Z) (3.4) B The potential energy usually does not depend on the momenta p and thus the integration over p in Eq. 3.3 can be performed analytically, resulting in 1 Z = E r k T d r N! 3N exp 3 ( ) / N B Λ (3.5) with Λ the thermal de Broglie wavelength

49 Simulation Tools 29 h Λ = (3.6) π mk T 2 B A similar integration can be performed for the numerator of Eq The probability for a given state with positional coordinates { r } is then simply proportional to the Boltzmann factor E ( r ) P ( r ) exp (3.7) kbt Finally, the ergodic theorem states that a measured equilibrium value of a physical equilibrium property A corresponds to an average over the ensemble, or = 3N 3 (, ) (, ) N A A r p P r p d rd p (3.8) With this theorem, all the necessary ingredients for the calculation of thermodynamic quantities within the CE are now available. In the GCE, the system can exchange particles with an infinite reservoir of constant composition. For the derivation of the probability distribution for this ensemble, the above result for the CE can be used. Consider a large system with N particles, volume V and temperature T, consisting of the actual system S with N particles and a reservoir R with the remaining N - N particles, as illustrated in Fig The probability to find N particles in the system and N - N in the reservoir after integration over all momenta, is given by P( N, V, T ) = ES ER ( N ) kbt kbt 3( N N ) c( N) c( N N) e e d r N E R ES B 3( N N ) B 3 ( ) ( ) k T k T N N = 1 c N c N N e d r e d r (3.9)

50 3 Chapter 3 System S N particles Reservoir R N N particles Figure 3.1. Representation of a GCE as a giant CE, consisting of the actual system S and the reservoir R. By inserting the first-order Taylor expansion for the Helmholtz energy of the reservoir F R and recognising that df is by definition the chemical potential µ dn one obtains dfr FR ( N - N) FR ( N) N = FR ( N) N µ (3.1) dn P( N, V, T ) = = N N = 1 N N = 1 c( N) e FR ( N N ) ES ( N ) kbt kbt FR ( N N ) ES kbt kbt 3N c( N) e e d r c( N) e FR ( N ) µ N ES ( N ) kbt kbt FR ( N ) µ N ES ( N ) kbt kbt 3N c( N) e e d r e e (3.11) Dividing this equation by exp(-f R (N )) leads to the observation that E ( r ) µ N P ( r, N ) c( N) exp (3.12) kbt Again, when the probability P for a given state is known, all thermodynamic properties can be calculated through Eq. 3.8.

51 3.2 Monte Carlo simulations Simulation Tools 31 The key concept for an accurate estimation of equilibrium property A (Eq. 3.8) is importance sampling [2]. Using general approaches (Simpson s rule,...) to calculate these integrals would lead to enormously long computational times. The time for evaluating these integrals can be significantly reduced, when states, which have a negligible probability of occurrence, are discarded. In doing so, the total phase space is reduced. Indeed, the one-dimensional integral I I b = f ( x) p( x) dx (3.13) a can numerically be approximated as N b a I f ( x ) p( x ) (3.14) N i = 1 i i In order to evaluate f(x i )p(x i ) in these points, uniformly distributed random numbers on the interval [a,b] can be generated. However, this integral can also be interpreted as an average of the function f(x) over the interval [a,b], weighted with the probability function p(x). By generating random points ξ, with a probability corresponding to p(ξ) instead of a uniform distribution, the calculation of I is simplified and reduces to I N b a f ( ξi ) (3.15) N i= 1 At first sight, the difference recognised between Eq and 3.15 is not fundamental, but it is however of a crucial significance. When generating points with probability p(ξ), only these points that have a non-zero contribution are taken into account. This more efficient sampling leads to maximal benefit when f(x) is a slowly varying function of x and p(x) is a function that is negligibly small in a substantial part of the interval [a,b]. The statistical-mechanical integral in Eq. 3.8 fulfills these conditions. One can benefit from the advantages of importance sampling if states can be generated in such a way that the relative occurrence of a particular state corresponds exactly to its probability in the ensemble. This goal is achieved through a first-order Markov process [3]. A Markov process of order N is defined as the process in which a new

52 32 Chapter 3 state is generated out of the N previous states. In a first order Markov process, a new state j is only influenced by the previous state i and by the transition probabilities π(i j). These transition probabilities must satisfy the following conditions [2] they have to be constant in time, they should depend on properties of the initial state i and final state j only and not on any other state, it should be possible to reach any final state in a finite number of steps (ergodicity). An expression for the transition probability can be obtained by considering the evolution of the state probability with time and realising that this state probability becomes constant at equilibrium dpi dt = Pj π ( j i) Pi π ( i j) = (3.16) j This condition is known as the condition of detailed balance. There exist many possible solutions for π(i j) that satisfy Eq The most stringent requirement is to apply a detailed balance condition for each individual term in the summation P π ( j i) P π ( i j) = (3.17) j which can be regarded as microscopic reversibility. From this microscopic reversibility, the ratio of transition probabilities can be determined i π ( i j) P = j π ( j i) P i (3.18) and depends on the ensemble under study. For the CE, this ratio is given by E E E kbt j i π ( i j) = kbt e = e π ( j i) (3.19) and E = E j E i the energy difference between the new state j and the old state i. Once again, several choices are possible for the individual transition probabilities.

53 Simulation Tools 33 An efficient algorithm is one with a high acceptance ratio for the considered transitions. Metropolis et al. [4] suggested such an efficient algorithm, by setting E kbt π ( i j) = e if E π ( i j) = 1 if E < (3.2) For the GCE, insertion of Eq into Eq leads to a ratio of transition probabilities given by π ( i j) c( N) = π ( j i) c( N) j i e E kbt µ (3.21) with µ = µ j µ i the difference in chemical potential between state j and state i. Again, the Metropolis scheme is adopted E µ c( N) j ( ) kbt c N j π ( i j) = e if E µ kbt ln c( N) i c( N) i c( N) j π ( i j) = 1 if E µ kbt ln < c( N) i (3.22) Having devised the probability for evolving from state i to state j, the next question is how to actually perform the transition. Indeed, if the difference between the two states is too large, the transition probability can become very small, resulting in an inefficient sampling of the phase space. If, on the contrary, the difference is too small, most moves will be accepted, but the time for computing the equilibration can be long. For the CE, two atoms of a different kind are selected at random and an exchange is considered. For the GCE, one atom is selected at random and one considers changing its nature. These atomic exchanges guarantee in most cases a sufficient sampling. For slowly converging systems, e.g. in the case of strongly repulsive interactions, special techniques exist to improve the efficiency of the Monte Carlo simulation [2]. Typically, equilibrium is reached after a number of trial sweeps, equal to about a thousand times the total number of lattice sites in a simulation slab (1 sweeps) [5]. However, this number is merely indicative and there are in fact cases where equilibrium is reached much faster or, on the contrary, much slower (e.g. in the presence of a metastable state). Attaining equilibrium also strongly depends on the material property under study. The sampling of the simulation slab can only start

54 34 Chapter 3 after equilibrium is attained. In order to get reliable statistics, a sufficient number of samples should be taken into account, but the sampled microstates should be uncorrelated. Therefore at least a few sweeps between subsequent samples are necessary. It is possible to calculate an autocorrelation function and to adjust the sampling frequency of the simulation accordingly, but this means an extra computational cost. Provided there is no correlation between successive values of the quantity Q i in the sampling process, the standard deviation σ of A can be calculated as σ = 1 ( A ) 2 i A (3.23) n 1 i In many cases the efficiency of the MC procedure can be enhanced by tayloring it to the specific problem at hand. This is particularly interesting since even in moderate simulations, typically between 1 6 to 1 7 Monte Carlo steps are usually needed. The detailed state of order in alloys can be quantified in a straightforward manner using MC simulations. The short-range order is related to the probability to find one kind of atom as a neighbour to another atom. The short-range order parameter for a binary alloy with composition x A and x B is frequently defined as [6] s = P AB A B ( P ) max x x x x AB A B (3.24) with P AB the probability for an AB-bond and (P AB ) max the maximum value for P AB in the perfectly ordered state. The value of s lies between and 1, corresponding to complete randomness and perfect order respectively. An alternative definition is the one by Warren and Cowley [7]. For the quantification of the long-range order (LRO) parameter, the lattice is subdivided into two (or more) sublattices, on which the various components can mix. The LRO parameter S is a measure for the number of atoms that reside on the correct sublattice. For a binary alloy with two sublattices α and β, the LRO parameter is defined as [6] N N A, α B, β x x A N N α β S = = 1 x 1 x A B B (3.25)

55 Simulation Tools 35 with N X,j the number of atoms A on sublattice j (α orβ) and N j the total number of sites on sublattice j. For the study of the segregation processes in binary alloys, either the CE or the GCE can be chosen. In the CE, the number of atoms of each element remains constant. Since the simulation slab is of limited dimensions, the surface to bulk ratio is unrealistically large and compositional rearrangements at the surface may have an exaggerated, unrealistic influence on the bulk composition. This can be remedied by introducing a larger slab, at the cost of longer simulation times, or by iteratively adjusting the content of the slab until, at equilibrium, the desired bulk composition is obtained. A third and more elegant method is the use of the GCE. The GCE can also be used for the estimation of a demixing region. Indeed, at equilibrium, the chemical potential of each species is equal in both phases. It will then be observed that for a small change in µ, the simulated bulk composition will change drastically. The two compositions at this discontinuity correspond to the boundaries of the demixing zone. In repeating this procedure at various temperatures, the phase boundary for demixing can be simulated. This approach is applied in Chapter 7 for the calculation of the phase diagram for the Pd-Rh and the Cu-Rh systems. Determination of the chemical potential in the GCE As mentioned, the use of the GCE requires the knowledge of the chemical potential for each species. This chemical potential fixes the bulk composition at a certain value. In practice, however, Monte Carlo simulations are performed for a given bulk composition. Therefore, the chemical potential must be determined beforehand. For the work presented in this thesis, an algorithm to iteratively calculate the chemical potentials for a given bulk composition was developed [8]. The chemical potential of each species i in the alloy can be written as µ = µ + RT ln x + RT ln γ (3.26) * i i i i * with µ i the standard chemical potential. The activity coefficient γ i accounts for () the system s non-idealities. As an initial guess µ i, the value for an ideal mixture can be used µ = µ + RT ln x (3.27) () * i i i

56 36 Chapter 3 In case of disordered alloys, this starting value corresponding to an ideal mixture, is a reasonable guess. For ordered alloys, on the other hand, this guess is rather poor. For (off-stoichiometric) ordered binary alloys a good initial guess is described in Refs. [8,9]. ( n) As long as the n-th iteration of the composition x i, obtained by the preliminary bulk MC simulations, does not correspond to the desired composition x i, a new value for µ i must be generated until convergence is reached. The correction scheme is based on the fact that, according to the thermodynamical stability criterion, the chemical potential increases monotonically with concentration. It is furthermore assumed that (certainly for a limited change in composition), the activity coefficient γ i varies smoothly and gradually with composition. This leads to the following correction after the n-th iteration step γ = γ ( n) ( n 1) i i i ( n) xi µ = µ + RT ln γ ( n+ 1) * ( n) i i i i x x (3.28) The flow scheme of this algorithm is summarised in Fig This algorithm is intensively used in a parametric study of surface segregation in ternary alloys [1], which was studied for the whole ternary phase diagram. 3.3 Molecular Dynamics While MC simulations are ideally suited for the calculation of equilibrium properties, they do not yield any information on the physical trajectory towards equilibrium. While this is presicely the strength and the reason for calculational economy of the method, this can turn into a disadvantage if one is interested in the kinetic behaviour of the system. As opposed to MC simulations, Molecular Dynamics (MD) simulations are ideally suited for the study of true kinetic processes.

57 Simulation Tools 37 Figure 3.2. Flow chart of the iterative algorithm to determine the chemical potentials µ i. In MD simulations, Newton s equations of motion are solved for each of the N particles in the system, according to d r dp 2 i i Fi = mi = dt 2 dt (3.29) with F i the force acting on atom i, m i the mass of particle i, and p i the momentum of atom i, The force can be evaluated from the potential energy E F i de = (3.3) dr i

58 38 Chapter 3 This leads to a coupled set of ordinary differential equations. Typical characteristics of these equations are: a) they are stiff, i.e. there may be long and short timescales and the algorithm must cope with both of them and b) calculating the forces is expensive, and should be limited as much as possible [11]. There exist many methods for the step-by-step numerical integration of the equations of motion. A popular MD method is the velocity Verlet algorithm [12], which proceeds as follows 1 1 pi ( t + δt) = pi ( t) + δt Fi ( t) ri ( t + δt) = ri ( t) + δt pi ( t + δt) / mi pi ( t + δt) = pi ( t + δt) + δt Fi ( t + δt) 2 2 (3.31) Important features of the Verlet algorithm are: i) it is exactly time reversible, ii) the total volume in phase space is conserved, limiting the drift of the total energy at small time steps, iii) it is of low order in time and iv) it requires only one (expensive) force evaluation per step. Constraints, such as constant temperature, can be applied in MD simulations by introducing a Lagrange multiplier for each corresponding constraint [13]. In contrast to MC simulations, MD is a deterministic technique: given a set of initial positions and velocities, the subsequent time evolution is in principle completely determined. During MD simulations, the complete trajectory in a 6N parameter space is calculated. However, sometimes, one is not directly interested in this detailed trajectory. MD simulations can also be used to calculate average ensemble quantities and therefore can be used to calculate equilibrium physical properties. In this way, MD simulations are complementary to MC simulations. Typical physical properties that can easily be calculated by MD simulations are the melting temperature and the coefficient of linear expansion. 3.4 Kinetic Monte Carlo Simulations Kinetic information can be combined with the stochastic nature of the MC method by using a hybrid simulation method, such as Kinetic Monte Carlo (KMC) simulations. This hybrid approach is of particular interest when equilibrium has

59 Simulation Tools 39 not yet been established, but time scales are too long for MD simulations. In these KMC simulations, a set of elementary steps is defined and a rate equation is assigned to each of these steps, commonly an Arrhenius-type equation with v i the attempt frequency and i i act Ei An intuitive KMC algorithm then proceeds as follows [14] 1) Make a list of all possible events i, calculate the rate (r i ) for each event, make a summation of all rates: r tot. kbt r = ν e (3.32) act E i the activation energy barrier for process i. 2) Choose at random one event from the list, accept the selected event if ξ < r i / r tot with ξ a random number, if the event is accepted, increase the time by 1/r tot. 3) Repeat steps 1) and 2) until the stop criterion has been met. The rate equations can be obtained in two ways. For several cases, attempt frequencies and activation energies are available from experiments. The rate for a specific event can also be calculated from transition-state theory [15], or from MD simulations. One drawback of the method is that only elementary steps, included in the event list, are taken into account. The omission of one crucial elementary step may lead to entirely different behaviour. 3.5 Conclusions In this chapter, the most important atomistic simulation methods were presented. Each of the methods has its own advantages and disadvantages. The choice for a particular simulation method therefore strongly depends on the specific problem. Monte Carlo simulations are ideally suited for the study of equilibrium processes, by discarding all information about the kinetics. On the other hand, Molecular Dynamics solve Newton s equations of motion and evolve towards equilibrium following the true kinetic path. However, for numerical reasons the time step must be taken very small, corresponding to maximum feasible simulation times of a few nanoseconds. Kinetic Monte Carlo simulations form the bridging between both, by using rate equations for each elementary process. The rate equations can

60 4 Chapter 3 be derived from MD simulations or from other theories, e.g. transition-state theory. One drawback is that all relevant processes must be known beforehand. References [1] D.A. McQuarry, Statistical Mechanics, Harper and Row, New York, [2] M.E.J. Newman and G.T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press, Oxford, [3] R.W. Gilks and S. Richardson, Markov chain Monte Carlo in practice, CRC Press, Boca Raton, [4] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 187. [5] S.M. Foiles, in: P.A. Dowben and A. Miller (Eds.), Surface segregation phenomena, CRC Press, Boca Raton, 199. [6] F.C. Nix and W. Shockley, Rev. Mod. Phys. 1 (1938) 1. [7] J.M. Cowley, Phys. Rev. 77 (195) 669. [8] C. Creemers, P. Deurinck, S. Helfensteyn and J. Luyten, Appl. Surf. Sci. 219 (23) 11. [9] S.M. Foiles and M.S. Daw, J. Mater. Res. 2 (1987) 5. [1] J. Luyten, S. Helfensteyn and C. Creemers, Appl. Surf. Sci , (23) 833. [11] M.P. Allen, in: N. Attig, K. Binder, H. Grubmüller and K. Kremer (Eds.), Computational Soft Matter, From Synthetic Polymers to Proteins, John von Neumann Institute for Computing, Jülich, 24. [12] W. C. Swope, H. C. Andersen, P. H. Berens and K. R. Wilson, J. Chem. Phys. 76 (1982) 637. [13] J-P. Ryckaert, G. Ciccotti and H. J.C. Berendsen, J. Comput. Phys. 23 (1977) 327. [14] W. M. Young and E.W. Elcock, Proc. Phys. Soc. 89 (1966) 735. [15] G. H. Vineyard, J. Phys. Chem. Solids 3 (1957) 121.

61 4 Energy models In this chapter, two different approaches are presented for the calculation of the total energy. The first belongs to the class of ab initio methods, which in principle do not rely on any experimental input data. In practice, a limited number of experimental data is still necessary for some approximations. However, in view of required calculation times, these methods suffer from serious limitations with respect to the total number of atoms that can realistically be modelled. Therefore, simplifications are sought for the ab initio methods by using parameterised expressions instead of solving the Schrödinger equation exactly. The specific model parameters are fitted to a set of well-known material properties and these methods are therefore called semi-empirical energy models. An important family of semi-empirical models is based on the Embedded Atom Method (EAM). Both ab initio methods and semi-empirical models are discussed in the next paragraphs, together with a critical comparison Density Functional Theory Theoretically, the energy E of any system consisting of electromagnetically interacting quantum particles can be calculated from Schrödinger s wave equation Hˆ Ψ ( α) = E Ψ( α) (4.1) with Ĥ the Hamilton operator and Ψ the many-body wavefunction. The variableα contains the position and spin coordinates of all particles in the system. The Hamilton operator Ĥ contains the kinetic energy of the nuclei ( T N ) and electrons 41

62 42 Chapter 4 ( T e ), as well as the electrostatic interaction energy between ions ( V NN ), ions and electrons ( V Ne ) and electrons ( V ee ) respectively [1], i.e. Hˆ = Tˆ + Tˆ + Vˆ + Vˆ + Vˆ N e Ne ee NN 2 2 N 2 2 Ne 2 ħ R ħ 1 i r e Z i i = 2 M 2 m 4πε R r i= 1 i i= 1 e i, j i j e 1 1 ZiZ je πε r r 2 4πε R R i j i j i j i j 2 (4.2) The wavefunction Ψ is a function of the positional and spin coordinates of all other particles in the system. Although, in principle, it provides a theoretical equation that is valid for each system, this coupling between the particles makes the Schrödinger equation practically insoluble, except for very simple systems containing only a few particles. Different approximations are necessary before the Schrödinger equation can be solved numerically. The first, frequently made approximation is the Born-Oppenheimer Approximation [2], which assumes the electrons to be in instantaneous equilibrium with the ions at fixed positions R i. Electrons indeed move much faster than ions, due to their much lower mass. Consequently, the electrons can be considered as moving in an external potential, that is generated by the ions at fixed positions (no kinetic energy), and the original wavefunction can be separated into two coupled equations Tˆ ˆ ˆ (, ) ˆ e + Vee + Vext Ri Θ α Ri = HeΘ ( α, Ri ) = ε n Θ( α, Ri ) (4.3) Hˆ + Vˆ R χ( R ) = E χ( R ) ( ( )) ( ) ( ) e NN i i i with Θ the electronic and χ the nuclear wave function respectively. The total energy now varies parametrically with the nuclear positions. Solving these two equations starts by assuming a fixed set of nuclear positions R i. The first equation must then be solved with these nuclear positions as variable parameters. Usually, only the ground state of the electron gas ε is important. Indeed, electronic excitations require high energies (on the order 1 4 K), much higher than the temperature of interest for metals. Subsequently, the constant contribution from

63 Energy models 43 V ˆNN must be added to H ˆ e. The so obtained total energy is a function of the three position coordinates of each of the N ions, i.e. 3N parameters. After exclusion of the translational and rotational degrees of freedom, a potential energy surface (PES) with 3N-5 parameters is obtained. Finally, the motion of the ions can be found by solving the equations of motion from classical mechanics on the constructed PES. It is evident that once a solution strategy is available for the electronic wavefunction, the remaining part becomes trivial. Hartree [3] made one of the earliest approximations in writing the many-electron wave function as a product of single-particle functions, ψ ( α,..., α ) = φ( α ) φ( α ) φ( α )... φ( α ) 1 N N (4.4) in which each single-particle wavefunction φ obeys a one-electron Schrödinger equation with a potential term arising from the average field of the other electrons, 2 2 N 2 Ne ħ 2 e Z j e ρ j r ( ) i + dr j φi = T + VH φi = εiφi 2me 4πε j= 1 R 4 j r πε (4.5) i j i= 1 rj r i Electrons, however are fermions and obey Pauli's exclusion principle, which implies that the wave function describing the electronic system must be antisymmetric upon exchange of any two electrons. A simple product of one-electron wavefunctions does not fulfill this requirement. The required anti-symmetry of the wave-function results in an extra contribution to the total energy of the system, called the exchange energy (V x ). This exchange energy can be taken into account when considering a Slater determinant, instead of the simple product of singleparticle wavefunctions [4]. This method is known as the Hartree-Fock solution to the Schrödinger equation. The inclusion of exchange effects leads to improved results for the total energy. The exchange energy, defined as the part that is present in the Hartree-Fock solution, but not in the Hartree solution can thus be written as V = E T V (4.6) x HF H Secondly, in reality, electrons are not moving independently from one another. Their motion is correlated, and this correlated motion results in a lowering of the energy with respect to the fictitious uncorrelated situation. The energy difference between correlated and uncorrelated situations is called the correlation energy and is given by

64 44 Chapter 4 V = T T (4.7) c By its construction, the Hartree-Fock method is not able to describe this correlation energy. Despite this shortcoming, the Hartree-Fock approximation is still an indispensable benchmark in molecular physics. Another breaktrough towards a solution of the Schrödinger equation came with the two theorems of Hohenberg and Kohn [5]. Instead of solving the Schrödinger equation for the wavefunction, a solution is sought for the electron density ρ( r ). The first theorem states that there exists a one-to-one correspondence between the ground state density ρ( r ) of a many-electron system and the external potential V ext. An immediate consequence is that any observable quantity is a unique functional of the electron density. The second theorem states that the ground state total energy functional H e [ρ] is of the form H F r V r dr [ ρ] = [ ρ] + ρ ( ) ( ) e HK ext (4.8) where the Hohenberg-Kohn density functional F HK is universal for any manyelectron system. H e [ρ] reaches its minimal value for the ground state density that corresponds to V ext. The Hohenberg-Kohn equation can be rewritten as F = T + V + V + V = T + V + V (4.9) HK H x c H xc in which V xc combines all exchange and correlation effects. Kohn and Sham [6] defined a hamiltonian H ˆ KS as follows Hˆ ˆ ˆ ˆ ˆ KS = T + VH + Vxc + Vext ħ e ρ ( r ') = m 4 r r ' r dr ' V i xc Vext e πε (4.1) This can be interpreted as the hamiltonian for a non-interacting electron gas subject to two external potentials, one from the exchange and correlation effects and one generated by the ions. The theorem of Kohn-Sham now states that the exact ground state density ρ r of an N electron system is ( )

65 Energy models 45 N ρ φ φ * ( r ) = ( r ) ( r ) i= 1 i i (4.11) where the single particle wave functions φ i are the N lowest energy solutions of the Kohn-Sham equation Hˆ KS φ i = ε i φ i (4.12) The Kohn-Sham wavefunctions φ i do not correspond to real wavefunctions of electrons, but instead, they describe mathematical quasi-particles in such a way that the overall density of these particles corresponds to the real electron density. Now only one single barrier remains, namely the precise expression of the exchange-correlation term. Nowadays, two widely accepted ways exist for taking this exchange-correlation energy into account. The first approximation is known as the Local Density Approximation (LDA). Within the LDA, it is postulated that E = ρ r ε ρ r dr LDA xc ( ) xc ( ( )) (4.13) in which ε xc stands for the exchange and correlation energy of an homogeneous electron gas. The most commonly used parameterisation is the one by Perdew and Wang [7], who fitted a function to all available information. The major part of this information came from Quantum Monte Carlo Simulations of the exchangecorrelation energy for a homogeneous electron gas, as performed by Ceperley and Alder [8]. The LDA approximation is expected to behave well for systems with a slowly varying density. However, it also performs very well for a large set of realistic cases. For systems with a strongly varying electron density, another approximation for the exchange-correlation energy is postulated, namely the Generalised Gradient Approximation (GGA). Within the GGA, ε xc is not only a function of the density itself, but also depends on the gradient of the density. Unfortunately, there is no unique way to introduce the gradient in ε xc. Candidate GGA functionals are therefore fitted to some experimental data for atoms and molecules from which then the best performing functional is retained. Contrary to LDA, GGA is strictly spoken not a purely ab initio method. However, some GGA functionals exist, that do not use any input of experimental information.

66 46 Chapter 4 In view of the symmetry of crystalline materials, the wavefunctions are not solved in terms of real space coordinates, but rather in reciprocal (k point) space. For numerical reasons, integrals are replaced by summations. The summation runs over a set of points within the Brillouin zone of the supercell. A frequently used set of special k points within the Brillouin zone is described by Monkhorst and Pack [9]. Strongly varying wavefunctions require that high frequencies be included. Again, for numerical reasons, a maximum frequency (cut-off) must be specified and frequencies beyond the cut-off frequency are not included in the wavefunction. Mainly these two parameters, the k point sampling and the cut-off frequency, determine the numerical convergence of the Density Functional Theory (DFT) calculations. Reasonable values must be determined beforehand in order to assure well converging calculations. In the region close to the nucleus, the density varies very strongly due to the strong bonding of the core electrons to the nucleus. An accurate description of these core electrons would require a high frequency cut-off and consequently long computational times. To circumvent this problem, the true behaviour of the core electrons close to the nucleus is sometimes replaced by a much softer pseudopotential [1,11]. Thanks to the DFT, quantum-mechanical calculations have become feasible for metallic systems with many electrons. However, the computational effort is still significant. The computer time needed for DFT scales with the total number of atoms to the third power. Therefore, surface segregation processes, which require the tracking of a few thousands of atoms, can not be modelled by quantummechanical methods. Semi-empirical models, which are computationally less demanding, provide an outcome. The most important class of semi-empirical models is known as Embedded Atom Methods (EAM) [12]. The basics of the EAM and its relation with DFT will be described in the next paragraph The Embedded Atom Method The fundamental equation within the Embedded Atom Method [13] is given by 1 E = E = F ρ + φ 2 i i ( h, i ) ij (4.14) i i i j i

67 Energy models 47 in which F i represents the energy release upon embedding an atom in the local electron density that is generated by the neighbouring atoms ( ρ h, i ) and φ ij the electrostatic repulsion energy between the screened nuclei. This equation can directly be derived from the density functional formalism [12,14], combined with only minor assumptions. From the density functional equation, the cohesive energy of a solid can indeed be written as e ZiZ j e Ziρ Ecoh = F ( ρ ) + dr 2 i, j i 4πε R ij i 4πε r Ri 1 e ρ( r ) ρ( r ) + dr dr E free atoms πε r12 2 (4.15) F(ρ) contains the electron kinetic energy, the exchange and correlation energy. E free atoms corresponds to the energy of the isolated atoms in vacuum. The remaining terms stand for the electrostatic interactions between nuclei, nuclei and electrons, and electrons respectively. Assuming that the density can be written as a superposition of atomic a densities ρ = ρ, one obtains with i i 1 E = F F + U a a a ρ ( ρ ) (4.16) coh i i ij i i 2 i, j i a a e n ( r ) n ( r ) Uij = dr dr 4πε (4.17) 2 a i 1 j r12 and n r r R Z r R a ( ) = ρ ( ) δ ( ) a i i i i i (4.18) with δ(x) the Dirac delta function. In the region around atom i, a background density ρb, i can be defined

68 48 Chapter 4 ρ ( r R ) = ρ (4.19) a b, i j j j i In most of this region, the atomic density of atom i itself dominates the total density. ρ b, i is therefore supposed to be small and only slowly varying. Next, the embedding energy upon placing an atom i in an electron gas of constant density ρ is defined as i a a ( ρ ) ( ρ ρ ) ( ρ ) ( ρ ) F = F + F F (4.2) i i i i i i Using this embedding energy in Eq. 4.16, one obtains with 1 E = F + U + E a ( ρ ) (4.21) coh i i ij error i 2 i, j i E = F F + + F a a ρ ( ρ ρ ) ( ρ ) (4.22) error i i i i i i i Setting this E error to zero leads to the desired expression for the electron density. If F ρ is negligible with respect to the other terms in Eq. 4.22, ( ) i F a a ( ρi ρb, i ) F ( ρi ρi ) + + (4.23) Thus, in case the background density varies slowly compared to ρ, the solution of E error = is obtained for form of Eq ρ i = ρ b, i, arriving in a natural way at the functional The assumed linear superposition of atomic densities is no longer valid when strongly localised chemical bonds are present. The EAM is therefore restricted to simple (alkali) metals and late transition metals, which show spherically symmetric valence electron shells. Many different formulations exist for the EAM. The most popular among them is the version by Foiles, Daw and Baskes, which will be further described [15]. a i

69 Energy models 49 The electron density ρ j of atom j is computed from the Hartree-Fock wave functions [16,17]. To account for possible electronic rearrangements upon alloying, the fixed total number of valence electrons N j of element j is allowed to redistribute over the s and d sublevels ρ = n ρ + n ρ = n ρ + ( N n ) ρ (4.24) j s, j s, j d, j d, j s, j s, j j s, j d, j where n s,j and n d,j denote the number of outer s and d electrons and N j the total number of valence electrons. ρ s,j and ρ d,j are the densities associated with the s and d ground state wave functions respectively. The pair interaction term φ ij is calculated as the electrostatic Coulomb repulsion between the screened, effective nuclear charges [25] 1 Zi ( Rij ) Zi ( Rij ) φij ( Rij ) = (4.25) 4πε R ij with Z i the effective charge of atom i and ε the permittivity of vacuum i ij Z ( R ) = Z e(1 + β R ) e α (4.26) νi i ij, i i ij R where Z,i denotes the number of outer electrons of atom i, and α i and β i are constants that depend on the nature of atom i. ν i is a parameter that is fixed empirically. Once the atomic electron densities ρ j and the pair interactions φ ij are known, the embedding energy function F i is derived by considering the pure element i. At equilibrium (lattice constant a ), the energy of this pure metal i is given by the negative of the sublimation energy ( E i ). In expanded or compressed form (lattice constant a) the energy of this configuration is given by Rose's universal equation of state [18] * i ( ) i (1 ) * a E a = E + a e (4.27) where a * denotes the relative deviation from the equilibrium lattice constant (a ) a = a / a 1 * E / 9B Ω i i i (4.28)

70 5 Chapter 4 with B i the bulk modulus of pure i and Ω i the equilibrium volume per atom i. The embedding function F i is now readily obtained by setting Eq equal to Eq This embedding function is assumed to be universally valid, which means that it depends only on the type of the embedded atom and the background electron density but not on the origin of this electron density. The same embedding function can thus be used to calculate the energy of an atom i in an alloy. The sum in Eq in principle extends over the whole lattice. In practice, however, it is restricted to second nearest neighbours. Foiles et al. [15] optimised the EAM parameters globally towards the group of the six elements Cu, Ag, Au, Ni, Pd and Pt. This was done by setting up a least squares optimisation to solve an overdetermined set of equations for the elastic constants and vacancy formation energy of each of the six elements as well as for the heats of solution in the dilute limits of all binary systems. Deurinck et al. [19,2] demonstrated that EAM often reproduces the driving forces for segregation more reliably if specific EAM parameters are calculated, specifically optimised for the binary alloy system under study The Modified Embedded Atom Method The EAM has been successfully applied to the face centered cubic (fcc) transition metals [12] with a nearly filled d-band. In an empirical way and not justified by strong physical arguments, Baskes [21] extended the main idea behind the EAM in order to make it applicable to materials with directional bonding as well. Similarly to the EAM, this modified embedded atom method (MEAM) defines the configurational energy E i of an atom i as the sum of energy contributions due to electrostatic interactions with its neighbours and the energy release upon embedding this atom in the local electron density generated by the other atoms of the system ρh, i 1 Ex = Ei = Fi ( ) + φij ( Rij ) i i Zi 2 j (4.29) where, h i ρ is the host electron density at the site of atom i due to the remaining atoms of the system, renormalised to the number of nearest neighbours Z i in the reference structure of a type-i atom, F i (ρ) is the embedding energy of atom i into the background electron density ρ, and φ ij (R ij ) is the core-core pair repulsion

71 Energy models 51 energy between atoms i and j separated by a distance R ij. In the original formulation by Baskes, the sum over i is restricted to nearest neighbours only, by an angularly dependent many-body screening function (see later). Baskes MEAM further follows the EAM concept, except for the improved approximation of the electron density. While the EAM uses a linear superposition of spherically averaged electron densities, in the MEAM, angular characteristics of the electron densities are included. Directional bonds are now taken into account in an empirical way by adding gradient and higher-order corrections to the simple EAM formulation. The host electron density of atom i, ρ h, i, is composed of the spherically () (1) symmetric partial electron density ρ i and angular dependent contributions ρ i, (2) ρ i and (3) ρ i. These partial electron densities have the following form: ρ = ρ ( R ) () a() i j ij j i = x R (1) 2 α a(1) ( ρi ) ij ρ j ( ij ) α j i = x x R R (2) 2 α β a(2) a(2) ( ρi ) ij ij ρ j ( ij ) ρ j ( ij ) α, β j i 3 j i 3 = x x x R x R (3) 2 α β γ a(3) α a(3) ( ρi ) ij ij ij ρ j ( ij ) ij ρ j ( ij ) α, β, γ j i 5 α, β, γ j i 2 2 (4.3) α α with xij = Rij / Rij and R α ij the components of the separation vector between atoms (3) i and j, projected on fixed (arbitrary) axis α. The second term in ρ i is a recent modification that did not appear in the original formulation. In the present form, the partial electron density functions are orthogonal Legendre functions [22,23]. Old parameter sets can be transformed to new ones, by replacing t (see further) by t + 3/ 5 t [23]. (1) (3) i i (1) i The atomic electron densities ρ are devised as exponential functions a ( l ) j ( l ) a( l) i ( Rij / Ri 1) ρ i fe β = (4.31) with the f and β ( l ) i adjustable parameters and R i the equilibrium distance of atom i in the reference structure.

