3.320: Lecture 22 (4/28/05) Ab-Initio Thermodynamics and Structure Prediction: Time Coarse-graining, Effective Hamiltonians and Cluster Expansions

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1 3.320: Lecture 22 (4/28/05) Ab-Initio Thermodynamics and Structure Prediction: Time Coarse-graining, Effective Hamiltonians and Cluster Expansions

2 Methods with multiple time scales: Coarse-grain fast one away 4/27/ Atomistic Modeling of Materials G. Ceder and N Marzari

3 Model Hamiltonians: Example of Relevant Lattice Models Cu-Au: Cu and Au on fcc sublattice

4 Configurational Disorder in Fixed Host LiCoO 2 : Li and vacancies Triangular lattice of Li and vacancies

5 Surface Adsorption e.g. O on Pt(111) Possible Hollow sites form a triangular lattice How to parameterize and equilibrate these models?

6 A practical basis to expand H({σ}) in i α ϕ α = σ i Examples of cluster functions Basis is complete

7 Cluster function basis is orthogonal suitable scalar product Note that for any basis function f ({σ} )g({σ} ) = 1 ϕ 2 N α = 0 {σ } 1 f {σ} 2 N {σ } ( ) g ({σ} ) orthogonality proof ϕ α ϕ β = 1 ϕ 2 N α {σ } ϕ β i α β ϕ α ϕ β = σ i 2 σ i σ i = δ αβ i α i β i β i α

8 Expand any function of configuration in cluster function basis: e.g. Energy E ({σ} ) = V α ϕ α α Coefficients V -> Effective Cluster Interactions H ({} σ ) = V 0 + V 1 σ i i V i, j σ i σ j i, j V i, j,k σ i σ j σ k + i, j,k 1 24 V i, j,k,l σ i σ j σ k σ l... i, j, k,l expanding the cluster functions into their spin products makes the expansion look like a generalized Ising model Now we are in a position to see what the formal definition of the interactions in the Ising-like model is

9 Definition of the Interactions E ({σ} ) = V α ϕ α α take scalar product with ϕ β V β = 1 2 N {σ } ϕ β E {σ} ( ) = 1 2 n β 1 ϕ β i=±1 2 N n β i β Work out example for pair interaction E ({σ} ) {σ } β Many direct and indirect methods have been developed for calculating the V α

10 H Practical Approach is to Determine them by fitting to the calculated energy of a large number of configurations. ({} σ ) = V 0 + V 1 σ i i V i, j σ i σ j i, j V i, j,k σ i σ j σ k + i, j,k Truncate Hamiltonian Expansion 1 24 V i, j,k,l σ i σ j σ k σ l... i, j, k,l First Principles method Calculate H({σ}) for several configurations {σ} (= structures) Fit Hamiltonian Expansion to the direct First Principles calculations Effective Cluster Interactions Ising-like model Phase diagram and thermodynamic quantities Monte Carlo simulation

11 CaO-MgO Phase Diagram

12 Calculated Mixing Energies as Input Potentials SCPIB Composition

13 Effective Cluster Interactions for CaO-MgO SCPIB Potentials Pairs Triplets Quadruplets

14 Calculated Phase Diagram 5000 T(K) Potentials SCPIB (no vibrations) SCPIB (with vibrations) Mole fraction MgO Figure after P. D. Tepesch et al. J. Am. Ceram. Soc. 79 (1996):

15 Calculating Metastable Phase in Li-Al 4/27/ Atomistic Modeling of Materials G. Ceder and N Marzari

16 1200 Calculated LIQUID T (K) 600 fcc B L1 2 L1 0 DO 3 bcc AI C Li (at + %) Li Figure by MIT OCW. After M. Sluiter et. al. Phys. Rev. B 42 (1990):

17 More Complicated Things: YBa 2 Cu 3 O z z Chain Plane T(K) T OI Oxygen Vacancy 500 Cu Ba Y OII OI c Figure after G. Ceder et al. Phys. Rev. B 41, (1990)

18 O Co Li x CoO 2 Li After A. Van der Ven et al. Phys. Rev. B 58, (1998) /27/ Atomistic Modeling of Materials G. Ceder and N Marzari

19 Surface adsorption: On on Pt(111) (similar to your lab assignment) LDA/GGA calculations on slab geometry

20 Adsorption energies for various O arrangements formation energy [mev] O-Pt system LDA GGA oxygen composition

21 Cluster Expansion 50 0 formation enrgy [mev] O-Pt system vasp C.E Effective Cluster Interactions ECI Pt-O Ru-O oxygen composition ECI [mev] cluster

22 Monte Carlo Simulation

23 Phase Diagram 4/27/ Atomistic Modeling of Materials G. Ceder and N Marzari

24 Pt,Ru Oxygen More complicated: Combined segregation and Adsorption Ru segregates to surface Layer of surface adsorption sites Top layer: Pt/Ru Environment δ i is +1 when oxygen is adsorbed at site i Parameterize the system with respect to occupation of these sites with Pt/Ru (σ i )or O/vacuum (δ i ) Pt bulk atoms Top layer: σ i = +1 or -1 if site is occupied by Pt(Ru) Technique used: Coupled Cluster Expansion (P.D. Tepesch, G.D. Garbulsky and G. Ceder, A Model for the Configurational Thermodynamics in Ionic Systems, Phys. Rev. Lett, 74: (1995) E(σ 1,σ 2,...,σ N,δ 1,δ 2,...,δ N ) = V 0 + V i σ i + V i δ i + V i, j σ i σ j + V i, j σ i δ j + V i, j δ i δ j /27/05 i i Atomistic i, Modeling j of Materials i, G. j Ceder and N Marzari i, j

