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Byeong-Joo Lee cmse.postech.ac.kr
Semi-Empirical Atomistic Simulationsin Materials Science and Engineering
Byeong-Joo Lee Byeong-Joo Lee
Pohang University of Science and TechnologyPohang University of Science and Technologycmse.postech.ac.kr cmse.postech.ac.kr
Byeong-Joo Lee cmse.postech.ac.kr
Process Condition
(Atomic Level) Structure Evolution(Atomic Level) Structure Evolution
Materials Properties
R&D in Materials Science and EngineeringR&D in Materials Science and Engineering
Simulation
Simulation
Byeong-Joo Lee cmse.postech.ac.kr
Background & PurposeBackground & Purpose
Many materials properties originate from atomistic structure evolution
• Experimental Examination of Atomistic Structure is now Feasible
→ Need of Atomic Level Simulations
Byeong-Joo Lee cmse.postech.ac.kr
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 cmse.postech.ac.kr
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 cmse.postech.ac.kr
Second Nearest Neighbor Modified EAM (2NN MEAM)Second Nearest Neighbor Modified EAM (2NN MEAM)
• Second Nearest-Neighbor Modified Embedded-Atom Method Potential Byeong-Joo Lee and M.I. Baskes, Phys. Rev. B. 62, 8564-8567 (2000). Formalism of the 2NN MEAM
• Second Nearest-Neighbor Modified Embedded Atom Method Potentials for BCC Transition Metals
B.-J. Lee, M.I. Baskes, H. Kim and Y. K. Cho, Phys. Rev. B. 64, 184102 (2001). Potential for Fe, Cr, Mo, W, V, Nb, Ta
• Semi-Empirical Atomic Potentials for the FCC metals Cu, Ag, Au, Ni, Pd, Pt, Al and Pb based on first and second nearest-neighbor modified embedded atom method
Byeong-Joo Lee, J.-H. Shim and M.I. Baskes, Phys. Rev. B. 68 144112 (2003). Potential for Cu, Ag, Au, Ni, Pd, Pt, Al, Pb
• Potential for other Elements: Ti, Zr, Si, Ge, C, O2
• Potential for Alloy Systems: Fe-Cr, Ni-W, Fe-Cu, Cu-Ni, Fe-Ni, Fe-Pt, Al2O3, SiO2
Byeong-Joo Lee cmse.postech.ac.kr
ContentsContents
• 2NN MEAM for Elements and Alloys2NN MEAM for Elements and Alloys
• ApplicationsApplications
▷ ▷ Phase Diagrams in Nano WorldPhase Diagrams in Nano World ▷ ▷ Computation of Grain Boundary EnergyComputation of Grain Boundary Energy ▷ ▷ Morphology Evolution of Nano StructuresMorphology Evolution of Nano Structures ▷ ▷ Mechanical Property of Nano Crystalline MaterialsMechanical Property of Nano Crystalline Materials ▷ ▷ Irradiation DefectsIrradiation Defects ▷ ▷ Quantum Dot/Wire Formation
Byeong-Joo Lee cmse.postech.ac.kr
Semi-Empirical Atomic Potentials Semi-Empirical Atomic Potentials – EAM/MEAM : General– EAM/MEAM : 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
Byeong-Joo Lee cmse.postech.ac.kr
Evaluation of MEAM Potential ParametersEvaluation of MEAM Potential Parameters
• 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 cmse.postech.ac.kr
MEAM for BCC Transition MetalsMEAM for BCC Transition Metals
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 cmse.postech.ac.kr
MEAM for BCC Transition Metals - Defect PropertiesMEAM for BCC Transition Metals - Defect Properties
Elem. Elem. EEvvff QQ self-interstitial self-interstitial
MEAM exp. MEAM exp. MEAM exp. MEAM exp. EEII structure structure
Fe 1.75 1.79 2.28 2.5 4.23 [110] dumbbell
Cr 1.91 1.80 2.61 3.1 3.90 [110] dumbbell
Mo 3.09 3.10 4.22 4.5 5.97 [110] dumbbell
W 3.95 3.95 5.56 5.5 8.98 [110] dumbbell
V 2.09 2.10 2.47 3.2 2.49 [110] dumbbell
Nb 2.75 2.75 3.32 3.6 2.56 [110] dumbbell
Ta 2.95 2.95 3.71 4.3 4.88 [110] dumbbell
Byeong-Joo Lee cmse.postech.ac.kr
MEAM for BCC Transition Metals - Thermal PropertiesMEAM for BCC Transition Metals - Thermal Properties
Ele. ε (0-100oC) CP(0-100oC) Tm ΔHm ΔVm/Vsolid
MEAM exp. MEAM exp. MEAM exp. MEAM exp. MEAM exp.