72 52 Chapter 4 The partial electron densities are combined into the total background density via ρ = ρ + Γ (4.32) () h, i i 1 i and ( ) 2 3 l l ρ i Γ i = ti () l= 1 ρi (4.33) with t ( l) i adjustable weights. Besides Eq. 4.32, some alternative formulations exist for the combination of partial electron densities into the total background density. Among them are the following expressions [24] ρ h, i = ρ () i 2 1+ exp( Γ ) i (4.34) or () Γi ρh, i = ρi exp 2 (4.35) which are all identical up to first order for small values of Γ i. The most commonly used formula, however, is still Eq When all weighting factors are set equal to zero, all angular dependent terms vanish and the MEAM reduces to the EAM. The embedding functions F i are devised as functions of the electron density F ρ ρ ρ = A E (4.36) h, i h, i h, i i ( ) i i ln( ) Zi Zi Zi with A i an adjustable parameter and reference state. E i the ground state energy of atom i in the Having now formulated an analytical expression for the embedding energy, the electrostatic pair interaction energy must be calculated from the equation of state for a reference structure. For pure metals, the most stable phase of the specified

73 Energy models 53 atomic species is generally taken as the reference state. The energy as a function of interatomic distance is again described by Rose s equation of state, so that 2 * * a ρ φii ( R) = Ei ( 1 + a ) e Fi ( ) Zi Zi (4.37) with ρ the background electron density for the reference structure. For the calculation of the electrostatic interaction between unlike atoms, a Roselike equation for the reference state of the alloy is constructed. Equiatomic binary compounds, where each i atom is entirely surrounded by neighbouring atoms j, show only pairwise interactions between unlike species and are thus ideal reference structures. Simple examples of such structures are B1 (sc, NaCl) or B2 (bcc, CsCl). φ ij (R) can then simply be derived from a() a() 1 1 Zij ρ j Zij ρ i Eij = ( Ei + E j ) = Fi ( ) + Fj ( ) + Zijφij ( R) 2 2 Zi Z j (4.38) with Z ij the number of nearest neighbours in the equiatomic reference structure. The energy E can be estimated by the equation of Rose, with sublimation energy form ij i j ij ij E = ( E + E ) / 2 + E ( E form ij equals the energy of formation), R ij the equilibrium nearest neighbour distance and α ij related to the bulk modulus of the ordered structure. φ ij (R) is then readily obtained from these two equations, once the host electron densities and the embedding functions are known. A problem often arises when no experimental or theoretical values are known for the properties of the binary reference structure. In that case, other reference structures can be considered, e.g. the L1 2 (fcc, Cu 3 Au) structure, resulting in somewhat more complicated formulas for the electrostatic repulsion energy [25]. The MEAM contains 8 adjustable parameters, which must be fitted to experimental properties. Apart from these 8 parameters, the MEAM contains 3 additional parameters, incorporated into Rose s equation of state, to ensure that the sublimation energy, bulk modulus and equilibrium volume of the reference structure are exactly reproduced. The input set of experimental data consists of two elastic shear constants, two structural energy differences (fcc-hcp and fccbcc) and the vacancy formation energy, but this is insufficient to determine all MEAM parameters. Consequently, Baskes introduced some arbitrariness in the parameterisation of the MEAM, due to the lack of consistent, easily accessible and reliable experimental results for a large class of materials. However, increased

74 54 Chapter 4 computational power nowadays allows first principles calculations on real and hypothetical model systems. These data can then complement the experimental results to a complete input data set that is necessary for a well-parameterised potential [26]. In the MEAM, the range of interactions is governed by a many-body radial screening function f c (x), based on a simple elliptical construction [27]. The issue whether to include other than nearest neighbours, is addressed by Baskes et al. [28] for the particular case of Al. It appears that, at least for Al, the question of whether a short range angular many-body potential is more appropriate than a long-range central many-body potential has not yet been resolved. Lee et al. introduced a MEAM formalism, that explicitly incorporates second nearest neighbours, first for bcc metals and later for fcc metals also [29,3]. The original viewpoint of the (M)EAM, namely the separation of the energy into a pair-wise electrostatic energy that is purely repulsive and an attractive embedding energy into the local electron density has become somewhat obsolete. Firstly, the nonunique separation of φ and F allows for various general forms for φ [12,31]. Furthermore, especially in the MEAM, the purely repulsive character of φ is completely lost, as F becomes zero and φ is negative for the pure metal in the equilibrium reference structure. Another viewpoint to the embedding function F was adopted by van Beurden et al. [26] They point out that F is a coordination function, rather than an attractive embedding energy. To illustrate this, one can write the energy of any monoatomic cubic structure with a center of symmetry as a() * * a Z ' ρ i Z ' a() ( ( ) ) ( ρ ) Z ' Ei ( R) = Ei 1+ a e + Fi Fi i Z Zi Z (4.42) The first term accounts for a purely linear behaviour of the energy with respect to the number of nearest neighbours Z, while the second term takes additional (nonlinear) effects into account. This viewpoint was confirmed by DFT calculations for Rh in different metastable cubic structures. Coordination function therefore would be a more accurate name for F Evaluation The EAM is a frequently used semi-empirical energy model, firmly based on the many-electron Schrödinger equation, but relying on parameterised expressions.

75 Energy models 55 The parameters are (in)directly related to material properties that can be determined experimentally or theoretically. In this way a sound compromise arises: the method has a sound theoretical basis but is also anchored to accurately known material properties. This also results in a good balance between accuracy and computational complexity. Its main asset is that it accounts for many-body interactions: the bond energies depend not only on the distance between atoms i and j, but also on the number and identity of their respective neighbours. EAM has been applied successfully to a variety of bulk and interface problems [32], especially with optimised parameters for the system under study. However, EAM suffers from a well-known deficiency in that it significantly underestimates the surface energies. This degree of underestimation is similar for all metals, so that in the end the difference in surface energy is fairly accurately reproduced. The reason for this underestimation can be found out and traced to the following. The purely pairwise (linear) electrostatic interaction energy φ contains 3 free parameters, i.e. α, β and ν. These parameters are determined by the elastic constants. The typical many-body character that is evidently of great importance at the surface must be accounted for by the embedding energy function F. This function has only one parameter left to be determined, namely n s. As experimental input for this non-linear behaviour of the energy with respect to the number of nearest neighbours, the vacancy formation energy is used. However, breaking the first bond is accompanied by the largest bond energy relaxation. n s characterises the bond energy relaxation upon the formation of a vacancy and this degree of bond relaxation is next extrapolated to the surface where more bonds (per atom) are broken. This leads to an excessive bond energy relaxation and consequently to surface energies that are typically 4% too low in value. Alternatively, when surface energies are used as input to determine n s, this effect leads to an overestimation of the vacancy formation energy. Furthermore, information about the (dilute) energy of mixing must also be described by this n s parameter. The EAM does not contain enough free parameters to describe the non-linear dependence of the energy. The MEAM, which has more free parameters, is therefore a better model and will be further used for the investigation of surface segregation phenomena. MEAM further surpasses EAM by accounting for the angular configuration between the different atoms. This angular dependence is however assumed rather than derived from explicit quantum-mechanical considerations. Nevertheless, MEAM has proven to produce (more) accurate results for a number of alloys [25,33,34]. The successful extrapolation of the semi-empirical potentials, which are fitted to bulk properties only, to a wide range of surface phenomena confirms that they indeed contain the necessary ingredients to describe well various subtleties of the metallic bond.

76 56 Chapter 4 References [1] S. Cottenier, Density Functional Theory and the family of (L)APW-methods: a step-by-step introduction, Instituut voor Kern- en Stralingsfysica, K.U. Leuven, Belgium, 22. [2] M. Born and R. Oppenheimer, Annalen der Physik 84 (1927) 457. [3] D. R. Hartree, Proc. Cambridge Philos. Soc. 24 (1928) 89. [4] J.C. Slater, Phys. Rev. 35 (193) 21. [5] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) 864B. [6] W. Kohn and L.J. Sham, Phys. Rev. 14 (1965) 1133A. [7] J.P. Perdew and Y. Wang, Phys. Rev. B 45 (1992) [8] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45 (198) 566. [9] H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13 (1976) [1] J.C. Philips, Phys. Rev. 112 (1958) 685. [11] M.T. Yin and M.L. Cohen, Phys. Rev. B 25 (1982) 743 [12] M.S. Daw, S.M. Foiles and M.I. Baskes, Mat. Sci. Rep. 9 (1993) 251. [13] M.S. Daw and M.I. Baskes, Phys. Rev. B 29 (1984) [14] M.S. Daw, Phys. Rev. B 39 (1989) [15] S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) [16] E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14 (1974) 177. [17] A.D. McLean and R.S. McLean, At. Data Nucl. Data Tables 26 (1981) 197. [18] J.H. Rose, J.R. Smith, F. Guinea and J. Ferrante, Phys. Rev. B 29 (1984) [19] P. Deurinck and C. Creemers, Surf. Sci. 419 (1998) 62. [2] P. Deurinck and C. Creemers, Surf. Sci. 441 (1999) 493. [21] M. I. Baskes, Phys. Rev. B 46 (1992) [22] M. I. Baskes, Mater. Sci. Eng., A 261 (1999) 165. [23] G. Wang, M. A. Van Hove, P.N. Ross and M. I. Baskes, J. Chem. Phys. 121 (24) 541. [24] M.I. Baskes, Mat. Chem. Phys. 5 (1997) 152. [25] M.I. Baskes, J.E. Angelo and C.L. Bisson, Modell. Sim. Mater. Sci. Eng. 2 (1994) 55. [26] P. van Beurden and G.J. Kramer, Phys. Rev. B 63 (21) [27] M.I. Baskes, Mater. Chem. Phys. 5 (152) [28] M.I. Baskes, M. Asta and S.G. Srinivasan, Phil. Mag. A 81 (21) 991. [29] B.-J. Lee, M.I. Baskes, H. Kim and Y.K. Cho, Phys. Rev. B 64 (21) [3] B.-J. Lee, J.H. Shim and M.I. Baskes, Phys. Rev. B 68 (23) [31] A.F. Voter and S.P. Chen, in Characterization of defects in materials, MRS Symposia Proceedings No. 82, R.W. Siegal, J.R. Weertman, and R. Sinclair (Eds.), Materials Research Society, Pittsburgh, 1987.

77 Energy models 57 [32] C. Creemers, P. Deurinck, S. Helfensteyn and J. Luyten, Appl. Surf. Sci. 219 (23) 11. [33] M.I. Baskes, J.S. Nelson and A.F. Wright, Phys. Rev. B 4 (1989) 685. [34] M.I. Baskes and R.A. Johnson, Mod. Simul. Mater. Sci. Eng. 2 (1994) 147.

78 58 Chapter 4

79 5 New parameterisation method for the MEAM Although the original MEAM has been successfully applied to the description of various bulk and surface phenomena, a few remarks can be formulated. For a proper and physically sound determination of the model parameters, a large set of consistent material properties must be available. At the time Baskes introduced his version of the MEAM, this large database of fundamental material properties was not available. The use of a set of values coming from different experimental techniques inevitably leads to the introduction of inconsistencies. Furthermore, for many elements, only a limited number of material properties is available. This lack of information leads to some arbitrariness in the assignment of the original ( l ) β i model parameters. Therefore, some of these parameters are taken equal for a certain class of materials. Another drawback of the use of experimental input data is the fact that the true equilibrium is always reflected in the experimental result. This means that the measured state corresponds to a minimal free energy with respect to all degrees of freedom. Indeed, by introducing a vacancy in a piece of material, a structural rearrangement occurs by the atoms in the neighbourhood of the hole. The precise details of this rearrangement are often not known, while in the fitting procedure, the atomic coordinates must be known explicitly. The creation of a vacancy is then calculated in a fixed lattice and compared to the measured value, which is assumed to be unrelaxed. Finally, although it is a striking feature of the MEAM to predict certain surface phenomena with remarkable accuracy, using only bulk input data, the model is not guaranteed to be accurate for all systems. With the current possibilities for the implementation of DFT, various properties can readily be calculated. The systematic use of DFT results eliminates all 59

80 6 Chapter 5 previous points of criticism. Indeed, by calculating a sufficiently large set of input data, the arbitrariness in the parameterisation is eliminated. Secondly, all input data are calculated within the same physical framework and do not suffer from inconsistencies between different data points. Thirdly, it is possible to separate electronic effects from structural rearrangements. And finally, the applicability to surface phenomena can be improved by also using calculated surface energies and surface relaxations as input. In this way, the MEAM is not used as a predictor of surface phenomena, by extrapolating bulk data, but it is used to interpolate between the different input data DFT calculations For the DFT calculations, the Vienna Ab Initio Simulation Package (VASP) [1,2], which makes use of periodic cells and a plane wave basis set, is used. The interaction between the core and valence electrons is described by ultrasoft Vanderbilt pseudopotentials with scalar relativistic corrections [3], generated by Kresse et al. [4]. For the Generalised Gradient Approximation (GGA), the PW91 [5] functional is used. In all calculations, the total energies are allowed to converge to within a few mev with respect to k point sampling and energy cutoff. The surface energy is calculated as follows γ ( ijk ) = E surf ( ijk ) 2 N N E s a ( ijk ) bulk (5.1) surf with E ( ijk ) the energy of a slab with two (ijk) oriented surfaces, N the total number of atoms in the slab, N s the number of surface atoms and a (ijk) the atomic area at the (ijk) surface. The calculation of the surface energy in this work is performed with 11-layer slabs. This number was sufficiently high for well-converged values for the surface energies (Appendix B). ads For the calculation of the adatom adsorption energy ( E ( ijk ) ), a slab with four surface atoms is used. This corresponds to a fractional coverage of.25 which is considered as an approximation for the dilute limit. In case the second nearest neighbour interactions do not contribute significantly to the total energy, the 25% fractional coverage indeed boils down to the dilute limit because the adsorbed

81 New parameterisation method for the MEAM 61 atoms only see substrate atoms and do not feel each other in any way. then calculated as E is ( ijk ) ads E ads ( ijk ) adatom surf free atom E( ijk ) ( fcc) E( ijk ) 2E = (5.2) 2 adatom with E( ) ( fcc ) the energy of a slab with an adatom adsorbed on an fcc site at ijk both surfaces and E free atom the energy of a free atom in vacuum. A subtle aspect of the surface behaviour of metals is the occurrence of a surface reconstruction at various orientations. A well-known reconstruction is the (2 1) missing-row (MR) reconstruction at the (11) surface. The energy of formation for any surface reconstruction (E reconstruction ) can be calculated as E reconstruction = E E N E 2 a r u bulk (1x1) (5.3) with E r and E u respectively the energy of a reconstructed and an unreconstructed slab, N the difference in the number of atoms between the two slabs ( N = for the MR reconstruction), E bulk the energy of an atom in the bulk and a (1x1) the atomic area for an unreconstructed slab. A negative energy of formation indicates that the reconstructed surface is stable, at least at lower temperatures. The (2 1) MR surface reconstruction at pure metal surfaces is experimentally observed only for Pt, Au and Ir [6-8] DFT-based MEAM parameterisation From the above arguments, it is clear that a new parameterisation scheme, based on first-principles calculations is extremely valuable for a more accurate description of surface phenomena. In this way, a state-of-the-art model is obtained, which combines the accuracy of quantum-mechanical calculations with the computational efficiency of the semi-empirical MEAM. A parameterisation scheme, which also includes surface properties, was proposed by van Beurden et al. [9]. The authors proposed a step-wise parameterisation. First, A i is fixed to a value of 1 and the electron density weighting parameters ( l) t are fitted to the unrelaxed (1) surface energy (γ (1) ), the (111) surface i energy (γ (111) ), the unrelaxed fcc-hcp stacking fault energy ( E hcp ) and the

82 62 Chapter 5 f unrelaxed vacancy formation energy ( E v ). Subsequently () (2) β i and β i are calculated using the relationship for the energy versus the interplanar distance in the [1] direction. The second derivative of this curve at equilibrium corresponds (1) to the elastic constant c 11. Finally, the remaining parameters β i and β (3) i are fitted to the variation of the (1) and (111) surface energy with respect to the first (2) interlayer spacing. It turns out that the contribution of β i to the energy for structures close to fcc is rather unimportant. For Pt, the authors indeed mention (2) fitting problems for β i, which tends to unphysical (negative) values. Although the above approach has led to improved MEAM parameters, the approach is further modified to obtain an even more consistent parameter set. Indeed, there is a certain degree of asymmetry in the procedure of van Beurden et ( l) al. The electron density weighting factors t are fitted from single data points ( ) only, while the exponential decay factors β l are fitted to entire curves. These ( l) ti parameters are prone to over-fitting and may thus adversely affect the determination of the subsequent parameters. Therefore, a new optimisation procedure is proposed, which remedies these undesired effects. First, instead of fitting the MEAM parameters in a step-wise manner, all MEAM parameters are calculated simultaneously from one global optimisation run. To this end, a goal function is defined with the squared sum of the absolute deviation between the MEAM and the DFT data. Proper weight factors are introduced for surfacespecific information. Each parameter can now take up as much physical information as possible, resulting in a more balanced and appropriately weighted fit. Secondly, in order to improve the stability of the fit, more DFT data are used as inputs. These extra inputs comprise the energy versus volume relationships for the hcp, bcc and sc structures. For the study of surface segregation in alloys, also MEAM parameters for the cross-potential must be derived, again entirely based on DFT input data. The energy versus volume relations are calculated for four intermetallic compounds: L1 2 (A 3 B), L1 (AB), L1 1 (AB) and L1 2 (B 3 A). An overview of the novel parameterisation scheme is presented in Fig. 5.1, together with the unit cells of the different intermetallic compounds. The range of interaction is restricted to first nearest neighbours only. This means that the angular screening function is not used: the first nearest neighbours are completely unscreened, while the second nearest neighbours are completely screened. Some criticism against the angular screening is given by B.J. Thijsse [1], as it introduces even more angular dependence and also a physics of its i i

83 New parameterisation method for the MEAM 63 own. This is confirmed by Lee et al. [11], who determined MEAM parameters using the same data set as Baskes, but left the screening constant as a free parameter to be determined. It was shown that for all fcc metals, except Ni, this value corresponds to complete screening of the second nearest neighbours in the fcc structure. It is believed that screening is more important for the bcc structure in which the ratio of nearest to next nearest neighbours does differ by only ± 15%, in contrast to ± 4% for the fcc structure. For the bcc structure, a proper screening constant is therefore applied when necessary for a satisfactory fit. According to this proposed scheme, new MEAM parameters are calculated for Cu, Pt, Pd and Rh as these metals are frequently used in catalysts, either as pure metals or as alloys. The resulting parameters for the pure metals are presented in Table 5.1 and for the binary cross-potential in Table 5.2. Figure 5.1 Graphical illustration of the proposed global optimisation of the MEAM parameters.

84 64 Chapter 5 E i R i α A i () β i Cu Pt Pd Rh (1) β i (2) β i Table 5.1. MEAM parameters for Cu, Pt, Rh and Pd obtained according to the novel proposed optimisation scheme. (3) β i (1) t i (2) t i (3) t i Cu Pt Pd Rh form A3 B Cu Pd Rh E (ev/atom) A B R (Å) α ij (-) f (-) A B f (-) form E (ev/atom) A3 B A B R (Å) α ij (-) f (-) A B f (-) form E (ev/atom) A3 B A B R (Å) α ij (-) f (-) A B f (-) form E (ev/atom) A3 B A B R (Å) α ij (-) f (-) A B f (-) Table 5.2. MEAM cross-potential parameters: the element in the column represents the majority component (A) in the A 3 B (L1 2 ) reference structure. The form first entry ( E ) is the heat of formation, the second ( R ) is the equilibrium A B 3 nearest neighbour distance and the third (α ij ) is related to the bulk modulus of the reference structure, the last two entries are the f values for A and B respectively. A B 3

85 New parameterisation method for the MEAM 65 Cu Pt MEAM DFT DFT MEAM Experimental Experimental E bcc (ev/atom) [13] [13] E hcp (ev/atom) [13] [13] E sc (ev/atom) f E v (ev/atom) [14] [15] [15] γ (1) (ev/å²) γ (111) (ev/å²) d 1 (%) ±.4 [16] -2.4±.8 [16] ±5.1 [16].2±2.6 [16] d 111 (%) ±.5 [16] -1.±.4 [17] -.7±.5 [18].9.7.5±.9 [16] 1.1±4.4 [16].±2.2 [16] 1.4±.9 [16] c 11 (Gpa) [19] [19] c 12 (Gpa) [19] [19] c 44 (Gpa) [19] [19] γ (11) (ev/å²) ads E (ev/atom) (1) ads E (111) (ev/atom) E {11} MR (ev/atom).1 - > < Table 5.3. MEAM predictions for various bulk and surface properties for Cu and Pt obtained with the parameters from Table 5.1 and compared to DFT and to experimental values. The values in the upper part of the table are used for fitting of the MEAM parameters, while the values in the lower part are independently predicted results. The properties represented in the table are the fcc-bcc E bcc, fcchcp E hcp and fcc-sc E sc energy differences, the unrelaxed vacancy formation f energy ( E v ), the unrelaxed surface energy for (1) and (111) surfaces (γ (1) and γ (111) ), the first interlayer relaxations d 1 and d 111, the elastic constants c 11, c 12 and c 44, the unrelaxed (11) surface energy γ (11), the unrelaxed adatom adsorption ads ads energy on fcc sites of a (1) and (111) surface ( E (1) and E (111) ), and finally the unrelaxed formation energy for the (2 1) missing-row reconstruction on {11} surfaces ( E MR{11} ).

86 66 Chapter 5 Pd Rh MEAM DFT DFT MEAM Experimental Experimental E bcc (ev/atom) [1] [1] E hcp (ev/atom) [1] [1] E sc (ev/atom) f [2] E v (ev/atom) [15] [15] γ (1) (ev/å²) γ (111) (ev/å²) d 1 (%) ±2.6 [16] 3.1±1.5 [16] ±1.1 [16].5±1.1 [16] d 111 (%) ±4.4 [16] -.9±1.3 [16] ±.9 [16].±4.4 [16] c 11 (Gpa) [19] [19] c 12 (Gpa) [19] [19] c 44 (Gpa) [19] [19] γ (11) (ev/å²) ads E (ev/atom) (1) ads E (111) (ev/atom) E {11} MR (ev/atom).6 - > Table 5.4. Same as Table 5.3. for Pd and Rh.16 - > The energy of bulk crystals in different structures, as well as the surface energies and the adatom adsorption energies, are very well reproduced with this new MEAM parameter set. From Tables 5.3 and 5.4, it can also be firmly concluded that all MEAM predictions for the formation energy of an MR reconstruction on a (11) surface are in full agreement with the experimental observations: Pt indeed features an MR reconstruction, while the other elements Cu, Pd and Rh do not. As these quantities cover a very broad range of coordination numbers, it can be stated that this MEAM implementation presents a reliable physical framework for the description of the energy as function of the coordination number.

87 New parameterisation method for the MEAM 67 The calculated equilibrium lattice constants ( R ) and energies of formation ( E ) for the various intermetallic compounds, used as input for the fit procedure, are given in Table 5.5. In order to further illustrate the improvement over van Beurden s approach, a matrix has been calculated in which the element on the i-th row and the j-th column indicates the relative change of property i when the value of parameter j is increased by 1%, while all other parameters are kept at their optimised value (Table 5.6). ij ij R ij DFT form E ij R ij MEAM form E ij Cu 3 Pt (L1 2 ) CuPt (L1 ) CuPt (L1 1 ) CuPt 3 (L1 2 ) Cu 3 Pd (L1 2 ) CuPd (L1 ) CuPd (L1 1 ) CuPd 3 (L1 2 ) Pt 3 Rh (L1 2 ) PtRh (L1 ) PtRh (L1 1 ) PtRh 3 (L1 2 ) Pt 3 Pd (L1 2 ) PtPd (L1 ) PtPd (L1 1 ) PtPd 3 (L1 2 ) Pd 3 Rh (L1 2 ) PdRh (L1 ) PdRh (L1 1 ) PdRh 3 (L1 2 ) Cu 3 Rh (L1 2 ) CuRh (L1 ) CuRh (L1 1 ) CuRh 3 (L1 2 ) Table 5.5 MEAM predictions for the energies of formation E form ij (ev/atom) and equilibrium nearest neighbour distance ij R (Å) for binary Cu-Pt, Cu-Pd, Pt-Rh, Pt- Pd, Pd-Rh and Cu-Rh intermetallic compounds, compared to DFT data.

88 68 Chapter 5 () β i (1) β i (2) β i (3) β i (1) t i (2) t i (3) t i A i γ γ E vac E st. fault 1 1 γ 1 (d 12 ) γ 111 (d 12 ) E(d [1] ) u E ( R ) hcp u E ( R ) 2 18 bcc u E ( R ) sc Table 5.6. Relative change (in %) of the input physical properties upon increasing the corresponding MEAM parameter by 1%. The underlined values correspond to the fit procedure by van Beurden et al. [9]. ( l) From these values, it is obvious that the weighting factors ti have a large influence on the fit of the surface energy as function of the first interlayer distance. In the non-global approach of van Beurden et al. [9], these parameters can not take up this information. Secondly, it can also be concluded that the input ( ) data are less sensitive to the β l parameters. However, e.g. the β (2) parameter is i fitted to more input data than in the original approach of [9]. The β (1) i and parameters, on the other hand, contain an equal amount of physical information in both parameterisation schemes. i (3) β i 5.3. Conclusions In this chapter, a new parameterisation scheme for the implementation of the MEAM is proposed, entirely based on ab initio DFT data. Instead of using a rather inconsistent and incomplete set of experimental input data, all input data are generated by an extended set of DFT calculations. A further improvement is achieved by replacing the step-by-step determination of van Beurden et al. by one global simultaneous fit with even more DFT data. This leads to a more consistent

89 New parameterisation method for the MEAM 69 set of model parameters. From a comparison to test data, also generated by DFT, it can be concluded that the newly derived parameters nicely describe the bulk and surface physical properties. By explicitly introducing surface-specific input data in the optimisation procedure, the new parameters are ideally suited to study surface segregation, surface ordering and even surface reconstructions in metallic alloys. References [1] G. Kresse and J. Furthmüller, Phys. Rev. B 54 (1996) [2] G. Kresse and J. Furthmüller, Comp. Mat. Sci. 6 (1996) 15. [3] D. Vanderbilt, Phys. Rev. B 41 (199) [4] G. Kresse and J. Hafner, J. Phys.: Condens. Matter 6 (1994) [5] J. Perdew and Y. Wang, Phys. Rev. B 45 (1992) [6] H. Niehus, Surf. Sci. 145 (1984) 47. [7] S.P. Withrow, J.H. Barrett and R.J. Culbertson, Surf. Sci. 163 (1985) L655. [8] C.M. Chan, Solid State Commun. 3 (1979) 47. [9] P. van Beurden and G.J. Kramer, Phys. Rev. B 63 (21) [1] P. van Beurden, On the surface reconstruction of Pt-Group metals: a theoretical study of adsorbate-induced dynamics, PhD thesis, Technische Universiteit Eindhoven, The Netherlands, 23. [11] B.J. Lee, J.H. Shim and M.I. Baskes, Phys. Rev. B 68 (23) [12] X. Huang, I.I. Naumov and K.M. Rabe, Phys. Rev. B 7 (24) [13] N. Saunders, A.P. Miodownik and A.T. Dinsdale, Calphad 12 (1988) 351. [14] P.A. Korzhavyi, I.A. Abrikosov, B. Johansson, A.V. Ruban and H.L. Skriver, Phys. Rev. B 59 (1999) [15] Y. Kraftmakher, Phys. Rep. 299 (1998) 79. [16] P.R. Watson, M.A. Van Hove and K.Hermann, Atlas of Surface Structures: Based on the NIST Surface Structure Database (SSD), American Chemical Society, Washington, [17] K.H. Chae, H.C. Lu and T. Gustafsson, Phys. Rev. B 54 (1996) [18] S.A. Lindgren, L. Wallden, J. Rundgren and P. Westrin, Phys. Rev. B 29 (1984) 576. [19] G. Simmons and H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook, MIT Press, Cambridge, [2] T. Korhonen, M.J. Puska and R.M. Nieminen, Phys. Rev. B 51 (1995) 9526.