25 Oxidation drags Ru to the surface C o C Ru C o, C Ru Oxygen Adsorbed 0.2 Ru in the Surface µ o Figure by MIT OCW. More oxidation strength

26 Low oxidation Oxidation drags Ru to the surface Figure by MIT OCW. More oxidation strength

27 Oxidation drags Ru to the surface Medium oxidation Figure by MIT OCW. More oxidation strength

28 Oxidation drags Ru to the surface High oxidation Figure by MIT OCW. More oxidation strength

29 Equilibration of Structure and Chemistry also key in other problems: Hydrogen Modified Al Fracture

30 For slow separation impurities can flow in Free energy of separation (and force displacement relation) depend on amount and arrangement of impurity Need to equilibrate both amount and arrangement of H on Al(111) for each separation

31 Lattice model for H on separating Al(111) surfaces H in tetrahedral sites Lattice model of tetrahedral sites on (111) V β V γ Interaction in-plane and between surfaces

32 Procedure Calculate energy of different H configurations on surface at different plane separations. Cluster expand H configuration energy at each plane separation Monte Carlo simulation at each plane separation Take derivatives of free energy (to get force) Construct grand potentials and construct equilibrium trajectory A. Van der Ven, G. Ceder, Acta Materialia 52, (2004)

33 Calculate Energy versus separation for various H concentrations and configurations Cluster expand Perform Monte Carlo at each separation

34 Interactions Inter-plane interaction in-plane interaction

35 First order transition: Separation at constant force due to impurity inflow Apply force with constant H chemical potential constant H concentration Ω(F,µ H ) = G µ H x Fh

36 Why is Ising like model such a good approximation for the real system. Look back at coarse-graining ideas Electronic Occupation Ψ Ψ Magnetic (electron spin) Vibrational Molecular Dynamics: can not reach configurational excitations Monte Carlo: too many energy evaluations required Configurational We can use lattice models for studying mixing and ordering or atoms in crystalline materials. But why is this a good approximation?

37 Coarse-graining: The concept Can we integrate partition function over fast degrees of freedom to obtain an effective Hamiltonian for the slower degrees of freedom? e.g. for an alloy: Can we find an effective free energy function for the substitutional arrangement of an alloy that includes the entropic effect of vibrations and electronic excitations? YES Use Monte Carlo, Molecular Dynamics, or analytical methods to integrate effect of temperature on fast degrees of freedom

38 Change coordinates r i i, r i {σ} = {σ 1, σ 2,σ 3,...σ Ν } Configurational arrangement {ν} = { r} i Vibrational state

39 Coarse-graining by reduction of degrees of freedom {σ} = {σ 1, σ 2,σ 3,...σ Ν } Configurational arrangement {ν} = Vibrational state Q = Q = {σ} {σ} υ exp( βe({σ},υ({σ}) ) ( ) exp βf({σ} Two approximations for F Partition Function of an Ising-like Model F({σ} = kt ln {υ} ( ) exp βe(υ({σ}) F is Effective Hamiltonian for {σ} degree of freedom G. Ceder, Computational Materials Science 1, (1993)

40 Approximations to F({σ}) determine which excitations (entropies) are included in the total free energy 1. Approximate F({σ}) by E({σ}) F({σ} = kt ln when doing Monte Carlo and free energy integration, only get configurational entropy {υ} ( ) exp βe(υ({σ}) 2. Approximate F({σ}) by E({σ}) -TS electronic ({σ}) when doing Monte Carlo and free energy integration, get configurational entropy and electronic 3. F({σ}) = E({σ}) TS electronic ({σ}) TS vib ({σ}) when doing Monte Carlo and free energy integration, get configurational entropy + electronic + vibrational

41 Summary The model on the time scale of the substitutional excitations is an Ising-like model (i.e. excitations are changes of occupation variables) The Hamiltonian of the Ising-like model is the free energy of the faster excitations (e.g. vibrations, electronic excitations). Only approximation is separation of time scales Cluster Expansion is a practical form for the Ising-like Hamiltonian

42 Can investigate effect of various approximations: Cd-Mg system After from M. Asta et al. Physical Review B 48, (1993) /27/ Atomistic Modeling of Materials G. Ceder and N Marzari

43 Calculated No vibrational entropy With vibrational entropy After M. Asta et al. Physical Review B 48, (1993)

44 Systems as 1994 Table removed for copyright reasons.

45 Simple Ternaries 4/27/ Atomistic Modeling of Materials G. Ceder and N Marzari After R. McCormack et al. Phys. Rev. B 51, (1995)

46 References 1. D. de Fontaine, in Solid State Physics H. Ehrenreich, D. Turnbull, Eds. (Academic Press, 1994), vol. 47, pp G. Ceder, A. Van der Ven, C. Marianetti, D. Morgan, Modeling and Simulation in Materials Science and Engineering 8, (2000) A. Zunger, in Statics and Dynamics of Alloy Phase Transformations P. E. A. Turchi, A. Goniss, Eds., pp (1994). 4. A. Van de Walle, G. Ceder, J. of Phase Equilibria 23, (2002) J. M. Sanchez, D. de Fontaine, Phys. Rev. B 25, (1982) J. M. Sanchez, F. Ducastelle, D. Gratias, Physica 128A, (1984)