Fe 12.4 12.1 26.1 25.5 2200 1811 13.2 13.8 3.4 3.5Cr 9.0 6.5 26.8 24.0 2050 2180 18.8 21.0 4.4 -Mo 5.3 5.1 25.9 24.1 3100 2896 20.1 37.5 3.0 -W 4.2 4.5 25.4 25.4 4600 3695 33.0 52.3 3.2 -V 8.7 8.3 26.1 25.4 1800 2183 11.7 21.5 1.3 -Nb 6.4 7.2 26.1 24.9 1900 2750 13.5 30.0 1.0 -Ta 5.8 6.5 25.7 25.7 3200 3290 22.3 36.6 2.1 -
Byeong-Joo Lee cmse.postech.ac.kr
MEAM for FCC Transition MetalsMEAM for FCC Transition Metals
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 1
7.0 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 cmse.postech.ac.kr
MEAM for Silicon, Germanium, CarbonMEAM for Silicon, Germanium, Carbon
C11 C12 C44 E(100) E(110) E(111) Evf Edia/fcc Edia/hcp Edia/bcc ε
(1012dyne/cm2) (erg/cm2) (eV) (eV) (0-100 oC)
Si 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
Ge 1.31 0.50 0.68 2213 1580 1290 3.04 0.43 0.42 0.40 3.00 1.32 0.50 0.68 900 - 0.43 0.42 0.40 - C 10.8 1.27 5.81 4762 4082 3333 3.19 4.52 4.52 4.03 7.20 10.8 1.27 5.77 - 7.0 4.59 4.48 4.28 1.00
Byeong-Joo Lee cmse.postech.ac.kr
Semi-Empirical Potentials for pure Fe Semi-Empirical Potentials for pure Fe 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
--
2.8665
--
1.192.05-.290.533.92
<111>cr0.69
-0.0520.0052.8673.61211.6
Byeong-Joo Lee cmse.postech.ac.kr
2NN MEAM for Aluminum2NN MEAM for Aluminum
Property MEAM Experiment/FPProperty MEAM Experiment/FP
B 0.794 0.794B 0.794 0.794 ((B/ B/ P)P)0K 0K 4.59 4.72 4.59 4.72 C11 1.143 1.143C11 1.143 1.143 C12 0.619 0.619C12 0.619 0.619 C44 0.316 0.316C44 0.316 0.316 E_fcc/bcc 0.12 0.10E_fcc/bcc 0.12 0.10 H_vac 0.68 0.68H_vac 0.68 0.68 QQDD 1.33 1.28 1.33 1.28 E(100)/(110)/(111) 848/948/629 1085E(100)/(110)/(111) 848/948/629 1085 % relaxation 1.8/-8.9/1.0 1.8/ -8.5±1.0 / 0.9±0.5% relaxation 1.8/-8.9/1.0 1.8/ -8.5±1.0 / 0.9±0.5 E_E_SF(111) SF(111) 141 < 166 141 < 166 MP 937 933MP 937 933 H / H / V_f 11.0/6.7 10.7/6.5V_f 11.0/6.7 10.7/6.5 Cp/Cp/ (0-100°C) 26.2/22.0 24.7/23.5 (0-100°C) 26.2/22.0 24.7/23.5
Byeong-Joo Lee cmse.postech.ac.kr
200K 850K 1000K200K 850K 1000K
MEAM for Fe-Cr Binary SystemMEAM for Fe-Cr Binary System
Byeong-Joo Lee cmse.postech.ac.kr
MEAM for Cu-Ni Binary SystemMEAM for Cu-Ni Binary System
Byeong-Joo Lee cmse.postech.ac.kr
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 cmse.postech.ac.kr
MEAM for Ni-W Binary System MEAM for Ni-W Binary System
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 cmse.postech.ac.kr
Size Effect on the Phase Diagram Size Effect on the Phase Diagram – T. Tanaka et al., Z. Metallkd. 2001.– T. Tanaka et al., Z. Metallkd. 2001.