90 7 Chapter 5

91 6 Surface segregation in disordered alloys In this chapter, the segregation behaviour is studied in Pt-Rh and Pt-Pd alloys, which feature complete solid solubility over the entire composition range. Both alloy systems are known for their interesting catalytical properties and are therefore widely used in technological applications. Whereas a pure Pt oxidation catalyst can clean a diesel engine s exhaust, gasoline engines are equipped with three-way Pt-Rh catalysts [1,2] that also reduce the NO x emissions to N 2. Moreover, Pt-Rh is the active catalyst in the Ostwald process for the production of nitric acid. Pt-Rh alloys also show a high resistance to corrosion and feature an outstanding thermal stability. Alloying of Pt and Pd can also significantly enhance their catalytic activity [3] or selectivity [4,5]. For example, the catalytic activity for the selective partial hydrogenation of 1,3-cyclooctadiene to cyclo-octene, defined as the uptake of hydrogen per mol per second, was studied for Pd Pt clusters with a 4:1 molar ratio, and a threefold increase with respect to pure Pd and even a factor twenty with respect to pure Pt was observed. 6.1 Introduction As pointed out in Chapter 2, three intrinsic driving forces for surface segregation can be distinguished: lowering of the surface energy, lowering of the mixing energy and the (partial) release of elastic bulk strain energy. Apart from these effects, the gaseous environment of a catalyst material can also influence and alter the surface segregation equilibrium. Preferential chemisorption can in some cases even cause a change in the segregating species. This phenomenon is indeed 71

92 72 Chapter 6 observed in the Pt-Rh system (see section 6.2). In the next sections, segregation will be studied in the disordered Pt-Rh and Pt-Pd alloy systems. 6.2 The Pt-Rh system * Material properties The material properties of Pt and Rh related to the driving forces for segregation are tabulated in Table 6.1 and the Pt-Rh phase diagram is shown in Fig. 6.1 [11]. Although Pt and Rh, which are both fcc metals, have almost identical heats of sublimation, the surface energies differ substantially, promoting Pt segregation. The reason for this unexpected effect lies in a different magnetic relaxation of the free atoms to their ground state [12]. Conceptually, the experimental heat of sublimation is the sum of two contributions. When a non-magnetic metal is vaporised by cutting bonds, we can consider that first a non-magnetic free atoms result. It is the energy cost of this first step that correlates with the surface energy. In a second, unrelated step the free atoms relax to their magnetic ground state. As the atomic radii differ only slightly (3%), release of strain energy is not considered an important driving force for segregation. Finally, the estimated heat of mixing [1] is very small and negative (attractive interactions). This is confirmed by a slightly upward curvature of the solid-liquid two phase region in the phase diagram. However, at lower temperatures, a demixing region is suggested in the phase diagram, pointing at strong repulsive interactions between Pt and Rh atoms. This demixing behaviour was first proposed by Raub [13], who extrapolated the behaviour of Pd-Ir, Pt-Ir and Pd-Rh to the Pt-Rh system. In doing so, Raub predicted an upper critical temperature for Pt-Rh of 133 K. Up to now, however, the existence of a miscibility gap is still a point of controversy. Lakis et al. [14-16] observed demixing in an indirect way in analytical electron microscopy experiments on alumina-supported Pt-Rh catalyst particles. On the basis of classical tight-binding d-band theory, Pt-Rh was also predicted to separate into two phases [17,18]. Wouda et al. [19] observed a limited degree of clustering with Scanning Tunneling Microscopy (STM), however not sufficient to explain a demixing region at low temperatures. Many other authors, on the contrary, found no evidence for this demixing behaviour [1,2-23]. In this work, the energy of formation for various ordered Pt-Rh compounds is calculated with DFT (see Table 5.5). These DFT calculations point at attractive interactions and are thus in * Part of this section has been published in J. Luyten, M. Schurmans, C. Creemers, B.S. Bunnik and G.J. Kramer, Surf. Sci. 61 (27)

93 Surface segregation in disordered alloys 73 Pt Rh Crystal structure fcc fcc Lattice parameter [6] [7] [6] [7] Atomic radius 1.38 [7] [7] Heat of sublimation [8] [8] Surface energy 2.23 [9] [1] 2.35 [9] 2.7 [1] Table 6.1. Material properties for Pt and Rh, relevant for surface segregation Figure 6.1. Phase diagram for the Pt-Rh system [11]. contradiction with the demixing behaviour as proposed by Raub (see next section). Surface segregation in Pt-Rh alloys: overview of literature data Surface segregation in Pt-Rh alloys has intensively been studied, both from an experimental and from a theoretical point of view. The first experimental determinations of the surface composition on Pt 1 Rh 9, Pt 5 Rh 5 and Pt 9 Rh 1 alloys were performed by Williams and Nelson [24]. With Ion Scattering Spectroscopy (ISS), they observed a Pt enrichment for all compositions studied. However, the surface composition showed an anomalous Pt increase at temperatures between 8 and 11 K, which is an indication of non-equilibrium at lower temperatures. Other workers [25-28], on the contrary, found Rh segregation. This Rh segregation was later attributed to the presence of small

94 74 Chapter 6 amounts of impurities (carbon, sulfur, oxygen) in the samples [29,32]. The measurements of Tsong et al. [28-3] were repeated on samples without impurities [31], and revealed a Pt enrichment of the surface layer, together with a Pt depleted second layer. A lot of other experiments were performed on several Pt-Rh samples [19,32-38]. The common conclusions from this extended collection of experimental results can be summarised as: i) Pt (strongly) segregates to the surface, ii) the second layer is enriched in Rh even at high temperatures, iii) in the case of vicinal surfaces, less coordinated sites are richer in Pt and iv) it is hard to achieve thermodynamic equilibrium. Even today, the explanation of the segregation behaviour in Pt-Rh from a theoretical point of view still is a challenge. Starting from the model of Williams and Nason [4] and deriving bond energies from the heat of sublimation of pure Pt and Rh leads to segregation of Rh. However, direct use of the surface energies for the calculation of the bond strengths, leads to Pt segregation. Earlier, the role of different surface vibrational properties of Pt and Rh was invoked as a possible mechanism to explain the experimental behaviour [38-4]. This role was later contested by Legrand et al. [41] in a Tight-Binding Ising Model (TBIM). Within the TBIM, Pt enrichment of the surface and an oscillating depth profile was obtained for all compositions. Other studies using pair interactions [42-44], semiempirical energy models [45, 46] or thermodynamic calculations [47] led to the same conclusions. Creemers et al. [48] estimated the segregation behaviour whithin the assumption of demixing below 133 K and obtained a rather good agreement with the experimentally determined temperature dependence of the Pt segregation. However, the experimental finding of a Rh enriched second layer is not consistent with the hypothesis of phase separation. Next, the MEAM with the newly determined parameters, will be combined with Monte Carlo (MC) simulations in order to predict the surface and second layer composition of Pt 25 Rh 75 alloys as a function of temperature. Simulation results First, the heat of mixing for bulk Pt-Rh alloys is calculated with MC/MEAM simulations. Since in a material s bulk the number of atoms of each kind is fixed, these simulations are performed in the Canonical Ensemble (CE) with a slab of 8 (2 2 2) atoms and periodic boundary conditions in the three dimensions. After equilibration steps, the ensemble is averaged during the next steps at an interval of 5 steps. This sampling interval guarantees that the subsequent sampling points are uncorrelated. The resulting curve is

95 Surface segregation in disordered alloys 75 presented in Fig The calculated heat of mixing is small and exothermic, resulting in solid solutions with only a limited preference for bonds between unlike atoms at all temperatures and compositions. This also contradicts the phase separation behaviour as proposed by Raub. The simulated value for the heat of mixing for Pt 5 Rh 5 (-1.6 kj/mol) is in good agreement with an estimation based on the Miedema model (-2. kj/mol) [1]. A first estimate for the extent of surface enrichment can be obtained by calculating the heat of segregation at infinite dilution. The segregation energy of an impurity atom A in a host matrix of atoms B is calculated as the energy difference between a slab with the impurity A placed at the surface and one with the impurity at the centre of the slab. A negative sign points at a favourable segregation of A. The MEAM results for the unrelaxed heats of segregation are presented in Table 6.2. From this table, it is clear that Pt has a strong tendency to segregate to the surface. Pt impurtity in Rh impurity in Rh(111) -.23 ev/atom Pt(111).29 ev/atom Rh(1) -.22 ev/atom Pt(1).25 ev/atom Rh(11) -.25 ev/atom Pt(11).3 ev/atom Table 6.2. Unrelaxed heats of segregation (ev/atom) at infinite dilution, calculated with the MEAM. -5 Heat of mixing (J/mol) MC/MEAM Miedema model Figure 6.2. MC/MEAM calculation of the heat of mixing for Pt-Rh alloys at 14 K. The estimation based on Miedema s model [1] is also shown. x Rh

96 76 Chapter Layer composition (at.%pt) Bulk Pt content Surface Second layer Third layer Figure 6.3. Layer composition (at.%pt) as a function of bulk Pt content for Pt- Rh(111) alloys at T = 14 K. Next, MC/MEAM simulations are performed within the CE for a slab, consisting of 216 atoms (54 layers, 4 atoms in each layer). The slabs have two free surfaces with periodic boundary conditions imposed in the directions parallel to the surface. Fig. 6.3 shows the surface composition of Pt x Rh 1-x (111) alloys as a function of the bulk Pt content. A similar systematic calculation of the surface segregation is not possible for the (1) and (11) surfaces, due to the formation of complex reconstructions at Pt rich surfaces. Indeed, pure Pt shows a missing-row reconstruction on the (11) surface and a quasi-hexagonal reconstruction on the (1) surface. In alloys, where Pt has a strong tendency to segregate to the surface, the strongly Pt enriched surfaces will also reconstruct. This is indeed observed for Pt 25 Rh 75 (11). Therefore, we focus our attention on particular cases for which experimental evidence is available. A lot of experimental work has been devoted to the segregation behaviour to the low index single crystal surfaces of Pt 25 Rh 75 alloys and to vicinal surfaces of Pt 25 Rh 75 (41) and Pt 5 Rh 5 (511) alloys.

97 Surface segregation in disordered alloys 77 a) Pt 25 Rh 75 (1) Siera et al. [49] studied the Pt segregation to the Pt 25 Rh 75 (1) surface by Auger Electron Spectroscopy (AES). Only the outermost layer was assumed to differ from the bulk, which leads to an underestimation of the Pt surface concentration in case of a (damped) oscillatory depth profile. They found a maximum Pt concentration of 53% Pt at 99 K, after which the Pt concentration dropped to 33% Pt at 13 K. Below 95 K, it was difficult to reach equilibrium. They also derived a segregation enthalpy of -7 ± 2 kcal/mol for temperatures above 1 K. Platzgummer et al. [5] studied the surface composition of a Pt 25 Rh 75 (1) alloy by a combination of Low Energy Electron Diffraction (LEED) and Low Energy Ion Scattering (LEIS). Subsequent heating and cooling led to a surface composition changing reversibly, with a decreasing Pt concentration at elevated temperatures. At the highest temperature (1223 K), the depth profile shows oscillatory behaviour: C 1 = 76 ± 2% Pt and C 2 = 4 ± 19% Pt. Finally, a STM study of Hebenstreit et al. [51] shows a surface composition of 74% Pt at 1173 K, with a slight preference for clustering of Pt atoms in the surface layer. The results from our MC/MEAM simulations are shown in Fig. 6.4, together with the experimental values from LEIS and STM. At low temperatures, the simulations reveal a pure Pt surface. Increasing the temperature leads to a decreasing Pt surface composition. This exothermic behaviour is expected for segregation in disordered alloys. At the highest temperatures, the simulations agree completely with the extremely surface sensitive STM and LEIS experiments. The experimental results below 11 K probably do not reflect equilibrium. The composition depth profile for a temperature of 12 K is shown in Fig The simulated second layer composition amounts to 19% Pt. Here also, agreement within the large experimental uncertainty (4±19%) is observed between the simulations and the available LEIS/LEED evidence.

98 78 Chapter 6 Surface composition (at.%pt) MC/MEAM exp (STM) exp (LEIS) Temperature (K) Figure 6.4. MC/MEAM results for the temperature dependent surface composition of a Pt 25 Rh 75 (1) alloy. Experimental STM and LEIS results are also indicated. Surface composition (at.%pt) MC/MEAM exp (LEIS) exp (STM) 1 Surface 2nd layer Layer Figure 6.5. Composition depth profile for a Pt 25 Rh 75 (1) alloy, obtained from MC/MEAM simulations at 12 K. The experimental LEIS and STM results are indicated for comparison.

99 Surface segregation in disordered alloys 79 The strong compositional oscillations are not a priori expected in a system with a negligible heat of mixing (Fig. 6.2). The fact that the MC/MEAM simulations completely reproduce these strong oscillations again demonstrates that the fitted MEAM potential contains all the essential physics for a quantitative description of surface segregation in this system. In order to quantify the degree of short-range order in the surface layer, a correlation matrix P is calculated. The element P i,j equals the probability for an atom i to have a j atom as a nearest neighbour within the surface layer minus the corresponding probability in the situation of ideal randomness. The latter corresponds to the nominal surface composition y j. For Pt 25 Rh 75 (1), the correlation matrix obtained from our MC/MEAM simulation at 12 K is PPt, Pt ypt PPt, Rh yrh.5.5 P = = PRh, Pt ypt PRh, Rh y Rh.1.1 As these values are close to zero, it can be conclude from the simulations that there is no significant preference for clustering nor for ordering of the Pt atoms in the surface layer. b) Pt 25 Rh 75 (111) Platzgummer et al. also measured the composition and order at the Pt 25 Rh 75 (111) surface using a combination of LEIS and LEED [5]. At 1373 K, they found an oscillatory depth profile with 71% Pt at the surface and 11% Pt in the second layer of an equilibrated sample. Below this temperature, thermodynamic equilibrium was hard to achieve. The same system was also studied with STM by Hebenstreit et al. [51]. They found a surface composition of 7% Pt at 1423 K. Furthermore, they observed a slight, but significant preference for unlike bonds. This result is in perfect agreement with the LEIS/LEED results of Platzgummer et al. A MEIS study by Brown et al. [52] revealed an oscillatory depth profile: C 1 = 69 ± 3% Pt and C 2 = 5 ± 3% Pt for the same alloy at 13 K. The surface composition is in very good agreement with the previous results, while the oscillations are somewhat more pronounced than from the LEIS/LEED experiments by Platzgummer et al. Finally, segregation to Pt 25 Rh 75 (111) surfaces was also studied by Siera et al. [49] using AES. Their surface composition is significantly lower than the LEIS/LEED and STM results. As explained above, this is probably a consequence of the interpretation of the AES results in terms of an enrichment of the outermost layer only.

100 8 Chapter 6 The MC/MEAM results for the surface composition as a function of temperature are presented in Fig. 6.6 together with the experimental values. At all temperatures, the Pt 25 Rh 75 (111) surface is only slightly richer in Pt than the Pt 25 Rh 75 (1) surface. At 14 K, the surface composition from the MC/MEAM simulations is somewhat higher than the experimental values. The depth profile is presented in Fig The simulations clearly agree with the oscillatory behaviour in the second layer, with only a slight overestimation of this second layer composition. Also for Pt 25 Rh 75 (111), the correlation matrix P, as defined in the previous paragraph, is calculated.3.3 P =.1.1 Again, the values in this matrix again exclude any short-range order. 1 Surface composition (at.%pt) MC/MEAM exp (STM) exp (MEIS) exp (LEIS) Temperature (K) Figure 6.6. MC/MEAM results for the temperature dependent surface composition of a Pt 25 Rh 75 (111) alloy. Experimental STM, MEIS and LEIS results are also indicated.

101 Surface segregation in disordered alloys 81 Surface composition (at.%pt) MC/MEAM exp (MEIS) exp (LEIS) exp (STM) Surface 2nd layer Layer Figure 6.7. Composition depth profile for a Pt 25 Rh 75 (111) alloy, obtained from MC/MEAM simulations at 13 K, compared to the experimental MEIS, LEIS and STM results. c) Pt 25 Rh 75 (11) Platzgummer et al. [53] also measured the segregation to the Pt 25 Rh 75 (11) surface with a combination of LEIS and LEED. They observed a (2 1) missingrow (MR) reconstruction at low temperatures, identical to the one on pure Pt(11), which deconstructs to a (1 1) structure at higher temperatures. At 1223 K, they found a surface composition of 8% Pt for the (2 1)-reconstructed surface. At 1373 K, the surface composition unexpectedly dropped to 48% Pt on an unreconstructed (1 1) surface. The same alloy has also been studied by Hebenstreit et al. [51] using STM. The STM experiments confirmed the formation of a (2 1) MR reconstruction. After annealing at 1173 K, they report a surface composition of 84% Pt, accompanied by a (2 1) MR reconstruction. The MC/MEAM simulation results are presented in Fig The simulated surface composition of the Pt 25 Rh 75 (11) (2 1) surface reaches a value of 78% Pt at 12 K, compared to the value of 84% Pt from STM and 8% Pt from LEIS at approximately the same temperature. A more severe discrepancy is found for the (1 1) surface: the MC/MEAM simulations predict a surface composition of 74% Pt at 14 K, in contrast to 48% Pt from LEIS. This LEIS value for the (1 1)

102 82 Chapter 6 surface composition is at least surprising in comparison with the corresponding (1) and (111) surface concentrations. Normally, at high temperatures, disordered alloys show a more pronounced segregation for the rougher (11) surface. The MC/MEAM simulations reflect this expected behaviour, in contrast to the experimental results. In order to test whether the MEAM potential is in agreement with this MR reconstruction at lower temperatures, two simulations were performed at very low temperatures (5 K), respectively for an ideal (1 1) terminated slab and for a MRreconstructed slab with the same number of Pt and Rh atoms. The energy of the reconstructed slab after equilibration is slightly lower, confirming the MR reconstruction at lower temperatures. The energy of formation for the (2 1) MR reconstruction is slightly negative for pure Pt also, confirming that for pure Pt this reconstructed surface is the most stable termination at lower temperatures (see Table 5.4 in Chapter 5). For pure Rh, on the contrary, this formation energy is positive, meaning that the equilibrium surface termination corresponds to an ideal (1 1) surface. These results are in complete agreement with the experimentally observed behaviour of the pure metals [54,55]. 1 MC/MEAM Surface composition (at.%pt) (2x1) (2x1) (2x1) MC/MEAM Exp. STM Exp. LEIS Exp. LEIS (1x1) (1x1) Temperature (K) Figure 6.8. MC/MEAM results for the temperature dependent surface composition of a (2 1) and (1 1) Pt 25 Rh 75 (11) alloy. Experimental STM and LEIS results are also shown.

103 Surface segregation in disordered alloys 83 d) Pt 25 Rh 75 (41) The Pt 25 Rh 75 (41) surface is presented in Fig. 6.9, after segregation at 8 K. This surface consists of (1) terraces, separated by double height steps. This leads to a variety of atomic sites with different coordination numbers Z = 6, 7, 8, 9 and 11. All these sites feature a different driving force for segregation and will therefore be characterised by a different composition. Moest et al. [32] have studied the terrace and step edge composition of a vicinal Pt 25 Rh 75 (41) surface between 4 and 7 C with LEIS. The authors observed a pronounced enrichment of the step edges with respect to the terrace sites. The terrace composition, however, is found to be significantly lower than the Pt 25 Rh 75 (41) counterpart (55% Pt at 4 C versus 8% Pt for (1)). The second layer is observed to be Pt depleted. The simulated composition of the different sites as a function of temperature is given in Fig. 6.1, and compared to the experimental results. The following observations can be made. Firstly, the simulated surface composition is nearly equal to the value of Pt 25 Rh 75 (1), as expected, but this is in contrast to the experimental values. Secondly, step edges are more enriched in Pt than the terraces. The MC/MEAM calculation for the step edge composition is in close agreement with the experimental result. Thirdly, the MC/MEAM simulations confirm the Pt depletion of the second layer. Figure 6.9. Schematic representation of the Pt 25 Rh 75 (41) surface, after segregation at 8 K (grey: Pt and dark: Rh).

104 84 Chapter 6 1 Surface composition (at.%pt) Z=6 Z=7 Z=8 Z=9 Z=11 LEIS (Z=8) LEIS (Z=6) Temperature (K) Figure 6.1. MC/MEAM calculation (open symbols) of the composition of the different sites in Pt 25 Rh 75 (41) as a function of temperature, compared to experimental LEIS evidence (solid symbols) [32]. e) Pt 5 Rh 5 (511) The Pt 5 Rh 5 (511) surface is schematically drawn in Fig This surface is characterised by step edges (Z = 7), terrace atoms (Z = 8) and step corner atoms (Z = 1) in equal numbers. Moest et al. [33] studied the Pt 5 Rh 5 (511) surface with LEIS at temperatures between room temperature and 13 K and observed a strong Pt enrichment. The Pt composition of the step edges is even higher than that of the terraces, which can be attributed to the lower coordination number of the step edges. Thirdly, the second layer is depleted in Pt. Two possible explanations are put forward to explain the large Pt enrichment of the surface. The first one relies on a bulk phase demixing, which is in contradiction with the observed second layer Pt depletion and also with our DFT results. In the second possible explanation, the formation of a surface reconstruction is proposed, obtained by removing a row of atoms at the center of the terraces, parallel to the step edges. However, based only on the number of nearest neighbours, this reconstruction would be considerably endothermic and would thus not be favoured.

105 Surface segregation in disordered alloys 85 Figure Schematic representation of the Pt 5 Rh 5 (511) vicinal surface after equilibration at 14 K (grey: Pt and dark: Rh) Surface composition (at. %Pt) Temperature (K) Z = 7 Z = 8 Z = 1 LEIS (Z=8) LEIS (Z=7) Figure MC/MEAM simulation results for the Pt 5 Rh 5 (511) surface composition as a function of temperature (open symbols), compared to the experimental LEIS [33] results (solid symbols). In Fig. 6.12, the MC/MEAM simulation results are shown, compared to the experimental LEIS evidence. These results confirm the principal experimental observations for this system. Firstly, the terraces are strongly enriched in Pt. Secondly, the step edges are even more enriched in Pt, due to their lower coordination. Thirdly, the simulated second layer composition at 3 K contains

106 86 Chapter 6 only 25% Pt, which indeed corresponds to a Pt depletion and which is in reasonable agreement with the corresponding value of 28% Pt for Pt 5 Rh 5 (1). No bulk phase demixing is observed in the simulations, in agreement with the simulated (slightly) exothermic heat of mixing. Also, in order to evaluate the hypothesis of a possible reconstruction as desribed above, MC/MEAM simulations are performed for the proposed reconstruction. As the reconstructed slab possesses a higher energy after equilibration at lower temperatures, this reconstruction is believed not to occur. It can therefore be stated that the experimentally observed strong Pt enrichment is an intrinsic property of the Pt 5 Rh 5 (511) surface without any further peculiarities. It is obtained without any of the proposed hypotheses in Ref. [33], either a bulk phase demixing or the formation of a surface reconstruction. 6.3 The Pt-Pd system Material properties The material properties, governing the three distinct driving forces for surface segregation in Pt-Pd alloys, are listed in Table 6.4. The phase diagram is given in Fig Also for this system, a two-phase demixing region is proposed by Raub [13], based on the observations of Pt, alloyed with neighbouring elements in the periodic table. Raub estimated the critical temperature of demixing based on a comparison of the difference in the melting points for the alloy constituents, however, without a solid theoretical foundation. A demixing behaviour would also be in contradiction with the nearly ideal melting line. Furthermore, experimental information [56] and ab initio data [23] do not support the idea of a demixing behaviour. From Table 6.4, it can be concluded that the lowering of the surface energy favours Pd segregation. As the mixing energy is small, this will not be a significant driving force for segregation. However, as it is slightly negative, an oscillatory depth profile is expected. Finally, Pt and Pd have almost identical atomic radii, which implies that the release of elastic strain energy will also be insignificant as a driving force.

107 Surface segregation in disordered alloys 87 Figure Phase diagram for the Pt-Pd system [11]. Pt Pd Crystal structure fcc fcc Lattice parameter (Å) [6] [7] [6] [7] Atomic radius (Å) 1.38 [7] [7] Heat of sublimation (kj/mole) [8] [8] Surface energy (J/m²) 2.23 [9] [1] 2.5 [9] 2. [1] Table 6.4. Material properties of pure Pt and Pd, relevant for surface segregation Segregation in Pt-Pd alloys: overview of literature data The first experimental study of surface segregation effects in Pt-Pd alloys was performed by Kuijers et al. [57] using AES. The authors observed a strong Pd enrichment of the surface, accompanied by a Pt enriched subsurface region. Du Plessis et al. [58] determined the surface concentration of various polycrystalline Pt-Pd alloys using AES and LEIS. Based on the different probing depth of the two techniques, the authors concluded that the surface is Pd enriched and the subsurface region Pd depleted. Van den Oetelaar et al. studied [59] surface segregation both in polycrystalline Pt 8 Pd 2 and Pt 2 Pd 8 alloys and in supported nanoparticles with LEIS. At thermodynamic equilibrium (7 8 C), the authors observed a Pd surface composition of 5% and 9% for polycrystalline Pt 8 Pd 2 and Pt 2 Pd 8 samples respectively under UHV conditions.

108 88 Chapter 6 For particles with a low metal dispersion, the authors found a surface enrichment that takes place to approximately the same extent as in the Pt-Pd bulk alloy. Radosavkic et al. [6] investigated Pt x Pd 1-x (111) alloys using a core level and valence band photoemission study and found a strong Pd enrichment, in agreement with the other experimental evidence. From a theoretical point of view, Hansen et al. [61] calculated the segregation behaviour in Pt 5 Pd 5 alloys with MC simulations and a total energy calculation based on potentials derived from linear muffin-tin orbital (LMTO) calculations. Rousset et al. calculated the surface composition for several Pd-containing alloys, using an equivalent medium approximation and a modified tight-binding scheme [62]. These authors found a strong Pd enrichment of the surface layer, together with a Pd depleted second layer. From the third layer on, the bulk composition is reached. Deng et al. [63] calculated the surface segregation in Pt-Pd alloys using MC simulations and an analytical embedded atom method. Pt Pd alloys with 3, 4, 5, 6 and 7 at.% palladium respectively were simulated at 8 K, clearly showing a Pd enriched surface and an ocillatory composition depth profile. From all experimental and theoretical information, it can be concluded that i) Pd strongly segregates to the surface, ii) the second layer is depleted in Pd, iii) from the third layer on, the bulk composition is reached. Simulation results The simulated heat of mixing for Pt-Pd alloys at 8 K is given in Fig. 6.14, together with one experimental value [64] at approximately the same temperature (773 K). Both the experimental value (-3.6 kj/mol) and the calculated heat (-2.3 kj/mol) of mixing point at slightly exothermic interactions. The preference for heteroatomic interactions is insufficient to induce the formation of ordered compounds at temperatures at which appreciable atomic mobility exists. This exothermic mixing behaviour is also incompatible with the assumed demixing behaviour proposed by Raub. Hansen et al. [61] report values for the segregation energies of Pd in Pt of -.149, -.129, and -.66 ev/atom in the first layer of (11), (1), and (111) surfaces respectively, calculated with Green s Function (GF)-LMTO. The unrelaxed heats of segregation, calculated with the MEAM are listed in Table 6.5. Both MEAM and GF-LMTO calculations agree on the sign of the segregation energies, although the MEAM values significantly overestimate the values of [61]. Therefore, a strong Pd enrichment of the surface can be expected.

109 Surface segregation in disordered alloys 89 Pd impurtity in Pt impurity in Pt(111) -.14 ev/atom Pd(111).11 ev/atom Pt(1) -.26 ev/atom Pd(1).27 ev/atom Pt(11) -.34 ev/atom Pd(11).34 ev/atom Table 6.5. Unrelaxed heats of segregation at infinite dilution, calculated with the MEAM. -5 Heat of mixing (J/mol) MC/MEAM exp [64] Figure MC/MEAM calculation for the heat of mixing of Pt-Pd alloys at T = 8 K, compared with one experimental value at T = 773 K. x Pd In Figs to 6.17, the MC/MEAM simulation results for the surface concentration as a function of bulk composition are shown for the three low index single crystal surfaces at 8 K. They are also compared with other theoretical work. In general, the MC/MEAM simulated surface composition is somewhat higher than the one from other theoretical calculations. This may be attributed to the underestimation of the surface energies by the EAM (see Section 4.4). The second layer composition, on the other hand, is in good agreement with the EAM calculations. From the third layer on, the nominal bulk composition is retained.

110 9 Chapter 6 Layer composition (at.%pd) Bulk Pd content (at.%pd) Surface Second layer Third layer Surface [63] Second layer [63] Third layer [63] Figure Pd layer concentration as a function of bulk Pd content for (111)- oriented slabs, calculated from MC/MEAM simulations at 8 K (solid symbols) and compared to the MC/EAM calculations of Deng et al. [63] (open symbols) Bulk Pd content (at.%pd) Surface Second layer Third layer Surface [63] Second layer [63] Third layer [63] Figure (1) surface Pd concentration as a function of bulk Pd content, calculated from MC/MEAM simulations at 8 K and compared to the MC/EAM calculations of Deng et al. [63].

111 Surface segregation in disordered alloys 91 1 Layer composition (at.%pd) Bulk Pd content (at.%pd) Surface Second layer Third layer Figure (11) layer Pd concentration as a function of bulk Pd content, calculated from MC/MEAM simulations at 8 K. 1 9 Surface composition (at.%pd) bulk MC/MEAM (111) MC/MEAM (1) MC/MEAM (11) exp [57] exp [59] (111) [62] (1) [62] Temperature (K) Figure Surface Pd content as a function of temperature for Pt 8 Pd 2, calculated by MC/MEAM simulation, and compared to other experimental and theoretical results.

