Byeong-Joo Lee calphad Interatomic Potentials for Ionic Systems Byeong-Joo Lee POSTECH-CMSE

Byeong-Joo Lee www.postech.ac.kr/~calphad

Interatomic Potentials Interatomic Potentials for Ionic Systemsfor Ionic Systems

Byeong-Joo LeeByeong-Joo Lee

POSTECH-CMSEPOSTECH-CMSE


BackgroundBackground• Importance of Ionic MaterialsImportance of Ionic Materials

Sensor, Battery, Devices, Metal Surfaces, etc.Sensor, Battery, Devices, Metal Surfaces, etc.

• Need to handle “ionic + covalent + metallic” materialsNeed to handle “ionic + covalent + metallic” materials

Interfacial Reaction between metals and SiO2 substrateInterfacial Reaction between metals and SiO2 substrate

Diffusion of metallic atoms in amorphous SiO2Diffusion of metallic atoms in amorphous SiO2

• Atomistic simulation on “ionic + covalent + metallic” Atomistic simulation on “ionic + covalent + metallic”

materialsmaterials

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Purpose and ScopePurpose and Scope

• Development of Interatomic Potential Model that covers Development of Interatomic Potential Model that covers

“ “ionic + covalent + metallic” materials, simultaneously.ionic + covalent + metallic” materials, simultaneously.

Review interatomic potentials for ionic and hybrid materials Review interatomic potentials for ionic and hybrid materials

Propose possible form of an interatomic potential formalismPropose possible form of an interatomic potential formalism


OutlineOutline • Interatomic Potential for Ionic Materials Interatomic Potential for Ionic Materials

Point Charge ModelPoint Charge Model

Polarization (Shell Model)Polarization (Shell Model)

• Many-Body PotentialsMany-Body Potentials

TersoffTersoff

EAM – MEAM – 2NN MEAMEAM – MEAM – 2NN MEAM

Many-body potentials used for ionic systemsMany-body potentials used for ionic systems

• Many-Body Potentials for Ionic MaterialsMany-Body Potentials for Ionic Materials

Charge Equilibration ModelCharge Equilibration Model

EAM + QeqEAM + Qeq

Tersoff + QeqTersoff + Qeq

• Proposal of New Interatomic Potential FormProposal of New Interatomic Potential Form


Interatomic Potential for Ionic Interatomic Potential for Ionic MaterialsMaterials

6/)/exp( ijijijijij

jiij rCrA

r

qq

Fixed Point Charge

Born-Mayer-Huggins

•Initially applied to liquid or glass, not crystals : probably, unable to reproduce crystal structures•1st MD on SiO2 glass Woodcock [5], 1976

•More information available with upgraded measuring techniques for crystal structures and dynamics

•1988: BMH + many-body interaction to reproduce O-Si-O bonding angle: (cosθjik - cosθojik)

2 [6]

•1988: BMH + modified Coulomb interaction considering excess charge distribution in oxygen + Ab Initio on SiO2 model clusters → α-quartz, α-cristobalite, Coesite, Stishovite, for the first time → TTAM [7]•1990: BMH + Ab Initio + Experimental Information on α-quartz → better description than TTAM → BKS [8]

•TTAM&BKS: representative Point Charge Potential for SiO2 during 1990s. (qSi = +2.4, qO = -1.2)•Limitation: use of Point Charge, pair-wise potentials not applicable to pure Si or Si/SiO2

•1994: Jiang & Brown: SW Si – BKS SiO2, ionization energy, charge variation, bond-softening function → behavior of O atom in Si [11] •2010: Soulairol & Cleri: SW Si – BKS SiO2 + different q for interface

•Initially applied to liquid or glass, not crystals : probably, unable to reproduce crystal structures•1st MD on SiO2 glass Woodcock [5], 1976

•More information available with upgraded measuring techniques for crystal structures and dynamics

•1988: BMH + many-body interaction to reproduce O-Si-O bonding angle: (cosθjik - cosθojik)

2 [6]

•1988: BMH + modified Coulomb interaction considering excess charge distribution in oxygen + Ab Initio on SiO2 model clusters → α-quartz, α-cristobalite, Coesite, Stishovite, for the first time → TTAM [7]•1990: BMH + Ab Initio + Experimental Information on α-quartz → better description than TTAM → BKS [8]

•TTAM&BKS: representative Point Charge Potential for SiO2 during 1990s. (qSi = +2.4, qO = -1.2)•Limitation: use of Point Charge, pair-wise potentials not applicable to pure Si or Si/SiO2

•1994: Jiang & Brown: SW Si – BKS SiO2, ionization energy, charge variation, bond-softening function → behavior of O atom in Si [11] •2010: Soulairol & Cleri: SW Si – BKS SiO2 + different q for interface

TTAMBKS


Interatomic Potential for Ionic Interatomic Potential for Ionic MaterialsMaterials

Fixed Point Charge + electronic polarization•Include dipole-charge, dipole-dipole interaction due to electronic polarization•Shell Model by Dick & Overhauser [13], 1958 Ion = core electron core + valence electron shell Deviation of Center of mass of Shell causes a dipole Shell connected to core by an artificial spring and interact through harmonic restoring

force

•Shell Model has been successful for diatomic molecule, alkali halides and also for Al2O3 [14]•BMH + polarization : representative approach during 1980s for alkali halides, binary, mixed

oxides [15]•Shell model: leading model for ionic materials in GULP [19]

•2002: Morse-Stretch pp + fixed point charge Coulomb + dipole polarization for Oxygen ions [17]

fitting (force-matching) on liquid SiO2 → better description for polymorphs than BKS

•Limitation: not applicable to pure Si or Si/SiO2, not describing variable charge •Next Step: Many-body + variable charge

•Include dipole-charge, dipole-dipole interaction due to electronic polarization•Shell Model by Dick & Overhauser [13], 1958 Ion = core electron core + valence electron shell Deviation of Center of mass of Shell causes a dipole Shell connected to core by an artificial spring and interact through harmonic restoring

force

•Shell Model has been successful for diatomic molecule, alkali halides and also for Al2O3 [14]•BMH + polarization : representative approach during 1980s for alkali halides, binary, mixed

oxides [15]•Shell model: leading model for ionic materials in GULP [19]

•2002: Morse-Stretch pp + fixed point charge Coulomb + dipole polarization for Oxygen ions [17]

fitting (force-matching) on liquid SiO2 → better description for polymorphs than BKS

•Limitation: not applicable to pure Si or Si/SiO2, not describing variable charge •Next Step: Many-body + variable charge


Many-Body Potential Many-Body Potential : EAM – 2NN MEAM

i ijijijii

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•Embedding energy of impurity atoms is determined by the electron density of the host (from first-principles)→ individual atoms are impurity atoms → EAM concept [29,30]

•How to compute F and Ф ? No specific function form was given in initial EAM → reason for so many EAMs•Rose universal equation of state [23] gives a guide [31]

•EAM : linear supposition for computation of electron density of a site → mainly for fcc

•Introduction of bonding directionality → Modified EAM (1nn interaction only) → applied to Si [32], bcc [33] and hcp [34], but stability problem

•Need to consider 2NN interactions to solve critical shortcomings in MEAM → 2NN MEAM [36,37]→ applicable to both metallic and covalent systems: metals, carbides, nitrides, Si, Ge, etc. [38-40]

•Embedding energy of impurity atoms is determined by the electron density of the host (from first-principles)→ individual atoms are impurity atoms → EAM concept [29,30]

•How to compute F and Ф ? No specific function form was given in initial EAM → reason for so many EAMs•Rose universal equation of state [23] gives a guide [31]

•EAM : linear supposition for computation of electron density of a site → mainly for fcc

•Introduction of bonding directionality → Modified EAM (1nn interaction only) → applied to Si [32], bcc [33] and hcp [34], but stability problem

•Need to consider 2NN interactions to solve critical shortcomings in MEAM → 2NN MEAM [36,37]→ applicable to both metallic and covalent systems: metals, carbides, nitrides, Si, Ge, etc. [38-40]


Many-Body Potential Many-Body Potential : Tersoff

jiji

ijijijijci

i rBbrArfEE,,

)exp()exp()(2

1

• 1985 Abell : Close relation between Morse-type pair potential and Rose universal behavior

→ replacement of Born-Mayer by Morse-Stretch

• Tersoff potential [24-26]

bij : bond order – 1nn interaction, bond length and angle, effect of local environments, etc.

• applied to C [27] and SiC [28] and extended to Brenner-REBO [87-89]

• for alloys : arithmetic mean to λ, μ and geometric mean to A, B, R, S

• 1985 Abell : Close relation between Morse-type pair potential and Rose universal behavior

→ replacement of Born-Mayer by Morse-Stretch

• Tersoff potential [24-26]

bij : bond order – 1nn interaction, bond length and angle, effect of local environments, etc.