Byeong-Joo Lee cmse.postech.ac.kr
Size Effect on the Melting Point Size Effect on the Melting Point - J.-H. Shim et al., Surface Science, 2002.- J.-H. Shim et al., Surface Science, 2002.
Surface Facet, Au Size dependence of Melting pointSurface Facet, Au Size dependence of Melting point
0 2 4 6 8 10 12 14 16400
500
600
700
800
900
1000
1100
1200
1300
1400
Present simulation, MEAM Lewis et al. [10], EAM Buffat and Borel [25], Experiment Sambles [26], Experiment
Tem
pera
ture
(K
)
Diameter (nm)
Byeong-Joo Lee cmse.postech.ac.kr
Grain Boundary Energy Grain Boundary Energy – Degree of Freedom– Degree of Freedom
• Misorientation : Misorientation : ▷ ▷ Difference in Crystallographic Directions of two GrainsDifference in Crystallographic Directions of two Grains
- Two degrees of freedom (- Two degrees of freedom (φφ and and θθ))
• Inclination AngleInclination Angle ▷ ▷ Location of Grain Boundary and Direction of Grain Boundary Location of Grain Boundary and Direction of Grain Boundary Normal - Three degrees of freedom (Normal - Three degrees of freedom (hklhkl))
Byeong-Joo Lee cmse.postech.ac.kr
Computation of Grain Boundary Energy Computation of Grain Boundary Energy – Problem in Periodicity
Byeong-Joo Lee cmse.postech.ac.kr
Computation of Grain Boundary Energy Computation of Grain Boundary Energy
Byeong-Joo Lee cmse.postech.ac.kr
Precipitation of Co in a Cu-Co Nano Powder Precipitation of Co in a Cu-Co Nano Powder - J.-H. Shim et al., JMR, 2002.- J.-H. Shim et al., JMR, 2002.
Byeong-Joo Lee cmse.postech.ac.kr
Pure W W + 0.4wt% Ni
Vaccum Annealing
Surface Transition and Alloying Effect Surface Transition and Alloying Effect – N.M. Hwang et al., 2000.– N.M. Hwang et al., 2000.
Byeong-Joo Lee cmse.postech.ac.kr
Surface Transition and Alloying Effect Surface Transition and Alloying Effect
Pure W W-20at%NiPure W W-20at%Ni
Byeong-Joo Lee cmse.postech.ac.kr
Stress Induced Crystallization of Amorphous MaterialsStress Induced Crystallization of Amorphous Materials
Byeong-Joo Lee cmse.postech.ac.kr
Effect of Hydrostatic Pressure Effect of Hydrostatic Pressure
Byeong-Joo Lee cmse.postech.ac.kr
Stress Effect
Hydrostatic Compressive Pressure
Amorphous Crystalline
Hydrostatic Tensile Pressure
Mechanism of Stress Induced Crystallization of Amorphous MaterialsMechanism of Stress Induced Crystallization of Amorphous Materials
Byeong-Joo Lee cmse.postech.ac.kr
Atomistic Approach to Mechanical PropertiesAtomistic Approach to Mechanical Properties
