R. Pasianot CAC-CNEA, Argentina and L. Malerba SCK•CEN, Belgium

1. IV Workshop on Multicale Modelling in Fe-Cr Alloys Stockholm – May 6-7, 2003 Design of crossed many-body empirical interatomic potentials for ferritic binary alloys based on phase diagram information SCK• CEN contribution to PERFECT / Physics Modelling Subproject R. Pasianot CAC-CNEA, Argentina and L. Malerba SCK•CEN, Belgium With the collaboration of A. Caro (LLNL) and E. Lopasso (CAB-CNEA)

2. No standard procedure currently exists to fit potentials for alloys • Potentials for binary alloys are often just superpositions of pure element potentials • They are fitted to only few alloy parameters, typically substitutional energy of one alloying atom, mixing enthalpy for one concentration, ordered compound energies … • Only by chance can this procedure guarantee the validity of the potential for a large range of concentrations and an acceptable prediction of the alloy thermodynamic functions • No defined procedure exists to include in the fitting the most important information about alloys: its phase diagram

3. Two parallel objectives for a task devoted to improve potentials (PERFECT) • Find alternative formalisms to EAM, more apt to treat, particularly, Fe and ferritic alloys • Elaborate optimised and possibly standard potential fitting procedures for binary alloys, including thermodynamic properties improved empirical potentials

4. SCK●CEN’s tasks in the PERFECT “ab initio and potentials” work-package 1. Apply and refine procedure by A. Caro to validate existing potentials on binary alloy phase diagrams Diagram Potential Phase 2. Elaborate procedure to fit potentials for binary alloys on the corresponding real phase diagram (test case: Fe-Cu) In collaboration with CNEA via Belgo-Argentine bilateral co-operation agreement

5. Summary of method to build phase diagrams from EAM potentials • Free energy per particle calculated using Gibbs-Duhem equation where h(T) is obtained from MD run and fitted to 2nd order polynomial in T • f(T0) calculated as difference between the actual system (described by EAM potential) and a reference system whose free energy is known (using hamiltonian switching method, via MD simulation), namely: • For solid: Einstein solid • For liquid: ideal gas • Same method extendable to alloys for different concentrations • Once f(c,T) is known for all involved phases, the phase diagram is constructed using the familiar common tangent method

6. LIQ Temperature, K LIQ Temperature, K at% Cu Phase diagrams from existing potentials are far from reality! Ackland97 Real (Fe-Cu) Ludwig98 Preliminary LIQ BCC FCC

7. Status of method to build phase diagrams from EAM potentials • Applied so far to Au-Ni1 and Fe-Cu2,3 systems • May encounter problems when some phases are not accessible to the potential, e.g. if they are not predicted to be stable (phases that do not exist in reality either) / case of LF potential • Solutions to this problem are being studied / the objective is to produce a robust tool for empirical interatomic potential validation (a posteriori) which should be extendable to any potential formalism 1E. Ogando Arregui, M. Caro and A. Caro, Phys. Rev. B66 (2002) 054201 2E. M. Lopasso, M. Caro, A. Caro and P. Turchi, Phys. Rev. B68 (2003) 214205 3A. Caro, P. Turchi, M. Caro and E. M. Lopasso, Phys. Rev. B, in press

8. Methodology to fit potential to PD:CVM applied to EAM fitting • The CVM formalism provides a solid theory for the construction of analytical expressions for the thermodynamic functions of an alloy, based on configurational considerations, as a function of a vector of variational variables called cluster correlation functions: ={1 2 … n}* • By minimising the relevant thermodynamic potential on the cluster correlation functions it was possible, applying the CVM, to build e.g. the phase diagram of even relatively complex systems (eg Al-Co-Fe)* • The CVM is otherwise typically used to treat ordered alloys • The theory is rather involved and for complex alloys becomes highly involved, but binary random alloys represent the simplest application case * See eg G. Inden and W. Pitsch “Atomic ordering”, in “Phase transformations in materials”, P. Haasen Ed, Weinheim: VCH (1991), chapter 9, p. 499