112 92 Chapter 6 For the Pt x Pd 1-x (11) alloys, MC/MEAM simulations are also performed for a (2 1) missing-row reconstructed slab. For all studied compositions, the (1 1) unreconstructed surface has a lower energy even at low temperatures. Therefore, it can be concluded that the (2 1) missing-row reconstruction is not stable for any investigated composition and this is in agreement with the experimental evidence. Fig shows the simulated surface composition for the low index surfaces as a function of temperature. The decreasing surface enrichment at higher temperatures is indicative for the exothermic segregation behaviour in disordered alloy systems. The simulations furthermore show that segregation is more pronounced to more open surfaces. 6.4 Conclusions In this chapter, the newly obtained MEAM parameters are used in MC simulations for the description of the bulk behaviour and surface segregation processes in two disordered alloy systems, Pt-Rh and Pt-Pd. For Pt 25 Rh 75, nice agreement with the experimental evidence is obtained for Pt 25 Rh 75 (111) and Pt 25 Rh 75 (1). For Pt 25 Rh 75 (11) the simulations confirm the missing-row reconstructed surface as the most stable termination at low temperatures. The simulated surface composition for this reconstructed surface also agrees reasonably well with the experimental value. On the other hand, the Pt concentration in the simulations for the (1 1) unreconstructed surface is much higher than the one reported from experiments. It should however be noted that the experimental value is unusually low in comparison with the equilibrium compositions of the less rough (1) and (111) surfaces. Also vicinal surfaces, for which experimental evidence is available, have been simulated. Again, the simulations reproduced all the experimental trends. For Pt 5 Rh 5 (511) a surface reconstruction was proposed in the literature as a possible explanation for the strong Pt enrichment of the surface. However, the simulations reproduce the experimentally observed enrichment without any assumed reconstruction. It was furthermore evidenced that the proposed reconstruction is not stable and would not occur. Also for Pt-Pd, nice agreement with all experimental and theoretical results is obtained. The missing-row reconstruction at the (11) surface, which is found in the Pt 25 Rh 75 system, is not stable in the Pt-Pd system and this is also in agreement with the experimental observations. It can therefore be concluded that the

113 Surface segregation in disordered alloys 93 potential developed here is very well suited for the study of disordered alloy systems. References [1] B.E. Nieuwenhuys, in: W.O. Haag, B.C. Gates and H. Knoezinger (Eds.), Advances in Catalysis, Vol.44, Academic Press, New York, [2] J. Siera, K.I. Tanaka, H. Hirano and B.E. Nieuwenhuys, in: J.N. Armor (Ed.), Environmental Catalysis, ACS Symposium Series, Vol. 552, American Chemical Society, Washington DC, 1994, 114. [3] N. Toshima, K. Kushihashi, T. Yonezawa and H. Hirai, Chem. Lett. (1989) [4] L.W. Gasser and J.A.T. Schwartz, US patent 4, , E.I. Dupont, [5] Z. Karpinski and T. Koscielski, J. Catal. 56 (1979) 43. [6] W.B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, Pergamon, Oxford, [7] J. Emsley, The Elements, Clarendon Press, Oxford, [8] D.R. Lide (Ed.), Handbook of Chemistry and Physics, CRC Press, Boca Raton, [9] W.R. Tyson and W.A. Miller, Surf. Sci. 62, 267 (1977). [1] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema and A.K. Niessen, in: F.R. de Boer and D.G. Pettifor (Eds.), Cohesion and Structure, Vol. 1, North-Holland, Amsterdam, [11] T.B. Massalski, H. Okamoto, P.R. Subramanian and L. Kacprzak, Binary Alloy Phase Diagrams, American Society for Metals, Ohio, 199. [12] M. Methfessel, D. Hennig and M. Scheffler, Phys. Rev. B 46 (1992) [13] E. Raub, J. Less-Common Met. 1 (1959) 3. [14] R.E. Lakis, C.E. Lyman and H.G. Stenger Jr., J. Catal. 154 (1995) 261. [15] C.E. Lymann, R.E. Lakis and H.G. Stenger Jr., Ultramicrosopy 58 (1995) 25. [16] C.E. Lymann, R.E. Lakis and H.G. Stenger Jr., B. Totdal and R. Prestvik, Microchim. Acta 132 (2) 31. [17] M. Cyrot and F. Cyrot-Lackmann, J. Phys. F 6 (1976) [18] M. Sluiter, P. Turchi and D. de Fontaine, J. Phys. F 17 (1987) [19] P.T. Wouda, B.E. Nieuwenhuys, M. Schmid and P. Varga, Surf. Sci. 359 (1996) 17. [2] K.T. Jacob, S. Priya and Y. Waseda, Metall. Mater. Trans. A 29 (1998) [21] Z.W. Lu, B.M. Klein and A. Zunger, J. Phase Equilibria 16 (1995) 261. [22] W.P.A. Jansen, J.M.H. Harmsen, A.W. Denier van der Gon, J.H.B.J. Hoebink, J.C. Schouten and H.H. Brongersma, J. Catal. 24 (21) 42. [23] H.L. Skriver,

114 94 Chapter 6 [24] F.L. Williams and G.C. Nelson, Appl. Surf. Sci. 3 (1979) 49. [25] F. C. M. J. M. Van Delft and B. E. Nieuwenhuys, Surf. Sci. 162 (1985) 538. [26] W. Athenstaedt and M. Leisch, Appl. Surf. Sci. 94/95 (1996) 43. [27] N. Sano and T. Sakurai, J. Vac. Sci. Technol. A 8 (199) [28] M. Ahmad and T.T. Tsong, J. Vac. Sci. Technol. A 3 (1985) 86. [29] T.T. Tsong, D.M. Ren and M. Ahmad, Phys. Rev. B 38 (1988) [3] M. Ahmad and T.T. Tsong, J. Chem. Phys. 83 (1985) 388. [31] D.M. Ren and T.T. Tsong, Surf. Sci. 184 (1987) L439. [32] B. Moest, P.T. Wouda, A.W. Denier van der Gon, M.C. Langelaar, H.H. Brongersma, B.E. Nieuwenhuys and D.O. Boerma, Surf. Sci. 473 (21) 159. [33] B. Moest, S. Helfensteyn, P. Deurinck, M. Nelis, A. W. Denier van der Gon, H. H. Brongersma, C. Creemers and B. E. Nieuwenhuys, Surf. Sci. 536 (23) 177. [34] Donald D. Beck, Craig L. DiMaggio and Galen B. Fisher, Surf. Sci. 297 (1993) 293. [35] Donald D. Beck, Craig L. DiMaggio and Galen B. Fisher, Surf. Sci. 297 (1993) 33. [36] R. Koller, Y. Gauthier, C. Klein, M. De Santis, M. Schmid and P. Varga, Surf. Sci. 53 (23) 121. [37] D. Beck, C.L. DiMaggio and G.B. Fisher, Surf. Sci. 297 (1993) 293. [38] A.D. van Langeveld and J.W. Niemantsverdriet, Surf. Sci. 178 (1986) 88. [39] F.C.M.J.M. van Delft, A.D. van Langeveld and B.E. Nieuwenhuys, Surf. Sci. 189/19 (1987) [4] A.D. van Langeveld and J.W. Niemantsverdriet, J. Vac. Sci. Technol. A 5 (1987) 558. [41] B. Legrand and G. Tréglia, Surf. Sci. 236 (199) 398. [42] J. Florencio, D.M. Ren and T.T. Tsong, Surf. Sci. Lett. 345 (1996) L29. [43] V. Drchal, A. Pasturel, R. Monnier, J. Kudrnovsky and P. Weinberger, Comp. Mat. Sci. 15 (1999) 144. [44] A.M. Schoeb, T.J. Reaker, L. Yang, X. Wu, T.S. King and A.E. DePristo, Surf. Sci. Lett. 278 (1992) L125. [45] D. Bazin, C. Mottet and G. Tréglia, Appl. Catal. A: General 2 (2) 47. [46] H. Deng, W. Hu, X. Shu, L. Zhao and B. Zhang, Acta Met. Sinica 37 (21) 467. [47] L.Z. Mezey and W. Hofer, Surf. Sci (1998) 845. [48] C. Creemers, S. Helfensteyn, J. Luyten, M. Schurmans and H.H. Brongersma, Surf. Rev. Lett. 11 (24) 341. [49] J. Siera, F.C.M.J.M. van Delft and B.E. Nieuwenhuys, Surf. Sci. 264 (1992) 435.

115 Surface segregation in disordered alloys 95 [5] E. Platzgummer, M. Sporn, R. Koller, S. Forsthuber, M. Schmid, W. Hofer and P. Varga, Surf. Sci. 419 (1999) 236. [51] E.L.D. Hebenstreit, W. Hebenstreit, M. Schmid and P. Varga, Surf. Sci. 441 (1999) 441. [52] D. Brown, P. D. Quinn, D. P. Woodruff, T. C. Q. Noakes and P. Bailey, Surf. Sci. 497 (22) 1. [53] E. Platzgummer, M. Sporn, R. Koller, M. Schmid, W. Hofer and P. Varga, Surf. Sci. 423 (1999) 134. [54] H. Niehus, Surf. Sci. 145 (1984) 47. [55] W. Nichtl, N. Bickel, L. Hammer, K. Heinz and K. Mueller, Surf. Sci. 188 (1987) L729. [56] M. Hansen and K. Anderko, Constitution of Binary Alloys, MacGraw-Hill, New York, [57] J. Kuijers, B. M. Tieman and V. Ponec, Surf. Sci. 75 (1978) 657. [58] J. du Plessis and G.N. Van Wyk, E. Taglauer, Surf. Sci. 22 (1989) 381. [59] L.C.A. van den Oetelaar, O.W. Nooij, S. Oerlemans, A.W. Denier van der Gon, H.H. Brongersma, L. Lefferts, A.G. Roosenbrand and J.A.R. van Veen, J Phys Chem. B 12 (1998) [6] D. Radosavkic, N. Barrett, R. Belkhou, N. Marsot and C. Guillot, Surf. Sci. 516 (22) 56. [61] P.L. Hansen, A.M. Molenbroek and A.V. Ruban, J Phys Chem B 11 (1997) [62] J.L. Rousset, J.C. Bertolini and P. Miegge, Phys Rev. B 53 (1996) [63] H. Deng, W. Hu, X. Shu, L. Zhao and B. Zhang, Surf. Sci. 517 (22) 177 [64] F.H. Hayes and O. Kubaschewski, Met. Sci. J. 5 (1971) 37.

116 96 Chapter 6

117 7 Segregation in phase separating alloy systems In the previous chapter, segregation in random solid solutions was treated for the two case studies Pt-Pd and Pt-Rh. In this chapter, alloy systems are studied that form solid solutions only at elevated temperatures, but separate into two distinct phases at lower temperatures. This is a consequence of considerable repulsive interactions. More in particular, segregation will be studied in the Pd-Rh and Cu- Rh alloy systems. The Pd 93 Rh 7 alloy is known as an important substrate for membrane catalysts, especially for the hydrogenation of acetylenic alcohols [1,2]. At low Rh concentrations the alloy is superior to many other catalysts with respect to activity and selectivity in the hydrogenation of dehydrolinalool into linalool. Also Cu-Rh has been studied extensively since 1975 [3] and has experienced renewed interest in the past few years [4 11]. From these studies one can conclude that for a variety of reactions, the activity of bimetallic Cu-Rh catalysts is better than that of pure Rh. For instance, addition of Cu to Rh(1) enhances the CO oxidation catalytic activity [1] and Cu-Rh/SiO 2 is more efficient than supported Rh in the catalytic conversion of methane to ethane [11] and in some dehydrogenation reactions [9]. 7.1 Introduction The interplay of the same driving forces for segregation as mentioned before now causes a completely different behaviour in alloy systems, which have a tendency to demix at lower temperatures. Segregation in random solid solutions is mainly 97

118 98 Chapter 7 driven by the first driving force, the difference in surface energy. At temperatures above the critical demixing temperature T c, these demixing systems also form solid solutions, however, with a considerable preference for bonds between like atoms. The strongly unfavourable heteroatomic interactions now may also become an important driving force as it tends to reinforce the first driving force for segregation of the minority component. At lower temperatures, a phase demixing occurs in the bulk. When considering a surface slab or a small particle, a coreshell configuration spontaneously develops, with the macroscopic phase with the lower surface energy located at the outside. Segregation now occurs from this phase. Therefore, segregation can be very pronounced in these systems, with possibly an abrupt change in surface composition at the critical demixing temperature. 7.2 The Pd-Rh system Material properties The phase diagram for the Pd-Rh system [12] is reproduced in Fig. 7.1 and the material properties that are of importance for surface segregation are given in Table 7.1. The surface energy of Pd is considerably lower than that of Rh. The difference in surface energy is therefore the stronger driving force in this system. Due to the repulsive interactions between the alloy constituents, demixing occurs below the critical temperature of 1118 K. These endothermic interatomic interactions are a driving force for the minority component to segregate to the surface. In the Pd-Rh system, the release of elastic strain energy is negligible as the difference in atomic radii is relatively small. Pd Rh Crystal structure fcc fcc Lattice parameter [13] [14] [13] [14] Atomic radius [14] [14] Heat of sublimation [15] [15] Surface energy 2.5[16] 2. [17] 2.35 [16] 2.7 [17] Table 7.1. Material properties for Pd and Rh that are relevant for surface segregation.

119 Segregation in phase separating alloy systems 99 Figure 7.1. Phase diagram for the Pd-Rh system [12]. Surface segregation in Pd-Rh alloys: overview of literature data Leiro et al. [18] studied surface segregation in a polycrystalline Pd 93 Rh 7 alloy using XPS and found evidence for Pd segregation. Batirev et al. investigated several Pd-rich polycrystalline Pd-Rh alloys using XPS and AES [19]. After annealing at temperatures of 7 à 75 C, a pronounced segregation of Pd was observed. Segregation in Pd-Rh alloys has also been experimentally investigated by Newton et al. [2], who, in agreement with the previous experiments, also observed a Pd enrichment of the surface. Furthermore, some theoretical work has already been performed on this system. Batirev et al. [18,19,21] used a tight-binding approach to estimate the (111) surface enrichment of Pd-Rh alloys at room temperature. Lovvik [22] investigated surface segregation in Pd-based alloys with DFT calculations. In his study, twelve metals are substituted at a 5% level in Pd(111). For Pd-Rh, these calculations point at a Pd enrichment of the surface with a monotonically decreasing composition depth profile. Simulation results First, in order to validate our MEAM parameters for this system, the phaseseparating part of the phase diagram is calculated with MC simulations. These simulations are performed in the Grand Canonical Ensemble, by systematically applying a slight deviation to the difference in chemical potential between the two components at a fixed temperature. A sudden change in composition occurs when the phase boundary is crossed. Indeed, thermodynamics dictates that at chemical

120 1 Chapter 7 equilibrium, the chemical potential of all species in the two coexisting phases must be equal. It can be seen from Fig. 7.2 that the simulated phase boundary agrees rather nicely with the experimental one. Next, MC/MEAM simulations are performed in order to investigate the surface segregation behaviour of Pd-Rh alloy systems. This is done in the Canonical Ensemble on a film-like slab of 28 atoms (7 layers, 4 atoms in each layer) with two surfaces. Starting from a random solid solution, the core-shell configuration spontaneously emanates from the simulations. The use of a relatively large number of atomic layers is necessary for the unhindered formation of this core-shell configuration. However, the time required for this configuration to build up may be rather long. In order to save computer time, a demixed coreshell film configuration is therefore chosen as the initial configuration. In Figs. 7.3 to 7.5, the simulated depth profiles are presented for Pd 5 Rh 5 (111), Pd 5 Rh 5 (1) and Pd 5 Rh 5 (11) at various temperatures. The demixed configuration at temperatures below the critical demixing temperature is clearly present. As Pd features the lower surface energy, the Pd-rich phase will be present at the outside of the film. Segregation now occurs from this phase, leading to a strongly Pd enriched surface. The combined effect of demixing and segregation leads to a pure Pd surface at low temperatures. The Pd composition monotonically decreases in the subsurface layers, as is expected in endothermic alloys. As both phases are very diluted, they can be approximated as random solutions. Segregation in demixing alloys can therefore again be described by the QCA. Indeed, as segregation occurs from the Pd-rich outer phase, the surface composition can be calculated by inserting the composition of this phase as the bulk composition in Eq Such a calculation has already been performed by Creemers et al. [23] in order to evaluate the segregation behaviour in Pt-Rh alloys under the hypothesis that this system would also demix. At higher temperatures, the whole film forms one single solid solution and segregation now occurs from this single phase. In Fig. 7.6 the Pd surface composition is shown for the three main single crystal surfaces of Pd 1 Rh 9 as a function of temperature. It appears that at the critical bulk demixing temperature, the surface composition changes more drastically due to the bulk phase transformation and hence the reduced bulk composition.

121 Segregation in phase separating alloy systems exp [12] MC/MEAM Temperature (K) (Pd,Rh) 8 (Pd) + (Rh) x Rh Figure 7.2. MC/MEAM simulated phase separation in the Pd-Rh system, compared to experimental observations [12]. 1 Layer composition (at.%pd) Pd-rich phase 5 K 7 K 9 K 13 K Rh-rich phase Layer Figure 7.3. MC/MEAM calculation of the composition depth profile in a Pd 5 Rh 5 (111) film at various temperatures.

122 12 Chapter 7 1 Layer composition (at.%pd) Pd-rich phase 5 K 7 K 9 K 13 K Rh-rich phase Layer Figure 7.4. MC/MEAM calculation of the composition depth profile in a Pd 5 Rh 5 (1) film at various temperatures. 1 Layer composition (at.%pd) Pd-rich phase 5 K 7 K 9 K 13 K Rh-rich phase Layer Figure 7.5. MC/MEAM calculation of the composition depth profile for Pd 5 Rh 5 (11) at various temperatures.

123 Segregation in phase separating alloy systems Surface composition (at.%pd) T c bulk Temperature (K) (111) (1) (11) Figure 7.6. MC/MEAM calculation of the Pd concentration at the outer surface of a Pd 1 Rh 9 film as a function of temperature for the three low index single crystal surfaces. 7.3 The Cu-Rh system Material properties The phase diagram for the Cu-Rh system is given in Fig The shape of the melting line and the demixing region suggest even stronger repulsions than in the Pd-Rh system. The material properties, relevant for surface segregation are tabulated in Table 7.2. The surface energy of Cu is much lower than that of Rh. The difference in surface energy of the two constituents is even more pronounced than in the Pd-Rh system. A large Cu enrichment of the surface is therefore expected. The critical temperature for demixing is 1423 K.

124 14 Chapter 7 Figure 7.7. Phase diagram for the Cu-Rh system [12]. Cu Rh Crystal structure fcc fcc Lattice parameter [13] [14] [13] [14] Atomic radius [14] [14] Heat of sublimation [15] [15] Surface energy [16] [17] 2.35 [16] 2.7 [17] Table 7.2. Material properties for Cu and Rh, relevant for surface segregation. Surface segregation in Cu-Rh alloys: overview of literature data Only little experimental information is available concerning the segregation to Cu-Rh alloy surfaces. From AES measurements, Sundaram et al. [24] and Reniers et al. [25] both report on a Cu enrichment of the surface layer. Simulation results Also for the Cu-Rh system, the bulk phase separation is first simulated in the Grand Canonical Ensemble using MC/MEAM simulations. The simulations (Fig. 7.8) nicely reproduce the asymmetry of the two-phase area. Especially at the Rhrich end, the simulated phase separation is in very good agreement with the one that is experimentally observed.

125 Segregation in phase separating alloy systems 15 2 exp [12] 18 MC/MEAM Temperature (K) (Cu) + (Rh) (Cu,Rh) x Rh Figure 7.8. MC/MEAM simulation of the phase diagram for the Cu-Rh system. Layer composition (at.%cu) K 9 K 11 K 15 K Layer Figure 7.9. MC/MEAM calculation of the composition depth profile in a Cu 5 Rh 5 (111) film at various temperatures.

126 16 Chapter 7 Layer composition (at.%cu) K 9 K 11 K 15 K Layer Figure 7.1. MC/MEAM calculation of the composition depth profile in a Cu 5 Rh 5 (1) film at various temperatures. Layer composition (at.%cu) K 9 K 11 K 15 K Layer Figure MC/MEAM calculation of the composition depth profile in a Cu 5 Rh 5 (11) film at various temperatures.

127 Segregation in phase separating alloy systems 17 Next, MC/MEAM simulations are performed for Cu 5 Rh 5 (111), Cu 5 Rh 5 (1) and Cu 5 Rh 5 (11) surfaces using the same procedure as for Pd-Rh. The resulting composition depth profiles are shown in Figs. 7.9 to Just as in the Pd-Rh system, a core-shell configuration develops. As Cu has the lower surface energy, the Cu-rich phase is at the outside of the simulation slab and segregation occurs from this phase. However, at all temperatures the composition depth profile shows a surprising departure from the expected monotonous decrease: a systematic oscillation in the composition is observed. As no experimental information is available, an attempt is made to validate this result by additional DFT calculations. To this end, the migration energy is estimated for a single Rh impurity in a host of pure Cu, both with the MEAM and with DFT. Within DFT, the calculations are performed for a slab consisting of 11 atomic layers with 4 atoms in each layer and with 9 vacuum layers on top. The k point sampling was performed on a Monkhorst-Pack grid [26]. The lattice constant corresponds to the one for pure Cu and no relaxations of the individual atomic positions are allowed. The Rh impurity is first placed at the center of the slab (fifth layer) and then moved layer by layer towards the surface. Within the MEAM, the migration energy is calculated with a slab consisting of 2 layers of 4 atoms each. The latter configuration is much more representative for the real migration energy, but is computationally not feasible with DFT. The results for the three low index surfaces are presented in Table 7.3. The segregation energy to the surface layer is always positive, in agreement with the higher surface energy of Rh. The migration energy to the second layer, however, depends on the surface orientation. For the (1) surface, the migration energy to the second layer is slightly negative, pointing at a preference of Rh to reside in the second layer and confirming the oscillatory depth profile of Fig However, the MEAM considerably overestimates this migration energy. For the (111) surface, the migration energy to the second layer is positive, contradicting the MEAM value and the oscillatory depth profile. Finally, for the (11) surface, the DFT shows that the Rh impurity prefers the second layer, while the MEAM shows a more favourable migration energy towards the third layer. All these results suggest that the surface induces complications in the Cu-Rh bond. The input data set for the determination of the alloy parameters for this particular system should take into account this specific additional surface information by possibly adding correction terms to the MEAM potential.

128 18 Chapter 7 Surface layer Second layer Third layer (1) (11) (111) Table 7.3. Migration energy of a Rh impurity in a Cu host matrix for different surface orientations. The upper value is calculated with DFT, while the lower value corresponds to the MEAM prediction. 7.4 Conclusions In this chapter, the newly developed MEAM parameters are used in conjunction with MC simulations for the description of the phase separating alloys Pd-Rh and Cu-Rh. First, in order to validate the parameters, the bulk phase demixing is simulated with the Grand Canonical Ensemble. For both Pd-Rh and Cu-Rh, nice agreement with the experimental phase diagram is obtained. Next segregation is studied in these alloy systems. In both cases, a core-shell configuration develops below the bulk critical temperature. The phase with the lower surface energy is then present at the outside of the film and segregation occurs from this phase. At temperatures above the bulk critical temperature, segregation occurs from one single solid solution phase. A priori, a monotonically decreasing composition depth profile is expected for these endothermic alloy systems. This is indeed the case for Pd-Rh. However, for Cu-Rh, an oscillatory composition depth profile is obtained. This surprising effect is checked by calculating the migration energy of a single Rh impurity in a Cu(1) host matrix with DFT. The migration energy to the second layer is exothermic and to the first layer it is endothermic, as expected. However, the DFT values for the migration energy in Cu(111) and Cu(11) do not support this oscillating depth profile. This points to a complicated influence of the surface on the binding energy of the constituent alloys, that is insufficiently taken into account by the present MEAM potential. References [1] Gmelin Handbook of Inorganic Chemistry, Pt, Supplement Vol. A1, Springer, Berlin, [2] V.M. Gryaznov, Platinum Met. Rev. 3 (1986) 68. [3] J.K.A. Clarke and A. Peter, J. Chem. Soc. Faraday Trans. 72 (1975) 1817.

129 Segregation in phase separating alloy systems 19 [4] B. Coq, R. Dutartre, F. Figueras and A. Rouco, J. Phys. Chem. 93 (1989) 494. [5] R. Khrishnamurthy, S.S.C. Chuang and K. Ghosal, Appl. Catal. 114 (1994) 19. [6] F. Solymosi and J. Cserenyi, Catal. Lett. 78 (1995) 882. [7] F.M.T. Mendes and M. Schmal, Appl. Catal. A 151 (1997) 393. [8] M. Fernandez-Garcia, A. Martinez-Arias, I. Rodriguez-Ramos, P. Ferreira- Aparicio and A. Guerrero-Ruiz, Langmuir 15 (1999) [9] P. Reyes, G. Pecchi and J.L.G. Fierro, Langmuir 17 (21) 522. [1] J. Szanyi and D.W. Goodman, J. Catal. 145 (1994) 58. [11] F. Solymosi and J. Cserenyi, Catal. Lett. 34 (1995) 343. [12] T.B. Massalski, H. Okamoto, P.R. Subramanian and L. Kacprzak, Binary Alloy Phase Diagrams, American Society for Metals, Ohio, 199. [13] W.B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, Pergamon, Oxford, [14] J. Emsley, The Elements, Clarendon Press, Oxford, [15] D.R. Lide (Ed.), Handbook of Chemistry and Physics, CRC Press, Boca Raton, [16] W.R. Tyson and W.A. Miller, Surf. Sci. 62, 267 (1977). [17] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema and A.K. Niessen, in: F.R. de Boer and D.G. Pettifor (Eds.), Cohesion and Structure, Vol. 1, North-Holland, Amsterdam, [18] J.A. Leiro, M.H. Heinonen and I.G. Batirev, Appl. Surf. Sci. 9 (1995) 515. [19] I.G. Batirev, A.N. Karavanov, J.A. Leiro, M. Heinonen and J. Juhanoja, Surf. Interf. Anal. 23 (1995) 5. [2] M.A. Newton, L. Jyoti, A.J. Dent, S. Diaz-Moreno, S.G. Fiddy and J. Evans, Chem. Phys. Chem. 5 (24) 156. [21] LG. Batirev and J.A. Leiro, J. Electron Spectrosc. Rel. Phen. 71 (1995) 79. [22] O.M. Lovvik, Surface Science 583 (25) 1. [23] C. Creemers, S. Helfensteyn, J. Luyten, M. Schurmans and H.H. Brongersma, Surf. Rev. Lett. 11 (24) 341. [24] V.S. Sundaram and R. Landers, Appl. Surf. Sci. 1 (1982) 567. [25] F. Reniers, M.P. Delpancke, A. Asskali, M. Jardinier-Offergeld and F. Bouillon, Appl. Surf. Sci. 81 (1994) 151. [26] H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13 (1976) 5188.