• applied to C [27] and SiC [28] and extended to Brenner-REBO [87-89]

• for alloys : arithmetic mean to λ, μ and geometric mean to A, B, R, S


• Umeno [14] : using Tersoff for SiO2 Independent fitting to λ, μ, A, B instead of mean values applicable to β-cristobalite, β-quartz which was difficult by BKS

• Kuo [15] : using MEAM for SiO2 applicable to α, β-quartz, α, β-cristobalite, β-tridymite

Many-Body Potential for Ionic Many-Body Potential for Ionic MaterialsMaterials


Charge Equilibration ModelCharge Equilibration Model

...)(0

2

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A

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A

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E

BA

ABBAA

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rnn

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0001 )...(

• 1991 Rappe & Goddard [48] : based on previous concepts on electronegativity, equilibration [49-57]. - equilibrium charge in molecules considering Coulomb interaction and penalty energy for charged isolated atoms (atomic self-energy)

IP & EA : ionization potential 과 electron affinity χ0 : electronegativity J0 : atomic hardness representing Coulomb repulsion between two electrons in an orbital

JAB : Coulomb interaction between A & B computed by a Coulomb integral on atomic charge density expressed for a Slater-type orbital

•Basic idea in Qeq model is to equalize the atomic chemical potential of all individual atoms (χ1 = χ2 = … = χN)

•First applied to SiO2 in 1999 [58] : Morse-Stretch pair potential + charge equilibration - Quartz-Stishovite phase transition & Silica glass

•Swamy & Gale [59] in 2000 : Titanium oxide system including rutile, anatase, brookite, TiO2-II, Ti2O3, monoclinic high- and low temp forms of Ti3O5, TiO, ramsdellite-type TiO2, g-Ti3O5, two Magneli phases: Ti4O7 and Ti6O11

• 1991 Rappe & Goddard [48] : based on previous concepts on electronegativity, equilibration [49-57]. - equilibrium charge in molecules considering Coulomb interaction and penalty energy for charged isolated atoms (atomic self-energy)

IP & EA : ionization potential 과 electron affinity χ0 : electronegativity J0 : atomic hardness representing Coulomb repulsion between two electrons in an orbital

JAB : Coulomb interaction between A & B computed by a Coulomb integral on atomic charge density expressed for a Slater-type orbital

•Basic idea in Qeq model is to equalize the atomic chemical potential of all individual atoms (χ1 = χ2 = … = χN)

•First applied to SiO2 in 1999 [58] : Morse-Stretch pair potential + charge equilibration - Quartz-Stishovite phase transition & Silica glass

•Swamy & Gale [59] in 2000 : Titanium oxide system including rutile, anatase, brookite, TiO2-II, Ti2O3, monoclinic high- and low temp forms of Ti3O5, TiO, ramsdellite-type TiO2, g-Ti3O5, two Magneli phases: Ti4O7 and Ti6O11


200

2

1)0()( iiiiiii qJqEqE

jiji

jiijiji

iies qqrVqEE,,

),,(2

1)(

jiji

ijjii

iies VqqqEE,,

0 2

1

jiji

ijjii

iies VqqqE,,2

1

0i

iqiesi qE /

)(),( rErqEE EAMestot

•1994 Streitz & Mintmire [60] : first Qeq approach for crystalline materials, EAM + Qeq for Al2O3•1994 Streitz & Mintmire [60] : first Qeq approach for crystalline materials, EAM + Qeq for Al2O3

•2004 Zhou [70] : solving charge stability problem, - extened to multicomponent oxides, O-Al-Ni-Co-Fe system [71]

•2007 Lazic [74] : MEAM (different from Baskes) + Qeq, not much is published

•2004 Zhou [70] : solving charge stability problem, - extened to multicomponent oxides, O-Al-Ni-Co-Fe system [71]

•2007 Lazic [74] : MEAM (different from Baskes) + Qeq, not much is published Oxidation of Al nano cluster [61,62]

Oxidation of Al nano cluster [61,62]

EAM + Charge EquilibrationEAM + Charge Equilibration


jiji

ijijijijci

i rBbrASRrfEE,,

)exp()exp(),,(2

1

2

22 iEiEi

iEiEi

i qe

AIq

e

AI

ijo

jiijLijIONij r

qqfE

4

- effective point charge with cutoff function in Coulomb potential, not with Ewald summation - Considering changes in ionic radius and short range interaction due to charge

- effective point charge with cutoff function in Coulomb potential, not with Ewald summation - Considering changes in ionic radius and short range interaction due to charge

•Crack propagation behavior of SiO2 with or without H2O•Adhesion strength on Al,Cu/TiN,W,SiO2 thin film interface [77]•Upgrade in parameter [78] & Formalism for Coulomb interaction [79]

•2007 Sinnott & Phillpot group [80] : confirm application to α, β-quartz, α, β-cristobalite, but stability problem. - atomic self-energy up to 4th order & introduction of bond-bending energy, (cosθOSiS - cosθo

OSiO)2, - COMB (Optimized Many-Body Potential), but cannot generate amorphous SiO2 & bad results for α-quartz •2010 modified version for SiO2 [81] : Slater 1s orbital type Coulomb integral & Ewald + another penalty term - applicable to α, β-quartz, α, β-cristobalite, β-tridymite, Coesite, Stishovite, generally worse than TTAM•2010 Hf/HfO2, Cu/Cu2O [83,84] : different bond-bending form depending on cation element

•Crack propagation behavior of SiO2 with or without H2O•Adhesion strength on Al,Cu/TiN,W,SiO2 thin film interface [77]•Upgrade in parameter [78] & Formalism for Coulomb interaction [79]

•2007 Sinnott & Phillpot group [80] : confirm application to α, β-quartz, α, β-cristobalite, but stability problem. - atomic self-energy up to 4th order & introduction of bond-bending energy, (cosθOSiS - cosθo

OSiO)2, - COMB (Optimized Many-Body Potential), but cannot generate amorphous SiO2 & bad results for α-quartz •2010 modified version for SiO2 [81] : Slater 1s orbital type Coulomb integral & Ewald + another penalty term - applicable to α, β-quartz, α, β-cristobalite, β-tridymite, Coesite, Stishovite, generally worse than TTAM•2010 Hf/HfO2, Cu/Cu2O [83,84] : different bond-bending form depending on cation element

Tersoff + Charge EquilibrationTersoff + Charge Equilibration

• 1996 Yasukawa [76] : introduce atomic energy ΣiΦi & Coulomb energy ½ΣiΣjEIONij


• Bond-Order : based on correlation between bond order & bond distance or bond energy describe bond dissociation → chemical reaction - including bonding angle, torsion, charge equilibration, van der Waals interaction, etc. - mainly for hydrocarbon system [85], but also to oxides, Si/SiO2 system [86]

• Most powerful : covering Hydrocarbon system like Brenner-REBO [87-89] and charge equilibration like COMB

• Number of parameters for Carbon, for example : 90s - how to determine the parameter values ? → 10 ~ 15 systems during up to now - retirement of Prof. Goddard → Dr. van Duin @ Penn State

Others : ReaxFFOthers : ReaxFF


SummarySummary

Up to now no interatomic potential for ionic + covalent + metallic alloy systems


Potential for Ionic+Covalent+Metallic Potential for Ionic+Covalent+Metallic MaterialsMaterials•Charge Effect ?

Correct physics : easy parameterization and good trasferability

•Point Charge vs. Charge Distribution ? TTAM that considered charge distribution could describe the SiO2 polymorphs for the first time

•Shell Model ? No publication for shell model + many-body potentialVariable charge can be superior to fixed charge, for bond dissociation, surface, interface, and other

defects

•Coulomb Integral ? COMB10 [81] is generally worse than BKS or TTAM for SiO2 polymorphsCoulomb intergral (COMB10 [81]) vs. effective point charge (Yasukawa 2010 [79]) ?

•Summation of Long Range Potential (1/r radial behavior) ? Ewald method [70], PPPM [75], direct summation method [82]

•Charge Equilibration Method ? Inverse matrix [60], Conjugate gradient method [70], Lagrangian dynamics [80]

•Manybody Potential ? - COMB had to change the functional form for bond-bending term, probably due to the limitation of

Tersoff. [Tersoff potential has never been applied to metallic alloy systems]- MEAM is also a kind of bond order potential, 2NN MEAM has been applied to both covalent and metallic alloy systems

•Conclusion 2NN MEAM + Qeq = Tersoff+Qeq + EAM+QeqPaying attention to charge stability and extension to multicomponent systems,and searching for the best solution for Coulomb integral, long range potential and charge equilibraion

•Charge Effect ? Correct physics : easy parameterization and good trasferability

•Point Charge vs. Charge Distribution ? TTAM that considered charge distribution could describe the SiO2 polymorphs for the first time

•Shell Model ? No publication for shell model + many-body potentialVariable charge can be superior to fixed charge, for bond dissociation, surface, interface, and other

defects

•Coulomb Integral ? COMB10 [81] is generally worse than BKS or TTAM for SiO2 polymorphsCoulomb intergral (COMB10 [81]) vs. effective point charge (Yasukawa 2010 [79]) ?

