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DEVELOPMENT OF SEMI-EMPIRICAL ATOMISTIC POTENTIALS MS-MEAM. M. I. Baskes Los Alamos National Laboratory and University of California, San Diego. OUTLINE. The Challenge The Concepts Bond energy Many body effects Transferability Reference state Screening The Models

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development of semi empirical atomistic potentials ms meam
DEVELOPMENT OF SEMI-EMPIRICAL ATOMISTIC POTENTIALS MS-MEAM

M. I. Baskes

Los Alamos National Laboratory

and

University of California, San Diego

outline
OUTLINE
  • The Challenge
  • The Concepts
    • Bond energy
    • Many body effects
    • Transferability
    • Reference state
    • Screening
  • The Models
    • Pair potentials
    • Embedded atom method (EAM)
    • Modified EAM (MEAM)
    • Multi-state MEAM (MS-MEAM
atomistic models have two purposes i
ATOMISTIC MODELS HAVE TWO PURPOSES (I)
  • Obtain understanding of physical processes
    • Model system (empirical) potentials
      • how specific properties affect collective behavior
        • dependence of yield strength on stacking fault energy
    • Semi-empirical potentials (fit to experimental data)
      • plasticity
      • phase transformations
    • First principles
      • diffusion

This is meant to be a short list of the many physical process that can be examined by atomistics

atomistic models have two purposes ii
ATOMISTIC MODELS HAVE TWO PURPOSES (II)
  • Obtain quantitative properties of specific materials
    • First principles
      • lattice constant
      • structural stability
      • elastic moduli
    • Semi-empirical potentials (fit to experimental data)
      • thermal expansion
      • melting point
      • yield strength
      • thermodynamics / free energy / phase diagrams

Many more examples exist

in order to achieve the second purpose we need a method that encompasses
IN ORDER TO ACHIEVE THE SECOND PURPOSE WE NEED A METHOD THAT ENCOMPASSES
  • Accuracy
    • thermodynamic properties must be known accurately to be useful (a few tenths of a percent of the cohesive energy)
  • Computational speed
    • analytic or tabular model
    • scales linearly with the number of atoms
    • parallel architecture

We only consider here models that have appropriate speed and then try to improve accuracy

concept bond energy
CONCEPT: BOND ENERGY
  • Every pair of atoms is connected by a bond (spring)
  • The bond energy depends on the separation of the atoms
  • The energy of a material is the sum of the bond energies
concept many body effects
CONCEPT: MANY BODY EFFECTS
  • All bonds are not equal
  • The bond energy also depends on the local environment (coordination)
    • Coordination / bond length / bond energy are correlated (Pauling)
concept transferability
CONCEPT: TRANSFERABILITY
  • The model will be accurate for all atomic environments
    • Volume (Nearest neighbor (NN) distance)
    • Coordination (crystal structure - number of NN)
    • Defects or strain (loss of symmetry)
concept reference state i
CONCEPT: REFERENCE STATE (I)
  • Reference structure
    • A specific crystal structure
    • Properties of the reference structure can be obtained from experiment or first principles calculations
      • Energy vs. volume (NN distance)
      • Elastic constants
      • Defect energies
    • Reference structures have high symmetry
    • Scaling
      • energy per atom of the equilibrium reference structure is -1
      • distance is scaled by the equilibrium NN distance
concept reference state ii
CONCEPT: REFERENCE STATE (II)
  • Reference path
    • A specific path connecting 2 reference structures
    • Properties along the reference path can be obtained from first principles calculations
      • Energy vs. distance along path
    • Reference paths encompass low symmetry states
    • Coordination changes along a reference path
  • Incorporation of many reference states will facilitate transferability
concept screening
CONCEPT: SCREENING
  • Atomic interactions have a finite range
    • Radial screening
      • at a cutoff distance the interactions go to zero (smoothly)
      • dependant on distance
      • independent of local geometry
    • Angular screening
      • intervening atoms reduce interactions to zero
      • dependent on local geometry
      • high compression
  • Necessary for computational scaling with the number of atoms

Fellows 11/18/2005

model pair potential
MODEL: PAIR POTENTIAL
  • Computation
  • Analytic or tabular
  • Scales with number of atoms
  • Parallel architecture

Accuracy

  • Transferable
    • Volume
    • Coordination
    • Defects/strain

i: all atoms

j: neighbors of atom i

independent of environment

radial screening

the embedded atom method is semi empirical

_

ρis obtained from a linear superposition of atomic densities

Fandϕare 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)