130 11 Chapter 7

131 8 Segregation in stoichiometric ordered alloys In the previous chapters, segregation was studied in systems that have a negligibly small preference for heteroatomic interactions (disordered) and in strongly endothermic alloys (phase separating). In this chapter, the remaining alloy type is studied, in which strong exothermic bonds exist between unlike atoms and as a consequence well-ordered compounds are formed at lower temperatures. Depending on the degree of exothermicity, these compounds either exist in only a narrow composition region around the exact stoichiometry or in a larger range, with a certain degree of solubility for an excess of one or the other component. As a case study, the Cu 3 Pt and Cu 3 Pd alloys are considered. The Cu 3 Pt alloy system is interesting from a technological point of view. Pure Pt shows interesting catalytic properties, but in alloys with Cu, the activity and selectivity are even improved for the oxidation of CO and for hydrocarbon reactions [1-3]. Also, the addition of Pd is responsible for a change in the reaction parameters compared with pure Cu(11) [4,5]. For example, the dehydrogenation rate of methoxy and formate species on the alloy surface is significantly increased compared to that on pure Cu. Cu-Pd alloys are also used for the denitrification of drinking water [6]. 8.1 Introduction As explained in chapter 2, the characteristics of surface segregation strongly depend on the kind of order in the alloy. In perfectly random systems, segregation corresponds to partial ordering and hence causes an entropy decrease. In order to 111

132 112 Chapter 8 be spontaneous, segregation must therefore be exothermic. Consequently, segregation in random solid solutions is energy-driven and can be very pronounced, but always diminishes with increasing temperature. In perfectly ordered systems on the other hand, the system gets somewhat randomised after segregation, leading to an increase of the configurational entropy. At the same time, the number of preferred unlike bonds decreases and the mixing energy becomes less exothermic as the system moves away from the minimum in configurational energy. As the formation of ordered systems is governed by rather strong exothermic interactions, this increase most often overpowers the decrease in surface energy and the (partial) release of elastic strain energy upon segregation, pushing the net balance for the segregation to the endothermic side. As a consequence, surface segregation is (mostly) entropy-driven, it is rather weak and occurs frequently as a local equilibrium within the first two surface layers. 8.2 The Cu 3 Pt system * Material properties Physical properties, related to the driving forces for surface segregation in the Cu 3 Pt system are summarised in Table 8.1 and in the phase diagram (Fig. 8.1) [7]. Pt Cu Crystal structure fcc fcc Lattice parameter (Å) [8] [9] [8] [9] Atomic radius (Å) 1.38 [9] [9] Heat of sublimation (kj/mole) [1] [1] Surface energy (J/m²) 2.23 [11] [12] [11] [12] Table 8.1 Material poperties of pure Cu and Pt * The content of this chapter has been published in J. Luyten, M. Schurmans, C. Creemers, B.S. Bunnik and G.J. Kramer, Surface Science 61 (27)

133 Segregation in stoichiometric ordered alloys 113 Figure 8.1 Phase diagram for the Cu-Pt system [7]. The two-phase melting domain shows a slight upward curvature, which is already an indication of attractive interactions between the Cu and Pt atoms. These exothermic interactions are manifest from the formation of ordered compounds at lower temperatures: Cu 3 Pt with a L1 2 structure and CuPt with a L1 1 structure. Somewhat surprisingly, the maximum order-disorder transition temperature for Cu 3 Pt does not occur at the stoichiometric composition, but at around 15 at.% Pt instead. From the material properties in Table 8.1, it can be inferred that the lower surface energy of Cu favours Cu segregation. Next, the exothermic nature of the Cu-Pt bond would also favour segregation of Cu (as Cu is the majority component in Cu 3 Pt) and finally, as Pt has the larger atomic radius, the release of strain energy favours Pt segregation. In the end, the net balance between these opposing driving forces can either favour Cu or Pt segregation. In the next sections, a detailed quantitative analysis of surface segregation to, and ordering at the three low index single crystal surfaces of Cu 3 Pt is presented, as studied with MC/MEAM. Surface segregation in Cu 3 Pt alloys: overview of literature data Shen et al. [13,14] performed quite a lot experimental work on Cu 3 Pt(111). They reported a surface layer with 8% Cu and a second layer that is Cu-depleted

134 114 Chapter 8 (69% Cu) after annealing for several weeks at a temperature close to the bulk order-disorder transition temperature (85 K). On the other hand, no evidence for long-range order in the surface region appeared in their experiments. Experiments in which Pt overlayers are grown on Cu(111) thin films suggest the formation of a Cu 3 Pt like surface alloy at lower temperatures [15]. Contradicting these results, a LEED study by Y. Gauthier et al. [16] revealed a different surface composition and an unusual depth profile (C 1 = 72% Cu, C 2 = 92% Cu, C 3 = 52% Cu, C 4 = 92% Cu). The authors attribute this discrepancy to a possible slight difference in sample preparation. Tight-Binding Ising Model (TBIM) calculations for surface segregation from different bulk samples were performed by the same authors to support this idea. Using a parameter set that stabilises the L1 2 structure, these calculations reveal a Cu enrichment of the surface layer and an equally large Cu depletion of the second layer, in good agreement with the LEIS experiments. However, using a second parameter set that stabilises the L1 1 structure at the equiatomic composition, two possible composition depth profiles are obtained, corresponding to different surface terminations (pure Cu or mixed Cu/Pt). The most stable surface termination (pure Cu layer) does not agree with the LEED results, while the metastable mixed Cu/Pt surface termination shows better agreement with the LEED results (C 1 = 52% Pt, C 2 = 98% Pt, C 3 = 52% Pt and C 4 = 98% Pt). Shen et al. [17] have studied the Cu 3 Pt(1) surface with LEIS in combination with LEED. Their measurements unambiguously showed a c(2 2) Cu-Pt underlayer below a (1 1) pure Cu termination. The first interlayer spacing was found to be slightly contracted by 3% with respect to the bulk value. They also concluded that, once the clean and well-annealed surface was formed, neither the c(2 2) LEED pattern, nor the LEIS Cu intensity changed during temperature cycling between 3 K and 85 K. Finally, Shen et al. [18] also investigated the segregation behaviour to the Cu 3 Pt(11) surface. In contrast with the results above for the (1) surface, the (11) surface is not bulk terminated by a pure Cu layer: under thermal equilibrium conditions at 8 K, the surface consists of 82% Cu, followed by a virtually pure Cu layer. The authors tentatively suggested that minor differences in bulk composition could have a large influence on the surface structure and composition. Simulation results First, in order to test the ability of the MEAM potential to accurately describe the bulk behaviour of Cu-Pt alloys, the heat of formation of various ordered Cu-Pt

135 Segregation in stoichiometric ordered alloys 115 compounds was calculated and compared to DFT calculations in Chapter 5. In general, the MEAM predictions for these structures are somewhat too exothermic. At the equitaomic composition, MEAM predicts the L1 structure to be slightly more stable than the L1 1 structure, in contradiction with the DFT data and the phase diagram. The MEAM energy difference between the two structures is however very small and this is not considered as an important shortcoming of our parameters, inasmuch we are not dealing with equiatomic compositions. Using a recently developed correction to the MEAM, the correct L1 1 ground state for CuPt could be reproduced [19]. In this modification to the MEAM, the ground state energies of ordered Cu-Pt compounds are adjusted with correction terms based on the Cluster Expansion Method [2] in order to obtain complete agreement of these formation energies with the DFT values. This correction is purely additive and does not influence the MEAM parameters derived in this work. MC simulations, performed with this corrected potential did not yield significant differences in the surface segregation behaviour of Cu 3 Pt [19], for which the energetics are (by definition) in complete agreement with the DFT results. The main effect is that it makes the potential more applicable for simulating all compositions of the phase diagram. Consequently, the study of segregation in ordered alloys is limited to this particular Cu 3 Pt composition. The order-disorder transition temperature obtained from MC/MEAM simulations is 725 K, which is an underestimation with respect to the experimentally measured 87 K [7]. Next, the detailed results of our MC/MEAM study of the segregation to, and the ordering at the three low index single crystal surfaces of Cu 3 Pt are presented and critically discussed. a) Cu 3 Pt(111) The MC/MEAM simulation results for the Cu 3 Pt(111) system at temperatures between 2 K and 11 K are presented in Fig Firstly, the MC/MEAM simulations show a Cu segregation at all temperatures, in agreement with the LEIS evidence and with the TBIM calculations with the L1 2 stabilising parameter set. At 85 K, the simulation results (C 1 = 82% Cu and C 2 = 72% Cu) agree quantitatively with the LEIS experiments. Secondly, at lower temperatures, the simulated surface composition shows endothermic behaviour (increasing surface composition with increasing temperature) up to the bulk order-disorder transition temperature (T c ). In this temperature regime, segregation is entropy-driven, in agreement with the general trends of surface segregation in ordered alloys. Above T c, the bulk becomes disordered (albeit with a substantial degree of short-range order) and segregation in disordered alloys is always exothermic. Our MC/MEAM simulations reproduce this expected behaviour for both ordered and disordered alloys.

136 116 Chapter 8 Surface composition (at.% Cu) ordered bulk T c MC/MEAM LEIS LEED disordered bulk Temperature (K) Figure 8.2 MC/MEAM simulation results for the surface composition of Cu 3 Pt(111) as a function of temperature, compared with the experimental LEIS [14,15] and LEED [16] results. In order to quantify the state of order in the surface layer, the probability of finding a Pt (Cu) atom as a nearest neighbour of a Cu (Pt) atom within the surface layer, P Cu,Pt (P Pt,Cu ), is calculated: P Cu,Pt =.2 and P Pt,Cu =.9. The values of these parameters for three limiting cases of perfect clustering, perfect ordering and perfect randomness are given in Table 8.2. From these numbers, it is clear that neither long-range order, nor Pt clustering occurs at the surface. This is in complete agreement with the LEIS evidence [13,14]. The authors estimated that only 1% of the Pt atoms in the surface have a Pt nearest neighbour, in perfect agreement with the simulated value (P Pt,Pt = 1 - P Pt,Cu ). perfect clustering perfect ordering perfect randomness P Cu,Pt 1/3.2 P Pt,Cu 1..8 Table 8.2. Values of the probabilities P Cu,Pt and P Pt,Cu for three limiting cases of perfect clustering, perfect ordering and perfect randomness.

137 Segregation in stoichiometric ordered alloys 117 MC/MEAM simulations have also been performed for slight deviations (excess of Pt) from the stoichiometric composition. These simulations reveal a slightly Ptenriched surface at temperatures below the order-disorder transition temperature and this agrees with the LEED results. This suggests that the surface Ptenrichment observed in the LEED experiments can presumably be attributed to a slight Pt excess in the bulk with respect to the exact stoichiometry. The simulated second layer composition, on the other hand, does always correspond to the bulk composition, and this does not agree with the strong oscillations in the LEED depth profile. b) Cu 3 Pt(1) MC/MEAM simulations are performed for the Cu 3 Pt(1) system at temperatures between 3 K and 12 K. At all temperatures, the surface consists of a pure Cu layer, followed by a well-ordered c(2 2) second layer. Below T c, this composition depth profile perfectly matches the bulk composition depth profile. Above T c, the oscillations die out towards the bulk of the simulation slab. Fig. 8.3 shows the depth profile, obtained from MC/MEAM simulations at temperatures of 5 K, 85 K and 12 K. The simulation results at lower temperatures (T<T c ) are in complete agreement with the experimental observations and with the segregation theory for ordered alloys [21]. However, the Cu 3 Pt surface alloy is stable up to much higher temperatures than its bulk counterpart. The stabilising force for such a surface alloy at high tempertatures is the large difference in surface energy between the components, leading to a pure Cu surface layer. In order to optimise the mixing energy, the second layer evolves to a c(2 2) ordered structure. This interplay between atomic order in, and surface segregation to different A 3 B alloy surfaces was already addressed by Moran-Lopez et al. [22] within the Bragg- Williams approximation.

138 118 Chapter Layer composition (at.% Cu) MC/MEAM, 5 K MC/MEAM, 85 K LEIS, 85 K MC/MEAM, 12 K Layer Figure 8.3 Comparison of the composition depth profile at 5 K, 85 K and 12 K for Cu 3 Pt(1) obtained by MC/MEAM simulations with the experimental LEIS [17] results. c) Cu 3 Pt(11) MC/MEAM simulations are performed for temperatures between 3 K and 12 K. For all temperatures, the surface consists of a pure Cu layer followed by a mixed Cu/Pt layer, similar to that in Cu 3 Pt(1). Below T c, this profile continues throughout the whole simulation slab. Above T c, the oscillations gradually diminish towards a constant bulk composition of 75% Cu. Fig. 8.4 shows the composition depth profile at 5 K, 8 K and 12 K with the pure Cu termination and a highly ordered second layer, which is in accordance with the observations on the (1) surface. Again, the surface alloy exists up to higher temperatures than bulk Cu 3 Pt. The results differ from the experimentally observed profile, which, however, has never been observed in analogous (ordering) systems. There are also no arguments suggesting a possible reconstruction of the Cu 3 Pt(11) surface as a possible explanation for this dicrepancy: pure Cu(11) does not reconstruct and the LEIS measurements on Cu 3 Pt(11) clearly point at a (1 1) surface termination [18]. The possibility of a (2 1) missing-row (MR)

139 Segregation in stoichiometric ordered alloys 119 recontruction has also been tested in the MC/MEAM simulations. To this end, a (1 1) slab and a (2 1) MR reconstructed slab are initially filled with exactly the same number of Cu and Pt atoms. Both slabs are equilibrated at very low temperatures (5 K) and the final energies are compared. As the final energy of the unreconstructed slab is.57 ev lower per surface atom, the MR reconstructed Cu 3 Pt(11) surface is never stable, in agreement with the experimental observations. In order to further investigate the discrepancy between the experimental and simulated results, additional comparative calculations are performed with DFT and the MEAM. For the ordered Cu 3 Pt(11) surface, two different terminations are possible. The first corresponds to a pure Cu surface layer, while the second features a mixed Cu-Pt surface termination. The surface energy (γ alloy ) for each termination is calculated as [23] γ alloy surf ( ) (, ) total ref ref form E Ni Ei N j E j Ni + N j ECuPt Ni N j = (8.1) a with E total the total energy of the slab, N i (N j ) the number of i (j) atoms, E ( E ) form the reference energy of a metal atom i (j) in its ground state structure (fcc), E CuPt the formation energy for Cu 3 Pt in the L1 2 structure and a surf the total area of the surface. The termination with the lowest surface energy will be preferred, certainly at lower temperatures. The slabs used in the DFT calculations consist of 11 atomic layers, with 9 vacuum layers on top of them. The sampling of the Brillouin zone is performed with a Monkhorst-Pack [24] mesh of k points. The calculations are performed for unrelaxed slabs only. The results for the two different surface terminations are presented in Table 8.3. From this table, ref i ref j DFT MEAM Pure Cu termination Mixed Cu/Pt termination Table 8.3 Surface energy (ev/å 2 ) for the two possible surface terminations of Cu 3 Pt(11), calculated with DFT and the MEAM.

140 12 Chapter Layer composition (at.% Cu) MC/MEAM, 5 K MC/MEAM, 8 K LEIS, 8 K MC/MEAM, 12 K Layer Figure 8.4 Comparison of the composition depth profile at 5 K, at 8 K and at 12 K for Cu 3 Pt(11) obtained by MC/MEAM simulations with the experi-mental LEIS [18] results. one can see that both the DFT and the MEAM agree on the pure Cu termination as the more stable one. The MC/MEAM simulation results also correspond to this Cu terminated surface, while the experimental result rather points at the mixed Cu/Pt surface termination. The experimental result indeed indicates that the preferred termination is a Cu/Pt mixed surface, with some additional Cu segregation up to 82% Cu and a second layer that corresponds to a pure Cu layer, which perfectly matches the bulk profile. The values in Table 8.3 suggest that the experimental sample was possibly in a metastable state, which was reproduced during different cycles of the experiment. The difference calculated by DFT is very small and we therefore also estimated it with DFT-LDA. The DFT-LDA surface energies for the pure Cu termination and for the mixed Cu/Pt surface amount to.127 ev/å 2 and.139 ev/å 2 respectively, resulting in an even more pronounced preference for the pure Cu terminated surface.

141 8.3 The Cu 3 Pd system Segregation in stoichiometric ordered alloys 121 Material properties Physical properties, related to the driving forces for surface segregation in the Cu- Pd alloy are summarised in Table 8.4 and in the phase diagram (Fig. 8.5) [7]. At higher temperatures, the Cu-Pd system features a continuous series of solid solutions, while at lower temperatures, a complex ordering behaviour is observed. The ordered phase Cu 3 Pd forms with the L1 2 -type structure over the broad composition range from 1 to 25% Pd. The maximum transformation temperature is 77 K and, similarly to the Cu-Pt situation, occurs at the off-stoichiometric composition of 15 at.% Pd. It is also observed that the L1 2 ordering type is present only in alloys with less than 2 at.% Pd. For alloys with 2 to 25% Pd, a slightly distorted tetragonal structure is observed. Secondly, the transformation from disordered (Cu,Pd) solid solutions to ordered CuPd occurs over the range 35 to 5 at.% Pd. In this composition range, Cu-Pd orders in a B2-type ordering, which corresponds to alternating pure layers in the [1] direction of a bcc lattice. The phase boundary for this structure shows a maximum at 87 K and also occurs at the off-stoichiometric composition of 4 at.% Pd. Other ordered compounds are also reported, corresponding to the stoichiometries Cu 5 Pd and Cu 5 Pd 3. Despite the exothermic interactions at lower temperatures, the melting line shows a downward curvature. Pd Cu Crystal structure fcc fcc Lattice parameter (Å) [8] [9] [8] [9] Atomic radius (Å) [9] [9] Heat of sublimation (kj/mole) [1] [1] Surface energy (J/m²) 2.5 [11] 2. [12] [11] [12] Table 8.4 Material poperties of pure Cu and Pd

142 122 Chapter 8 Figure 8.5. Phase diagram for the Cu-Pd system [7]. From Table 8.4, it is seen that Cu features the lower surface energy, which forms a driving force for Cu segregation. The difference in surface energy is however smaller than in the Cu-Pt case. Next, Pd has the larger atomic radius, which should promote Pd segregation. This difference is comparable to Cu-Pt. Finally, segregation will be counterbalanced by the exothermic interatomic interactions. Surface segregation in Cu 3 Pd alloys: overview of literature data The surface composition of Cu-Pd alloys was studied by van Langeveld et al. [25] with AES and with calculations based on a CBE model of surface segregation. The measurements were performed on thin films with a bulk composition varying from to 1% Pd. The data derived from the experimental results are in good agreement with the calculated values, showing that the surfaces of clean and equilibrated Cu-Pd alloys are slightly enriched in Cu. This Cu enrichment was later confirmed by several authors [26-33]. Bergmans et al. [31] performed a detailed analysis with LEIS and LEED of the Cu 85 Pd 15 (11) surface at room temperature after annealing and equilibrating at 6 K. The composition and the structure of the outermost atom layers were determined: an oscillating concentration profile with a slight Cu enrichment in the first layer (89 ± 2% Cu) and a strong Cu depletion in the second layer (6 ± 8% Cu). In accordance with earlier work [32,33], a (2 1) LEED pattern was observed, caused by a strong ordering in the second layer.

143 Segregation in stoichiometric ordered alloys 123 Lovvik [34] determined the (111) surface segregation energy for several Pd-based (95% Pd) transition metal alloys. For the Cu-Pd system, Cu was found to preferentially enrich the surface, while Pd has a tendency to reside in the second layer, in agreement with the experimental findings. Gallis et al. [35], on the contrary, concluded from TBIM calculations that in the disordered state, Pd segregates for x Cu <.31, whereas Cu segregates for x Cu >.31. Deurinck et al. [36] performed MC/EAM simulations for the segregation to the Cu 3 Pd(11) surface with parameters that were specifically optimised for the Cu- Pd alloy system. These simulations revealed a behaviour that was in good agreement with the experimental results on Cu 85 Pd 15 (11), i.c. Cu segregation to the surface, an oscillating concentration profile, and a second atomic layer exhibiting a substantial degree of ordering that accounts for the observed (2 1) LEED pattern. Bozzolo et al. [37] calculated the surface energies of the different (1) and (11) terminations of Cu 3 Pd alloys with the Bozzolo-Ferrante-Smith (BFS) model. For the (1) surface, the pure Cu layer is predicted as the preferred termination, while for Cu 3 Pd(11) the pure Cu termination and the mixed Cu/Pd layer feature almost the same energy. Simulation results The heats of formation and equilibrium lattice parameters for various ordered Cu- Pd compounds are given in Table 5.5 (Chapter 5). The calculated equilibrium lattice parameters are in good agreement with the DFT calculations, while the heats of formation show slight deviations. Overall, the interactions are less attractive in the Cu 3 Pd alloy than in Cu 3 Pt. MC/MEAM simulations for the segregation in Cu 3 Pd alloys lead to a Pd enrichment of the (111) surface and a preference for mixed Cu/Pd terminations at the (1) and (11) surface. This is in contradiction with the experimental results, pointing at a slight Cu enrichment. In order to elucidate the origin of this discrepancy, additional DFT calculations were performed for the various surface terminations of Cu 3 Pd alloys. The results are shown in Table 8.5. Cu 3 Pd(1) Cu 3 Pd(11) Termination MEAM DFT MEAM DFT Cu/Pd Cu Table 8.5. MEAM and DFT calculation of the surface energy (ev/atom) of mixed Cu/Pd and pure Cu termination in Cu 3 Pd(1) and Cu 3 Pd(11) alloys.

144 124 Chapter 8 The DFT values in Table 8.5 are in good agreement with the MEAM results and support the simulation results with a preference for the mixed Cu/Pd termination. On the basis of these values, it can be concluded that the MEAM once again nicely reproduces the DFT results. However, it seems that, exceptionally, the DFT data do not directly support the experimentally observed segregation behaviour. This discrepancy is possibly a deficiency of the DFT implementation, either due to the approximation for the exchange-correlation energy or due to the use of ultrasoft pseudo-potentials. Indeed, it is suggested that current implementations of DFT are prone to an intrinsic surface error [38]. This intrinsic surface error is larger for GGA than for LDA. Furthermore, within GGA, the intrinsic surface error is not even equal for all functionals. Recently, Mattson et al. [39] proved the non-equivalence of the Perdew-Wang [4] and Perdew-Burke-Ernzerhof [41] functionals. A benchmark of calculated surface energies for Cu and Pd within different DFT approximations (LDA and GGA) and for GGA, with different exchange-correlation functionals could shed some light on the reliability of the present DFT results. It can be seen that the surface energies of Cu and Pd (Tables 5.3 and 5.4) calculated by DFT and used as an input for the MEAM parameterisation procedure, are very comparable in magnitude. A small intrinsic surface error on these numbers may drastically influence the predicted segregation behaviour. Another reason for the discrepancy can possibly be found in the fact that the DFT calculations are performed for unrelaxed systems at K. The lattice constants correspond to the minima in the energy versus volume relations, without further relaxation of the individual atomic positions. For the Cu-Pd system, the influence of atomic relaxations may however not be negligible [42]. All experiments on the contrary are performed at elevated temperatures. Both thermal dilation and vibrational entropy contributions could alter the driving force for segregation. In this perspective, the DFT and MEAM data of Table 8.5 are not necessarily in contradiction with the experimental results. 8.4 Conclusions In this chapter, segregation in the ordering systems Cu 3 Pt and Cu 3 Pd is studied. For the Cu-Pt system, the L1 1 bulk structure is not reproduced as the most stable bulk structure at the equiatomic composition. In order to resolve this deficiency, a very recent additive (bulk) correction to the MEAM, based on the CEM is proposed. The segregation results for Cu 3 Pt and Cu 3 Pd are however not affected

145 Segregation in stoichiometric ordered alloys 125 by this correction, as at this composition, the energetics are in complete agreement with the DFT results. The simulation results for the surface composition of Cu 3 Pt(111) alloys are in complete agreement with the LEIS experiments and contradict the LEED results. The LEED results for the Pt surface enrichment can be rationalised in light of a slight off-stoichiometric effect. The reported strong oscillations in the composition depth profile are however not observed in the MC/MEAM simulations. Also for the Cu 3 Pt(1) surface, complete agreement with the experimental LEIS results is found. Finally, for the Cu 3 Pt(11) surface, the simulation results deviate from the experimental evidence. It must however be pointed out that the experimental results for the latter surface are somewhat peculiar and perhaps correspond to a metastable state. The possibility of a MR reconstruction is evaluated but it is apparently not stable and does not yet explain the deviation from the experimental results. Both DFT and the MEAM agree on the pure Cu surface layer as the more stable one. For the Cu 3 Pd alloy system, the MEAM once again works well as interpolating model between the DFT data. However, exceptionally, the DFT data seem to predict surface terminations that disagree with the experimentally observed segregation behaviour. Further research is necessary to elucidate why the DFT data apparently deviate from the experimental data. References [1] H.C. de Jongste and V. Ponec, J. Cat. 63 (198) 389. [2] R. Linke, U. Schneider, H. Busse, C. Becker, U. Schröder, G. R. Castro and K. Wandelt, Surf. Sci (1994) 47. [3] J.T. Kummer, J. Catal. 38 (1975) 166. [4] M.A. Newton, S.M. Francis, Y. Li, D. Law and M. Bowker, Surf. Sci. 259 (1991) 45. [5] A. Newton and M. Bowker, Surf. Sci (1994) 445. [6] Y. Yoshinaga, T. Akita, I. Mikami and T. Okura, J. Catal. 174 (1998) 72. [7] P.R. Subramanian and D.E. Laughlin, in Binary Alloy Phase Diagrams, T.B. Massalski Editor in Chief, 2 nd Edition, Vol. 2, American Society for Metals, Ohio, 199. [8] W.B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, Pergamon, Oxford, [9] J. Emsley, The Elements, Clarendon Press, Oxford, [1] C.J. Smith (Ed.), Metal Reference Book, 5th edition, Butterworths, London, [11] W.R. Tyson and W.A. Miller, Surf. Sci. 62 (1977) 267.

146 126 Chapter 8 [12] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema and A.K. Niessen, in: F.R. de Boer and D.G. Pettifor (Eds.), Cohesion and Structure, Vol. 1, North-Holland, Amsterdam, [13] Y.G. Shen, D.J. O'Connor, K. Wandelt and R.J. MacDonald, Surf. Sci. 328 (1995) 21. [14] Y.G. Shen, D.J. O Connor, K. Wandelt and R.J. MacDonald, Surf. Sci (1995) 746. [15] Y.G. Shen, D.J. O'Connor and R.J. MacDonald, Nucl. Instrum. Meth. B 135 (1998) 361. [16] Y. Gauthier, A. Senhaji, B. Legrand, G. Tréglia, C. Becker and K. Wandelt, Surf. Sci. 527 (23) 71. [17] Y.G. Shen, D.J. O Connor and K. Wandelt, Surf. Sci. 46 (1998) 23. [18] Y.G. Shen, D.J. O Connor and K. Wandelt, Surf. Sci. 41 (1998) 1. [19] M. Schurmans, J. Luyten, C. Creemers, R. Declerck, and M. Waroquier, accepted for publication in Physical Review B. [2] J.M. Sanchez, F. Ducastelle and D. Gratias, Physica A 128 (1984) 334. [21] W.M.H. Sachtler and R.A. Van Santen, Adv. Catalysis 26 (1977) 69. [22] J.L. Moran-Lopez and K.H. Benneman, Phys. Rev. B 15 (1977) [23] S. Müller, J. Phys. Condens. Matter 15 (23) R1429 [24] H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13 (1976) [25] A.D. van Langeveld, H.A.C.M. Hendrickx and B. E. Nieuwenhuys, Thin Solid Films 19 (1983) 179. [26] T.S.S. Kumar and M.S. Hegde, Appl. Surf. Sci. 2 (1985) 29. [27] G.A. Kok, A. Noordermeer and B.E. Nieuwenhuys, Surf. Sci. 152 (1985) 55. [28] A. Noordermeer, G.A. Kok and B. E. Nieuwenhuys, Surf. Sci. 172 (1986) 349. [29] D.C. Peacock, Appl. Surf. Sci. 27 (1986) 58. [3] F. Reniers, M. P. Delplancke, A. Asskali, M. Jardinier-Offergeld and F. Bouillon, Appl. Surf. Sci. 81 (1994) 151. [31] R.H. Bergmans, M. van de Grift, A.W. Denier van der Gon and H.H. Brongersma, Surf. Sci. 345 (1996) 33. [32] D.J. Holmes, D.A. King and C.J. Barnes, Surf. Sci. 227 (199) 179. [33] M.A. Newton, S.M. Francis, Y. Li, D. Law and M. Bowker, Surf. Sci. 259 (1991) 45. [34] O.M. Lovvik, Surf. Sci. 583 (25) 1. [35] C. Gallis, B. Legrand, A. Safil, G. Treglia, P. Hecquet and B. Salanon, Surf. Sci (1996) 588. [36] P. Deurinck and C. Creemers, Surf. Sci. 345 (1996) 33. [37] G. Bozzolo, J.E. Garcesa, R.D. Noebe, P. Abel and H.O. Mosca, Progr. Surf. Sci. 73 (23) 79.

147 Segregation in stoichiometric ordered alloys 127 [38] T.R. Mattson and A.E. Mattson, Phys. Rev. B 66 (22) [39] A.E. Mattsson, R. Armiento, P.A. Schultz and T.R. Mattsson, Phys. Rev. B 73 (26) [4] J.P. Perdew and Y. Wang, Phys. Rev. B 45 (1992) [41] J.P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77 (1996) [42] R.J. Cole, N.J. Brooks, P. Weightman, S.M. Francis and M. Bowker, Surf. Rev. Lett. 3 (1996) 1763.

148 128 Chapter 8

149 9 Segregation in off-stoichiometric ordered alloys * In the previous chapter, segregation is in stoichiometric ordered alloys is studied. A slight deviation from this ideal stoichiometric composition causes completely different segregation behaviour. This is illustrated in this chapter for the Pt 8 Co 2 (111) alloy system. Pt-Co alloys are frequently used for catalytic purposes [1,2]. Pt-Co particles, deposited on a carbon substrate, indeed show a high catalytic activity for the oxidation of methanol and are therefore well suited for use in fuel cells [3,4]. Pt-Co alloys are also magnetic and show an improved chemical resistance compared to pure Co [5,6] Material properties The phase diagram for the Pt-Co system [7] is given in Fig At lower temperatures, the ground state crystal structure of Co corresponds to a hexagonal close packed (hcp) lattice (ε-co), in contrast to Pt, which features a face centered cubic (fcc) lattice structure. At 7 K, pure Co transforms from the hcp to the fcc structure (α-co). Moreover, pure Co exhibits considerable magnetic interactions, resulting in a Curie temperature of 13 K. The overall strength of the magnetic interactions decreases with increasing Pt content, resulting in a Curie-temperature of 85 K for PtCo and of 52 K for Pt 3 Co. Two intermetallic compounds are * Part of this chapter has appeared in M.A. Vasylyev, V.A. Tinkov, A.G. Blaschuk, J. Luyten and C. Creemers, Appl. Surf. Sci. 253 (26)

150 13 Chapter 9 Figure 9.1. Phase diagram for the Pt-Co system. [7] Pt Co Ground state crystal structure fcc hcp Lattice parameter [8] [9] a = 2.57 [9] c/a =.994 [9] Atomic radius 1.38 [9] 1.25 [9] Heat of sublimation [1] 427 [1] Surface energy 2.23 [11] [12] [11] 2.55 [12] Table 9.1. Poperties of pure Pt and Co formed: PtCo with the L1 structure and Pt 3 Co with the L1 2 structure. The orderdisorder transition temperatures for both compounds are 11 K and 123 K respectively. Other material properties, which are relevant for surface segregation, are tabulated in Table Experimental and theoretical evidence The experimental starting point is a recent study of Vasylyev et al. [13,14] on Pt 8 Co 2 (111) using Ion Scattering Spectroscopy (ISS). At low temperatures, the surface region consists of a pure Pt layer, followed by a strongly oscillating,

151 Segregation in off-stoichiometric ordered alloys 131 sandwich-like layer stacking (C 1 = 98% Pt, C 2 = 32% Pt, C 3 = 94% Pt, C 4 = 7% Pt). At higher temperatures, a normal segregation profile is observed in which the surface layer still is strongly enriched in Pt and the second layer is Pt depleted (C 1 = 96% Pt, C 2 = 6% Pt, C 3 = 83% Pt, C 4 = 79% Pt). Gauthier et al. [15,16] also investigated the surface structure and composition depth profile at the (111) surface of a disordered Pt 8 Co 2 alloy with LEED. The samples were annealed at temperatures in the range of K, which is slightly higher than the bulk order-disorder transition temperature for this composition. The authors found an oscillating depth profile: a pure Pt surface layer followed by a strongly Pt depleted second layer and only a slight Pt enrichment in the third layer. They also found weak relaxations of the interlayer spacing: d 12 =.3% and d 23 =.9%. Additional experimental work on this system comprises other single crystal surfaces, namely Pt 25 Co 75 (111) [17], Pt 25 Co 75 (1) [18] and Pt 25 Co 75 (11) [19,2], the Pt 9 Co 1 (11) [21]), the effect of CO adsorption [22], as well a lot of work on deposited thin films. Beside experiments, also theoretical work has already been performed on the Pt- Co alloy. Mezey et al. [23] used a modern thermodynamic calculation of interface properties with a second moment approximation (MTCIP-2A) and calculated the surface composition of a Pt 8 Co 2 (111) alloy. They obtained good agreement with the experimental LEED results of Gauthier et al. [15]. 9.3 Simulation results and discussion The simulation slab consists of 24 layers with 576 atoms in each layer and with periodic boundary conditions to mimic an infinite piece of material. Two free surfaces are created by adding three layers of vacuum at the top and at the bottom of the simulation block. The MC simulations are allowed to equilibrate during MC steps. After this equilibration, samples are taken during the next MC steps with an interval of MC steps. The parameters for the CBE model are given in Table 9.2. The binding energy between like atoms is calculated from the experimental surface energies [12] of the pure elements. The value of the alloying parameter is chosen so as to correctly reproduce the experimental value of the order-disorder transition temperature of 123 K for Pt 3 Co.

