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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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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Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
Interatomic Potential for Ionic Interatomic Potential for Ionic MaterialsMaterials
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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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
Many-Body Potential Many-Body Potential : EAM – 2NN MEAM
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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]
Byeong-Joo Lee www.postech.ac.kr/~calphad
Many-Body Potential Many-Body Potential : Tersoff
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• 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
Byeong-Joo Lee www.postech.ac.kr/~calphad
• 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
Byeong-Joo Lee www.postech.ac.kr/~calphad
Charge Equilibration ModelCharge Equilibration Model
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• 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
Byeong-Joo Lee www.postech.ac.kr/~calphad
200
2
1)0()( iiiiiii qJqEqE
jiji
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jiji
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jiji
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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
Byeong-Joo Lee www.postech.ac.kr/~calphad
jiji
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- 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
Byeong-Joo Lee www.postech.ac.kr/~calphad
• 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
Byeong-Joo Lee www.postech.ac.kr/~calphad
SummarySummary
Up to now no interatomic potential for ionic + covalent + metallic alloy systems
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
ReferencesReferences
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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).
1. B. J. Alder and T. E. Wainwright, ”Studies in Molecular Dynamics. I. General Method,” J. Chem. Phys. 31, 459 (1959).2. L. V. Woodcock, “Isothermal Molecular Dynamics Calculations for Liquid Salts,” Chem. Phys. Lett. 10, 257 (1971).3. A. Rahman, R. H. Fowler and A. H. Narten, ”Structure and Motion in Liquid BeF2, LiBeF3, and LiF from Molecular Dynamics Calculations,” J. Chem. Phys. 57, 3010 (1972).4. L. V. Woodcock, Advances in Molten Salts Chemistry, Vol. 3 Chap. 1, pp.1-75, Plenum, New York (1975).5. L.V. Woodcock, C.A. Angell and P. Cheeseman, “Molecular Dynamics Studies of the Vitreous State: Simple Ionic Systems and Silica,” J. Chem. Phys. 65, 1565 (1976).6. B.P. Feuston and S.H. Garofalini, “Empirical Three-Body Potential for Vitreous Silica,” J. Chem. Phys. 89, 5818 (1988).7. S. Tsuneyuki, M. Tsukada, H. Aoki and Y. Matsui, “First-Principles Interatomic Potential of Silica Applied to Molecular Dynamics,” Phys. Rev. Lett. 61, 869 (1988).8. B.W.H. van Beest, G.J. Kramer and R.A. van Santen, “Force Fields for Silicas and Aluminophosphates Based on Ab Initio Calculations,” Phys. Rev. Lett. 64, 1955 (1990). 9. C. R. A. Catlow and A.M. Stoneham, “Ionicity in Solids,” J. Phys. C: Solid State Phys. 16, 4321 (1983).10. J. R. Tessman, A. H. Kahn and W. Shockley, “Electronic Polarizabilities of Ions in Crystals,” Phys. Rev. 92, 890 (1953).11. Z. Jiang and R. A. Brown, “Modeling Oxygen Defects in Silicon Crystals using an Empirical Interatomic Potential,” Chem. Engin. Sci. 49, 2991 (1994).12. R. Soulairol and F. Cleri, “Interface Structure of Silicon Nanocrystals Embedded in an Amorphous Silica Matrix,” Solid State Sciences 12, 163 (2010). 13. B .G. Dick and A. W. Overhauser, “Theory of the Dielectric Constants of Alkali Halide Crystals,” Phys. Rev. 112, 90 (1958).14. G. J. Dienes, D. O. Welch, C. R. Fischer, R. D. Hatcher, O. Lazareth and M. Samberg, “Shell-Model Calculation of Some Point-Defect Properties in α-Al2O3,” Phys. Rev. B 11, 3060 (1975). 