•Summation of Long Range Potential (1/r radial behavior) ? Ewald method [70], PPPM [75], direct summation method [82]

•Charge Equilibration Method ? Inverse matrix [60], Conjugate gradient method [70], Lagrangian dynamics [80]

•Manybody Potential ? - COMB had to change the functional form for bond-bending term, probably due to the limitation of

Tersoff. [Tersoff potential has never been applied to metallic alloy systems]- MEAM is also a kind of bond order potential, 2NN MEAM has been applied to both covalent and metallic alloy systems

•Conclusion 2NN MEAM + Qeq = Tersoff+Qeq + EAM+QeqPaying attention to charge stability and extension to multicomponent systems,and searching for the best solution for Coulomb integral, long range potential and charge equilibraion


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Mulliken, “A New Electroaffinity Scale; Together with Data on Valence States and on Valence Ionization Potentials and Electron Affinities,” J. Chem. Phys. 2, 782 (1934).50. R. T. Sanderson, “Partial Charges on Atoms in Organic Compounds,” Science 11, 207 (1955).51. L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaca, New York, (1960).52. R. P. Iczkowsky and J. L. Margrave, “Electronegativity,” J. Am. Chem. Soc. 83, 3547 (1961).53. R. G. Parr, R. A. Donnelly, M. Levy and W. E. Palke, “Electronegativity: The Density Functional Viewpoint,” J. Chem. Phys. 68, 3801 (1978).54. P. Politzer and H. Weinstein, “Some Relations between Electronic Distribution and Electronegativity,” J. Chem. Phys. 71, 4218 (1979).55. R. G. Parr and R. G. Pearson, “Absolute Hardness: Companion Parameter to Absolute Electronegativity,” J. Am. Chem. Soc. 105, 7512 (1983).56. W. J. Mortier, K. van Genechten and J. Gasteiger, “Electronegativity Equalization: Application and Parametrization,” J. Am. Chem. Soc. 107, 829 (1985).57. W. J. Mortier, S. K. Ghosh and S. Shankar, “Electronegativity-Equalization Method for the Calculation of Atomic Charges in Molecules”, J. Am. Chem. Soc. 108, 4315 (1986).58. E. Demiralp, T. Cagin and W.A. Goddard, “Morse Stretch Potential Charge Equilibrium Force Field for Ceramics: Application to the Quartz-Stishovite Phase Transition and to Silica Amorphous”, Phys. Rev. Lett. 82, 1708 (1999).59. V. Swamy, J. D. Gale, “Transferable Variable-Charge Interatomic Potential for Atomistic Simulation of Titanium Oxides”, Phys. Rev. B 62, 5406 (2000).60. F. H. Streitz and J. W. Mintmire, “Electrostatic Potentials for Metal-Oxide Surfaces and Interfaces,” Phys. Rev. B 50, 11996 (1994). 61. T. Campbell, R. K. Kalia, A. Nakano, P. Vashishta, S. Ogata and S. Rodgers, “Dynamics of Oxidation of Aluminum Nanoclusters using Variable Charge Molecular-Dynamics Simulations on Parallel Computers,” Phys. Rev. Lett. 82, 4866 (1999).62. T. Campbell, G. Aral, S. Ogata, R. K. Kalia, A. Nakano and P. Vashishta, “Oxidation of aluminum Nanoclusters,” Phys. Rev. B 71, 205413 (2005).63. N. Rosen, “Calculation of Interaction between Atoms with s-Electrons,” Phys. Rev. 38, 255 (1931).64. C. C. J. Roothaan, “A Study of Two-Center Integrals Useful in Calculations on Molecular Structure. I,” J. Chem. Phys. 19, 1445 (1951).65. S. W. de Leeuw, J. W. Perram and E. R. Smith, “Simulation of Electrostatic Systems in Periodic Boundary Conditions. I. Lattice Sums and Dielectric Constants,” Proc. R. Soc. London Ser. A 373, 27 (1980).66. E. R. Smith, “Electrostatic Energy in Ionic Crystals,” Proc. R. Soc. London Ser. A 375, 475 (1981).67. D. E. Parry, “The Electrostatic Potential in the Surface Region of an Ionic Crystal,” Surf. Sci. 49, 433 (1975).68. J. Hautman and M. L. Klein, “An Ewald Summation Method for Planar Surfaces and Interfaces,” Mol. Phys. 75, 379 (1992).69. D. M. Heyes, “Surface Stress of Point Charge Lattices,” Surf. Sci. Lett. 293, L857 (1993).70. X. W. Zhou, H. N. G. Wadley, J.-S. Filhol, M. N. Neurock, “Modified Charge Transfer–Embedded Atom Method Potential for Metal/Metal Oxide Systems,” Phys. Rev. B 69, 035402 (2004).71. X. W. Zhou, H. N. G. Wadley, “A Charge Transfer Ionic–Embedded Atom Method Potential for the O–Al–Ni–Co–Fe System,” J. Phys.: Condens. Matter 17, 3619 (2005).72. J. R. Smith, H. Schlosser, W. Leaf, J. Ferrante and J. H. Rose, “Connection between Energy Relations of Solids and Molecules,” Phys. Rev. A 39, 514 (1989).73. J. Ferrante, H. Schlosser and J. H. Rose, “Global expression for representing diatomic Potential-energy curves,” Phys. Rev. A 43, 3487 (1991).74. I. Lazic, M. Ernst, B. Thijsse, “Atomistic Simulation Methods for Studying Self Healing Mechanisms in Al/Al2O3,” Proceedings of the First International Conference on Self Healing Materials, 18-20 April 2007, Noordwijk aan Zee, The Netherlands75. R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York, 1981.76. A. Yasukawa, “Using an Extended Tersoff Interatomic Potential to Analyze the Static-Fatigue Strength of SiO2 under Atmospheric Influence,” JSME Int. J., Ser. A 39, 313 (1996).77. T. Iwasaki and H. Miura, “Molecular dynamics analysis of adhesion strength of interfaces between thin films,” J. Mater. Res. 16, 1789 (2001).78. A. Yasukawa, in Japan Society of Mechanical Engineers, p.71, Sept. 19, 2003, Hitachi City, Ibaraki, Japan.79. A. Yasukawa, “Atomistic Simulation of Environment-Assisted Crack Propagation Behavior of SiO2, ” J. Solid Mech. Mater. Engin. 4, 599 (2010). 80. J. Yu, S. B. Sinnott, S. R. Phillpot, “Charge optimized many-body Potential for the Si/SiO2 system,” Phys. Rev. B 75, 085311 (2007)81. T.-R. Shan, B. D. Devine, M. Hawkins, A. Asthagiri, S. R. Phillpot and S. B. Sinnott, “Second-Generation Charge-Optimized Many-Body Potential for Si/SiO2 and amorphous Silica,” Phys. Rev. B 82, 235302 (2010). 82. D. Wolf, P. Keblinski, S. R. Phillpot, and J. Eggebrecht, “Exact Method for the Simulation of Coulombic Systems by Spherically Truncated, Pairwise r-1 Summation,” J. Chem. Phys. 110, 8254 (1999).83. T.-R. Shan, B. D. Devine, T. W. Kemper, S. B. Sinnott and S. R. Phillpot, “Charge Optimized Many-Body Potential for the Hafnium/Hafnium Oxide System,” Phys. Rev. B 81, 125328 (2010).84. B. D. Devine, T.-R. Shan, S. B. Sinnott, and S. R. Phillpot, “Charge Optimized Many-Body Potential for the Copper/Copper Oxide System,” 2011 (unpublished)85. A. C. T. van Duin, S. Dasgupta, F. Lorant and W. A. Goddard III, “ReaxFF: A Reactive Force Field for Hydrocarbons,” J. Phys. Chem. A 105, 9396 (2001).86. A. C. T. van Duin, A. Strachan, S. Stewman, Q. Zhang, X. Xu and W. A. Goddard III, “ReaxFFSiO Reactive Force Field for Silicon and Silicon Oxide Systems,” J. Phys. Chem. A 107, 3803 (2003).87. D. W. Brenner, ”Empirical Potential for Hydrocarbons for Use in Simulating the Chemical Vapor Deposition of Diamond Films,” Phys. Rev. B 42, 9458 (1990).88. S. J. Stuart, A. B. Tutein and J. A. Harrison, “A reactive Potential for Hydrocarbons with Intermolecular Interactions,” J. Chem. Phys. 112, 6472 (2000).89. D. W Brenner, O. A Shenderova, J. A Harrison, S. J. Stuart, B. Ni and S. B. Sinnott, “A second-generation reactive empirical bond order (REBO) Potential energy expression for Hydrocarbons,” J. Phys.: Condens. Matter 14, 783 (2002).