152 132 Chapter 9 At that time, new MEAM parameters were not available yet. Therefore, in this chapter, the original parameters of Baskes for Pt [24] and Co [25] are used (see ( ) Table 9.3). In [25], the partial electron weighting factors t l i for Co are defined for the exponential expression, as defined in Eq In order to use only one ( l) single expression, namely the square root form of Eq. 4.32, the original t i parameters for Co are transformed to new parameters. The values for the Pt-Co cross potential [26] are tabulated in Table 9.4. The heats of formation for the two ordered compounds in the phase diagram show good agreement with firstprinciples calculations [27] and experimental values. The MC/MEAM simulations show only a slight overestimation of this transition temperature, T c (MEAM) = 115 K. As pointed out in Chapter 2, four driving forces can be distinguished: lowering of the surface energy, lowering of the mixing energy, (partial) release of the elastic strain energy around solute atoms and, finally, segregation driven by preferential chemisorption. Under the experimental UHV conditions, the latter can be neglected. In ordering alloys, the pronounced preference for the energetically favourable AB bonds requires specific interpretation of these driving forces in view of the ordered lattice. ε PtPt (J/mol) ε CoCo (J/mol) α PtCo (J/mol) Table 9.2. CBE parameters for the Pt-Co system E i R α A () i β i i (1) β i (2) β i (3) β i Pt Co Table 9.3. MEAM parameters for Pt and Co (1) t i (2) t i (3) t i form E ij (ev/atom) R ij (Å) α ij Cu Pt f f Table 9.4. MEAM parameters for the Pt-Co cross potential

153 Segregation in off-stoichiometric ordered alloys 133 PtCo (LI ) lattice constant heat of formation Pt 3 Co (LI 2 ) lattice constant heat of formation MEAM DFT experiment ev/at 2.71 Å [27] ev/at [27] 2.79 Å [27] -.75 ev/at [27] 2.65 Å [8] ev/at [29] 2.71 Å [8] ev/at [28] Table 9.5. Calculated lattice parameters and heats of formation for the Pt-Co intermetallic compounds, compared to DFT calculations and experimental values. As shown in the previous chapter, in a strictly stoichiometric compound, atoms end up on the wrong sublattice after segregation, leading to less energy-favourable AB bonds. Therefore, surface segregation in such compounds is often endothermic and a substantial gain in surface energy is needed in order to overcome this and to generate an appreciable surface enrichment [29]. In offstoichiometric ordered compounds such as Pt 8 Co 2, on the other hand, the exchange of excess Co from the surface layer partially restores the ideal order in the bulk. In such alloys, the more stable starting configuration is the one in which the excess A (B) atoms in the bulk are randomly distributed over the β (α) sublattice. Following this reasoning, it can readily be derived that, neglecting strain energy effects, the segregation energy ( E) of a Pt atom on the Co sublattice to the (111) surface equals as opposed to E = a γ + 3 α (9.1) atom E = a γ 3.4 α (9.2) atom for segregation from the Pt sublattice. The first terms in these expressions represent the gain in surface energy, while the second terms stand for the gain in mixing energy. As α is negative for ordering alloys, even without a difference in surface energy, lowering of the mixing energy contributes exothermically to the segregation energy. As the bulk is a quasi-infinite source of Pt atoms on the Co sublattice, this E does not depend on the degree of surface enrichment and will normally lead to a quasi 1% segregation. This is confirmed by MC/CBE simulations performed on a hypothetical alloy of two elements with equal surface energies. In the case of Pt 8 Co 2 (111), this off-stoichiometric effect [3] once more drives Pt to the surface. Finally, for (dilute) disordered solutions of A in B,

154 134 Chapter 9 the (partial) release of the strain energy around the solute atoms in the bulk, causes the solute atoms to segregate to the surface, especially if they are larger than the atoms of the host matrix. In exactly stoichiometric ordered compounds, this elastic strain energy is negligible, as the lattice parameter of the compound has adjusted to the ordered stacking. In off-stoichiometric compounds, however, the excess atoms are subjected to some degree of strain. Partial release of this strain upon segregation again results in a tendency for Pt to segregate. In view of the small difference in surface energy between Pt and Co (Table 9.1), lowering of the surface energy contributes only slightly to the total segregation enthalpy. The tendency for Pt segregation is however enhanced by the second (and the third) effect as Pt 8 Co 2 is over-stoichiometric in Pt with respect to the stoichiometric Pt 75 Co 25 L1 2 -case: 2% of the Co lattice sites are occupied by Pt atoms. The segregation of this excess Pt lowers the mixing energy and possibly also the strain energy in the bulk. The MC/CBE and MC/MEAM simulation results for the surface composition are plotted as a function of temperature in Fig Both the experiments and the MC simulations produce evidence for an exothermic segregation of Pt, which causes the surface enrichment to become less at higher temperatures. At lower temperatures, a surface composition of 1% Pt is observed in the ISS experiments and this is confirmed by MC simulations. Also at 673 K and 973 K, the simulated surface composition is in good agreement with the experimental ISS observations. However, the segregation enrichment as a function of temperature varies more gradually in the MC simulations than in the experimental results. The MC/MEAM simulations show a pure Pt surface layer at all simulated temperatures. The composition depth profile at two different temperatures (7 K and 1 K) is shown in Fig. 9.3 and Fig The MC/CBE simulations only show a slightly oscillating composition depth profile. The MC/MEAM simulations at 1 K, on the contrary, show a second layer of 61% Pt, in reasonable agreement with the LEED results (48 ± 1% Pt) and the MTCIP-2A calculations (57% Pt). The third layer is again enriched in Pt (9% Pt), also in complete agreement with the LEED experiments and MTCIP-2A calculations. From the fourth layer on, the bulk composition of 8% Pt is achieved. At lower temperatures, the ISS experiments of Vasylyev et al. [13] show a sandwich-like structure, with a second layer that is strongly depleted in Pt. The MC/MEAM simulations confirm the existence of such a sandwich-like structure, but the predicted depletion of the second layer is less pronounced (C 1 = 1% Pt, C 2 = 7% Pt, C 3 = 95% Pt, C 4 = 76% Pt).

155 Segregation in off-stoichiometric ordered alloys Surface composition (at.% Pt) Experiment MC/CBE MC/MEAM Temperature (K) Figure 9.2. MC/CBE and MC/MEAM simulation results for the surface composition of Pt 8 Co 2 (111) as a function of temperature, compared to experimental results. 1 9 Layer composition (at.% Pt) MEAM CBE ISS Layer Figure 9.3. Composition depth profile at 7 K from MC/CBE and MC/MEAM simulations, compared to experimental ISS results.

156 136 Chapter Layer composition (at.% Pt) Layer MEAM CBE MTCIP-2A LEED ISS Figure 9.4. Composition depth profile at 1 K obtained from MC/CBE and MC/MEAM simulations, compared to experimental LEED and ISS results and theoretical MTCIP-2A calculations. The results for the interlayer spacing calculated with the MEAM are compared to the experimental values by LEED [15]. The MEAM calculations evidence a slight outward relaxation of the first layer (+2.26%) and second layer (+.45%), in contradiction with the experiments (-.3% and +.9% respectively). The outward relaxation by MEAM is presumably driven by the pure Pt surface layer. The larger Pt atoms are somewhat frustrated by the smaller lattice constant of bulk Pt 8 Co 2 and expand slightly to relax this strain energy. The discrepancy that still exists between the MC/MEAM simulations and the experimental results is possibly due to magnetic interactions. Indeed, these magnetic interactions introduce an positive energy of mixing which drives the system towards clustering of the magnetic species [32]. By studying Pt 75 Co 25 films, Rooney et al. [32] indeed observed a significant increase of the Curietemperature by more than 2 K, which could only be the consequence of significant Co clustering, presumably driven by magnetic energy.

157 9.4 Conclusions Segregation in off-stoichiometric ordered alloys 137 Despite the rather crude approximations made in the CBE model, interesting conclusions can already be drawn from this oversimplified model. The most important feature of the CBE model is the prediction of the so-called offstoichiometric effect, leading to a pure Pt surface layer. The temperature dependent surface composition, obtained from the MC/CBE simulations agrees well with ISS experiments. On the other hand, the model predicts only slight oscillations, if present, in contradiction with the LEED and EELS results. The more sophisticated MEAM is necessary in order to account for the various subleties at a metal surface. Our MC/MEAM simulations indeed confirm both the LEED results at high temperatures and the sandwich-like structure at low temperatures, although the predicted second layer depletion is by far not as strong as in the experiments. References [1] X. Zhang and K.Y. Chan, J. Mater. Chem. 12 (22) 123. [2] N. Toshima and T. Yonezawa, New. J. Chem. 22 (1998) [3] S.T. Bromley, G. Sankar, C.R.A. Catlow, T. Maschmeyer, B.F.G. Johnson and J.M. Thomas, Chem. Phys. Lett. 34 (21) 524. [4] X. Zhang, K.Y. Tsang and K.Y. Chan, J. Electroanal. Chem. 573 (24) 1. [5] D. Weller, A. Moser, L. Folks, M.E. Best, W. Lee, M.F. Toney, M. Schwickert, J.U. Thiele and M.F. Doerner, IEEE Trans. Magn. 36 (2) 1. [6] G. Moraitis, H. Dreyssee and M.A. Khan, Phys. Rev. B 54 (1996) [7] T.B. Massalski, H. Okamoto, P.R. Subramanian and L. Kacprzak, Binary Alloy Phase Diagrams, American Society for Metals, Ohio, 199. [8] W.B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, Pergamon, Oxford, [9] J. Emsley, The Elements, Clarendon Press, Oxford (1991). [1] In: D.R. Lide, Editor, Handbook of Chemistry and Physics, CRC Press, Boca Raton, [11] W.R. Tyson and W.A. Miller, Surf. Sci. 62 (1977) 267. [12] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema and A.K. Niessen, in: F.R. de Boer and D.G. Pettifor (Eds.), Cohesion and structure, Vol. 1, North-Holland, Amsterdam, [13] M.A. Vasylyev, S.P. Chenakin and V.A. Tinkov, Vacuum 78 (25) 19. [14] M.A. Vasylyev, V.A. Tinkov, A.G. Blaschuk, J. Luyten and C. Creemers, Appl. Surf. Sci. 253 (26) 181.

158 138 Chapter 9 [15] Y. Gauthier, R. Baudoing-Savois, J. M. Bugnard, U. Bardi and A. Atrei, Surf. Sci. 276 (1992) 1. [16] Y. Gauthier, Surf. Rev. Lett. 3 (1996) [17] Y. Gauthier, R. Baudoing-Savois, J.M. Bugnard, W. Hebenstreit, M. Schmid and P. Varga, Surf. Sci. 466 (2) 155. [18] Y. Gauthier, P. Dolle, R. Baudoing-Savois, W. Hebenstreit, E. Platzgummer, M. Schmid and P. Varga, Surf. Sci. 396 (1998) 137. [19] J.M. Bugnard, R. Baudoing-Savois and Y. Gauthier, Surf. Sci. 281 (1993) 62. [2] J.M. Bugnard, Y. Gauthier and R. Baudoing-Savois, Surf. Sci. 344 (1995) 42. [21] E. Platzgummer, M. Sporn, R. Koller, M. Schmid, W. Hofer and P. Varga, Surf. Sci. 453 (2) 214. [22] K.S. Shpiro, N.S. Telegina, V.M. Gryaznov, K.M. Minachev and Y. Rudny, Catal. Lett. 12 (1992) 375. [23] L.Z. Mezey and W. Hofer, Surf. Sci (1996) 15. [24] M.I. Baskes, Phys. Rev. B 46 (1992) [25] M.I. Baskes and R.A. Johnson, Model. Simul. Mat. Sci. Eng. 2 (1994) 147. [26] H.G. Kim and H.M. Lee, Z. Metallk. 96 (25) 211. [27] H.L. Skriver, < [28] R. Orinani and W.K. Murphy, Acta Metall. 1 (1962) 879. [29] R.A. Van Santen and W.M.H. Sachtler, J. Catal. 33 (1974) 22. [3] A.V. Ruban, Phys. Rev. B 65 (22) [31] L. Rubinovich and M. Polak, Surf. Sci. Rep. 38 (2) 127. [32] P.W. Rooney, A.L. Shapiro, M.Q. Tran and F. Hellman, Phys. Rev. Lett. 75 (1995) 1843 and references therein.

159 1 Segregation in ternary alloys In the previous chapters, segregation in binary alloy systems was studied. Surface segregation could be qualitatively predicted by considering the material properties and the phase diagram. Adding a third component to a binary alloy system can be compared to the introduction of a third dimension to the two-dimensional space. An extra variable is now needed in order to uniquely characterise the composition. Crossing the binary barrier therefore presents an interesting scientific challenge. Additionally, beside the academic challenge of crossing the binary barrier, ternary (and higher order) systems also have important technological applications as catalyst materials. As an example, for three-way catalysts, not only binary Pt- Rh or Pd-Rh alloys are used but also ternary Pt-Pd-Rh alloys. The same alloy system is also a good catalyst for the production of nitric acid in the Ostwald process [1,2]. In this chapter, segregation in the ternary Pt-Pd-Rh alloy will be studied by MC simulations using the newly parameterised MEAM Introduction For reasons of complexity, also from an experimental point of view, the vast majority of research on surface segregation has been addressed to binary alloy systems and only little effort has been devoted to ternary systems. However, ternary alloys are very interesting as they may exhibit a behaviour that is specific for higher order systems and that can never be observed in binary systems. Pioneering work with respect to surface segregation in ternary alloys has been performed by Guttmann [3] and McLean [4], based on statistical thermodynamics 139

160 14 Segregation in ternary alloys of adsorption isotherms. Hoffmann et al. [5] refined Guttmann s approach and gave an analytic derivation, within the Quasi-Chemical Approximation and the Constant Bond Energy (CBE) model, resulting in the following set of coupled equations A, C ya x H C RT ln ( Ksegr ) = RT ln = yc xa RT B, C yb x H C RT ln ( Ksegr ) = RT ln = yc xb RT A, C segr B, C segr (1.1) with ( γ γ ) 2α, ( ) (.5) Zl ( yb xb ) Zv xb ( γ γ ) 2α, ( ) (.5) Z ( y x ) Z x H = a + Z x y + Z x A, C segr at A C A C l A A v A + α ' H = a + Z x y + Z x B, C segr at B C B C l B B v B + α ' l A A v A (1.2) and α a ternary alloy parameter α = α α α (1.3) ' A, B A, C B, C The first two terms in Eq. 1.2 correspond to the QCA/CBE segregation in a binary alloy (see Chapter 2), while the third term introduces the effect of a third component. Although this QCA/CBE approach has serious limitations, the equations can be solved to get first estimates for the surface enrichment. However, the QCA treatment omits the formation of SRO and LRO in the alloy as well as the accompanying non-ideal mixing entropy effects, while the CBE model does not properly account for the many-body character of the metallic bond. Despite these shortcomings, the effects that are typical for higher order alloy systems can already be described within this approach and the different driving forces for surface segregation become apparent. As can be seen from Eq. 1.2, these driving forces are intimately related to the driving forces for segregation in binary alloys, but the presence of the third component introduces additional competitions between these forces.

161 Chapter In a similar way as in binary alloys, a surface-active component is defined as a component with a relatively low surface energy. The most surface-active component is the component with the lowest surface energy. Typical higher-order segregation effects are now co-segregation and blocking. Co-segregation points at the simultaneous segregation of two surface-active components, while the third component remains in the bulk. Co-segregation can be enhanced by attractive interactions between the surface-active components or by strong repulsive interactions of the two surface-active components with the third component in the bulk. Also, a surface-active component can be prevented from segregating by strong attractive forces with a component with a high surface energy. In order to optimise the mixing energy in the bulk, the component with the intermediate surface energy will segregate to the surface. This phenomenon, in which the segregation of a surface-active component is impeded by attractions with a component of a high surface energy in the bulk, is called blocking. In a recent explorative study, Luyten et al. [6] have performed extensive MC/CBE simulations to illustrate the complexity of the interplay of driving forces in ternary alloys for the whole ternary phase diagram. From the above considerations, it becomes clear that a straightforward prediction of surface segregation is less trivial than in the binary case and MC simulations are even more needed. A theoretical study of surface segregation in Pt-Pd-Rh alloys has already been performed by Chen et al. [7], however, their potential did not reproduce the experimentally observed demixing behaviour for the Pd-Rh binary subsystem, which makes the results for the ternary system questionable The Pt-Pd-Rh system To our knowledge, no experimental phase diagram or thermodynamic data are available for the ternary Pt-Pd-Rh system. An estimate thus needs to be made by extrapolating from the binaries. This extrapolation can be done based on expressions for the Gibbs free energy. The calculation of phase diagrams from the Gibbs energy of the phases is commonly referred to as CALPHAD (Calculation of Phase Diagrams) [8]. The commercially available software package Thermo-Calc [9] contains databases for a large set of binary and higher order alloys systems. Regarding the Pt-Pd-Rh system, the most recent database for solutions, the SGTE solutions database SSOL4 [1], contains only binary interaction parameters for the Pd-Rh system, while for Pt-Rh and Pt-Pd these parameters are assumed zero. Ternary interactions are also neglected due to lack of experimental data. From this database, a section of the phase diagram at 6 K can be calculated. As the binary Pd-Rh alloy separates into two phases at lower temperatures, it is expected that

162 142 Segregation in ternary alloys also in the ternary phase diagram, a miscibility gap will develop near that binary axis (see Fig. 1.1). However, for the reasons mentioned above, this section should be considered approximative. A qualitative estimate of the interplay of driving forces for segregation in the ternary Pt-Pd-Rh system can be made by considering the material properties for Pd, Pt and Rh. Pd has the lower surface energy and is a surface-active component. Rh features the higher surface energy and is therefore expected to be anchored in the bulk. The endothermic nature of the Pd-Rh bond presents an extra driving force for the segregation of Pd. Finally, Pt has the possibility to co-segregate to the surface, assisted by the exothermic nature of the Pt-Pd bond. It can however also be retained in the bulk due to its attraction to Rh, with which it also forms exothermic bonds. The subtle balance of these driving forces is now further unraveled by MC/MEAM simulations. Figure 1.1. MC/MEAM simulation for the Pt-Pd-Rh phase diagram at 6 K (ο), compared to the ThermoCalc calculation (full line).

163 1.3. Simulation results Chapter The MEAM with the newly derived parameters for the binary subsystems is used for all calculations. Ternary effects of mixing are partly incorporated in the model through the definition of the partial electron densities (Eq. 4.3). In order to test whether the MEAM can be extended to the Pt-Pd-Rh system without introducing specific ternary parameters, the energies of formation and the equilibrium nearest neighbour distances of a slab with alternating layers in the [111], [1] and [11] direction are first calculated with both DFT and the MEAM. The results of these calculations are presented in Table 1.1. It can be deduced that the MEAM nicely reproduces both the nearest neighbour distance at equilibrium and the energy of formation. It is therefore concluded that it seems at least a reasonable first estimation to neglect ternary interaction parameters for this system. The simulation slab consists of 24 (6 layers, 4 atoms in each layer) atoms for bulk calculations and of 28 (7 layers, 4 atoms in each layer) atoms for simulations, in which a surface is involved. The simulations are performed on a fixed lattice in the Canonical Ensemble. 1 7 MC steps are performed in order to equilibrate the sample, followed by another 1 7 MC steps for averaging the ensemble properties at an interval of 1 6 MC steps. As the lattice constants for the three components are rather similar, no important strain effects are expected. The lattice constant of the ternary alloy is therefore approximated by a linear average of the lattice constants of the pure components with the composition according to Vegard s rule. For all simulations, the initial slab is filled as a randomly disordered alloy. Next, the phase diagram is calculated with MC/MEAM simulations (Fig. 1.1) and compared to the Thermo-Calc calculation. From the MC/MEAM simulations, complete solid solubility is predicted at 6 K if x Pt ca..4. For x Pt <.4, the major part of the phase diagram is subject to a two-phase demixing, with a Pd-rich and a Rh-rich phase in equilibrium. It is the first attempt to construct a reliable phase diagram for this alloy system. The miscibility gap is slightly smaller than was predicted by Thermo-Calc. This is a logical consequence of the omission in Thermo-Calc of Pt-Rh and Pt-Pd interactions and thus of the compatibilising effect of Pt, that will be explained in the next paragraph. As these interactions are explicitly taken into account by the MEAM potential, the miscibility gap predicted by the MC/MEAM simulations is believed to be the more accurate.

164 144 Segregation in ternary alloys R ij DFT form E ij R ij MEAM form E ij [111] [1] [11] Table 1.1. DFT and MEAM predictions for the energies of formation E (ev/atom) and equilibrium nearest neighbour distance R ij (Å) for ternary alloys with alternating layers in the [111], [1] and [11] directions. form ij Figure 1.2. Contour plot of the heat of mixing (ev/atom) at 14K, calculated by MC/MEAM simulations for the Pt-Pd-Rh ternary alloy.

165 Chapter (1) Pd + Pt co-segregation (3) Pd segregation (2) Figure 1.3. Segregation map for the Pt-Pd-Rh system, obtained from our MC/MEAM simulations at 12 K. The squares in the figure indicate the three compositions which will be studied more in detail. Fig. 1.2 shows a contour plot with the results for the actual heat of mixing calculated by MC/MEAM simulations at 14 K. Close to the binary axes, the expected behaviour is reproduced: Pt-Rh as well as Pt-Pd are slightly exothermic, while Pd-Rh alloys show endothermic mixing. This endothermic mixing will lead to phase separation at lower temperatures. It can be seen from Fig. 1.2 that the system becomes less endothermic by introducing a certain amount of Pt. Indeed, the Pt atoms act as a compatibiliser between the Pd and Rh atoms, by forming exothermic Pt-Pd and Pt-Rh bonds at the expense of unfavourable Pd-Rh bonds. Fig. 1.3 shows the distinct composition regions for surface enrichment in a specific component at 12 K. A strong Pd segregation is observed for all compositions. This is expected on the basis of the low surface energy of Pd and the repulsive Pd-Rh interactions. Close to the binary Pt-Rh axis (lower Pd concentrations), a domain with co-segregation of Pd and Pt is identified. Outside this domain, Pd segregates abundantly. The three squares, indicated in the figure, correspond to the three compositions which will be studied in more detail.

166 146 Segregation in ternary alloys Next, the temperature dependence of the segregation is studied for three particular ternary compositions. The first composition corresponds to Pt 83 Pd 15 Rh 2 (1), which was experimentally characterised by Yuantao et al. [11]. The second composition Pt 2 Pd 4 Rh 4 (2) lies in the region of phase separation at lower temperatures and the third composition Pt 4 Pd 1 Rh 5 (3) lies in the region where co-segregation of Pd and Pt is observed in the MC/MEAM simulations. The study is limited to the (111) surface only. In Fig. 1.4, the MC/MEAM simulated depth profile of a Pt 83 Pd 15 Rh 2 (111) at 1 K alloy is shown and compared to the experimental values of [11]. According to Fig. 1.1, this nominal bulk composition exhibits complete solid solubility at all temperature and lies outside the region for co-segregation. Therefore, only a strong Pd segregation is expected. The agreement between the simulations and experimental results is quite good, taking the error bar on the experiments (8% on the Pt surface composition) into account. In our simulations, the surface contains 77% Pt, 23% Pd and no Rh. Furthermore, the influence of the surface is clearly restricted to one single layer only. The Pd surface enrichment is presented as a function of temperature in Fig At all temperatures, the simulated Rh presence at the surface is negligibly small. Furthermore, the Pd surface composition decreases with increasing temperatures. This is in agreement with the expected exothermic behaviour for segregation in disordered alloys. 9 8 Layer composition (at.%) Rh Pt Pd exp Rh [11] exp Pt [11] exp Pd [11] Layer Figure 1.4. MC/MEAM composition depth profile of a Rh 2 Pt 83 Pd 15 (111) alloy at 1 K, compared to the experimental values by Yuantao et al. [11].

167 Chapter Surface composition (at.%) Rh Pt Pd Temperature (K) Figure 1.5. MC/MEAM simulation results for the surface composition of Rh 2 Pt 83 Pd 15 (111) alloys as a function of temperature. The second composition, Pt 2 Pd 4 Rh 4, lies in the two-phase region of the phase diagram. For this case, a behaviour similar to binary, phase separating Pd-Rh alloys is expected. This involves the formation of a core-shell configuration with the phase of lower surface energy, i.e. the Pd-rich phase, located at the outside of the simulation film. Segregation then occurs from this phase. This behaviour is indeed confirmed by the MC/MEAM calculations. The composition depth profiles at 6 K and at 12 K are shown in Figures 1.6 and 1.7 respectively. At 6 K, the core-shell configuration clearly emanates from the simulations. Segregation occurs from the Pd-rich subsurface phase, while the Rh-rich phase is located in the center of the simulation slab. The second layer shows a slight oscillation for Pt and Pd. At 12 K, the bulk forms one single solid solution. Due to strong Pd segregation extending to 3 à 4 atomic layers, the concentrations of Pd and Rh in the center of the slab deviate slightly from the nominal composition. The Pt concentration profile shows a slight oscillation in the second layer, while the Pd and Rh composition vary monotonously towards the bulk. The explanation for this behaviour can again be found in the nature of the chemical interactions. The surface is rich in Pd and as Pt forms exothermic bonds with Pd, the Pt

168 148 Segregation in ternary alloys Layer composition (at.%) Rh Pt Pd Layer Figure 1.6. MC/MEAM result for the composition depth profile of the Pt 2 Pd 4 Rh 4 (111) alloy at 6 K Rh Pt Pd Layer composition Layer Figure 1.7. MC/MEAM result for the composition depth profile of the Pt 2 Pd 4 Rh 4 (111) alloy at 12 K.

169 Chapter composition shows a slight oscillation, in agreement with the results for the binary Pt-Pd alloys. The main components Pd and Rh form endothermic bonds and therefore the Pd composition is expected to decrease monotonically towards the bulk. This is also in agreement with the results for binary Pd-Rh alloys at higher temperatures. Finally, Pt and Rh form exothermic bonds, but as the Rh and Pd compositions are twice as large as the Pt concentration, the endothermic Pd-Rhlike behaviour will dominate, resulting in the depth profile as shown in Fig The phenomenon of co-segregation is studied in more detail by considering the Pt 4 Pd 1 Rh 5 (111) alloy. The surface composition, obtained from the MC/MEAM simulations is plotted as a function of temperature in Fig The cosegregation of Pd and Rh occurs above 9 K only. Below 9 K, the Pt surface concentration is lower than the bulk composition and Pd is the only segregating component. In order to check the consistency of the QCA/CBE approach with the MC/MEAM simulations, we have calculated the segregation energy in both cases. For the QCA/CBE, this has been done by the straightforward application of Eq The CBE parameters are derived from the surface energies, which are in turn calculated with DFT and from the results for the heat of mixing obtained from MC/MEAM simulations. For the MC/MEAM simulations, the well-known van t Hoff equation is used ln( K ) H = R ( 1/ T ) i, j i, j segr segr (1.4.) Fig. 1.9 shows a plot of K i, j ln( segr ) versus 1/T for segregation of Pt and Pd. The respective enthalpies for segregation are derived from the slope of these curves., The values for i j are given in Table 1.2. It can be seen that both approaches H segr agree qualitatively on the magnitude of the segregation enthalpy. The difference between the two approaches can, on the one hand, be ascribed to non-ideal mixing both in the bulk and in the surface layer, which is not accounted for by the QCA and, on the other hand, to the rough estimates for the constant bond energy parameters. From this table, it can also be concluded that segregation of both Pt and Pd is an exothermic process. The much larger segregation energy for Pd causes a suppression of the Pt segregation. Upon increasing the temperature, this pronounced Pd segregation diminishes and the Pt concentration increases. This explains why co-segregation occurs at temperatures above 9 K only.

170 15 Segregation in ternary alloys Surface composition (at.%) Rh Pt Pd Temperature (K) Figure 1.8. Surface composition for the Pt 4 Pd 1 Rh 5 (111) alloy as a function of temperature, as obtained from the MC/MEAM simulations Pd,Rh Pt,Rh 8 ln(k segr ) /T (1/K) Figure 1.9. van t Hoff plot (ln(k segr ) versus 1/T) with the MC/MEAM segregation results for the Pt 4 Pd 1 Rh 5 (111) alloy.

171 Pd, Rh H segr Chapter QCA/CBE MC/MEAM kj/mole kj/mole Pt, Rh H segr kj/mole kj/mole Table 1.2 Calculated segregation energies from the QCA/CBE approach and from MC/MEAM simulation results Conclusions Ternary alloys can exhibit a segregation behaviour that is typical for multicomponent systems and is non-existent in binary alloys. Indeed, co-segregation, site competition and blocking are typical phenomena that may occur in ternary alloys. Furthermore, ternary alloys also have important applications as catalysts in the chemical process industry. In this chapter, segregation is therefore studied for the ternary alloy system Pt-Pd-Rh, with the previously derived MEAM potential for the pure elements and for the binary alloys. First, the ternary bulk phase diagram is constructed from MC/MEAM simulations. To our knowledge, this is the first attempt to construct a phase diagram for this system. The simulated phase diagram is believed to be more accurate than the prediction from the SSOL4 database calculated by ThermoCalc. Subsequently, the heat of mixing for the whole of the ternary diagram at 14 K is also determined. Next, MC/MEAM simulations are performed in order to study the segregation behaviour to the (111) surface as a function of bulk composition. For all compositions, a strong Pd segregation is observed. Furthermore, a region of Pd-Pt co-segregation could be identified close to the binary Pt-Rh axis of the phase diagram. The segregation behaviour is then further studied as a function of temperature for three particular compositions of special interest. For Pt 83 Pd 15 Rh 2, good agreement with the experimental data [11] is obtained. The second alloy Pt 2 Pd 4 Rh 4 behaves in a way that is analogous to the binary Pd-Rh system: at low temperatures a core-shell configuration develops with the Pd-rich phase of lower surface energy in the subsurface layers and it is from this phase that segregation occurs. Finally, the Pt 4 Pd 1 Rh 5 alloy system shows co-segregation of Pt and Pd at higher temperatures. At low temperatures, only Pd segregates to the surface, since it is the most exothermic process. At higher temperatures, the Pd segregation diminishes and the less exothermic segregation of Pt becomes possible. The present chapter shows that the MEAM is very well suited for the study of ordering

172 152 Segregation in ternary alloys and segregation in ternary alloys. Due to the lack of experimental data, only part of the calculations could be validated. However, if strong discrepancies between the simulated bulk or surface behaviour and future experimental work would be observed, a possible suggestion could be in the refinement of the MEAM with ternary interaction parameters. This could then yield a more accurate description of the ternary alloy energetics. References [1] V.I. Chernyshov and I.M. Kisil, Platinum Metals Rev. 37 (1993) 136. [2] P.A. Kozub, G.I. Gryn and I.I. Goncharov, Platinum Metals Rev. 44 (2) 74. [3] M. Guttmann, Surf. Sci. 53 (1975) 213. [4] M. Guttmann and D. McLean, in W.C. Johnson, J.M. Blakely (Eds.), Interfacial Segregation, ASM, Metals Park, [5] M.A. Hoffmann, P. Wynblatt, Metall. Trans. A 2 (1989) 215. [6] J. Luyten, S. Helfensteyn and C. Creemers, Appl. Surf. Sci (23) 833. [7] Y. Chen, S. Liao and H. Deng, Appl. Surf. Sci. 253 (27) 674. [8] L. Kaufman and H. Bernstein, Computer calculation of phase diagrams, Academic Press, New York, 197. [9] J.O. Andersson, T. Helander, L. Höglund, P.F. Shi and B. Sundman, Calphad 26 (22) 273. [1] [11] cited in Ref. [7] as N. Yuantao, Y. Zhengfen, Precious Metals China 2 (1999) 1.