J. Schiøtz et al.,J. Schiøtz et al.,Nature, 1998. Nature, 1998.
Byeong-Joo Lee cmse.postech.ac.kr
Response of pure Al to severe shear strainResponse of pure Al to severe shear strain
Byeong-Joo Lee cmse.postech.ac.kr
Irradiation Effect on Fe and Fe-Cu AlloysIrradiation Effect on Fe and Fe-Cu Alloys
Byeong-Joo Lee cmse.postech.ac.kr
Energetics of Dot Formation during Thin Film Growths : Ge on SiEnergetics of Dot Formation during Thin Film Growths : Ge on Si
Coverage (Dot-size) EII-EI EIII-EI
Double-Layer (5nm) 0.010 eV/atom 0.025 eV/atom Triple-Layer (5nm) 0.017 eV/atom 0.031 eV/atom Triple-Layer (10nm) 0.007 eV/atom 0.016 eV/atom
Double-Layer (10nm) 0.004 eV/atom : (100) side Double-Layer (10nm) 0.002 eV/atom : (110) side Double-Layer (12nm) 0.003 eV/atom : (100) side
I II III
Byeong-Joo Lee cmse.postech.ac.kr
SummarySummary
Performance of 2NN MEAM for Elements and Alloys
• Phase Diagrams in Nano World• Computation of Grain Boundary Energy• Morphology Evolution of Nano Structures• Mechanical Property of Nano Crystalline Materials• Irradiation Defects• Quantum Dot/Wire Formation
Byeong-Joo Lee cmse.postech.ac.kr
Semi-Empirical Atomic Potentials Semi-Empirical Atomic Potentials – EAM/MEAM : General– EAM/MEAM : 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
Byeong-Joo Lee cmse.postech.ac.kr
Semi-Empirical Atomic Potentials Semi-Empirical Atomic Potentials – EAM : Electron Density– EAM : Electron Density
i ijijijii rFE
)(
)(2
1)(
Byeong-Joo Lee cmse.postech.ac.kr
Semi-Empirical Atomic Potentials Semi-Empirical Atomic Potentials – MEAM : Electron Density– MEAM : Electron Density
+ 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
Byeong-Joo Lee cmse.postech.ac.kr
Semi-Empirical Atomic Potentials Semi-Empirical Atomic Potentials – MEAM : Electron Density– MEAM : Electron Density
+ 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 cmse.postech.ac.kr
Semi-Empirical Atomic Potentials Semi-Empirical Atomic Potentials – MEAM : Embedding Function– MEAM : 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)
Byeong-Joo Lee cmse.postech.ac.kr
Semi-Empirical Atomic Potentials Semi-Empirical Atomic Potentials – MEAM : Universal EOS– MEAM : 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
/
Byeong-Joo Lee cmse.postech.ac.kr
Semi-Empirical Atomic Potentials Semi-Empirical Atomic Potentials – MEAM : Universal EOS– MEAM : Universal EOS
i ijijijii rFE
)(
)(2
1)(
*
)1()()())(( *21 a
cuo
eaErErrF
))](()([2
)(1
rFrEZ
rou
Byeong-Joo Lee cmse.postech.ac.kr
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 cmse.postech.ac.kr
Semi-Empirical Atomic Potentials Semi-Empirical Atomic Potentials – 2NNMEAM – 2NNMEAM
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 cmse.postech.ac.kr
Semi-Empirical Atomic Potentials Semi-Empirical Atomic Potentials – 2NNMEAM for Alloy Systems– 2NNMEAM for Alloy Systems
i ijijijii rFE
)(
)(2
1)(
Byeong-Joo Lee cmse.postech.ac.kr
Evaluation of MEAM Potential ParametersEvaluation of MEAM Potential Parameters