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S. Daw and M. I. Baskes, “Embedded-Atom Method: Derivation and Application to Impurities, Surfaces, and Other Defects in Metals,” Phys. Rev. B 29, 6443 (1984).31. S. M. Foiles, M. I. Baskes and M. S. Daw, “Embedded-Atom Method Functions for the fcc Metals Cu, Ag, Au, Ni, Pd, Pt and Their Alloys,” Phys. Rev. B 33, 7983 (1986).32. M. I. Baskes, J. S. Nelson and A. F. Wright, “Semiempirical Modified Embedded-Atom Potentials for Silicon and Germanium,” Phys. Rev. B 40, 6085 (1989).33. M. I. Baskes, “Modified Embedded-Atom Method Potentials for Cubic Materials and Impurities,” Phys. Rev. B 46, 2727 (1992).34. M. I. Baskes and R. A. Johnson, “Modified Embedded Atom Method Potentials for HCP Metals,” Modelling Simul. Mater. Sci. Eng. 2, 147 (1994).35. M. I. Baskes, “Determination of Modified Embedded Atom Method Parameters for Nickel,” Mater. Chem. Phys. 50, 152 (1997).36. B.-J. Lee, M. I. Baskes, “Second Nearest-Neighbor Modified Embedded-Atom Method Potential,” Phys. Rev. B 62, 8564 (2000).37. B.-J. Lee, M. I. Baskes, H. Kim, Y. K. Cho, “Second Nearest-Neighbor Modified Embedded Atom Method Potentials for BCC Transition Metals,” Phys. Rev. B 64, 184102 (2001).38. H.-K. Kim, W.-S. Jung, B.-J. Lee, “Modified Embedded-Atom Method Interatomic Potentials for the Nb-C, Nb-N, Fe-Nb-C and Fe-Nb-N systems,” J. Mater. Res. 25, 1288 (2010). 39. B.-J. Lee, “A Semi-Empirical Atomistic Approach in Materials Research,” J. Phase Equilib. Diff. 30, 509 (2009).40. B.-J. Lee, W.-S. Ko, H.-K. Kim, E.-H. Kim, “Overview: The modified embedded-atom method Interatomic Potentials and recent progress in atomistic Simulations,” CALPHAD 34, 510 (2010).41. F. H. Stillinger and A. Weber, “Computer simulation of local order in condensed phases of silicon,” Phys. Rev. B 31, 5262 (1985). 42. T. Watanabe, H. Fujiwara, H. Noguchi, T. Hoshino and I. Ohdomari, “Novel Interatomic Potential Energy Function for Si, O Mixed Systems,” Jpn. J. Appl. Phys. 38, L366 (1999).43. T. Watanabe, D. Yamasaki, K. Tatsumura and I. Ohdomari, “Improved Interatomic Potential for stressed Si, O mixed systems,” Appl. Surf. Sci. 234, 207 (2004).44. Y. Umeno, T. Kitamura, K. Date, M. Hayashi, and T. Iwasaki, “Optimization of Interatomic Potential for Si/SiO2 system based on force matching,” Comput. Mater. Sci. 25, 447 (2002).45. S.R. Billeter, A. Curioni, D. Fischer and W. Andreoni, “Ab Initio Derived Augmented Tersoff Potential for Silicon Oxynitride Compounds and Their Interfaces with Silicon,” Phys. Rev. B 73, 155329 (2006).46. M. I. Baskes, “Modified Embedded Atom Method Calculations of Interfaces,” Report number: SAND--96-8484C, Sandia National Laboratories, Livermore, 1996.47. C.-L. Kuo and P. Clancy, “Development of Atomistic MEAM Potentials for the Silicon–Oxygen–Gold Ternary System,” Modelling Simul. Mater. Sci. Eng. 13, 1309 (2005).48. A. K. Rappe and W. A. Goddard III, “Charge Equilibration for Molecular Dynamics Simulations,” J. Phys. Chem. 95, 3358 (1991).49. R. S. Mulliken, “A New Electroaffinity Scale; Together with Data on Valence States and on Valence Ionization Potentials and Electron Affinities,” J. Chem. Phys. 2, 782 (1934).50. R. T. Sanderson, “Partial Charges on Atoms in Organic Compounds,” Science 11, 207 (1955).51. L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaca, New York, (1960).52. R. P. Iczkowsky and J. L. Margrave, “Electronegativity,” J. Am. Chem. Soc. 83, 3547 (1961).53. R. G. Parr, R. A. Donnelly, M. Levy and W. E. Palke, “Electronegativity: The Density Functional Viewpoint,” J. Chem. Phys. 68, 3801 (1978).54. P. Politzer and H. Weinstein, “Some Relations between Electronic Distribution and Electronegativity,” J. Chem. Phys. 71, 4218 (1979).55. R. G. Parr and R. G. Pearson, “Absolute Hardness: Companion Parameter to Absolute Electronegativity,” J. Am. Chem. Soc. 105, 7512 (1983).56. W. J. Mortier, K. van Genechten and J. Gasteiger, “Electronegativity Equalization: Application and Parametrization,” J. Am. Chem. Soc. 107, 829 (1985).57. W. J. Mortier, S. K. Ghosh and S. Shankar, “Electronegativity-Equalization Method for the Calculation of Atomic Charges in Molecules”, J. Am. Chem. Soc. 108, 4315 (1986).58. E. Demiralp, T. Cagin and W.A. Goddard, “Morse Stretch Potential Charge Equilibrium Force Field for Ceramics: Application to the Quartz-Stishovite Phase Transition and to Silica Amorphous”, Phys. Rev. Lett. 82, 1708 (1999).59. V. Swamy, J. D. Gale, “Transferable Variable-Charge Interatomic Potential for Atomistic Simulation of Titanium Oxides”, Phys. Rev. B 62, 5406 (2000).60. F. H. Streitz and J. W. Mintmire, “Electrostatic Potentials for Metal-Oxide Surfaces and Interfaces,” Phys. Rev. B 50, 11996 (1994). 61. T. Campbell, R. K. Kalia, A. Nakano, P. Vashishta, S. Ogata and S. Rodgers, “Dynamics of Oxidation of Aluminum Nanoclusters using Variable Charge Molecular-Dynamics Simulations on Parallel Computers,” Phys. Rev. Lett. 82, 4866 (1999).62. T. Campbell, G. Aral, S. Ogata, R. K. Kalia, A. Nakano and P. Vashishta, “Oxidation of aluminum Nanoclusters,” Phys. Rev. B 71, 205413 (2005).63. N. Rosen, “Calculation of Interaction between Atoms with s-Electrons,” Phys. Rev. 38, 255 (1931).64. C. C. J. Roothaan, “A Study of Two-Center Integrals Useful in Calculations on Molecular Structure. I,” J. Chem. Phys. 19, 1445 (1951).65. S. W. de Leeuw, J. W. Perram and E. R. Smith, “Simulation of Electrostatic Systems in Periodic Boundary Conditions. I. Lattice Sums and Dielectric Constants,” Proc. R. Soc. London Ser. A 373, 27 (1980).66. E. R. Smith, “Electrostatic Energy in Ionic Crystals,” Proc. R. Soc. London Ser. A 375, 475 (1981).67. D. E. Parry, “The Electrostatic Potential in the Surface Region of an Ionic Crystal,” Surf. Sci. 49, 433 (1975).68. J. Hautman and M. L. Klein, “An Ewald Summation Method for Planar Surfaces and Interfaces,” Mol. Phys. 75, 379 (1992).69. D. M. Heyes, “Surface Stress of Point Charge Lattices,” Surf. Sci. Lett. 293, L857 (1993).70. X. W. Zhou, H. N. G. Wadley, J.-S. Filhol, M. N. Neurock, “Modified Charge Transfer–Embedded Atom Method Potential for Metal/Metal Oxide Systems,” Phys. Rev. B 69, 035402 (2004).71. X. W. Zhou, H. N. G. Wadley, “A Charge Transfer Ionic–Embedded Atom Method Potential for the O–Al–Ni–Co–Fe System,” J. Phys.: Condens. Matter 17, 3619 (2005).72. J. R. Smith, H. Schlosser, W. Leaf, J. Ferrante and J. H. Rose, “Connection between Energy Relations of Solids and Molecules,” Phys. Rev. A 39, 514 (1989).73. J. Ferrante, H. Schlosser and J. H. Rose, “Global expression for representing diatomic Potential-energy curves,” Phys. Rev. A 43, 3487 (1991).74. I. Lazic, M. Ernst, B. Thijsse, “Atomistic Simulation Methods for Studying Self Healing Mechanisms in Al/Al2O3,” Proceedings of the First International Conference on Self Healing Materials, 18-20 April 2007, Noordwijk aan Zee, The Netherlands75. R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York, 1981.76. A. Yasukawa, “Using an Extended Tersoff Interatomic Potential to Analyze the Static-Fatigue Strength of SiO2 under Atmospheric Influence,” JSME Int. J., Ser. A 39, 313 (1996).77. T. Iwasaki and H. Miura, “Molecular dynamics analysis of adhesion strength of interfaces between thin films,” J. Mater. Res. 16, 1789 (2001).78. A. Yasukawa, in Japan Society of Mechanical Engineers, p.71, Sept. 19, 2003, Hitachi City, Ibaraki, Japan.79. A. Yasukawa, “Atomistic Simulation of Environment-Assisted Crack Propagation Behavior of SiO2, ” J. Solid Mech. Mater. Engin. 4, 599 (2010). 80. J. Yu, S. B. Sinnott, S. R. Phillpot, “Charge optimized many-body Potential for the Si/SiO2 system,” Phys. Rev. B 75, 085311 (2007)81. T.-R. Shan, B. D. Devine, M. Hawkins, A. Asthagiri, S. R. Phillpot and S. B. Sinnott, “Second-Generation Charge-Optimized Many-Body Potential for Si/SiO2 and amorphous Silica,” Phys. Rev. B 82, 235302 (2010). 82. D. Wolf, P. Keblinski, S. R. Phillpot, and J. Eggebrecht, “Exact Method for the Simulation of Coulombic Systems by Spherically Truncated, Pairwise r-1 Summation,” J. Chem. Phys. 110, 8254 (1999).83. T.-R. Shan, B. D. Devine, T. W. Kemper, S. B. Sinnott and S. R. Phillpot, “Charge Optimized Many-Body Potential for the Hafnium/Hafnium Oxide System,” Phys. Rev. B 81, 125328 (2010).84. B. D. Devine, T.-R. Shan, S. B. Sinnott, and S. R. Phillpot, “Charge Optimized Many-Body Potential for the Copper/Copper Oxide System,” 2011 (unpublished)85. A. C. T. van Duin, S. Dasgupta, F. Lorant and W. A. Goddard III, “ReaxFF: A Reactive Force Field for Hydrocarbons,” J. Phys. Chem. A 105, 9396 (2001).86. A. C. T. van Duin, A. Strachan, S. Stewman, Q. Zhang, X. Xu and W. A. Goddard III, “ReaxFFSiO Reactive Force Field for Silicon and Silicon Oxide Systems,” J. Phys. Chem. A 107, 3803 (2003).87. D. W. Brenner, ”Empirical Potential for Hydrocarbons for Use in Simulating the Chemical Vapor Deposition of Diamond Films,” Phys. Rev. B 42, 9458 (1990).88. S. J. Stuart, A. B. Tutein and J. A. Harrison, “A reactive Potential for Hydrocarbons with Intermolecular Interactions,” J. Chem. Phys. 112, 6472 (2000).89. D. W Brenner, O. A Shenderova, J. A Harrison, S. J. Stuart, B. Ni and S. B. Sinnott, “A second-generation reactive empirical bond order (REBO) Potential energy expression for Hydrocarbons,” J. Phys.: Condens. Matter 14, 783 (2002).