173 11 Catalyst nanoparticles In order to understand the catalytic properties of transition metal alloys, segregation to low index and vicinal single crystal surfaces of binary and ternary alloys was studied in the previous chapters. However, a gap still exists between these highly idealised single crystal surfaces and the form these materials acquire when used in a chemical reactor. Indeed, for practical applications, small transition metal or alloy particles (2 to 5 nm), containing a few hundreds of atoms, are deposited on a porous substrate in order to increase the active surface area and for a better reaction speed and selectivity. In this chapter, first the stability of two competing low energy structures for pure metal particles is investigated, namely the cubo-octahedron (CUBO) and icosahedron (ICO). Then follows a study of the surface segregation in binary alloy particles, again for both mentioned structures. This study pertains to free (unsupported) particles, as no interaction with a substrate is taken into account Introduction It is well-known that reducing the size of a piece of material is accompanied by a drastic change in its thermodynamic properties. It has indeed been observed experimentally that the melting point of small pure metal particles is significantly lower than that of their bulk counterpart [1]. Nanoparticles of ordered alloys can also have a reduced order-disorder transition temperature [2]. In the case of abundant segregation, one can even observe the formation of a different phase in the core [3]. An important factor distinguishing clusters from other lowdimensional structures is the possibility of achieving non-crystallographic (icosahedral, decahedral, or other) arrangements. Although the behaviour of 153

174 154 Chapter 11 extended low index crystal surfaces is still indicative for the behaviour of nanoparticles, these phenomena show that insights from experimental or theoretical results on quasi-infinite samples can not directly be transferred to the nanometer scale. On the other hand, a detailed understanding of the behaviour of such particles is essential for a better understanding of their catalytic properties. The validity of the MEAM potential for the study of metallic nanosystems becomes an issue. The MEAM is fitted to bulk and surface properties of periodic crystal structures. It is also well-known that, in addition to the common ferromagnetic metals, some transition metals that are non-magnetic in bulk form may become magnetic when the dimensionality is reduced, as in ultrathin films [4 6], in nanowires [7 9], and in clusters [1 13]. These altered magnetic properties are not included in our MEAM potential and this possibly limits its applicability for the study of very small clusters. This effect is however believed to vanish for particles with more than 1 atoms. For Rh clusters e.g., the magnetic moment equals the bulk value of zero from N = 6 atoms on. Therefore, our study will not consider particles that are smaller than 1 atoms. The energetically most stable structures are often called magic sizes. Magic sizes may either correspond to the completion of a geometrically perfect structure (geometric magic sizes) or to the closing of an electronic shell (electronic magic sizes). In this work, only geometric magic sizes are discussed [14]. In particles with a total number of atoms corresponding to geometric magic numbers, different shells can be defined. The atoms in a given shell are all located at a certain distance from the center atom. The composition profile as function of the shell number provides information that is similar to the composition depth profile for extended surfaces. Two competing geometrically closed-shell particle shapes, containing exactly the same number of atoms are drawn in Fig for small particles of fcc metals. The CUBO features vertex sites (Z = 5), edges (Z = 7), {1} facets (Z = 8) and {111} facets (Z = 9). This CUBO can directly be cut from an undistorted fcc lattice. The second particle shape (ICO) features vertex sites (Z = 6), edges (Z = 8) and {111} facets (Z = 9). As the CUBO and ICO are two frequently observed particle shapes for pure metals and alloys, no attempt has been undertaken to search for more stable (lower energy) particle shapes. For example, the decahedron, which is sometimes also observed, is not considered in this study. The search for the most stable particle shape is usually performed through genetic algorithms. However, the energy landscape of particle shapes shows many local minima with small barriers between them. It is believed that at temperatures that are not too low, a mixture of various particles exists, containing both stable and

175 Catalyst nanoparticles 155 metastable particle shapes. It is therefore instructive to study the segregation in both CUBO and ICO structures simultaneously. The distribution of atomic sites as a function of the coordination number Z is given in Table 11.1 for a CUBO and an ICO respectively. The distribution of the incompletely coordinated sites (Z<12) is entirely different for the two structures. It can be deduced from Table 11.1 that the average coordination of these surface atoms is higher for the ICO. Therefore, the surface energy is generally lower for the ICO. But in contrast to the CUBO, the ICO is not commensurate with a fcc structure and the atoms in the core of the particle have a slightly higher energy. The interplay of the icosahedral stacking fault energy (with respect to the fcc stacking) and the surface energy will determine which structure will be favoured. It is expected that for a small number of atoms, minimisation of the surface energy is most often the more important driving force, resulting in the ICO as the lower energy shape. With increasing size of the particle, the ratio of core to surface atoms increases and the CUBO should become the more stable shape. The stability of the CUBO with respect to the ICO is first studied for pure Cu, Pt, Pd and Rh particles with the MEAM and the newly optimised parameters. Next, surface segregation and ordering is investigated as a function of temperature for binary alloy nanoparticles with different kinds of interactions, again for both the CUBO and the ICO. In alloy nanoparticles, segregation to specific sites at the surface inherently influences segregation to other sites. This coupling, together with the ordering tendency of the bulk leads to an intriguing competition between the different sites. Pt-Pd is selected as a prototype for disordered alloys, Pd-Rh as a model for demixing alloys and Cu 3 Pt for ordering alloys. Figure 11.1 CUBO (left) and ICO (right), both with 39 atoms.

176 156 Chapter 11 Z CUBO ICO total average Z Table Distribution of atomic sites as a function of the coordination number Z for a CUBO and an ICO with 39 atoms Computational set-up All Monte Carlo simulations are performed in the Canonical Ensemble, with a fixed number of atoms of each species. The initial configuration is equilibrated in 1 6 MC steps, after which the ensemble is sampled during the next 1 6 MC steps at an interval of 1 4 MC steps. For the pure metal particles, MC simulations are performed in which the atomic positions are fully relaxed. In each MC step, an atom is selected at random. A random displacement is selected in each direction (x, y and z). In order to optimise the acceptance probability, the maximum displacement in each direction is chosen as.3 Å Pure Cu, Pt, Rh and Pd nanoparticles Overview of theoretical and experimental literature data The energetics of small Cu clusters has been treated extensively in a review by Alonso [15]. For larger sizes, up to 1 atoms, global optimisation studies by semi-empirical potentials have been performed. Doye et al. [16] used the Sutton- Chen [17] potential, while Darby et al. [18] used the Gupta [19] potential. In both cases, the best structure at a total number of atoms N = 13 is the ICO, in agreement with DFT calculations, and there is no indication in favour of disordered structures at geometric magic numbers. Baletto et al. [2] compared the energies of the ICO, the decahedron, and the CUBO at larger sizes, finding that the crossover from icosahedral to decahedral structures occurs at around 1

177 Catalyst nanoparticles 157 atoms and that decahedron and cubo-octahedral structures are in close competition up to 3 atoms at least. This behaviour is in agreement with the inert gas aggregation experiments of Reinhard et al. [21,22], who were able to identify a prevalence of small icosahedra, intermediate-size decahedra, and large fcc clusters, with a wide interval of sizes in which decahedra and fcc clusters coexisted. The crossover size between the ICO and the decahedron was in agreement with the calculations. The energetic stability of small Pt clusters has been addressed by Doye and Wales [16] and Massen et al. [23], who found a low-symmetry structure at N = 55. Baletto [2], on the contrary, found the ICO structure also at this size, though with a narrow crossover range among ICO, decahedral, and CUBO structures. Ab initio results are more sparse. Fortunelli [24] performed calculations for N = 13, 38, and 55. At N = 13, they found that the Ino decahedron (a decahedron in which the edges are cut to expose {1}-like facets) has a lower energy than the ICO and the CUBO, but a structure originating from the symmetry breaking of the CUBO is even lower in energy. At N = 55 the order of the structures is reversed, and the ICO becomes more stable than the Ino decahedron and the CUBO. Fortunelli et al. [24] attributed the results at N = 13 to the small size of the molecule, while the behaviour at N = 55 is intermediate between finite molecules and fully metallic systems. There are fewer studies on Rh particles than on the other transition metals Cu, Pt and Pd. Besides their catalytic properties, these clusters are very interesting because their lowest-lying isomers could have nonzero spin, thus becoming magnetic [25]. Barreteau et al. [26] performed calculations within a tight-binding description, combined with an empirical potential for inhomogeneous relaxations and determined the critical size at which the CUBO and ICO clusters have roughly the same energy. It appeared that this size is similar for Rh and Pd. The CUBO structure is found to be unstable for N = 13 but, for 55 N 561 this shape is metastable, explaining the coexistence of both geometries at intermediate sizes. The energetics of small Pd clusters have been investigated by several groups employing different methods. Analogous to Rh particles, small Pd particles also have nonzero spin [27,28] and exhibit magnetic properties. At larger sizes, ab initio results are restricted to the comparison of a small set of selected structures at a few special sizes. Kumar et al. [29] have compared ICO with CUBO and Ino decahedron clusters at N = 55 and 147, finding that slightly distorted ICO structures have in both cases the lowest energy. Nava et al. [3] also found that a Jahn-Teller distorted ICO is lower in energy than a perfect ICO at N = 55. Moreover, Nava et al. [3] found that the ICO and CUBO are very close in energy

178 158 Chapter 11 at both N = 147 and N = 39, with the ICO prevailing at 147 and CUBO at 39 atoms. By tight-binding calculations, Barreteau et al. [26,31] found that for 39 atoms the ICO is still lower in energy than the CUBO. Global optimisation studies up to N 1 by the Sutton-Chen potential confirmed the optimisation with the Gupta potential by Massen et al. [23]. At larger sizes, a comparison of the different structural motifs shows that Pd behaves in a similar way as Pt (with rather small crossover sizes so that fcc clusters are already in close competition with the other motifs at N 1) when modelled by the Rosato potential, while it behaves very similarly to Ag when treated by means of EAM potentials [14]. Calculations by Jennison et al. [32] support the results from the Rosato potential, finding that fcc structures are more favoured in Pd than in Ag. By Transmission Electron Microscopy (TEM), José-Yacamán [33] observed thiol-passivated Pd nanoparticles in the range of 1 to 5 nm diameter, and saw a variety of structures, ranging from simple and twinned fcc, over ICO and decahedral to amorphous structures. They were able to observe rather large ICO clusters, explaining their presence as probably being due to kinetic trapping effects. A general conclusion that can be drawn from the whole of these theoretical results is that all structures are very close in energy. Furthermore, when preparing nanoparticles experimentally, (initial) growth conditions can have a large influence on the final particle shape distribution. It is therefore worthwhile to compare the behaviour of both the ICO and CUBO structures, and not to limit the study to the locally most stable shape only. Simulation results Fig shows the MEAM calculated energy upon uniformly expanding or compressing ICO and CUBO nanoparticles with 39 atoms of pure Cu, Pt, Rh and Pd. A few interesting conclusions can be drawn from this graph. First, of the four metals studied, only Pd prefers the CUBO particle shape. The energy difference between the structures amounts to.1 ev/atom in favour of the CUBO. For Rh, the energy difference between the two particle structures is very small (.4 ev/atom). For Pt, this energy difference is somewhat larger,.9 ev/atom. For Cu particles, the ICO is clearly preferred by more than.2 ev/atom. Secondly, ICO particles show an equilibrium nearest neighbour distance that is significantly larger than in the bulk metal by ±.4 Å. It must however be noted that for the ICO, not all interatomic distances are equal: radial (intershell) bonds are compressed, while intrashell bonds are expanded. The interatomic distance, shown in Fig corresponds to the larger intrashell distance. The intershell distance is smaller by approximately 5%. CUBO particles feature an equilibrium separation that is slightly smaller than the bulk counterpart, but remains larger

179 Catalyst nanoparticles 159 than the intershell distance in the ICO. Table 11.2 summarises the equilibrium nearest neighbour distances and the energy differences for Cu, Pt, Pd and Rh Energy (ev/atom), Pt and Rh Nearest neighbour distance (A) Energy (ev/atom), Cu and Pd Pt, ICO Pt, CUBO Rh, ICO Rh, CUBO Cu, ICO Cu, CUBO Pd, ICO Pd, CUBO Figure MEAM prediction for the energy as a function of the nearest neighbour distance of Cu, Pt, Rh and Pd CUBO and ICO particles with 39 atoms. The energy should be read from the right axis for Cu and Pd, and from the left axis for Pt and Rh. CUBO ICO bulk CUBO ICO R R R E E E Cu Pt Rh Pd Table MEAM calculation of the Cu, Pt, Rh and Pd equilibrium nearest CUBO ICO neighbour distance (Å) in CUBO ( R ) and ICO ( R ) particles, compared to bulk the bulk value ( R ), together with the cohesive energy (ev/atom) of a CUBO CUBO ICO ( E ) and an ICO ( E ) particle. The last column lists the energy difference ( E) between a CUBO and ICO particle CUBO ICO E = E E. A negative sign corresponds to the CUBO being the most stable particle shape.

180 16 Chapter 11 The great importance of the surface in small particles can further be evidenced by the following intuitive empirical relationship for the average energy per atom in a nanoparticle (E/N) as a function of the total number of atoms (N) [34] E N cluster sub 1/3 = Ei + N (11.1) sub with Ei the bulk cohesive energy and a constant. In this equation, represents the average excess energy per surface atom, the total number of surface atoms being proportional to N 2/3. This constant depends on the shape of the particle and also on the anisotropy (orientation dependence) of the surface energy. In Fig. 11.3, the average atomic energy is plotted against N -1/3. For each metal, the plot is constructed for the most stable shape only. For Cu, Pt and Rh, this corresponds to the ICO, while for Pd the CUBO is chosen, in accordance with the results from Fig and Table The expected linear behaviour, corresponding to Eq is clearly confirmed Energy (ev/atom), Pt and Rh l = 1 l = Energy (ev/atom), Cu and Pd Pt, ICO Rh, ICO Cu, ICO Pd, ICO Cu (9x9,l) CUBO N -1/3 Figure Mean atomic energy versus N -1/3 for nanoparticles of Cu (ICO), Pt (ICO), Rh (ICO) and Pd (CUBO). For Cu and Pd, the energy should be read from the right axis, while for Pt and Rh from the left axis. Also shown is the Cu (9, l) CUBO for various values of l (see below).

181 Catalyst nanoparticles 161 For a geometric magic size CUBO, which always contains an equal number of atoms as the ICO, the ratio of {111} to {1} facets is rather low. This ratio can be further optimised by evaluating the energy of a octahedron (without {1} facets) and gradually increasing the size of the {1} facets. The following notation for a unique identification of these particles is now proposed: a (m,l) CUBO corresponds to a CUBO in which the basal plane of the original octahedron consisted of a square of m m atoms and the {1} facets are squares with l l atoms. Fig shows the results for a Cu particle, starting from an octahedron with m=9. For the pure octahedron, the vertex atoms have a very low coordination (Z = 4) and therefore have a high energy. By cutting these edge atoms, thereby creating a {1} facet, the total energy can decrease. However, when the size l of the {1} facets becomes too large, the total energy increases again. In between, a structure with an optimal ratio of {111} to {1} facet sites can be found. For m = 9, the CUBO s with l = 2 and l = 3 show an equal energy. For CUBO particles with m = 8, l = 2 is more favourable, while for m = 1, l = 3 is more favourable. The equal stablity can be rationalised by considering the distribution l = Energy (ev/atom) l = l = l = l = 3 l = 3 l = N Figure MEAM prediction of the energy of a Cu (9, l) CUBO with different (l l) truncations. The two structures of (nearly) equal energy are shown as inset.

182 162 Chapter 11 m = 9 Cu Pt Rh Pd l MEAM Eq Table 11.3 Energy (ev/atom) of (9,2) and (9,3) CUBO s calculated with the MEAM and with Eq for Cu, Pt, Rh and Pd. of atomic sites of the two particles. In a first approximation, the energy is proportional to Z [35]. The average of Z for l=2 amounts to 3.151, while for l=3 this value is Both particle shapes thus yield almost the same value and hence an almost equal stability is expected. The energy per atom, calculated by the MEAM can indeed fairly be reproduced by sub E i E = a Z = Z (11.2) 12 with a the proportionality constant. This observation further leads to the conclusion that this should be a generic feature for metallic particles. This is supported by Table 11.3, that shows the calculations for Cu, Pt, Rh and Pd. The numbers in Table 11.3 indicate which truncation is preferred when starting from an octahedron with m m atoms in its basal plane. This does not necessarily correspond to the absolute minimum in energy for the respective total number of atoms. For optimising the particle shape for a given number of atoms, the value of in Eq must be minimised. Therefore, the points of Fig are added to Fig and tabulated in Table Now, it can be seen that l = 3 and l = 4 lie on the straight line for the Cu ICO and feature therefore a comparable stability. These optimal truncations can be compared with the optimal particle shape, obtained from a Wulff construction [36], which was also developed in order to find the equilibrium shape of macroscopic crystals by minimising the surface energy at fixed volume. If the excess energy of edges and vertices is neglected, an approximative criterion is obtained γ γ d = (11.3) d (1) (1) (111) (111)

183 Catalyst nanoparticles 163 with γ (hkl) the (hkl) surface energy and d (hkl) the distance from the center of the particle to the {hkl} facet. The ratio of surface energies in the l.h.s. of Eq is given in Table 11.4, together with the results for the (9, l) CUBO as a function l. For Cu, Rh and Pd, the ratio of surface energies is nearly equal (1.15), while for Pt this ratio is slightly higher (1.3). This ratio for Cu, Rh and Pd lies in between the truncations l=3 and l=4, and these particles also feature almost the same value for. For Pt, the surface energy ratio almost exactly corresponds to l=3 and therefore this particle shape has the smallest value for. It can be seen that the Wulff criterion confirms the particle shape estimations from the MEAM. In all previous calculations, the total energy was always optimised with respect to the nearest neighbour distance by isotropically expanding or compressing the particle. In order to check whether important non-isotropic relaxations manifest themselves, Molecular Dynamics (MD) simulations are performed in which the atomic positions can fully relax. For the MD simulations, the newly MEAM potential for Cu has been implemented within the CAMELION package of B.J. Thijsse [37]. A fingerprint of the atomic configuration is given by the radial distribution function (RDF) which shows the variation of the probability to find a neighbouring atom as a function of interatomic distance. Fig compares the RDF for the Cu ICO with 147 atoms, as obtained from the MD/MEAM simulations at 1 K, with the unrelaxed ICO. It can be seen that the peak at the nearest neighbour distance is reproduced in the MD/MEAM simulations. Secondly, the next nearest neighbours are located at a somewhat shorter distance with respect to the unrelaxed ICO. A calculation of the average separation for the first three shells reveals a slight contraction of 3%. This contraction causes more substantial deviations at higher interatomic distances. Overall, relaxations are limited and have only a minor influence on the final energy. m = 9 Cu Pt Rh Pd γ (1) /γ (111) l d (1) /d (111) Table Ratio of distance from center to {1} and {111} facets for a (9,l) CUBO as a function of l and the corresponding stability factor for Cu, Pt, Rh and Pd.

184 164 Chapter 11 The effect of structural rearrangements is also quantified by MC/MEAM simulations, in which the atoms are allowed to move from the fcc lattice points, for a Cu particle with 147 atoms. Starting from a CUBO, the final result after MC steps (Fig. 11.6) closely resembles the ICO, which is indeed the more stable particle shape. The original {1} facets fold into two separate {111} facets, which are energetically preferred. This result shows that MC simulations allow for the determination of structural phase transitions for small systems. However, other techniques, e.g. genetic algortithms, are much more efficient in finding equilibrium structures than the classical MC simulations. Unrelaxed MD at 1 K Arbitrary units Interatomic distance (A) Figure Radial distribution function for the Cu ICO with 147 atoms, as compared to the unrelaxed ICO.

185 Catalyst nanoparticles 165 Figure Snapshot of a MC/MEAM simulation result after MC steps, starting from a CUBO with 39 atoms Nanoparticles of disordered Pt-Pd alloys Rousset et al. [38,39] characterised free and supported Pt-Pd nanoparticles with Time of Flight (TOF) mass spectrometry, High Resolution TEM and Energy Dispersive X-ray (EDX) analysis. It was concluded that the supported Pt-Pd particles are well crystallised with CUBO shapes. Later, Rousset et al. [4] also studied the segregation behaviour in Pt-Pd nanoparticles with LEIS. From the LEIS measurements, it was concluded that the cluster surfaces are strongly enriched in Pd. For their theoretical approach of the equilibrium cluster structure, Rousset et al. [4] used MC simulations combined with a Modified Tight-Binding (MTB) scheme. The MC/MTB simulations additionally showed that Pd segregation in small Pt-Pd particles takes place preferentially to the low coordinated sites. MC/MEAM simulations are performed for small (39 atoms) and large (923 atoms) CUBO and ICO particles. This is done for three particular particle compositions: 12, 25 and 5% Pd respectively. Fig shows the simulated site occupancy for the various surface sites at the Pt 232 Pd 77 CUBO and ICO. A strong Pd segregation is observed, in agreement with the experimental and theoretical evidence [4]. At lower temperatures, the formation of a core-shell nanoparticle develops with the Pd atoms in the surface layer and the Pt atoms in the core of the particle. Furthermore, the extent of surface enrichment is most pronounced to the lower coordinated sites as the segregation energy is indeed most exothermic for

186 166 Chapter 11 these lower coordinated sites. Due to the limited number of Pd atoms in the particle, segregation to the higher coordinated surface sites is suppressed. As the temperature increases, segregation to the lower coordinated surface sites diminishes in favour of the segregation to the higher coordinated surface sites, which is relatively less exothermic. This explains the observed behaviour as a function of temperature in Fig Fig shows the Pd composition as a function of the shell number at 12 K. Shell 1 is the outer surface layer. The depth profile is very similar for the CUBO and the ICO. For the Pt 272 Pd 37 particle, all Pd atoms are located at the surface and consequently, the second shell is completely depleted in Pd. Increasing the Pd content leads to oscillating depth profiles, in agreement with the results for segregation to the low index surfaces (Chapter 6). The results for the larger particle (923 atoms) are completely analogous and will therefore not be shown. 1 9 Site composition (at. % Pd) Temperature (K) Z = 5, CUBO Z = 7, CUBO Z = 8, CUBO Z = 9, CUBO Z = 6 ICO Z = 8 ICO Z = 9, ICO Figure Simulated site-specific composition for the surface sites at a Pd 77 Pt 232 CUBO and corresponding ICO as a function of temperature.

187 Catalyst nanoparticles 167 Composition (at.% Pd) x = 37, CUBO x = 77, CUBO x = 155, CUBO x = 37, ICO x = 77, ICO x = 155, ICO Shell number Figure Pt composition as a function of shell number at 12 K for the Pt 39 x Pd x CUBO and ICO. Shell number 1 corresponds to the outermost surface layer Nanoparticles of demixing Pd-Rh alloys Fig shows the occupancy for the sites with varying coordination as a function of temperature for the Pd 155 Rh 154 CUBO. The results for the ICO are completely comparable to those for the CUBO. This is expected as endothermic Pd-Rh alloys strive to achieve a core-shell configuration. For this system, the rule of thumb again holds that, the lower the coordination of a specific site, the stronger the enrichment in the component with the lowest surface energy, i.c. Pd. Due to this competition between the sites at low temperatures, segregation to the {111} facets is not complete. Upon increasing the temperature, the exothermic segregation to the lower coordinated sites diminishes and the Pd composition of the {111} facets increases, up to a maximum at 7 K. For the other particles Pd 77 Rh 232 and Pd 37 Rh 272, this competition does not occur. As a consequence, in these cases the Pd composition for the vertex and edges decreases monotonically, while the Pd concentration in the {1} facets and {111} facets increases monotonically with increasing temperature.

188 168 Chapter Composition (at.% Pd) Z = 5 Z = 7 Z = 8 Z = 9 Z = Temperature (K) Figure Simulated site composition for Pd 155 Rh 154 CUBO as a function of temperature x = 37 x = 77 x = 155 Composition (at.% Pd) Shell number Figure Pd composition as a function of shell number at 12 K for the Pd x Rh 39-x CUBO. Shell number 1 corresponds to the outermost surface layer.

189 Catalyst nanoparticles 169 Fig shows the Pd composition as a function of the shell number at 12 K. The core-shell configuration emanates from the simulations: the outermost shell is strongly Pd enriched, while the core of the particle is strongly Rh enriched. The core-shell configuration is triggered by the endothermic nature of the Pd-Rh bond, while the low surface energy for Pd compared to Rh causes the Pd-rich phase to be the outer phase. The typical monotonous depth profile for phase separating systems is observed, in agreement with the results for the Pd-Rh single crystal surfaces in Chapter Nanoparticles of ordering Cu 3 Pt alloys MC/MEAM simulations are performed for CUBO and ICO of 39 and 923 atoms of Cu 3 Pt alloys. After equilibration at 3 K, the final energy of the 39 ICO is.2 ev/atom lower than the corresponding CUBO. This suggests that the ICO is the more stable of the two structures, in agreement with the results for the pure metals. Fig shows the simulated site composition for the different coordination numbers as a function of temperature for the ICO and CUBO. It is clear that for both the CUBO and ICO, the lowest coordinated vertices are almost completely enriched in Cu. Also, the edges are considerably enriched in Cu. For the CUBO, the {1} facets favour a similar Cu enrichment as the edges. The Cu atoms, responsible for this enrichment, can segregate either from the {111} facets or from the core of the particle. At lower temperatures, the core of the particle shows a high degree of order due to the exothermic Cu-Pt interactions. In this case, Cu atoms are not easily exchanged between the particle core and its surface. This can be rationalised as follows: in stoichiometric ordered alloys, strain energy effects are often negligible as the crystal has spontaneously adjusted its lattice parameter to the ordered stacking. Apart from this strain effect, the net energy balance for segregation to extended surfaces of perfectly ordered alloys is made up by two terms: i) a gain in surface energy (exothermic) at the expense of ii) a less exothermic mixing energy. As the attractive interactions in ordered alloys are relatively strong, the lower surface energy can in general not make up for the less favourable mixing energy in the bulk. This balance is more favourable if the exchange occurs between nearest neighbours in the first two layers. However, an appreciable lowering of the surface energy is still necessary to overcome the lost mixing energy. Exactly the same arguments hold for the particle: at lower temperatures, the core of the particle is highly ordered and this will hamper the Cu segregation. On the other hand, Cu has a strong tendency to occupy the sites with a lower coordination. In contrast to the case of extended surfaces, where the bulk is the only reservoir for segregating species, the Cu atoms, segregating to the vertices, edges and {1} facets can now also originate from the {111} facets,

190 17 Chapter 11 where fewer Cu-Pt bonds have to be broken. However, this causes a Pt enrichment of the {111} facets and thus an increase of the surface energy. A delicate balance between the ordering tendency and the surface energy is thus honoured. From our simulations, it can be concluded that, at lower temperatures, segregation mainly occurs from the {111} facet sites. The resulting Pt enrichment of these {111} facets contrasts with the Cu enrichment on the extended Cu 3 Pt(111) surfaces (see Chapter 8). Increasing the temperature alters the segregation mechanism. At higher temperatures, the core of the particle becomes more randomised. The decreasing number of bonds between unlike atoms raises the energy of the core and makes the exchange of Cu atoms from this core to low-coordinated surface sites more favourable. Pt atoms that originally resided on the low-coordinated surface sites are now also exchanged with Cu atoms from the core of the particle. Also the ICO is characterised by these two different segregation mechanisms. At lower temperatures, the {111} facet sites are depleted in Cu, while the core remains quasi unaffected. At higher temperatures, segregation occurs more and more from the particle core and the Cu concentration of the {111} gradually increases. 1 9 Site composition (at.% Cu) Temperature (K) Z = 5, CUBO Z = 7, CUBO Z = 8, CUBO Z = 9, CUBO Z = 12, CUBO Z = 6, ICO Z = 8, ICO Z = 9, ICO Z = 12, ICO Figure Site occupation for the various coordination numbers of Cu 231 Pt 78 CUBO and ICO particles as function of temperature

191 Catalyst nanoparticles 171 The Cu composition as a function of the shell number for the CUBO and ICO at 4 distinct temperatures is presented in Fig For the CUBO, the depth profile is oscillating at all temperatures. Due to the high ratio of {1} to {111} facet sites for the 39 CUBO, segregation to these {1} facet sites will dominate the segregation behaviour in this particle. At lower temperatures, the {1} facet sites strive at becoming a pure Cu layer, in agreement with the results on Cu 3 Pt(1). Analogous to the extended Cu 3 Pt(1) surface, the second layer orders in a c(2 2) pattern, leading to a composition of the second shell of approximately 5%. The somewhat higher Cu composition, observed in the simulations, is due to the contribution of the atoms which are located below the {111} facets and these aim at a stoichiometric composition of 75% Cu. At higher temperatures, the orderdisorder transition occurs and consequently, the driving force for the formation of alternating Cu and Cu/Pt shells diminishes. This leads to less strong oscillations, as observed in Fig As far as the 39 ICO is concerned, it can be observed that the Cu surface composition remains quasi constant as a function of temperature. The decreasing Cu concentration of the edges is compensated for by the increasing Cu concentration of the {111} facets. A second important feature is the monotonic concentration decrease over the shells at 3 K, while from 6 K on, the depth profile becomes oscillating. The reason for this is that at 3 K, the {111} facets are strongly Cu depleted. In order to form an oscillating depth profile, due to the exothermic Cu-Pt interactions, the second shell below these {111} facets is preferentially occupied by Cu. On the other hand, the edge and vertex sites are Cu enriched and these sites drive the second layer towards a Pt enrichment. The net balance for the second shell can therefore lead to both a Cu or a Pt enrichment. For the 39 ICO at low temperatures, the monotonous depth profile is preferred over the oscillating profile. Increasing the temperature leads to a gradual increase of the Cu content of the {111} facets and therefore to a smaller driving force for the second shell towards a Cu enrichment. This causes a change in the depth profile from monotonous to oscillating between 5 K and 6 K. Increasing the size of the magic number CUBO and ICO up to 923 atoms does not alter the global picture. The only qualitative difference for the larger ICO is the depth profile, which now becomes oscillating at all temperatures. This can be attributed to the fact that the particle is now large enough for all surface sites to become enriched in Cu.

192 172 Chapter Composition (at.% Cu) Shell number T = 3 K, CUBO T = 6 K, CUBO T = 9 K, CUBO T = 12 K, CUBO T = 3 K, ICO T = 6 K, ICO T = 9 K, ICO T = 12 K, ICO Figure Shell composition at 3 K, 6 K, 9 K and 12 K for the CUBO and ICO Cu 231 Pt 78 particle. Shell number 1 corresponds to the outermost surface layer. Next, segregation is also studied in cubo-octahedra of larger sizes with 1126, 2556, 4794, 8678 and atoms, for which the ratio of {111} facets to {1} facets is increased. Fig shows the composition of the different sites of the smallest nanoparticle (N = 1126) as a function of temperature. Again, the two distinct segregation mechanisms become apparent. At higher temperatures, the Cu concentration at the {111} facets increases by normal segregation and the core of the particle becomes slightly depleted in Cu. Table 11.3 shows the segregation energies for the atomic exchange of a Pt atom at a vertex site with a Cu atom from either the particle core or from a {111} facet. The more exothermic segregation energy shows that an atomic exchange coming from a {111} facet is more favourable. Therefore, the Cu enrichment at the low-coordinated sites does not originate from the particle core, but from the nearby {111} facets instead. The composition of the particle core is quasi unaffected and remains close to the original value. Upon further increasing the size of the nanoparticles, the ratio of {111} facet sites to all the other low-coordinated sites also increases. In larger particles, the exchange mechanism, that was dominant in small particles at lower temperatures,

193 Catalyst nanoparticles 173 i.e. between the sites with Z < 9 and the {111} facets, does no longer cause a significant Pt enrichment of the latter. Fig shows the simulation results for the four larger particles at 3 K. At this low temperature, the largest particle (d) now shows almost perfectly p(2 2)-ordered {111} facets, while the {1} facets are still pure Cu. Fig shows how the short-range order parameter (SRO) as a function of temperature for two different particles compares to the SRO in a quasi-infinite bulk. While for the large particle, the order-disorder transition temperature coincides with the bulk value, the smaller particle shows a more gradual orderdisorder transition at slightly lower temperatures. These results therefore confirm the well-known lowering of the transition temperature in smaller particles. Pt exchange with Cu atom from Segregation energy (ev/atom) core -.6 {111} facet -.45 Table Segregation energies of Cu atoms to vertex sites from core sites and from {111} facet sites Composition (at.% Cu) Temperature (K) Z = 6 Z = 7 Z = 8 Z = 9 Z = 12 Figure MC/MEAM simulation for the site-specific composition in a Cu 3 Pt CUBO nanoparticle with 1126 atoms as a function of temperature.

