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Interatomic Potentials for Ionic Systems

Interatomic Potentials for Ionic Systems. Byeong-Joo Lee POSTECH-CMSE. Background. Importance of Ionic Materials Sensor, Battery, Devices, Metal Surfaces, etc. Need to handle “ionic + covalent + metallic” materials Interfacial Reaction between metals and SiO2 substrate

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Interatomic Potentials for Ionic Systems

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  1. Interatomic Potentials for Ionic Systems Byeong-Joo Lee POSTECH-CMSE

  2. Background • Importance of Ionic Materials • Sensor, Battery, Devices, Metal Surfaces, etc. • Need to handle “ionic + covalent + metallic” materials • Interfacial Reaction between metals and SiO2 substrate • Diffusion of metallic atoms in amorphous SiO2 • Atomistic simulation on “ionic + covalent + metallic” materials • ???

  3. Purpose and Scope • Development of Interatomic Potential Model that covers • “ionic + covalent + metallic” materials, simultaneously. • Review interatomic potentials for ionic and hybrid materials • Propose possible form of an interatomic potential formalism

  4. Outline • Interatomic Potential for Ionic Materials • Point Charge Model • Polarization (Shell Model) • Many-Body Potentials • Tersoff • EAM – MEAM – 2NN MEAM • Many-body potentials used for ionic systems • Many-Body Potentials for Ionic Materials • Charge Equilibration Model • EAM + Qeq • Tersoff + Qeq • Proposal of New Interatomic Potential Form

  5. Interatomic Potential for Ionic Materials Fixed Point Charge Born-Mayer-Huggins TTAM BKS • Initially applied to liquid orglass, 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 onα-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

  6. Interatomic Potential for Ionic Materials Fixed Point Charge + electronic polarization • Include dipole-charge, dipole-dipole interaction due to electronic polarization • Shell Model by Dick & Overhauser [13], 1958 • Ion = core electroncore + valence electron shell • Deviation of Center of mass of Shell causes adipole • Shell connected to core by an artificial spring and interact throughharmonic restoring force • Shell Model has been successful for diatomic molecule, alkali halidesand 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

  7. Many-Body Potential : EAM – 2NN MEAM • 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]

  8. Many-Body Potential : Tersoff • 1985Abell : Close relation between Morse-typepair 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

  9. Many-Body Potential for Ionic Materials • 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

  10. Charge Equilibration Model • 1991Rappe &Goddard [48] : based on previous concepts on electronegativity, equilibration [49-57]. • - equilibrium charge in molecules • consideringCoulomb interaction and penalty energy for charged isolated atoms (atomic self-energy) • IP &EA : ionization potential과 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 onatomic charge density expressed for aSlater-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

  11. EAM + Charge Equilibration • 1994Streitz & 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 Oxidation of Al nano cluster[61,62]

  12. Tersoff + Charge Equilibration • 1996 Yasukawa [76] : introduce atomic energy ΣiΦi& Coulomb energy ½ΣiΣjEIONij - effective point charge with cutoff functionin 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 - cosθoOSiO)2, • - COMB (Optimized Many-Body Potential), but cannot generate amorphous SiO2 & bad results for α-quartz • 2010modified version for SiO2 [81] : Slater 1s orbital type Coulomb integral &Ewald + anotherpenalty term • - applicable to α, β-quartz, α, β-cristobalite, β-tridymite, Coesite, Stishovite, generally worse thanTTAM • 2010 Hf/HfO2, Cu/Cu2O [83,84] : different bond-bending form depending on cation element

  13. Others : ReaxFF • 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

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

  15. Potential for Ionic+Covalent+Metallic Materials • Charge Effect ? • Correct physics : easy parameterization and goodtrasferability • Point Charge vs. Charge Distribution ? • TTAM that considered charge distribution could describe the SiO2polymorphs for the first time • Shell Model ? • No publication for shell model + many-body potential • Variable 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 polymorphs • Coulomb 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 potentialhas 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+Qeq • Paying 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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  17. Atomistic Simulations- MEAM & Applications Byeong-Joo Lee Dept. of MSE Pohang University of Science and Technology (POSTECH) calphad@postech.ac.kr

  18. Pair Potentials (~1980) ▷ Elastic Constants are NOT correctly reproduced Many Body Potentials (1980's) ▷ Embedded Atom Method (EAM: 1983) ▷ Finnis and Sinclair Potential (1984) ▷ Glue Model (1986) ▷ Equilivalent-Crystal Model (1987) Semi-Empirical Atomic Potentials - Historical Background

  19. EAM Potentials (1983, M.S. Daw and M.I. Baskes) ▷ Successful mainly for FCC elements - many other many-body potentials show similar performance 1NN MEAM Potentials (1987,1992, M.I. Baskes) ▷ Show Possibility for description of various structures - important to be able to describe multi-component system 2NN MEAM Potentials (2000, B.-J. Lee & M.I. Baskes) ▷ Applicable to fcc, bcc, hcp, diamond structures and their alloys Semi-Empirical Atomic Potentials – History of Development