Atomistic SimulationsAtomistic Simulations - MEAM & - MEAM & ApplicationsApplications

Byeong-Joo LeeByeong-Joo Lee

Dept. of MSEDept. of MSE

Pohang University Pohang University

of Science and Technologyof Science and Technology

(POSTECH)(POSTECH)

[email protected]@postech.ac.kr


Semi-Empirical Atomic Potentials - Historical BackgroundSemi-Empirical Atomic Potentials - Historical Background

• Pair Potentials (~1980)Pair Potentials (~1980)

▷ ▷ Elastic Constants are NOT correctly reproducedElastic Constants are NOT correctly reproduced

• Many Body Potentials (1980's)Many Body Potentials (1980's) ▷ ▷ Embedded Atom Method (EAM: 1983)Embedded Atom Method (EAM: 1983)

▷ ▷ Finnis and Sinclair Potential (1984)Finnis and Sinclair Potential (1984)

▷ ▷ Glue Model (1986)Glue Model (1986)

▷ ▷ Equilivalent-Crystal Model (1987)Equilivalent-Crystal Model (1987)


Semi-Empirical Atomic Potentials – History of DevelopmentSemi-Empirical Atomic Potentials – History of Development

• EAM Potentials (1983, M.S. Daw and M.I. Baskes)EAM Potentials (1983, M.S. Daw and M.I. Baskes) ▷ ▷ Successful mainly for FCC elementsSuccessful mainly for FCC elements - many other many-body potentials show similar performance- many other many-body potentials show similar performance

• 1NN MEAM Potentials (1987,1992, M.I. Baskes)1NN MEAM Potentials (1987,1992, M.I. Baskes) ▷ ▷ Show Possibility for description of various structuresShow Possibility for description of various structures - important to be able to describe multi-component system- important to be able to describe multi-component system

• 2NN MEAM Potentials (2000, B.-J. Lee & M.I. Baskes)2NN MEAM Potentials (2000, B.-J. Lee & M.I. Baskes) ▷ ▷ Applicable to fcc, bcc, hcp, diamond structures and their alloysApplicable to fcc, bcc, hcp, diamond structures and their alloys


EAM/MEAMEAM/MEAM – General – General

i ijijijii rFE

)(

)(2

1)(

E E : Total Potential Energy: Total Potential Energy F F : Embedding Energy: Embedding Energy : Electron Density (Considering Bonding Directionality): Electron Density (Considering Bonding Directionality) : Pair Interaction Energy: Pair Interaction Energy


EAM/MEAM EAM/MEAM – Embedding Function– Embedding Function

i ijijijii rFE

)(

)(2

1)(

F AEc o o( ) ln

eea

eq

e n

n

r

rrr ln

)(

)'(2

M.I. Baskes et al., Phys. Rev. B, 40, 6085 (1989)


EAM/MEAM EAM/MEAM – Universal EOS– Universal EOS

i ijijijii rFE

)(

)(2

1)(

J.H. Rose et al., Phys. Rev. B, 29, 2963 (1984)

** )1()( ac

u eaErE

)1/(* erra

91 2

B

Ec

/


EAM/MEAM EAM/MEAM – Electron Density for EAM– Electron Density for EAM

i ijijijii rFE

)(

)(2

1)(


EAM/MEAM EAM/MEAM – Electron Density for MEAM– Electron Density for MEAM

+ Angular contribution

( ) ( )( ) ( ) i ja

ijj i

R0 2 0

2

( ) ( )( ) ( )

iij

ijja

ijj i

R

RR1 2 1

2

( ) ( ) ( )( )

,

( ) ( )

iij ij

ijja

ijj i

ja

ijj i

R R

RR R2 2

22

2

2

21

3

( ) ( ) ( )( )

, ,

( ) ( )

iij ij ij

ijja

ijj i

ij

ijja

ijj i

R R R

RR

R

RR3 2

33

2

3

2

3

5


EAM/MEAM EAM/MEAM – Electron Density for MEAM– Electron Density for MEAM

+ Angular contribution

with ti(0) =1

i i G ( ) ( )0

ti

h

h

ih

i

( )( )

( )1

3

0

2

Ge

( )

2

1

3

0

2)0(

)(3

1

)(2)0(2)()(2 )(1)()()(h i

hi

h

hii

hi

hii tt


EAM/MEAM EAM/MEAM – 1– 1stst Nearest Neighbor MEAM Nearest Neighbor MEAM

i ijijijii rFE

)(

)(2

1)(

*

)1()()())(( *21 a

cuo

eaErErrF

))](()([2

)(1

rFrEZ

rou


1NN MEAM vs. 2NN MEAM 1NN MEAM vs. 2NN MEAM –– Many-Body ScreeningMany-Body Screening

Cmax

Cmin

i j

xC

y Rij2 2

21 1

2

C

X X X X

X Xik kj ik kj

ik kj

2 1

1

2

2

( ) ( )

( )

Xik=(Rik/Rij)2 and Xkj=(Rkj/Rij)2

S fC C

C Cikj c

min

max min

fc(x) = 1 x 1

1 1 4 2 ( )x

0 x 0

S Sij ikjk i j

,

0 x 1


2NN MEAM 2NN MEAM – Computation of pair-wise potential– Computation of pair-wise potential

i ijijijii rFE

)(

)(2

1)(

Eu (R) F(o(R))