194 174 Chapter 11 (a) (b) (c) (d) Figure Snapshot at the end of the MC simulation run at 3 K for the various Cu 3 Pt particle sizes with a) 2556 atoms, b) 4794 atoms, c) N = 8678 atoms and d) atoms. The dark atoms represent Cu atoms and the grey atoms stand for Pt atoms.

195 Catalyst nanoparticles 175 Short range order parameter N=1126 N=16786 bulk Temperature (K) Figure Evolution of short-range order in the CUBO Cu 3 Pt particles with 1126 and atoms particle, compared to the behaviour in a quasi-infinite bulk Conclusions In this chapter, an attempt has been made to bridge part of the material gap by studying the surface segregation and ordering in free nanoparticles. First, the stability of CUBO and ICO pure metal magic number particles is studied with the MEAM potentials. For very small particles, the ICO seems to be the most stable particle shape for Cu, Pt and Rh, while for Pd the CUBO turns out to be more stable. However, with an optimal ratio of d (1) /d (111), the CUBO shows a stability that is comparable to the ICO. Next, surface segregation in alloy nanoparticles is studied for disordered Pt-Pd CUBO and ICO particles. For both particles, a strong Pd segregation and an oscillating depth profile are observed, in agreement with the results on Pt-Pd single crystal surfaces. Subsequently, phase separating Pd-Rh particles are studied. They show a pronounced segregation of the element with the lower surface energy, preferentially to the surface sites with the lower coordination. Finally, surface segregation and ordering is studied in ordering Cu 3 Pt particles. At low temperatures, the Cu atoms move from the {111} facets to the less coordinated sites, leaving the core of the particle quasi undisturbed. This leads to

196 176 Chapter 11 a Pt enrichment of these {111} facets, which is in contrast with the results on extended Cu 3 Pt(111) surfaces. The vast majority of the results obtained in this chapter can not unambiguously be validated by experiments or by theoretical calculations. It would be worthwhile to also consider the stability of these small metal particles with ab initio data. Furthermore, for very small sizes, magnetism may become an important issue, which is not included in the present MEAM potential. It would be useful to develop a formalism for inclusion of magnetic effects in the MEAM. References [1] Buffat Ph. and J.-P. Borel, Phys. Rev. A 13 (1976) [2] R. Vallée, M. Wautelet, J.P. Dauchot and M. Hecq, Nanotechnology 12 (21) 68. [3] M. Müller and K. Albe, Phys. Rev. B 72 (25) [4] M.J. Zhu, D.M. Bylander and L. Kleinman, Phys. Rev. B 42 (199) [5] R. Wu and A.J. Freeman, Phys. Rev. B 45 (1992) [6] S. Blugel, Phys. Rev. Lett. 68 (1992) 851. [7] V. Bellini, N. Papanikolaou, R. Zeller and P.H. Dederichs, Phys. Rev. B 64 (21) [8] D. Spisak and J. Hafner, Phys. Rev. B 67 (23) [9] A. Bergara, J.B. Neaton and N.W. Ashcroft, Int. J. Quantum Chem. 91 (23) 239. [1] A.J. Cox, J.G. Louderback and L.A. Bloomfield, Phys. Rev. Lett. 71 (1993) 923. [11] A.J. Cox, J.G. Louderback, S.E. Apsel and L.A. Bloomfield, Phys. Rev. B 49 (1994) [12] G. Gantefor and W. Eberhardt, Phys. Rev. Lett. 76 (1996) 4975 [13] Y. Jinlong, F. Toigo and W. Kelin, Phys. Rev. B 5 (1994) 7915 [14] F. Baletto and R. Ferrando, Rev. Mod. Phys. 77 (25) 371. [15] J.A. Alonso, Chem. Rev. (Washington, D.C.) 1 (2) 637. [16] J.P.K. Doye and D.J. Wales, New J. Chem. 22 (1998) 733. [17] A.P. Sutton, J. Chen, Philos. Mag. Lett. 61 (199) 139. [18] S. Darby, T.V. Mortimer-Jones, R.L. Johnston and C. Roberts, J. Chem. Phys. 116 (22) [19] R.P. Gupta, Phys. Rev. B 23 (1981) [2] F. Baletto, C. Mottet, R. Ferrando, Chem. Phys. Lett. 354 (22) 82. [21] D. Reinhard, B.D. Hall, P. Berthoud, S. Valkealahti and R. Monot, Phys. Rev. Lett. 79 (1997) 1459.

197 Catalyst nanoparticles 177 [22] D. Reinhard, B.D. Hall, P. Berthoud, S. Valkealahti and R. Monot, Phys. Rev. B 58 (1998) [23] C. Massen, T.V. Mortimer-Jones and R.L. Johnston, J. Chem. Soc. Dalton Trans. 23 (22) [24] A. Fortunelli and E. Aprà, 23, J. Phys. Chem. 17 (23) [25] T. Futschek, M. Marsman and J. Hafner, J. Phys.: Condens. Matter 17 (25) [26] C. Barreteau, M.C. Desjonquères and D. Spanjaard, Eur. Phys. J. D 11 (2) 395. [27] B.V. Reddy, S.N. Khanna and B.I. Dunlap, Phys. Rev. Lett. 7 (1993) [28] N. Watari and S. Ohnishi, Phys. Rev. B 58 (1998) [29] V. Kumar and Y. Kawazoe, Phys. Rev. B 66 (22) [3] P. Nava, M. Sierka and R. Ahlrichs, Phys. Chem. Chem. Phys. 5 (23) [31] C. Barreteau, R. Guirado-López, D. Spanjaard, M.C. Desjonquères and A.M. Oles, Phys. Rev. B 61 (2) [32] D.R. Jennison, P.A. Schultz and M.P. Sears, J. Chem. Phys. 16 (1997) [33] M. José-Yacamán, M. Marín-Almazo and J.A. Ascencio, J. Mol. Catal. A: Chem. 61 (21) 173. [34] J.A. Northby, J. Xie, D.L. Freeman and J.D. Doll, Z. Phys. D 12 (1989) 69. [35] M. Scheffler and C. Stampfl, in K. Horn and M. Scheffler (Eds.), Handbook of Surface Science, Elsevier, Amsterdam, 2. [36] C.H.P. Lupis, Chemical Thermodynamics of Materials, North-Holland, [37] B.J. Thijsse et. al., Camelion-(M)EAM: CAMELION is a joint product of ComMaS and CRI->SIS, both at Delft University of Technology, Faculty of Applied Sciences, Laboratory of Materials Science. Contact person is Prof. Dr. B.J. Thijsse ( B.J.Thijsse@tnw.tudelft.nl). [38] J.L. Rousset, A.M. Cadrot, F. Santos Aires, A. Renouprez, P. Melinon, A. Perez, M. Pellarin, J. L. Vialle and M. Broyer, J. Chem. Phys. 12 (1995) [39] J.L. Rousset, F. Santos Aires, F. Bornette, M. Cattenot, M. Pellarin, L. Stievano and A. Renouprez, Appl. Surf. Sci. 164 (2) 163. [4] J.L. Rousset, A.J. Renouprez and A.M. Cadrot, Phys. Rev. B 58 (1998) 215.

198 178 Chapter 11

199 12 Simulation of CO adsorption and CO oxidation on Cu 3 Pt(111) The alloy systems studied in the previous chapters are frequently used as catalyst materials. Numerous surface science studies have been performed in order to understand the segregation behaviour and ordering at these alloy surfaces. However, these materials are mostly characterised in an ultra high vacuum environment. The influence of a gaseous ambient species is seldom taken into account in surface science studies. Due to preferential chemisorption, the presence of adsorbates may have a drastic influence on the surface composition, order and structure. One of the numerous striking examples is the CO-induced lifting of the quasi-hexagonal reconstruction Pt(1) [1]. Nowadays, the STM technique is able to visualise chemical reactions in a direct way. Complementary to this experimental work, a lot of computer simulation studies have been performed. In these studies, mostly simple systems, e.g. a chemical reaction on a pure metal surface, and simple models are used. Having shown in the previous chapters that the MEAM provides a solid framework for an accurate description of the metallic bond, we now continue with the simulation of metallic alloys in the presence of one or more adsorbates, i.c. the CO chemisorption and, even further, with the particular reaction of CO oxidation on Cu 3 Pt(111) surfaces. Not only the thermodynamics of adsorption and segregation is studied with equilibrium MC simulations, but also the kinetic aspects of the chemical reaction are treated by Kinetic Monte Carlo (KMC) simulations. 179

200 18 Chapter Introduction With the introduction of an adsorbate, three different types of chemical bonding come into play: the metal-metal bond, the metal-adsorbate bond and the interactions between the adsorbates. As shown in the previous chapters, the metalmetal bond is correctly described by the MEAM with surface-specific parameters. Regarding the metal-adsorbate interaction, two different adsorption mechanisms can be distinguished: physisorption and chemisorption. Physisorption is a weak interaction, characterised by the lack of a true chemical bond between the surface and the adsorbate, i.e. no electrons are shared [2]. The net attractive force stems from an equilibrium between the repulsive force, acting at short distances and the attractive Van der Waals force at larger distances. Typical values for the heat of physisorption are 2-4 kj/mol. The more interesting adsorption phenomenon, from a catalytic point of view, is chemisorption. In this case, the adsorption energy becomes comparable to chemical binding energies ( 1-2 kj/mol). The potential energy surface for chemisorption features pronounced mimina at well-defined high-symmetry sites at the surface. The strong binding of the adsorbate with the metal surface may weaken the intramolecular bond energy and consequently dissociation of the molecule is made possible. In contrast with associative or molecular adsorption, this dissociative adsorption is one of the main reasons for the success of metals and metallic alloys in catalysis. The presence of the adsorbate on the surface also influences the metal-metal bonds. As a consequence of the perturbed electron distribution in the neighbourhood of the adsorbate, adsorption of a second adsorbate may be facilitated or complicated and surface reconstructions may occur. Finally, lateral interactions can occur between the different adsorbed species. These lateral interactions originate from weak attractive Van der Waals interactions, weak repulsive dipole-dipole interactions and strong repulsive interactions due to closed shell orbital overlap at very short separations. These lateral interactions often result in ordered adsorbate layers for various coverages. Compared to chemisorption on pure metal surfaces, an extra effect comes into play when considering adsorption on alloy surfaces. Upon alloying, the chemical environment of the metal atoms changes, which in turn may influence the adsorption properties. This effect is usually referred to as a ligand effect [3-5]. A first qualitative estimate of the effect of an adsorbate on the surface composition (see also Chapter 2) can be obtained with the formalism of the Quasi- Chemical Approximation (QCA), in which the influence of the adsorbing secies is

201 Simulation of CO adsorption and CO oxidation on Cu 3 Pt(111) 181 considered as an extra driving force. As a consequence of simplifications, only one single analytical equation has to be solved. However, within this QCA/CBE framework, many over-simplifications are made: i) the metal-metal interactions can not simply be described by a constant value, ii) lateral interactions between the adsorbate atoms and molecules are in general not negligible, iii) the assumption of ideal mixing entropy is inadequate and iv) ligand effects are not taken into account. A more appropriate simulation framework will be developed in the next section. Besides the thermodynamic aspects of surface segregation and adsorption, also kinetic aspects are relevant. Indeed, a chemical reaction often takes place in a steady-state regime without reaching the true thermodynamic equilibrium. These kinetic aspects can be treated by KMC simulations, as explained in Chapter 3. The KMC algorithm requires parameters (attempt frequency and activation energy) for the rate equations of each individual process. Often, a good estimate of these characteristics can be obtained from experimental results or from theoretically obtained values. The attempt frequency for adsorption can be extracted from the kinetic theory of gases. The rate of surface collisions r collisions is given by r collisions pi = (12.1) 2π m k T i B with p i the partial pressure and m i the mass of the impinging atom or molecule. The effective rate of adsorption is then the rate of surface collisions multiplied by the sticking probability S (T) r adsorption pi = S ( ) T (12.2) 2π m k T i B with kbt S ( T ) = S e (12.3) act Eads S is a sticking coefficient prefactor and energy for adsorption. act E ads corresponds to the activation

202 182 Chapter 12 The activation energy and attempt frequency for the chemical reaction can be derived from experimental results, or alternatively obtained from theoretical DFT data. Diffusion of adsorbates is usually a very fast process, as a consequence of the low activation energy barriers. In order not to spend too much computer time on surface diffusion, the adsorbates are usually assumed to be in equilibrium. In practice, after each event in the KMC, a number of diffusion steps is allowed before the next MC step is performed. In our KMC program, the diffusion steps are performed according to the Metropolis scheme for equilibrium MC simulations. Diffusion into the substrate features a much higher activation energy barrier. A kinetic study of the deposition of one monolayer (ML) of Cu atoms on a Pt(111) substrate revealed an activation energy barrier of 1.11 ±.12 ev, assuming an attempt frequency of k B T/h [6]. The interdiffusion of Cu and Pt starts at approximately 525 K. The interdiffusion of Pt and Cu is therefore completely neglected in all kinetic simulations below 45 K. Regarding the desorption process, the kinetic parameters can be derived from Temperature Programmed Desorption (TPD) experiments. During a TPD experiment, one or more gases are adsorbed on a cold surface. The crystal is then heated with the crystal temperature increasing linearly with time. A mass spectrometer monitors the evolution of the desorbing products. In a TPD plot, the rate of desorption is plotted against the temperature. Redhead [7] derived an analytical equation for the TPD process, assuming an Arrhenius-type rate equation r dθ E N dt k T act d n des = = ν desθ exp s B (12.4) with r d the desorption rate, N s the number of surface atoms, θ the fractional coverage, the attempt frequency, n the order of the desorption process and ν des act E des the activation energy for desorption. The temperature T varies linearly with time t T = T + β t (12.5) H with β Η the heating rate and T the starting temperature. Fig shows the desorption rate as a function of time during a TPD experiment, according to

203 Simulation of CO adsorption and CO oxidation on Cu 3 Pt(111) 183 Eq The temperature T p at the maximum in the curve corresponds to the point where the following criterion is fulfilled act Edes 3 k T (12.6) B p The whole TPD experiment can be simulated by KMC simulations [8]. In Fig. 12.1, the simulated curve is compared to the analytical solution with the same kinetic parameters and neglecting lateral interactions and surface diffusion. KMC simulations have the advantage that lateral interactions and surface diffusion can straightforwardly be included, which is difficult in the analytical solution method Computational set-up At present, it is impossible to capture the different nature of all chemical interactions in one single model. Therefore, a combination of different models will be constructed in order to describe the metal-adsorbate system as accurately as possible Eq KMC Rate of desorption (s -1 ) Temperature (K) Figure Equivalence of the analytical solution to Eq and the KMC simulation result with the same input parameters.

204 184 Chapter 12 The MEAM has been proven to be very successful in describing the surface segregation to free surfaces in vacuum. The MEAM contains an explicit manybody character, which is essential for the metallic bond. However, this makes the model unsuitable for the description of metal-adsorbate or adsorbate-adsorbate interactions. Van Beurden [9] indeed mentions the inherent preference of the MEAM for highly coordinated adsorption sites in his MEAM study on the CO induced lifting of the quasi-hexagonal reconstruction on Pt(1) [9]. A second shortcoming is the short-range over which the MEAM potential acts and the difficulty to extend it to larger distances. Adsorbate-adsorbate interactions indeed do act over much larger distances than metal-metal interactions. The MEAM is therefore used only for the metal-metal interactions. In order to describe the metal-adsorbate interactions in an accurate way, we rely on a more empirical description, based on DFT data, fitted to an extension of the regular solution model. The metal (M) adsorbate(x) bond energy ε M-X is devised as ε m l l α ( x ) (12.7) = M X M X M l= l with α M X adjustabe coefficients and x M the local composition in the neigbourhood of metal M. Setting m equal to zero corresponds to an adsorption energy that is independent of the local chemical environment of the metal atom. A first order expansion corresponds to the sub-regular solution model. By taking the first and higher order terms into account, the ligand effect, i.e. the dependence of the adsorption energy on the atomic environment, can be modelled. The model parameters for the substrate-adsorbate interactions are given in Table 12.1 (see also Section 12.3). Finally, for the lateral interactions, a simple dipole-dipole repulsion energy relation is used E C = (12.8) R lat i, j 3 with C a constant and R the separation distance between the two adsorbates. This equation stems from the dipole-dipole interactions, which are the dominant lateral interactions at larger distances. At short distances, a Pauli repulsion between the occupied orbitals of the adsorbed molecules causes a much stronger repulsion. This interaction is approximated as a hard sphere interaction below a cut-off distance of 1.4 Å. This choice for the lateral adsorbate interaction function is discussed in Section 12.3.

205 Simulation of CO adsorption and CO oxidation on Cu 3 Pt(111) 185 CO O Pt Cu Pt Cu fcc -.85 ev -.8 ev.33 ev. ev hcp -.82 ev -.8 ev.78 ev.12 ev bridge -.9 ev -.32 ev.96 ev.4 ev top -.91 ev -.46 ev 2. ev 2. ev Table Implemented adsorption energies for CO and O on Cu 3 Pt(111) 12.3 Overview of experimental and theoretical results In this section, the available literature on CO and O 2 adsorption on Cu 3 Pt (111), as well as on the reaction mechanism of the CO oxidation reaction is systematically reviewed. As information on these points is not always available for the Cu 3 Pt(111) alloy, the most important results on pure Cu(111) and pure Pt(111) are also discussed. CO/Cu 3 Pt(111) Castro et al. [1] studied the interaction of CO with ordered and disordered Cu 3 Pt(111) using AES, LEED, TPD, UPS and work function measurements. The specific preparation method for the disordered surface resulted in a surface with a Pt enrichment (6% Pt). From the UPS measurements, the authors conclude that the CO preferentially adsorbs on Pt sites. The maximum peak temperatures in the TPD experiment occur at 33 K and 3 K for the ordered and disordered surface respectively (see Fig. 12.2). This suggests that CO binds more strongly to the ordered surface than to the disordered one. Assuming a pre-exponential factor of 1 13 s -1 act, activation energies for desorption ( E des ) of 83 kj/mol and of 77 kj/mol are estimated for the ordered and disordered surface. The same authors [11] report on one single adsorption state, corresponding to CO adsorbed on Pt sites up to a saturation coverage of.5 ML. At higher coverages, three new desorption states are recognised and assigned to CO adsorption on Cu ( E = 3 kj/mol) and to CO adsorption on mixed CuPt sites ( E = 42 kj/mol act des and E act des = 6 kj/mol). The CO peak on Cu 3 Pt is shifted by about 1 K downwards with respect to that on pure Pt(111). act des

206 186 Chapter 12 Shen et al. [12] investigated CO adsorption at 2 K on a Cu enriched and a Pt enriched Cu 3 Pt(111) surface. The Pt enriched surface is obtained by sputtering and annealing at 5 K. Preferential sputtering of the lighter element, i.e. Cu, causes a Cu depletion of the subsurface layers. Due to the limited diffusion at low temperatures, no atomic exchanges occur and the surface remains Pt enriched. The composition of the Pt enriched surface was determined as Cu 68 Pt 32. During the initial stages of the chemisorption process, CO resides preferentially on Pt sites. After adsorption of CO, the Pt intensity peak was no longer visible on the Cu enriched surface. On the other hand, the Pt peak on the Pt enriched surface falls to about 15% of its clean surface intensity. On this surface, the Cu intensity also decreases. By comparing the peak intensities before and after CO desorption, the authors exclude a Pt segregation. Lateral repulsions possibly drive the CO molecules to Cu-Pt mixed sites. Finally, Becker et al. [13] have performed a High Resolution Electron Energy Loss Spectroscopy (HREELS) study of CO adsorption on Cu 3 Pt(111) at 1 K. At low coverages, the authors observed the adsorption of CO molecules on Pt top sites, with already indications of a bridge site and a second top site occupation. The number of occupied bridge sites remained relatively small throughout the experiments. Figure Experimentally observed TPD spectrum for CO desorption from ordered Cu 3 Pt(111) and disordered, Pt enriched Cu 4 Pt 6 (111) surfaces [1].

207 Simulation of CO adsorption and CO oxidation on Cu 3 Pt(111) 187 From a theoretical viewpoint, one study concluded that the CO-Pt bond is stronger on Cu 3 Pt(111) than on Pt(111) [14]. This result is however in contradiction with the experimental results mentioned above and with yet another theoretical study, which found a weaker bond strength for CO/Cu 3 Pt(111) [15]. A review of all all theoretical and experimental work regarding the CO adsorption on Pt(111) is given in [16]. Numerous DFT studies experienced serious problems to predict the top site as most stable adsorption site on Pt(111). A recent study by Orita et al. [17] was able to predict the right site preference, although the adsorption energies (Table 12.2) are too large compared to experiments [18]. Therefore, we determined the CO adsorption energy on a Pt as follows: CO adsorption on a Pt top site in Cu 3 Pt(111) is set to.91 ev, which corresponds to a subtraction of.26 ev from the experimentally determined adsorption energy [18] of CO on pure Pt(111), in order to account for the 1 K peak shift between CO desorption from Pt(111) and from Cu 3 Pt(111). From [17], a difference between CO on a Pt top site in Cu 3 Pt(111) and Pt top site on Pt(111) atoms amounts to.8 ev. By adding this.8 ev to the.91 ev, a CO adsorption energy of.99 ev is obtained. From this reference, the calculated differences from [17] are implemented for the bridge, fcc and hcp sites. Experimental results for the CO adsorption energy on Cu(111) fall in the range of ev [19-21]. Successful DFT calculations, which predict the correct site preference for CO on Cu(111) are performed in [22-24]. The differences in adsorption energy of CO on top sites with that on bridge sites are reported to be of the order of ev/atom and with the hollow sites of the order of ev/atom. For the present calculations, the CO adsorption energy on a Cu top site is fixed to the experimental value of -.46 ev, and subsequently the site differences as found in these DFT studies are implemented. It can be deduced from all experimental and theoretical evidence that Cu(111) is less active for CO adsorption, as compared to Pt(111). Adsorption site Pt(111) fcc [17] hcp [17] bridge [17] top [17] Table Overview of CO binding energies (ev/atom) with the various adsorption sites on a Pt(111) surface.

208 188 Chapter 12.8 Lateral interaction energy (ev) CO-CO CO-O O-O Interatomic distance (A) Figure Lateral CO-CO, O-O and CO-O interaction energies, used in the MC simulations. Lateral repulsions between adsorbed CO molecules on Pt(111) are evidenced by several theoretical calculations [25-31] and by experiments, showing a decrease in heat of adsorption on increasing the coverage [32]. Petrova et al. [33] used oscillating functions for the CO-CO lateral interaction energy, which can originate from indirect interactions [34,35], mediated by the substrate electrons. The lateral interactions among CO molecules, used in this thesis, are represented in Fig O 2 /Cu 3 Pt(111) Concerning the oxygen adsorption on Cu 3 Pt(111), no experimental information is available. Also from a theoretical point of view, there is only one study [36] that reports on the oxygen binding energy to the Cu 3 Pt(111) system. Therefore, we will review the most important experimental and theoretical results of oxygen adsorption on Cu(111) and Pt(111) and derive parameters from a comparison between them. Molecularly adsorbed oxygen has been detected on the Cu(111) surface using different techniques [37-4]. These molecular precursors dissociate at around

209 Simulation of CO adsorption and CO oxidation on Cu 3 Pt(111) K and no evidence was found for their existence at room temperature. Also, it was found that, after dissociation, the oxygen atoms populate the threefold hollow sites [41,42] with an estimated saturation coverage ranging from.3 ML [43] to.5 ML [42,44,45]. The experimental results for O 2 adsorption on Pt(111) are summarised by Zhdanov et al. [46]. First, at low temperatures (below 1 K), oxygen adsorption is molecular [47]. With increasing temperature, adsorbed O 2 molecules dissociate or desorb at 16 K. After dissociation, adsorbed O atoms form p(2 2) islands [48] even at coverages as low as.4 ML. The p(2 2) islands are reported to be stable up to temperatures of 45-5 K. Secondly, STM studies [49,5] report that the distance between oxygen atoms at 16 K is significantly larger than twice the nearest neighbour distance, indicating that the O-O interaction on Pt(111) is strongly repulsive. Thirdly, at room temperature, oxygen forms a p(2 2) structure at.25 ML with adsorption on fcc hollow sites [51-53]. Finally, accurate measurements of the oxygen diffusion on Pt(111) are lacking. Field emission data indicate that the activation energy for this process is 113 kj/mol at relatively high coverages [54]. Analysis of STM results [55] in a small temperature range of K give 41 kj/mol and an attempt frequency of cm²/s. DFT calculations also predict a limited diffusion barrier of 54 kj/mol [56]. Some indirect data concerning oxygen diffusion can be obtained from a study of CO oxidation on Pt(111) [57], which indicate that two differently prepared overlayers with equal initial coverages display a different kinetic behaviour at 35 K. Taking into account that CO diffusion is fairly rapid, one can from this observation conclude that oxygen is not able to reach an equilibrium distribution at 35 K during the period of the experiment ( 1 s). This would point at a diffusion activation energy of at least 92 kj/mol [58]. Regarding the site preference, Xu et al. [59] performed a theoretical DFT analysis of the adsorption and dissociation of oxygen on Cu(111). The adsorption energy of atomic oxygen on a relaxed surface amounts to ev/atom on a fcc site, while the hcp site features a.12 ev/atom higher binding energy (see Table 12.3). Adsorption on the top site was unstable with respect to small perturbations and on the bridge site impossible (on the relaxed surface). The diffusion of an O atom on Cu(111) most likely starts from a threefold hollow site over a bridge site to another threefold hollow site, with a diffusion barrier estimated at.45 ev. DFT calculations for the binding energies of O on Pt(111) agree on the fcc hollow site as the lowest energy state, while the hcp hollow site is significantly less stable by ±.45 ev [6]. The O chemisorption energy on bridge and top sites are.63 ev and 2. ev higher than on the fcc site. The value of.63 ev also agrees

210 19 Chapter 12 nicely with the calculated barrier of O movement from the fcc site to the bridge site [36]. The bond energies of atomic O with the Cu and Pt sites on Cu 3 Pt (111) are derived from these data. The recombination of two oxygen atoms and subsequent desorption of O 2 is not taken into account. Consequently, the absolute value of the adsorption energies are of no importance, but only the differences in binding site energies matter. Therefore, the adsorption energy of O on a Cu fcc hollow site in Cu 3 Pt (111) is set equal to zero and all other energies are referred to this value. According to [59], the binding energy of O on a hcp site is.12 ev higher and on the bridge site it is.4 ev higher, which agrees with the activation energy barrier for the O movement from fcc to bridge (see next section). The difference between an O atom on a fcc hollow site on Pt(111) and Cu(111) amounts to.33 ev [36]. With this reference, the differences among the sites are implemented according to [6]. Zhdanov [46] perfomed KMC simulations with varying values for the third next nearest neighbour lateral interactions and only found reasonable agreement with experiment by setting this third nearest neighbour interaction energy to zero. Assuming strongly repulsive nearest neighbour and next nearest neighbour interactions, Yeo et al. [61] obtained a third next nearest neighbour interaction energy of 21 kj/mol, which is slightly larger than the values assumed by Zhdanov. Tang et al. [62] performed DFT calculations for various O structures on Pt(111) and determined Effective Cluster Interactions (ECI) based on them. The fitted ECI point at repulsive interactions between first and second nearest neighbours (.24 ev and.4 ev respectively), while the third nearest neighbour interaction is slightly negative (-.6 ev). As the latter value is rather small and in view of the accuracy of current DFT implementations, it is hard to decide on the precise nature of these interactions at this interatomic distance. On the other hand, Nagasaka et al. [63] found an attractive O-O interaction of.27 ev, in agreement with experimental values by the same authors [64]. The lateral interactions implemented in our simulations are shown in Fig Adsorption site Pt(111) Cu(111) fcc [6] (relaxed) [59] (unrelaxed) [59] hcp [6] (relaxed)[59] (unrelaxed) [59] bridge [6] (unrelaxed) [59] top [6] (unrelaxed) [59] Table Adsorption energies for atomic oxygen on Cu(111) and Pt(111), according to DFT calculations.

211 Simulation of CO adsorption and CO oxidation on Cu 3 Pt(111) 191 CO oxidation on Cu 3 Pt(111) The lateral interactions between chemisorbed O and CO on Pt(111) were calculated with DFT by Bleakly et al. [65]. They found a small difference of.4 ev between CO chemisorbed on a clean Pt surface and a O(2 2) precovered Pt(111) surface, pointing at only weak O-CO repulsions. Within the O island the Pt most far top site is preferred. Nagasaka et al. [63] also calculated lateral interaction energies on Pt(111) and confirmed the small repulsive interactions. The lateral interactions for CO-O are also plotted in Fig Based on these considerations, we have adopted weak CO-O repulsions, with a.4 ev energy difference for CO chemisorption on a clean and oxygen p(2 2) precovered surface. The reaction pathway for the CO oxidation on Cu 3 Pt (111) is described by Zhang et al. [36] and includes activation of the adsorbed O atoms from the fcc hollow site to a bridge site, and activation of the CO on the top site. The authors state that first, the CO molecule moves quite freely from its initial top site to a slightly offtop position, with little energy change, while the O atom vibrates around its threefold hollow position. Secondly the O atom becomes activated and moves to a bridge site to achieve the transition state with the CO on the off-top site. During the second period, the energy changes drastically. It was concluded that the O activation from the hollow site to a bridge site is the most important step in the reaction. The activation energy for the reaction is determined by the difference in binding energy of O to the fcc and to the bridge site. This difference amounts to.38 ev on Cu 3 Pt(111). The oxidation mechanism on Cu(111) and Pt(111) is quite similar to that on Cu 3 Pt [66]. Fig illustrates this reaction mechanism. Figure Reaction mechanism for CO oxidation on Cu 3 Pt(111), as found in [36], with a) the original state and b) the transition state. Large atoms represent metal atoms (dark: Cu and grey: Pt), while small atoms stand for adsorbates (dark: CO and grey: O).