  20. EAM/MEAM – General E : Total Potential Energy F : Embedding Energy : Electron Density (Considering Bonding Directionality)  : Pair Interaction Energy

  21. EAM/MEAM – Embedding Function M.I. Baskes et al., Phys. Rev. B, 40, 6085 (1989)

  22. EAM/MEAM – Universal EOS J.H. Rose et al., Phys. Rev. B, 29, 2963 (1984)

  23. EAM/MEAM – Electron Density for EAM

  24. EAM/MEAM – Electron Density for MEAM + Angular contribution

  25. EAM/MEAM – Electron Density for MEAM + Angular contribution with ti(0) =1

  26. EAM/MEAM – 1st Nearest Neighbor MEAM

  27. 1NN MEAM vs. 2NN MEAM –Many-Body Screening Xik=(Rik/Rij)2 and Xkj=(Rkj/Rij)2 Cmax Cmin j i fc(x) = 1 x 1 0  x  1 0x  0

  28. 2NN MEAM – Computation of pair-wise potential

  29. Evaluation of MEAM Potential Parameters for Elements Ec, Re, B, A, d,  (0),  (1),  (2),  (3), t(1), t(2), t(3), Cmax, Cmin ▷ Cohesive Energy of Stable and Metastable Structure ▷ Nearest Neighbor Distance ▷ Bulk Modulus, Elastic Constants (C11, C12, C44) ▷ Stacking Fault Energy ▷ Vacancy Formation Energy ▷ Surface Energy

  30. Elastic Constants ▷ B, C11, C12, C44, ... Defect Energy ▷ Surface Energy ▷ Heat of Vacancy Formation, … Structural Energy ▷ Energy and Lattice Parameters in Different Structures Thermal Property ▷ Specific Heat ▷ Thermal Expansion Coefficient ▷ Melting Temperature, ... Semi-Empirical Atomic Potentials - Performance

  31. Elem. C11 C12 C44 E(100) E(110) E(111) Evf Ebcc/fcc Efcc/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 BCC Transition Metals – B.-J. Lee et al., PRB, 2001

  32. Elem. C11 C12 C44 E(100) E(110) E(111) Evf Ebcc/fcc Efcc/hcp ε(0-100oC) 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 FCC Transition Metals – B.-J. Lee et al., PRB, 2003

  33. C11 C12 C44E(100)E(110)E(111)EvfEdia/fccEdia/hcpEdia/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 MEAM for Silicon

  34. 2NN MEAM Interatomic Potentials – for Al and Fe

  35. 2NN MEAM – 2NNMEAM for Alloy Systems

  36. 2NN MEAM for Alloy Systems – Optimization of Potential Parameter, Fe-Pt

  37. 2NN MEAM for Fe-Cr Binary System – B.-J. Lee et al., CALPHAD, 2001 200K 850K 1000K

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

  39. MEAM for Ni-Si Binary System Dilute Heat of Solution (eV/atom) Si in (Ni) -1.50 (-1.37) Ni in (Si) +0.50 Ni3Si 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) NiSi2 0.28 (0.28) 5.391 (5.406) 1.93 (1.60) 2.39 1.69 0.70 (0.58) 0.32 8.0 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)

  40. MEAM for Co-Pt Binary System - 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 970-980 760-770 Temperature (K) 1000 1100 840

  41. MEAM for Ni-W Binary System –J.-H. Shim et al., 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

  42. MEAM for Ni-W Binary System a (Å) c (Å) Ec(eV) B(Gpa) Ni4W (D1a) 5.73 3.553 5.36 292 5.733.5535.40293 Ni3W (L12) 3.62 - 5.58 319 3.58 - 5.65287 Ni3W (D019) 2.56 4.05 5.59 316 2.53 - 5.42289 NiW3 (L12) 3.86 - 7.29 316 3.84 - 7.55283 NiW3 (D019) 2.76 4.44 7.36 321 2.76 - 7.70304

  43. Fe ▷ Finnis-Sinclair – modified by Calder and Bacon (1993) Fe-Cu ▷ Osetsky (1996) Fe: Pair-Potential, Osetsky (1995) Cu: Pair-Potential, Osetsky (1995) ▷ Ackland, Bacon, Calder (1997) Fe: F-S type, Ackland et al. (1997) Cu: F-S type, Ackland, Tichy, Vitek, Finnis (1987) ▷ Ludwig, Farkas,.. (1998) → C.S. Becquart, C. Domain, Fe: EAM, Simonelli, Pasianot, Savino(1993) Cu: EAM, Voter (1993) Empirical Potentials for Multicomponent Systems

  44. History 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

  45. History 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

  46. 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 Fe, C, N, Fe-C and Fe-N systems

  47. 2NN MEAM for pure Fe- PRB 64, 184102 (2001); 71, 184205 (2005)

  48. MEAM for Carbon– Physical Property of Diamond

  49. MEAM for Carbon– Physical Property of Graphite

  50. MEAM for Carbon– for several structures

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