Z1

2(R)

Z2 S

2(aR)

o

(R) Z1a(0) (R) Z 2S

a(0) (aR)

Eu (R) F(o(R))

Z1

2 (R)

(R) (R)Z2 S

Z1

(aR)

(R) (R) ( 1)n

n1 Z2S

Z1

n

(an R)


Evaluation of MEAM Potential Parameters for ElementsEvaluation of MEAM Potential Parameters for Elements

EEcc, R, Ree, B, A, d, , B, A, d, (0)(0), , (1)(1), , (2)(2), , (3)(3), t, t(1)(1), t, t(2)(2), t, t(3)(3), C, Cmaxmax, C, Cminmin

▷ ▷ Cohesive Energy of Stable and Metastable StructureCohesive Energy of Stable and Metastable Structure

▷ ▷ Nearest Neighbor Distance Nearest Neighbor Distance

▷ ▷ Bulk Modulus, Elastic Constants (C11, C12, C44)Bulk Modulus, Elastic Constants (C11, C12, C44)

▷ ▷ Stacking Fault EnergyStacking Fault Energy

▷ ▷ Vacancy Formation EnergyVacancy Formation Energy

▷ ▷ Surface EnergySurface Energy


Semi-Empirical Atomic Potentials - PerformanceSemi-Empirical Atomic Potentials - Performance

• Elastic ConstantsElastic Constants ▷ ▷ B, C11, C12, C44, ... B, C11, C12, C44, ...

• Defect EnergyDefect Energy ▷ ▷ Surface EnergySurface Energy

▷ ▷ Heat of Vacancy Formation, …Heat of Vacancy Formation, …

• Structural EnergyStructural Energy ▷ ▷ Energy and Lattice Parameters in Different StructuresEnergy and Lattice Parameters in Different Structures

• Thermal PropertyThermal Property ▷ ▷ Specific HeatSpecific Heat

▷ ▷ Thermal Expansion CoefficientThermal Expansion Coefficient

▷ ▷ Melting Temperature, ...Melting Temperature, ...


MEAM for BCC Transition Metals MEAM for BCC Transition Metals – B.-J. Lee et al., PRB, 2001

Elem. C11 C12 C44Elem. C11 C12 C44 EE(100) (100) E E(110) (110) E E(111)(111) E Evvff E Ebcc/fccbcc/fcc E Efcc/hcpfcc/hcp

Fe 2.430 1.380 1.219 2510 2356 2668 1.75 0.069 -0.023

2.431 1.381 1.219 2360* 1.79 0.082 -0.023

Cr 3.909 0.897 1.034 2300 2198 2501 1.91 0.070 -0.02

3.910 0.896 1.032 2200* 1.80 0.075 -0.029

Mo 4.649 1.655 1.088 3130 2885 3373 3.09 0.167 -0.038

4.647 1.615 1.089 2900* 3.10 0.158 -0.038

W 5.326 2.050 1.631 3900 3427 4341 3.95 0.263 -0.047

5.326 2.050 1.631 2990* 3.95 0.200 -0.047

V 2.323 1.194 0.460 2778 2636 2931 2.09 0.084 -0.011

2.324 1.194 0.460 2600* 2.10 0.078 -0.036

Nb 2.527 1.331 0.319 2715 2490 2923 2.75 0.176 -0.012

2.527 1.332 0.310 2300* 2.75 0.140 -0.036

Ta 2.664 1.581 0.875 3035 2778 3247 2.95 0.148 -0.023

2.663 1.582 0.874 2780* 2.95 0.166 -0.041


MEAM for FCC Transition Metals MEAM for FCC Transition Metals – B.-J. Lee et al., PRB, 2003

Elem. C11 C12 C44Elem. C11 C12 C44 E E(100) (100) E E(110) (110) E E(111)(111) EEvvff E Ebcc/fccbcc/fcc E Efcc/hcp fcc/hcp εε (0-100(0-100ooC)C)

Cu 1.762 1.249 0.818 1382 1451 1185 1.11 -0.08 0.007 17.0 1.762 1.249 0.818 1770 1.03-1.30 -0.04 0.006

17.0 Ag 1.315 0.973 0.511 983 1010 842 0.94 -0.08 0.005 18.9 1.315 0.973 0.511 1320 1.1 -0.04 0.003 19.1 Au 2.015 1.697 0.454 1138 1179 928 0.90 -0.06 0.009 14.2 2.016 1.697 0.454 1540 0.9 -0.04 0.003 14.1 Ni 2.612 1.508 1.317 1943 2057 1606 1.51 -0.16 0.02 12.6 2.612 1.508 1.317 2240 1.6 -0.09 0.02 13.3 Pd 2.342 1.761 0.712 1743 1786 1435 1.50 -0.17 0.02 11.0 2.341 1.761 0.712 2043 1.4,1.7 -0.11 0.02 11.0 Pt 3.581 2.535 0.775 2288 2328 1710 1.50 -0.28 0.02 9.2 3.580 2.536 0.774 2691 1.35,1.5 -0.16 0.03 9.0 Al 1.143 0.619 0.316 848 948 629 0.68 -0.12 0.03 22.0 1.143 0.619 0.316 1085 0.68 -0.10 0.06 23.5 Pb 0.556 0.454 0.194 426 440 375 0.58 -0.04 0.003 30.1 0.555 0.454 0.194 534 0.58 -0.02 0.003 29.0


MEAM for SiliconMEAM for Silicon

C11 C12 C44 E(100) E(110) E(111) Evf Edia/fcc Edia/hcp Edia/bcc ε

(1012dyne/cm2) (erg/cm2) (eV) (eV) (0-100oC)

1.67 0.65 0.80 2631 1766 1442 3.67 0.57 0.55 0.52 2.65

1.68 0.65 0.80 1135* 3.3-4.3 0.57 0.55 0.53 2.69


2NN MEAM Interatomic Potentials 2NN MEAM Interatomic Potentials – for Al and Fe

Property MEAM-Al (exp.) MEAM-Fe (exp.)

C11 (1012 dyne/cm2)

C12 (1012 dyne/cm2)

C44 (1012 dyne/cm2)

Evf (eV)

QD (eV)

EIf (eV)

1.143 (1.143)0.619 (0.619)0.316 (0.316)

0.68 (0.68)1.33 (1.33)

2.49 (-)

2.430 (2.431)1.380 (1.381)1.219 (1.219)

1.75 (1.79)2.28 (2.5)4.20 (-)

E(100) (mJ/m2)

E(110) (mJ/m2)

E(111) (mJ/m2)

d(100) (%)

d(110) (%)

d(111) (%)

848 (1085a)948 (1085a)629 (1085a)+1.8 (+1.8)

-8.9 (-8.5±1.0) +1.0 (0.9±0.5)

2510 (2360a)2356 (2360a)2668 (2360a)

-1.1 (-0.2, -1.5)-1.5 (0)

-10.5 (-16.9)

Ebcc/fcc (eV/atom)

Efcc/hcp (eV/atom)

0.12 (0.10b)0.03 (0.06b)

0.048 (0.082b)-0.018 (-0.023b)

(0-100oC) (10-6/K)

Cp (0-100oC) (J/mol·K)

m.p. (K)

Hm (KJ/mol)

Vm (%)

22.0 (23.5)26.2 (24.7)937 (933)11.0 (10.7)

6.7 (6.5)

12.4 (12.1)26.1 (25.5)

2000 (1811)13.2 (13.8)

4.0 (3.5)


2NN MEAM 2NN MEAM – – 2NNMEAM for Alloy Systems2NNMEAM for Alloy Systems

i ijijijii rFE

)(

)(2

1)(


2NN MEAM for Alloy Systems 2NN MEAM for Alloy Systems – Optimization of Potential Parameter, Fe-Pt– Optimization of Potential Parameter, Fe-Pt

Selected value Procedure for the determination

ΔEc -0.4600 Fitting to ΔH or Ttr

re 2.7181 Fitting to lattice parameter

B 2.6201 Fitting to bulk modulus

d 0.25dFe+0.75dPt Assumption

Cmin(Fe-Pt-Fe) 0.36 ( = CminFe) Assumption

Cmin(Pt-Fe-Pt) 1.53 ( = CminPt) Assumption

Cmin(Fe-Fe-Pt) [0.5(CminFe)1/2 + 0.5(Cmin

Pt) 1/2 ]2 Assumption

Cmin(Fe-Pt-Pt) [0.5(CminFe)1/2 + 0.5(Cmin

Pt) 1/2 ]2 Assumption

ρ0 ρ0Fe = ρ0

Pt = 1.0 A temporary assumption


200K 850K 1000K200K 850K 1000K

2NN MEAM for Fe-Cr Binary System 2NN MEAM for Fe-Cr Binary System – B.-J. Lee et al., CALPHAD, 2001


