3.320: Lecture 21 (4/28/05)Methods with multiple time scales:Coarse-grain fast one awaySurface AdsorptionA practical basis to expand H({s}) inCluster function basis is orthogonalExpand any function of configuration in cluster function basis: e.g. EnergyDefinition of the InteractionsPractical Approach is to Determine them by fitting to the calculated energy of a large number of configurations.CaO-MgO Phase DiagramCalculated Mixing Energies as InputEffective Cluster Interactions for CaO-MgOCalculated Phase DiagramCalculating Metastable Phase in Li-AlCalculatedMore Complicated Things: YBa2Cu3OzLixCoO2Surface adsorption: On on Pt(111) (similar to your lab assignment)Adsorption energies for various O arrangementsCluster ExpansionMonte Carlo SimulationPhase DiagramMore complicated: Combined segregation and AdsorptionOxidation drags Ru to the surfaceOxidation drags Ru to the surfaceOxidation drags Ru to the surfaceOxidation drags Ru to the surfaceEquilibration of Structure and Chemistry also key in other problems: Hydrogen Modified Al FractureFor slow separation impurities can flow inLattice model for H on separating Al(111) surfacesProcedureCalculate Energy versus separation for various H concentrations and configurationsInteractionsApply force with constant H chemical potentialWhy is Ising – like model such a good approximation for the real system. Look back at coarse-graining ideasCoarse-graining: The conceptChange coordinatesApproximations to F({s}) determine which excitations (entropies) are included in the total free energySummaryCan investigate effect of various approximations:Cd-Mg systemCalculatedSystems as 1994Simple TernariesReferences4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N Marzari3.320: Lecture 22 (4/28/05)Ab-Initio Thermodynamics and Structure Prediction: Time Coarse-graining, Effective Hamiltonians and Cluster Expansions4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariMethods with multiple time scales:Coarse-grain fast one awayModel Hamiltonians: Example of Relevant Lattice ModelsCu-Au: Cu and Au on fcc sublatticeConfigurational Disorder in Fixed HostLiCoO2: Li and vacanciesTriangular lattice of Li and vacanciesSurface Adsorptione.g. O on Pt(111)Possible “Hollow” sites form a triangular latticeHow to parameterize and equilibrate these models ?A practical basis to expand H({σ}) inExamples of cluster functionsϕα=σii∈α∏Basis is completeCluster function basis is orthogonalf {σ}()g {σ}()=12Nf {σ}(){σ}∑g {σ}()suitable scalar product12Nϕα= 0{σ}∑Note that for any basis functionorthogonality proofϕαϕβ=12Nϕα{σ}∑ϕβϕαϕβ=σi2i∈α∩β∏σii∈αi∉β∏σii∈βi∉α∏=δαβExpand any function of configuration in cluster function basis: e.g. EnergyE {σ}()= Vαϕαα∑Coefficients V -> Effective Cluster InteractionsHσ{}() = V0+ V1 σi + 12 i∑Vi , jσii, j∑σj + 16Vi, j ,kσii, j ,k∑σjσk + 124Vi, j ,k ,lσii , j, k,l∑σjσkσl ...expanding the cluster functions into their spin products makes the expansion look like a generalized Ising modelNow we are in a position to see what the formal definition of the interactions in the Ising-like model isDefinition of the InteractionsE {σ}()= Vαϕαα∑take scalar product with ϕβVβ=12N{σ}∑ϕβE {σ}()=12nβi=±1i∈β∑ϕβ12N −nβ{σ}−β∑E {σ}()Work out example for pair interactionMany direct and indirect methods have been developed for calculating the Vα4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariPractical Approach is to Determine them by fitting to the calculated energy of a large number of configurations.Hσ{}() = V0+ V1 σi + 12 i∑Vi , jσii, j∑σj + 16Vi, j ,kσii, j ,k∑σjσk + 124Vi, j ,k ,lσii , j, k,l∑σjσkσl ...Truncate Hamiltonian ExpansionCalculate H({σ}) for several configurations {σ} (= structures) Fit Hamiltonian Expansion to the direct First Principles calculationsIsing-like modelPhase diagram and thermodynamic quantitiesFirst PrinciplesmethodMonte CarlosimulationEffective Cluster InteractionsCaO-MgOPhase Diagram4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N Marzari4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCalculated Mixing Energies as Inputzzzzzzzzzzzzzzzzzzzzz00.10.20.30.40.50.60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1CompositionzPotentialsSCPIB4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariEffective Cluster Interactions for CaO-MgOzzzzzzzzzzz123456-0.05-0.04-0.03-0.02-0.0100.010.02zSCPIBPotentials13212Pairs Triplets Quadruplets4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCalculated Phase Diagram0 0.25 0.5 0.75 1500040003000200010000Mole fraction MgOT(K)PotentialsSCPIB (no vibrations)SCPIB (with vibrations)Figure after P. D. Tepesch et al. J. Am. Ceram. Soc. 79 (1996): 2033-2040.4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCalculating Metastable Phase in Li-Al4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCalculatedAfter M. Sluiter et. al. Phys. Rev. B 42 (1990): 10460.100080060040020001200T (K)0 0.2 0.4 0.6 0.8 1CLiLIQUIDB32L12L10DO3bcc(at+ %)AIfccLiFigure by MIT OCW.More Complicated Things: YBa2Cu3OzOxygenVacancyCuBaYChain Plane0.2 0.30.40.50 0.16.06.26.4 6.6 6.8 7.0z10005000T(K)OIITOIOIcFigure after G. Ceder et al. Phys. Rev. B 41, (1990) 8698-8701.4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N Marzari4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariLixCoO2OCoLiAfter A. Van der Ven et al. Phys. Rev. B 58, (1998) 2975-2987.4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariSurface adsorption: On on Pt(111) (similar to your lab assignment)LDA/GGA calculations on slab geometry4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariAdsorption energies for various O arrangements0.0 0.2 0.4 0.6 0.8 1.0-400-300-200-1000formation energy [meV]oxygen compositionO-Pt system LDA GGA4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N MarzariCluster Expansion0.00.20.40.60.81.0-400-350-300-250-200-150-100-50050formation enrgy [meV]oxygen compositionO-Pt system vasp C.E234567-1001020304050607080ECI [meV]clusterECI Pt-O Ru-OEffective Cluster Interactions4/27/05 3.320 Atomistic Modeling of Materials G. Ceder and N