THE EMBEDDED ATOM METHOD IS SEMI-EMPIRICAL

embedding

energy

host electron

density

pair interaction

UBER

model eam
MODEL: EAM
  • Computation
  • Analytic or tabular
  • Scales with number of atoms
  • Parallel architecture

Accuracy

  • Transferable
    • Volume
    • Coordination
    • Defects/strain

i: all atoms

j: neighbors of atom i

radial screening

depends on environment

complex materials require the addition of angular forces





θ











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



model meam
MODEL: MEAM
  • Computation
  • Analytic or tabular
  • Scales with number of atoms
  • Parallel architecture

Accuracy

  • Transferable
    • Volume
    • Coordination
    • Defects/strain
  • Environmental dependence of bonding
  • Angular screening
  • Assumed functional forms
    • embedding function
    • electron density
    • background electron density
    • screening
modified embedded atom method meam

Background Electron Density

Universal Binding Energy Relationship

UBER

Embedding Function

Pair Potential

MODIFIED EMBEDDED ATOM METHOD (MEAM)

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

concept of the screening ellipse leads to a simple screening model
CONCEPT OF THE SCREENING ELLIPSE LEADS TO A SIMPLE SCREENING MODEL

screening ellipse defined by C

2y/rik

Cmin and Cmax set limits of screening

2x/rik

goes from 0 to 1 smoothly

multi state meam ms meam
MULTI-STATE MEAM (MS-MEAM)
  • Same Functional Form as MEAM
  • Multiple Reference States
  • Environmental Dependence of Bonds
  • Angular Screening
  • Assumed Functional Forms
    • Asymptotic embedding function
    • Background electron density
model ms meam
MODEL: MS-MEAM
  • Computation
  • Analytic or tabular
  • Scales with number of atoms
  • Parallel architecture

Accuracy

  • Transferable
    • Volume
    • Coordination
    • Defects

(we hope!)

  • Same functional form as MEAM
  • Multiple reference states
  • Environmental dependence of bonds
  • Angular screening
  • Assumed functional forms
    • asymptotic embedding function
    • background electron density
multi state modified embedded atom method ms meam

Screening

MULTI-STATE MODIFIED EMBEDDED ATOM METHOD (MS-MEAM)

Basic Ansatz

Embedding Function

Background Electron Density

first application of ms meam has been completed
FIRST APPLICATION OF MS-MEAM HAS BEEN COMPLETED
  • Cu Chosen as Model Material
  • VASP/GGA-PW Used for First Principles Energy Calculations
    • ~1000 E/V points calculated
  • M. I. Baskes, S. G. Srinivasan, S. M. Valone, and R. G. Hoagland, Multistate modified embedded atom method, PHYSICAL REVIEW B 75, 094113 2007
ms meam embedding function
MS-MEAM EMBEDDING FUNCTION

fcc equilibrium density

ms meam electron densities
MS-MEAM ELECTRON DENSITIES
  • simple smooth functions
  • negative square densities
need to have two sets of electron densities
NEED TO HAVE TWO SETS OF ELECTRON DENSITIES
  • magnetism
  • electronic states
  • charges
ms meam screening functions
MS-MEAM SCREENING FUNCTIONS

screening ellipse

k

i

j

Fellows 11/18/2005

ms meam is predictive for energy vs nn distance
MS-MEAM IS PREDICTIVE FORENERGY vs. NN DISTANCE

* used in development of functions

coordination 1-12

energy differences for equal coordination structures are small
ENERGY DIFFERENCES FOR EQUAL COORDINATION STRUCTURES ARE SMALL

* used in development of functions

Fellows 11/18/2005

homogeneous transformations used to determine screening functions

BCC  SC  FCC

(trigonal)

2D-HEX  2D-SQ

FCC  BCC (Bain)

HOMOGENEOUS TRANSFORMATIONS USED TO DETERMINE SCREENING FUNCTIONS
conclusions
CONCLUSIONS
  • MS-MEAM Has The Potential to be a Fast, Accurate Method of Calculating Atomistic Interactions
  • Consider MS-MEAM to be a Method for Interpolation/Extrapolation of a FP Data Base
  • There is No Fitting – Just Direct Calculation From the Data Base
  • This Method Could Enable Quantitative Thermodynamic Predictions of Multi-component, Multi-phase Materials