9. Methodology to fit potential to PD:CVM applied to EAM fitting • The CVM formalism can be used to express not only the energy of a system, but also the entropy, thereby giving access to the alloy free energy • By coupling the CVM formalism to the EAM scheme the energy of a system can be expressed as a function of: • The variational variables,  • The parameters of the potential, a • Species concentrations and alloy volume • Via a double, iterative minimisation, it is hence expected that the potential can be fitted to reproduce experimentally available thermodynamic functions for the relevant alloy

10. Basic concepts of the CVM • Cluster = set of lattice points 1, …, r, generally considered to be representative of a crystalline system with N>>r sites • Cluster configuration = vector of integral numbers expressing the configuration of a cluster (each possible value of the site operatork corresponds to a possible chemical species): r={1 2 … r} • Cluster probability, p(r) = probability that a cluster of size r is in a certain configuration r (simplest example: point probability in random alloy = xi) • Cluster correlation functions = average values of the product of the site occupation operators for the corresponding cluster – used as variational variables on which the termodynamic potentials can be minimised

11. 2 7 1 5 4 3 6 4 3 2 1 Basic clusters for bcc and fcc • bcc • irregular tetrahedron • includes up to 2nn distance • can describe large number of ordered alloys • fcc • regular tetrahedron and octahedron • tetrahedron includes up to 1nn distance only • octahedron extends to 2nn

12. Cluster probabilities Cluster of size r, with K chemical species: Total number of configurational probas Number of independent probas Binary alloys: probas are linear functions of : (for rules to build M see Inden & Pitsch) • Clusters of relevance: • Pairs (3p) ● Tetrahedron (bcc – 6p, fcc – 8p) • Triangles (8p) ● Octahedron (fcc - 10p)

13. Example from CVM: expression for configurational entropy The regular solution theory corresponds to a zero approximation of the CVM

14. Energy in the EAM/CVM formalism • The EAM energy can be expressed in the CVM formalism as a function of the probability of: • pairs (pair component) • pairs and triangles (many-body component) 2nd order Taylor development of embedding function around equilibrium

15. Electronic density and embedding function in the alloy embedding function depends only on element electronic density on i is superposition of effects of surrounding atoms effect of tj on ti decided by coefficient which can be fitting parameter or estimated to remain fixed one coefficient is enough, which can be a fitting parameter or estimated to remain fixed gauge choice

16. Thermodynamic functions • For p=T=0, E=F=G=H an expression for the mixing enthalpy can be obtained: h= h(v,x,,a) • For p=0, T=T0, G=F=H-TS=E-TS  expression for the free energy and, by derivation, for the chemical potential, can be obtained j=e+xi de/dxj

17. Fitting procedure • A self-consistent minimisation procedure of thermodynamic functions on variational variables (linear) and object function on the potential parameters (quadratic form or more complex) must be set up • Possible thermodynamic fitting “parameters”: • Mixing enthalpy curve nodes • Common tangent of the free energy for a certain T, imposed in correspondence with a certain phase concentration, taken from the phase diagram • In addition: • Classical parameters, such as heat of solution, defect binding of formation energies, bulk modulus, etc

18. Application to Fe-Cu • No development on pure metal potential components is foreseen for the moment • For pure Fe, Mendeleev potential1 will be used • For pure Cu, Mishin potential2 will be used • For crossed potential, fitting on e.g. • Mixing enthalpy (from ab initio calculations) • Solubility limit (from phase diagram) • Cu-V binding energy (from ab initio calculations) • Validation by building corresponding phase diagram 1M.I. Mendeleev et al., Phil. Mag. 83(35) (2003) 3977-3994 2Y. Mishin et al., Phys. Rev. B 63 (2001) 224106

19. Summary and Discussion • While new formalisms for pure bcc metal potentials are being studied, so far little effort has been devoted to rationalise the fitting procedure of alloy potentials • The most important information concerning an alloy is its phase diagram • A procedure for the validation of an interatomic potential by building the corresponding phase diagram has been developed and is being refined • The CVM theory provides a formalism to express analytically the free energy of an alloy • By introducing the CVM formalism in the EAM scheme, analytical expressions for the configurational free energy of the alloy can be built and used to fit the potential to points of the phase diagram, eg imposing common tangent to the free energy of the two phases • The case study is currently Fe-Cu, using existing EAM potentials for pure Fe and pure Cu • If the procedure is successful, it can be in principle extended to both other systems within the EAM approximation and, with more or less theoretical difficulties, to other interatomic potential formalisms