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Operated by Los Alamos National Security, LLC for NNSA

HOW TO DEVELOP MODIFIED EMBEDDED ATOM METHOD POTENTIALS

M. I. BaskesLos Alamos National Laboratory

andUniversity of California, San Diego



OUTLINE

Modified EAM (MEAM) Single Element Alloys



THE EMBEDDED ATOM METHOD IS SEMI-EMPIRICAL

distance

is obtained from a linear superposition of atomic densitiesF and ϕ are obtained by fitting to the following properties: Universal Binding Energy Relationship (UBER) (lattice constant, bulk modulus, cohesive energy) Shear moduli Vacancy formation energy Structural energy differences (hcp/fcc, bcc/fcc)

_

UBER

E = ∑i

⎝⎜⎜⎜⎛

⎠⎟⎟⎟⎞

Fi(_ρi) +

12∑j≠i

jij(R ij)

pair interactionhost electrondensity

embeddingenergy

ρρ



COMPLEX MATERIALS REQUIRETHE ADDITION OF ANGULAR FORCES

EAM uses a linear superposition of spherically averaged electron densities

MEAM allows the background electron density to depend on the local symmetry

ρkl( )2 = ρl Rik( ) ρl Rkj( )

j∑ Pl

0 cosθikj( )( )i

∑

θ



MODIFIED EMBEDDED ATOM METHOD (MEAM)

12 parameters + angular screening for the pair potential and electron densities

b*=β l( ) Rre

−1⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

€

j(R) = 2Z

uE (R) − F(ρ 0(R)){ }S

Pair Potential

a* =α Rre

−1⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟ 2α =9ΩB

cE

Universal Binding Energy RelationshipUBER

uE (R) =−Ec 1+a* +δa*3 reR

⎛ ⎝ ⎜ ⎞

⎠ ⎟ e−a*

Embedding Function

F(ρ) =AEcρ lnρ €

ρl = e−b*S

Background Electron Density

ρkl( )2 = ρl Rik( ) ρl Rkj( )

j∑ Pl

0 cosθikj( )( )i

∑Γ = t(l)

l=1

3∑ ρ(l ) ρ(0)( )2ρ=ρ(0) 1+Γ



CONCEPT OF THE SCREENING ELLIPSE LEADS TO A SIMPLE SCREENING MODEL

€

Sik = Sijkj≠ i,k∏

x2 + 1C y

2 = 12 ρik( )

2

C =2 Xij+ Xjk( )− Xij−X jk( )

2−1

1− Xij_ Xjk( )2Xij =

ρijρik

⎛ ⎝ ⎜ ⎞

⎠ ⎟2

goes from 0 to 1 smoothly

Sijk = ⎫ ⎬ ⎪

⎭ ⎪

0 C≤Cmin

ϕ( C−CminCmax −Cmin

) Cmin <C < Cmax

1 C≥Cmax

f x( )

screening ellipse defined by C

Cmin and Cmax set limits of screening

2y/r ik

2x/rik



RECIPE FOR SINGLE ELEMENT

Choose Reference Structure Assemble Data Base Associate/Fit Parameters with Data Base Validate Iterate



CHOOSE REFERENCE STRUCTURE

• Simple Crystal Structure– fcc– bcc– hcp– diamond cubic

• Usually Equilibrium Ground State• Must Have Data• Relevant to Simulations



ASSEMBLE DATA BASE• Experiment (reference structure)

– Cohesive energy– Lattice constant– Elastic constants– Vacancy formation energy– Stacking fault energy– Thermal expansion– Other phases

• First Principles (reference structure)– Energy vs. volume– Other phase relative energies– Transformation path



ASSOCIATE/FIT PARAMETERS WITH DATA BASE (I)• UEOS

– Cohesive energy (Ec)– Lattice constant (re)– Bulk modulus (α)– Thermal expansion (δ)

• Partial Electron Density Weights– Vacancy formation energy (t1)– Shear elastic constants (t2)– Stacking fault energy (t3)



ASSOCIATE/FIT PARAMETERS WITH DATA BASE (II)

• Embedding Energy Strength (A)– Energy of other phases

• Atomic Electron Density Decay– Energy of other phases (β0)– Surface relaxation (β1)– Shear elastic constants (β2)– c/a (β3)

• Angular Screening– Energy of other phases – Shear elastic constants



VALIDATE (I –Reference Phase)