MEAM for Cu-Ni Binary System MEAM for Cu-Ni Binary System – B.-J. Lee and J.-H. Shim, CALPHAD, 2004


MEAM for Ni-Si Binary SystemMEAM for Ni-Si Binary System

Enthalpy of Formation (eV/atom)Lattice constant (Å)Bulk Modulus (100 GPa)C11 (100 GPa)C12 (100 GPa)C11-C12 (100 GPa)C44 (100 GPa)(100) fracture energy (J·m-2)

NiNi33SiSi 0.36 (0.36) 3.504 (3.504) 2.64 3.67 (3.63-3.75) 2.13 (2.00-2.05) 1.54 1.96 (1.67-1.72) 5.3 (7.2)

NiSiNiSi22

0.28 (0.28) 5.391 (5.406) 1.93 (1.60) 2.39 1.69 0.70 (0.58) 0.32 8.0

Dilute Heat of Solution (eV/atom)

Si in (Ni)Si in (Ni) -1.50 (-1.37) Ni in (Si)Ni in (Si) +0.50


MEAM for Co-Pt Binary System MEAM for Co-Pt Binary System - S.I. Park et al., Scripta Mater., 2001.- S.I. Park et al., Scripta Mater., 2001.

Property Pt3Co PtCo PtCo3

Cohesive Energy 5.500 5.215 4.873(eV/atom) 5.555±0.017 5.228±0.005

Lattice Constant a=3.833 3.754, c/a=.98 3.625(Å) a=3.831 3.745, c/a=.98 3.668

Transition 1070-1080 970-980 760-770Temperature (K) 1000 1100 840


MEAM for Ni-W Binary System MEAM for Ni-W Binary System – J.-H. Shim et al., J. Mater. Res.,J. Mater. Res., 2003

Property fcc (XW=0.11) Ni4W

Cohesive Energy 4.922 5.36 (fcc, 5.27) (eV/atom) 4.925 5.40

Lattice Constant a=3.57 a=5.73, c=3.553(Å) a=3.56 a=5.73, c=3.553


a (Å) c (Å) Ec(eV) B(Gpa)

Ni4W (D1a) 5.73 3.553 5.36 292 5.73 3.553 5.40 293

Ni3W (L12) 3.62 - 5.58 319 3.58 - 5.65 287

Ni3W (D019) 2.56 4.05 5.59 316 2.53 - 5.42 289

NiW3 (L12) 3.86 - 7.29 316 3.84 - 7.55 283

NiW3 (D019) 2.76 4.44 7.36 321 2.76 - 7.70 304

MEAM for Ni-W Binary SystemMEAM for Ni-W Binary System


Empirical Potentials for Multicomponent SystemsEmpirical Potentials for Multicomponent Systems

• FeFe ▷ ▷ Finnis-Sinclair – modified by Calder and Bacon (1993)Finnis-Sinclair – modified by Calder and Bacon (1993)

• Fe-CuFe-Cu ▷ ▷ Osetsky (1996)Osetsky (1996)

• Fe: Pair-Potential, Osetsky (1995)Fe: Pair-Potential, Osetsky (1995)• Cu: Pair-Potential, Osetsky (1995)Cu: Pair-Potential, Osetsky (1995)

▷ ▷ Ackland, Bacon, Calder (1997)Ackland, Bacon, Calder (1997)• Fe: F-S type, Ackland et al. (1997)Fe: F-S type, Ackland et al. (1997)• Cu: F-S type, Ackland, Tichy, Vitek, Finnis (1987)Cu: F-S type, Ackland, Tichy, Vitek, Finnis (1987)

▷ ▷ Ludwig, Farkas,.. (1998) Ludwig, Farkas,.. (1998) →→ C.S. C.S. Becquart, C. Domain,Becquart, C. Domain, • Fe: EAM, Fe: EAM, Simonelli, Pasianot, SavinoSimonelli, Pasianot, Savino (1993)(1993)• Cu: EAM, Voter (1993)Cu: EAM, Voter (1993)


History of Fe-C Alloy PotentialHistory of Fe-C Alloy Potential

• R.A. Johnson, G.J. Dienes, A.C. Damask, Acta Metall. 12, 1215 (1964).

• metal-metal: pairwise interaction• metal-carbon: pairwise interaction• can consider only one carbon atoms, not applicable to carbides

• V. Rosato, Acta Metall. 37, 2759 (1989).

• metal-metal: many-body interaction • metal-carbon: pairwise interaction • can consider only one carbon atoms, not applicable to carbides

• M. Ruda, D. Farkas, and J. Abriata, Scr. Mater. 46, 349 (2002).

• metal-metal: many-body interaction (EAM)• metal-carbon: many-body interaction (EAM)• carbon-carbon: many-body interaction (EAM)• unacceptable results


History of Carbon PotentialHistory of Carbon Potential

• J. Tersoff, Phys. Rev. Lett. 61 (1988) 2879.• structural properties (cohesive energies, bond lengths of various polytypes)• elastic properties (elastic constants of diamond) • point defect properties (vacancy formation and migration energies, and interstitial formation energies in diamond and graphite) • applicable to monolayer of graphite • applicable to only Diamond Structures (C, Si, Ge, SiC, …)

• D.W. Brenner, Phys. Rev. B 42 (1990) 9458; J. Phys.: Condens. Matter 14 (2002) 783.

• modification of Tersoff formalism to better describe hydrocarbons

• M.I. Heggie, J. Phys.: Condens. Matter 3 (1991) 3065.• E.P. Andribet et al., Nucl. Instr. & Meth. in Phys. Res. B 115 (1996) 501.

• To better describe graphite structure than Tersoff• Only for graphite


(2NN) MEAM for Fe, C, N, Fe-C and Fe-N systems(2NN) MEAM for Fe, C, N, Fe-C and Fe-N systems

• Fe, Cr, Mo, W, V, Nb, TaSecond Nearest-Neighbor Modified Embedded Atom Method Potentials for BCC Transition MetalsByeong-Joo Lee, M.I. Baskes, Hanchul Kim and Yang Koo Cho,

Phys. Rev. B. 64, 184102 (2001).

• CA Modified Embedded Atom Method Interatomic Potential for CarbonByeong-Joo Lee and Jin Wook Lee, CALPHAD 29, 7-16 (2005).

• Fe-CA Modified Embedded Atom Method Interatomic Potential for the Fe-C System Byeong-Joo Lee, Acta Materialia 54, 701-711 (2006).

• Fe-NA Modified Embedded-Atom Method Interatomic Potential for the Fe-N System: A Comparative Study with the Fe-C system

Byeong-Joo Lee, T-H Lee and S-J Kim, Acta Materialia 4597-4607 (2006).


2NN MEAM for pure Fe2NN MEAM for pure Fe - PRB 64, 184102 (2001); 71, 184205 (2005)- PRB 64, 184102 (2001); 71, 184205 (2005) MEAM F-S type EAM F-S type pair potential expt. This work Calder-Bacon Simonelli Ackland Osetsky

C11C12C44Ev

f

∆Vvf/Ω

Evm

EIf

∆VIf/Ω

E(100)E(110)E(111)∆d(100)∆d(110)∆d(111)∆Ebcc/fcc

∆Ehcp/fcc

a(bcc)a(fcc)

ε(0-100oC)Cp (0-100oC)

m.p.∆Hm

∆Vm

2.431a

1.381a

1.219a

< 2b

< -.4c

0.55d

-<110>due

-

2360f

-0.2, -1.5g

0g

-16.9g

-0.082h

-0.023h

2.8665i

-12.1j

25.5j

1811h

13.8h

3.5j

2.4301.3801.2191.75-.410.534.20

<110>du1.70251023562668-1.1-1.5

-10.5-0.048-0.0182.86373.61112.526.1200012.93.3

2.434k

1.381k

1.221k

1.83-.210.91k

4.85<110>du

1.33

1920k

-0.054m

0.0m

2.866

2.421.471.121.63

0.663.54

<111>cr

-0.027 0.0

2.8664

2200

2.431.451.16 1.70-.180.784.87

<110>du1.76

1812n

1585n

2269n

--

-0.054-

2.86653.690

235821.0

--

1.192.05-.290.533.92

<111>cr0.69

-0.0520.0052.8673.61211.6


MEAM for CarbonMEAM for Carbon – Physical Property of Diamond– Physical Property of Diamond

MEAM Tersoff exp./calc.