• 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 cmse.postech.ac.kr
MEAM for CopperMEAM for Copper
Property MEAM Experiment/FPProperty MEAM Experiment/FP
B 1.420 1.420B 1.420 1.420 ((B/ B/ P)P)0K 0K 4.95 5.25 4.95 5.25 C11 1.762 1.762C11 1.762 1.762 C12 1.249 1.249C12 1.249 1.249 C44 0.818 0.818C44 0.818 0.818 E_fcc/bcc 0.08 0.04E_fcc/bcc 0.08 0.04 H_vac 1.11 1.03-1.19H_vac 1.11 1.03-1.19 QQDD 2.19 2.09 2.19 2.09 E(100)/(110)/(111) 1382/1451/1185 1770E(100)/(110)/(111) 1382/1451/1185 1770
% relaxation -2.5/-9.3/-3.1 -1.1% relaxation -2.5/-9.3/-3.1 -1.1±±.4/-8.5.4/-8.5~-10~-10/-0.7±.5/-0.7±.5 E_E_SF(111) SF(111) 42 40 42 40~48, ~48, <80<80 MP 1602 1358MP 1602 1358 H / H / V_f 18.6/7.7 13.3/4.2V_f 18.6/7.7 13.3/4.2 Cp/Cp/ (0-100°C) 24.9/17.0 24.5/17.0 (0-100°C) 24.9/17.0 24.5/17.0
Byeong-Joo Lee cmse.postech.ac.kr
MEAM for TungstenMEAM for Tungsten
Property MEAM Experiment/FPProperty MEAM Experiment/FP
B 3.14 3.14B 3.14 3.14 ((B/ B/ P)P)0K 0K 4.79 4.50 4.79 4.50 C11 5.326 5.326C11 5.326 5.326 C12 2.050 2.050C12 2.050 2.050 C44 1.631 1.631C44 1.631 1.631 E_fcc/bcc 0.26 0.20E_fcc/bcc 0.26 0.20
E_hcp/fcc -0.047 -0.047E_hcp/fcc -0.047 -0.047 H_vac 3.95 3.95H_vac 3.95 3.95 QQDD 5.56 5.5 5.56 5.5 E(100)/(110)/(111) 3900/3427/4341 2990, 3468E(100)/(110)/(111) 3900/3427/4341 2990, 3468 % relaxation -3.2/-3.0/-13.2 -3.1/ -5.9/ -15.1% relaxation -3.2/-3.0/-13.2 -3.1/ -5.9/ -15.1 MP MP ∼∼4600 36954600 3695
H H //V_fV_f 33.0/3.2 52.3/ - 33.0/3.2 52.3/ - Cp/Cp/ (0-100°C) 25.4/4.2 25.4/4.5 (0-100°C) 25.4/4.2 25.4/4.5
Byeong-Joo Lee cmse.postech.ac.kr
2NN MEAM for Nickel2NN MEAM for Nickel
Property MEAM Experiment/FPProperty MEAM Experiment/FP
B 1.876 1.876B 1.876 1.876 ((B/ B/ P)P)0K 0K 4.90 4.89 4.90 4.89 C11 2.612 2.612C11 2.612 2.612 C12 1.508 1.508C12 1.508 1.508 C44 1.317 1.317C44 1.317 1.317 E_fcc/bcc 0.16 0.11E_fcc/bcc 0.16 0.11 H_vac 1.51 1.60H_vac 1.51 1.60 QQDD 2.98 2.87 2.98 2.87 E(100)/(110)/(111) 1943/2057/1606 2240E(100)/(110)/(111) 1943/2057/1606 2240 % relaxation -1.8/-6.0/-1.4 -3.2% relaxation -1.8/-6.0/-1.4 -3.21.1/ -8 ±1 / -1 ±11.1/ -8 ±1 / -1 ±1 E_E_SF(111) SF(111) 125 125 125 125 MP 2013 1728MP 2013 1728 H / H / V_f 24.6/9.1 17.5/4.5V_f 24.6/9.1 17.5/4.5 Cp/Cp/ (0-100°C) 25.4/12.6 26.5/13.3 (0-100°C) 25.4/12.6 26.5/13.3
Byeong-Joo Lee cmse.postech.ac.kr
2NN MEAM for Nickel2NN MEAM for Nickel
Property MEAM Experiment/FPProperty MEAM Experiment/FP
B 1.876 1.876B 1.876 1.876 ((B/ B/ P)P)0K 0K 4.90 4.89 4.90 4.89 C11 2.611 2.612C11 2.611 2.612 C12 1.508 1.508C12 1.508 1.508 C44 1.317 1.317C44 1.317 1.317 E_fcc/bcc 0.19 0.11E_fcc/bcc 0.19 0.11 H_vac 1.60 1.60H_vac 1.60 1.60 QQDD 2.92 2.90 2.92 2.90 E(100)/(110)/(111) 2715/2660/2238 2240E(100)/(110)/(111) 2715/2660/2238 2240 % relaxation -0.8/-6.3/+0.2 -3.2% relaxation -0.8/-6.3/+0.2 -3.21.1/ -8 ±1 / -1 ±11.1/ -8 ±1 / -1 ±1 E_E_SF(111) SF(111) 125 125 125 125 MP 1900 1728MP 1900 1728 H / H / V_f 16.5/7.4 17.5/4.5V_f 16.5/7.4 17.5/4.5 Cp/Cp/ (0-100°C) 25.6/12.7 26.5/13.3 (0-100°C) 25.6/12.7 26.5/13.3