Atomistic potentials for metallic systems Recent advances and existing challenges

Atomistic potentials for metallic systemsRecent advances and existing challenges Y. Mishin George Mason University, Virginia, USA CECAM Workshop Ab initio meets classical simulations…(Lyon, France) 10/18/2005

Outline • What is EAM? • Functional form, properties and areas of application • Potential generation procedures: science or art? • EAM success stories • FCC metals • Intermetallic compounds • EAM challenges • “Difficult” properties • BCC transition metals • Thermodynamic properties and phase diagrams • Angular-dependent potentials • MEAM • Angular-dependent potentials (ADP) • Future work

Many-body interactions Embedded-atom potential (EAM) Daw and Baskes (1984) and Finnis and Sinclair (1984) • Potential functions contain adjustable parameters • Classical (Newtonian) forces  Energy minimization and MD • Expression for unrelaxed Evf • Expressions for elastic constants cij • Expressions for the dynamical matrix  Phonon dispersion and DOS • Expressions for the stress tensor • Fast calculation of E in Monte Carlo simulations

The workhorse of atomistic simulations Extremely fast calculation of energies and forces! Fast MD and Monte Carlo simulations (~106 atoms, ~10 ns) • Mechanical properties • Dislocations (core structure, Peierls stresses, dynamics) • Grain boundaries (structure, energy, motion, segregation) • Fracture cracks • Diffusion • Direct MD simulations (especially grain boundaries and surfaces) • Combine harmonic TST with KMC (barriers by NEB) • Thermodynamics • Quasi-harmonic thermodynamics • Grain-canonical Monte Carlo • Calculation of phase diagrams etc, etc, etc…

Potential generation procedures Parameterization of functions • Elemental metal: 3 functions, binary system: 7 functions, etc • No physical meaning (“Any function is good as long as it works”) • Forms of functions Vij(r) and i(r): • Analytical: smooth and reliable but lack flexibilitity • Cubic splines: very flexible but can give surprises • Smooth cutoff with Rc covering 3-5 coordination shells • Direct fit of F() or inversion of (modified) Rose’s UEOS E(V) • Fitting: minimization of the weighted mean squared deviation from target properties • Weights give a powerful tool for controlling the quality of potential • Multi-dimensional minimization (simplex method) • Simulated annealing

Potential generation procedures Fitting and testing for elemental metals • Experimental data: • E0, a0, cij, Evf, SF energy • Sometimes Evm, Eif, surface energies, phonons, thermal expansion • Ab initio data: • Structural energies: E(V) for alternative structures • Homogeneous deformation paths; e.g. Bain path, trigonal path (FCC-SC-BCC), twinning deformation path (FCC-FCC or BCC-BCC) • Forces in snapshots drawn from MD (solid and liquid). Force matching method (Ercolessi and Adams, 1994) In modern potentials the ab initio part of the database strongly dominates over experiment. Some potentials use only 2-3 experimental numbers and the rest ab initio

Transferability of potentials Ability to give reasonable results between and beyond fitted points in configuration space. This is the most meaningful measure of quality of potentials Equilibrium ??? ???? ?? Configuration space Fitted points • Ab initio data sample a larger area of configuration space than experiments can do • Split the database into the fitting and testing sets! • The more properties you test the more flaws you find Potential generation is largely based on experience, intuition, tricks of the trade, and luck. It is currently art rather then science

Potential generation procedures Fitting and testing for binary systems • Typical scheme: • Take/generate accurate potentials for metals A and B • Fit the cross-interaction function VAB(r) • Use transformation coefficients as additional parameters • For systems with compounds fit to (test against): • Experimental properties of a chosen compound (E0, a0, cij, planar faults; sometime surface energies, phonons, etc) • Ab initio E(V) functions for a set of compounds with different structures and compositions across the diagram. Most of them do not exist on the diagram. One compound is not enough! • Other ab initio data (e.g. point defect formation energies) • For systems without intermediate phases (e.g. Cu-Ni, eutectic systems) • Fit to ab initio E(V) functions for a set of non-existent compounds with different structures and stoichiometries across the diagram

EAM success stories Excellent potentials exist for several FCC metals (e.g. Cu, Ag, Ni and Al) Accurately reproduce E0, a0, cij, phonon dispersion curves, thermal expansion, Evf, Evm, interstitials, SF energy, gamma surfaces, structural energies, high-pressure p(V), etc. Melting properties (not used in the fit!): Cu Tm = 1327 K (exper. 1357 K); Hm = 12.1 kJ/mol (exper. 11.5 kJ/mol) Ag Tm = 1265 K (exper. 1235 K); Hm = 12.2 kJ/mol (exper. 11.7 kJ/mol) Thermal expansion: Cu Ag