C11

C12

C44

Evf

Evsplit

E I(T)

f

E I(H)

f

E I(110)db

f

E I(100)db

f

10.791.276.233.357.23

unstableunstable

12.79.3

10.9a

1.2a

6.4a

4.3a

9.7a

19.6a

20.9a

-10.0a

10.80c

1.27c

5.77c

7.2d

9.1d

23.6d

--

16.7d

Eideal(100)Eideal(110)Eideal(111) E1×1(100)E2×1(100)E1×1(111)

Δd1-2(111) Δd1-2(111)Δd1-2(111)Δd1-2(111)

881157154666712457202069-17.9+9.5-52.5+21.1

7565b

4949b 4040b

6639b

2772b -15.9b +2.0b -39. 8b +4.3b

9850e, 9250g

6540g

7960f, 5340g

9190e

5370e

6270f

-49f

+9f

ε (300-1200 K)Cp (300-1200 K)

825.5

1~6h

5~22h


MEAM + LJ Tersoff exp./calc.

Biso

C11

C12

C33

C44

C13

Evf

Evsplit

E InterLayer

f

2.3810.99-0.450.38

0.00030.0003

6.29.54.9

-12.1a

-1.9a

---

7.1a

10.8a

-

2.86b

10.60c

1.80c

0.365c

0.04c

0.15c, -0.12d, -0.005e

7.6f

9.2f

7.0g

E (0001) 84

rhombohedral graphite simple graphite hexagonal graphite exp.

Egra/dia -0.01 +0.01 -0.003 -0.02a

Lattice parameter, a 2.45 2.45 2.45 2.46~2.47b

Lattice parameter, c 6.63 6.95 6.66 6.71~6.93b

MEAM for CarbonMEAM for Carbon – Physical Property of Graphite– Physical Property of Graphite


MEAM for CarbonMEAM for Carbon – for several structures– for several structures


MEAM (+ LJ) exp./calc.

ΔE of grapheneΔE of (10,10) CNTΔE of (17,0) CNTΔE of C60 bucky ball

Young’s Modulus of (10,10) CNTVacancy formation energy in (8,0) CNT

0.0370.050.050.5910.45.71

0.02a, 0.045b

0.086a, 0.10b

0.088a, 0.10b

0.46a, 0.47b

10.02c

5.59d

MEAM for CarbonMEAM for Carbon – Nanotubes and Fullerenes– Nanotubes and Fullerenes


N2 Re (Å) Ec (eV/atom)

Exp. 1.10 -4.88

Bond-order 1.11 -4.96

2NN MEAM 1.10 -4.88

N3 Re (Å) Angle (degree)

Ec (eV/atom)

Bond-order 1.272 180 -3.712

2NN MEAM 1.116 180 -3.45

▪ Bond length and Cohesive energy for N2

▪ Bond length, Bond angle and cohesive Energy for N3

MEAM for NMEAM for N22


in BCC Fe MEAM expt./calc.

Dilute Heat of Solution of Carbon (eV)Migration Energy Barrier of Carbon (eV)Vacancy-Carbon Binding Energy (eV)Vacancy-Carbon Binding Distance (ao)

Dilute Heat of Solution of Nitrogen (eV)Migration Energy Barrier of Nitrogen (eV)Vacancy- Nitrogen Binding Energy (eV)Vacancy- Nitrogen Binding Distance (ao)

1.220.820.900.43

0.330.780.640.42

1.1a 0.88b, 0.86c,0.81-0.83d

0.41e, 0.85f, 1.05g, 1.1b, 0.44j

0.365k, 0.40j

0.32 0.76~0.80

0.67j

0.45j

2NN MEAM for Fe-C & Fe-N2NN MEAM for Fe-C & Fe-N – in BCC Fe– in BCC Fe

Carbon in O site vacancy-carbon carbon-carbon SIA-carbon vacancy-two carbon


in FCC Fe MEAM expt./calc.

Dilute Heat of Solution of Carbon (eV)Migration Energy Barrier of Carbon (eV)Vacancy-Carbon Binding Energy (eV)Carbon-Carbon Binding Energy (eV)

Dilute Heat of Solution of Nitrogen (eV)Migration Energy Barrier of Nitrogen (eV)Vacancy- Nitrogen Binding Energy (eV)Nitrogen - Nitrogen Binding Energy (eV)

0.301.520.67

-0.12 <110>-0.35 <100>

-0.481.360.23

-0.31 <110>-0.10 <100>

0.36a, 0.12b

1.4c,1.53d

0.37 ~ 0.41e

<110> alignment is less repulsive

-0.531.75

<100> alignment is less repulsive

2NN MEAM for Fe-C2NN MEAM for Fe-C & Fe-N& Fe-N – in FCC Fe– in FCC Fe


2NN MEAM for Fe-N2NN MEAM for Fe-N – in Fe– in Fe44NN

ΔΔHHf f = +3.1 ~ -40 kJ/mol (-10.5) = +3.1 ~ -40 kJ/mol (-10.5) MEAM: -6.8 kJ/mol MEAM: -6.8 kJ/mol

a = 3.80 Å a = 3.80 Å MEAM: 3.80 ÅMEAM: 3.80 Å


2NN MEAM for Fe-N2NN MEAM for Fe-N – in Fe– in Fe22NN

• Identification of the most stable atomic structure of FeIdentification of the most stable atomic structure of Fe22NN• ΔΔHHf f = - 5.7 kJ/mol = - 5.7 kJ/mol MEAM: -20.9 kJ/mol MEAM: -20.9 kJ/mol • a = 2.76 a = 2.76 Å, c = 4.42 Å Å, c = 4.42 Å MEAM: a = 2.81 MEAM: a = 2.81 Å, c = 4.32 ÅÅ, c = 4.32 Å


Atomistic Simulation Atomistic Simulation – Interatomic Potentials and Applications

Interatomic Potential:Performance of 2NN MEAM for Elements and Alloys

• Fundamental Properties of Structural Materials• Elastic Property• Defect (Point, Dislocation, Grain Bd./Interface) Property• Phase Transformations• Deformation/Fracture Mechanism

• Fundamental Properties of Nano Materials• Thermodynamic Property• Atomic/Nano Structural Evolution

• Fundamental Properties of Amorphous Materials• Irradiation Defects, etc.


Second Nearest Neighbor Modified EAM (2NN MEAM) Pure Elements

•Fe, Cr, Mo, W, V, Nb, Ta, Li Phys. Rev. B. 64, 184102 (2001); MSMSE 20, 035005 (2012) . •Cu, Ag, Au, Ni, Pd, Pt, Al, Pb Phys. Rev. B. 68, 144112 (2003). •Ti, Zr & Mg Phys. Rev. B. 74, 014101 (2006); CALPHAD 33, 650-57 (2009). •Mn, P Acta Materialia 57, 474-482 (2009).; J. Phys.: Condensed Matters (2012), in press.•C, Si, Ge, In CALPHAD 29, 7-16 (2005); 31, 95-104 (2007); 32, 34-42 (2008); 32, 82-88 (2008)

Multicomponent Systems•Fe-C, Fe-N, Fe-H Acta Materialia 54, 701-711 (2006); 54, 4597-4607 (2006); 55, 6779-6788 (2007). •Fe-Ti & Fe-Nb Scripta Materialia 59, 595-598 (2008).•Fe-Ti-C & Fe-Ti-N Acta Materialia 56 , 3481-3489 (2008); Acta Materialia 57 , 3140-3147 (2009).•Fe-Nb-C & Fe-Nb-N J. Materials Research 25, 1288-1297 (2010).•Al-H & Ni-H, V-H J. Materials Research 26, 1552-1560 (2011); CALPHAD 35, 302-307 (2011).•Fe-Mn Acta Materialia 57, 474-482 (2009).•Fe-Cr CALPHAD 25, 527-534 (2001). •Fe-Cu Phys. Rev. B. 71, 184205 (2005). •Fe-Pt J. Materials Research 21, 199-208 (2006). •Fe-Al J. Phys.: Condensed Matters 22, 175702 (2010)•Fe-P J. Phys.: Condensed Matters (2012), in press.•Al-Ni CALPHAD 31, 53 (2007). •Co-Cu J. Materials Research 17, 925-928 (2002). •Co-Pt Scripta Materialia 45, 495-502 (2001). •Cu-Ni CALPHAD 28, 125-132 (2004). •Ni-W J. Materials Research 18, 1863-1867 (2003). •Cu-Ti Mater. Sci. and Eng. A 449-451, 733 (2007).•Cu-Zr J. Materials Research 23, 1095 (2008).•Cu-Zr-Ag Scripta Materialia 61, 801 (2009). •Mg-Al , Mg-Li CALPHAD 33, 650-57 (2009); MSMSE 20, 035005 (2012) . •Ga-In-N J. Phys.: Condensed Matter 21, 325801 (2009).

Documents

Byeong-Joo Lee calphad Interatomic Potentials for Ionic Systems Byeong-Joo Lee POSTECH-CMSE