• Thermal properties– Expansion– Specific heat

• Surfaces– Energy– Relaxation– Reconstruction

• Point defects– Vacancy mobility– Interstitial

• Formation energy• Geometry• Migration energy



VALIDATE (II – Other Phases)

• Liquid– Heat of fusion– Density– Melting point

• Solid– Relative stability– Lattice constants– Internal relaxation– Elastic constants– Transformation paths



EXPERIMENTAL DATA BASE FOR FE

Cohesive Energy (eV) 4.29a Lattice Constant (300K) (Å) 2.866Bulk Modulus (GPa) 167C44 (GPa) 117

C’ (GPa) 47.5a γ Transformation Temperature (P=0) (K) 1185γ δ TransformationTemperature (P=0) (K)

1657

a ε Transformation Pressure (0K) (GPa) 6-15Vacancy Formation Energy (eV) 1.7Thermal Expansion (500K) µm·m−1·K−1 14.4



MEAM PARAMETERS ARE CORRELATED WITH PHYSICAL PROPERTIES

Physical Property MEAM ParameterCohesive Energy Ec

a Lattice Constant re

Bulk Modulus αThermal Expansion (500K) δC44 t2, β2

C’ t2, β2

a γ Transformation Temperature (P=0) A, β0,Cmin

γ δ TransformationTemperature (P=0) A, β0,Cmin

a ε Transformation Pressure (0K) t3, β3

Vacancy Formation Energy t1, β1



MEAM PARAMETERS FOR FE

Ec (eV) 4.29 a 5.0729re (Å) 2.469 δ 0.3A 0.6 β0 4.045t1 -1.6 β1 2t2 12 β2 0.8t3 -0.05 β3 1Cmax 1.9 Cmin 0.7



MEAM REPRODUCES DATA BASE EXTREMELY WELL

Physical Property Experiment MEAMCohesive Energy (eV) 4.29 4.29a Lattice Constant (300K) (Å) 2.866 2.866Bulk Modulus (GPa) 167 170C44 (GPa) 117 116

C’ (GPa) 47.5 49a γ Transformation Temperature (P=0) (K) 1185 1175γ δ TransformationTemperature (P=0) (K) 1657 1750a ε Transformation Pressure (0K) (GPa) 6-15 11Vacancy Formation Energy (eV) 1.7 1.7Thermal Expansion (500K) µm·m−1·K−1 14.4 14.3



RECIPE FOR A BINARY SYSTEM

Choose Reference Structure Assemble Data Base Associate/Fit Parameters with Data Base Validate Iterate



CHOOSE REFERENCE STRUCTURE

• Simple Crystal Structure– Preferably with 1NN of opposite type– B1 (rock salt –NaCl)

• Equilibrium Ground State if Simple• Must Have Data• Relevant to Simulations

– stoichiometry– Structure

• Derive Analytic Expression for Cross Pair Potential



ASSEMBLE DATA BASE

• Experiment or First Principles (reference structure)– Cohesive energy– Lattice constant– Elastic constants– Thermal expansion

• Other Phases– Cohesive energy



ASSOCIATE/FIT PARAMETERS WITH DATA BASE• UEOS (reference structure)

– Cohesive energy (Ec)– Lattice constant (re)– Bulk modulus (α)– Thermal expansion (δ)

• Electron Density Scaling– Elastic constants (ρa

0)– Other cohesive energies (ρa

0, Cmin, Cmax)



VALIDATE (I –Reference Phase)

• Thermal properties– Expansion– Specific heat

• Surfaces– Energy– Relaxation– Reconstruction– Segregation



VALIDATE (II –Reference Phase)

• Point defects– Vacancy (on A and B sub-lattices) mobility– Interstitial

• Formation energy• Geometry• Migration energy



VALIDATE (III – Other Phases)

• Liquid– Heat of fusion– Density– Melting point

• Solid– Relative stability– Lattice constants– Internal relaxation– Elastic constants– Transformation paths– Dilute heats of solution– Short range order



ISSUES FOR MULTI-COMPONENT SYSTEMS There are a limited number of free parameters for

multi-components• For AB choose ρaB

0

• For AC choose ρaC0

• There is no ρ to choose for BC

There are Cmin(A,B,C) and Cmax(A,B,C) to choose• All combinations of A, B, C



TAKE AWAY

MEAM is Parameterized to Facilitate Relatively Easy Function Determination

MEAM Can Quantitatively Reproduce the Fe Data Base Including Phase Transformations

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