Phonon dispersion curves for FCC metals Cu Ag Similarly good agreement for Ni and Al

EAM potentials for intermetallic compounds Ti-Al L10-TiAl (Phys.Rev. B 68, 024102 (2003)) Ni-Al B2-NiAl (Phys.Rev. B 65, 224114 (2002)) Ni-Al Ni3Al (Acta Mater. 52, 1451 (2004))

EAM potential for Ni3Al* Ni-Al phase diagram* • Accurate lattice properties of Ni3Al, B2-NiAl • Thermodynamics and phase diagram • Point defects and diffusion in Ni3Al • Generalized stacking faults in Ni3Al • /’ interphase boundaries • ’ particles in  matrix • Dislocations in Ni3Al [110] screw dislocation in Ni3Al /’ alloy at T = 700 K (100) /’ interface at T = 700 K* Nye tensor distribution * Y. Mishin, Acta Mater. 52, 1451 (2004)

EAM challenges There are “difficult” properties which consistently defy EAM: • Surface energies are always too low (10-20%) • Vacancy migration energy is too low (unless included in the fit) • Melting points are often way off experiment (unless liquid included in the fit) Binary phase diagrams are often incorrect even qualitatively. There have been only a few attempts to compute them Cu-Ag interactions were fit to ab initio E(V) functions of several compounds with different structures and stoichiometries. No experimental data! The diagram has been calculated by grand-canonical Monte Carlo simulations

EAM challenges (continued) EAM does not work well for BCC transition metals. This central-force model cannot capture the covalent component of bonding • Modified EAM (MEAM) (Baskes, 1987) • Tensor electron density • Short-range interactions (1-2 coordination shells) • Screening procedure • In principle can work for transition metals and even covalent solids (Si, Ge) • Much slower than EAM • No extensive use of ab initio data • No extensive testing like for EAM-FCC

Angular-dependent potentials (ADP) • An extension of EAM to include non-central interactions • Can apply to BCC transition metals • Energy is penalized for dipole and quadrupole distortions: Angular-dependent terms Regular EAM

ADP method detail Potential functions • EAM: Vij, i, Fi metal: 3 functions, binary: 7 functions • ADP: Vij, i, Fi, Uij, Wij metal: 5 functions, binary: 13 functions Must be fit to a large ab initio database History • First proposed for the Fe-Ni system (Acta Mater. 53, 4029 (2005) ) • More general than the embedded-defect method (Pasianot et al, 1991) • Similar to but much faster than the MEAM (Baskes, 1987) Computational overhead: about a factor of ~2 slower than EAM

ADP fitting database: Fe-Ni system • Pure Ni and Fe: • Experimental a0, E0,cij, Ev, SF(Ni), s, for FCC-Ni and BCC-Fe • Ab initio energies E(V) of FCC, BCC, SC and Diamond • Fe-Ni alloys: • Ab initio energies E(V) of L12-Fe3Ni, D03-Fe3Ni, L10-FeNi, B2-FeNi, B1-FeNi, L12-Ni3Fe. No experimental data!

Ab initio ADP Structural energies of Fe-Ni alloys

ADP potential for BCC Ta* • Accurate lattice properties, including elastic constants, thermal expansion, high-pressure behavior • Defects: vacancy formation and migration, surface energies (!) • Structural energies, including A15, omega, -U • Reasonable agreement with ab initio for • homogeneous deformation paths • Gamma surfaces High-pressure EOS Homogeneous twinning path *With A.Y. Lozovoi (QUB)

Gamma surfaces for BCC Ta <100>{110} <110>{110} <111>{211} <111>{110}

½<111> screw dislocation core in Ta Contour plot of the screw component of the Nye tensor Computed with the new ADP potential The compact core with mirror symmetry across (211) planes is in agreement with ab initio calculations (Rao and Woodward, 2002) Working on the Cu-Ta system…

Future work • Application of EAM potentials for binary phase diagram construction. This might be more accurate than CE calculations • Development of ADP potentials for binary systems involving BCC transition metals • Potentials for ternary systems – almost unexplored area • Global potential database (at least for metallic systems). Possible problems: • Uniform format of potential files • Uniform testing procedures (which properties to test, which agreement is considered “good”?) • Quality control: only “high-quality” potentials can be included

Atomistic potentials for metallic systems Recent advances and existing challenges