15. G. V. Lewis and C. R. A. Catlow, “Potential Models for Ionic Oxides,” J. Phys. C: Solid State Phys. 18, 1149-1161 (1985).16. P. Vashishta, R. K. Kalia, J. P. Rino and I. Ebbsjö, “Interaction Potential for SiO2: A Molecular-Dynamics Study of Structural Correlations,” Phys. Rev. B 41, 12197 (1990).17. P. Tangney and S. Scandolo, “An Ab Initio Parametrized Interatomic Force Field for Silica,” J. Chem. Phys. 117, 8898 (2002).18. D. Herzbach, K. Binder and M.H. Müser, “Comparison of Model Potentials for Molecular-Dynamics Simulations of Silica,” J. Chem. Phys. 123, 124711 (2005).19. J. D. Gale and A.L. Rohl, “The General Utility Lattice Program (GULP),” Mol. Simul. 29, 291 (2003).20. J. H. Rose, F. Ferrante and J. R. Smith, “Universal Binding Energy Curves for Metals and Bimetallic Interfaces,” Phys. Rev. Lett. 47, 675 (1981).21. J. R. Smith, F. Ferrante and J. H. Rose, “Universal Binding-Energy Relation in Chemisorptions,” Phys. Rev. B 25, 1419 (1982).22. J. H. Rose, F. Ferrante and J. R. Smith, “Universal features of bonding in metals,” Phys. Rev. B 28, 1835 (1983).23. J. H. Rose, J. R. Smith, F. Guinea and F. Ferrante, “Universal Features of the Equation of State of Metals,” Phys. Rev. B 29, 2963 (1984).24. J. Tersoff, “New Empirical Model for the Structural Properties of Silicon,” Phys. Rev. Lett. 56, 632 (1986)25. J. Tersoff, “New Empirical Approach for the Structure and Energy of Covalent Systems,” Phys. Rev. B 37, 6991 (1988).26. J. Tersoff, “Empirical Interatomic Potential for silicon with improved elastic properties,” Phys. Rev. B 38, 9902 (1988).27. J. Tersoff, “Empirical Interatomic Potential for Carbon with Applications to Amorphous Carbon,” Phys. Rev. Lett. 61, 2879 (1988).28. J. Tersoff, ”Modeling Solid-State Chemistry: Interatomic Potentials for Multicomponent Systems,” Phys. Rev. B 39, 5566 (1989). 29. M. S. Daw and M. I. Baskes, “Semiempirical, Quantum Mechanical Calculation of Hydrogen Embrittlement in Metals,” Phys. Rev. Lett. 50, 1285 (1983).30. M. 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).
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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)
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
EAM/MEAMEAM/MEAM – General – General
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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
Byeong-Joo Lee www.postech.ac.kr/~calphad
EAM/MEAM EAM/MEAM – Embedding Function– Embedding Function
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Byeong-Joo Lee www.postech.ac.kr/~calphad
EAM/MEAM EAM/MEAM – Universal EOS– Universal EOS
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Byeong-Joo Lee www.postech.ac.kr/~calphad
EAM/MEAM EAM/MEAM – Electron Density for EAM– Electron Density for EAM
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Byeong-Joo Lee www.postech.ac.kr/~calphad
EAM/MEAM EAM/MEAM – Electron Density for MEAM– Electron Density for MEAM
+ Angular contribution
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3
5
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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)
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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, ...
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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)
Byeong-Joo Lee www.postech.ac.kr/~calphad
2NN MEAM 2NN MEAM – – 2NNMEAM for Alloy Systems2NNMEAM for Alloy Systems
i ijijijii rFE
)(
)(2
1)(
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
MEAM for Cu-Ni Binary System MEAM for Cu-Ni Binary System – B.-J. Lee and J.-H. Shim, CALPHAD, 2004
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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)
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
(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).
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
MEAM for CarbonMEAM for Carbon – for several structures– for several structures
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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
Byeong-Joo Lee www.postech.ac.kr/~calphad
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.
Byeong-Joo Lee www.postech.ac.kr/~calphad
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).