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Atomistic and Mesoscale Simulations

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Atomistic and mesoscale simulations of free solidification in comparison

Solidification of an undercooled Lennard-Jones system is considered by atomistic and mesoscale simulations. The influence of the parameters of a Nosé–Hoover thermostat on the temperature profile in the molecular dynamics box during the free solidification of the sample is analyzed. Direct comparison of the temperature profiles and of the interface dynamics in molecular dynamics with phase-field simulations is given.

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Atomistic and Mesoscale Simulations

  1. 1. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 1 Atomistic and Mesoscale Simulations Quantum Mechanics Molecular Dynamics Quasi-continuum Mesoscale parameters Domain walls Defects Domain patterns Grains Ferroelectric film Connect from First Principles (QM) to Materials Applications (Films)
  2. 2. Atomistic and Mesoscale Simulations of Ferroelectric Materials Atomistic (quantum mechanics-> force fields -> molecular dynamics) William A. Goddard III, Materials and Process Simulation Center Mesoscale (molecular dynamics -> mesoscale -> macro models) Tahir Cagin, Materials and Process Simulation Center, Caltech Current Team: Alejandro Strachan, Qingsong Zhang, Mustafa Uludogan 2
  3. 3. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 3 Multi scale modeling EOS of different phases, vacancy energy, surface energy hours minutes seconds microsec nanosec picosec femtosec time distanceÅ nm micron mm cm meters MESO Continuum (FEM) QM MD MD simulations of domain walls, cracks dislocations Work/volume; Fracture; Fatigue ELECTRONS ATOMS DOMAINS GRIDS First principles Force Fields Constitutive relations: mechanism of domain wall motion, plastic deformation 3
  4. 4. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 4 Atomistic simulations: Goddard • First Principles Force Fields [Reax FF describes chemistry (reactions)] Based on accurate ab-initio (DFT-GGA) data: - EOS of different structures in a wide pressure range - Crystal defects (domain walls, surfaces, etc.) - Atomic charges and polarizability of relevant molecules - Bond dissociation and transition states for reactions • Molecular Dynamics simulations using ReaxFF - obtain structures, energetics, and mobility of domain walls (180º and 90º) - Deformation of samples containing domain walls (uniaxial) - Dislocation activity in domain walls - Oxygen migration at surfaces and on domain walls - Crack nucleation and propagation along domain walls 4
  5. 5. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 5 Mesoscale modeling: Cagin • First Principles Mesoscale Models Based on accurate Molecular Dynamics data: - Provides constituitive equation describing behavior over wide range of stress and temperature - Describes deformation and mobility of domain walls as function stress and Temp - Includes effect of Chemistry, charge transfer, polarizability - Includes effect of Dislocations, Oxygen migration, Crack nucleation and propagation • Couple to Macroscopic Continuum models of deformation and failure 5
  6. 6. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 6 Strategy for First Principles Force Field for BaTiO3 • Ab-initio data for various crystalline phases (different coordinations) in a wide pressure range (10% expansion to 50% compression) - BaO: B1 (NaCl - coord: 6), B8 (NiAs - coord 6), B2 (CsCl -coord: 8), B4 (GaAs - coord 4) - TiO2: Rutile; Anatase, Brookite (Coord 6), quartz (coord 4) - BaTiO3: cubic, tetragonal, orthorhombic, and rhombohedral - 180º domain wall energy and structure • Ab-initio charges (Mulliken) and polarizabilities of molecules for finite systems - Clusters of TiO2, BaO • Bond dissociation curves - Single and double bonds: Ti-O, Ba-O, O-O Require that One FF reproduces all the ab-initio data (ReaxFF) Not obvious that this is possible, but we have demonstrated it 6
  7. 7. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 7 ReaxFF FeaturesFeatures •Based on First Principles •Allows Bond Order to change as distances and coordination change •Completely defined in terms of geometry •Depends only on the elements (do not treat O differently in H2CO or BaTiO3) •Describes Reactions, Surfaces,Defects •Allows charges to flow depending on environment and applied fields (QEq) •Allows each atom to Polarize in response to local and applied fields Energy termsEnergy terms •Valence based Terms (EBO) • Bond distance → Bond order • Bond order → Bond energy •Nonbond Terms •Extended Shell Terms (ESH) •Pauli Repulsion and Dispersion Terms (EMS) •Pauli Principle and Valency (sum of Bond Orders gives valency) MSSHBO EEEE ++= 7
  8. 8. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 8 Valence Terms IDEA: Bond Distance →Bond Order Bond Order → Bond Energy BO r M n n r r E E BO E E BO D BO BO BO Val E D ij ij r r n ij o BO ij VB ij i j i oc i i ij VB ij VB ij VB ij i ij j i i i oc i oc oc i ij o ( ) ( ) exp( ) ( ) ( ) ( ) exp( ) ( ) [exp( ) ] ( ) = ⋅ ⋅ − ⋅ = + = − ⋅ ⋅ − ⋅ = − = ⋅ ⋅ − ≠ ≠ ∑ ∑ ∑ ∆ ∆ ∆ ∆ α α 1 Bond order vs. bond distance Bond energy vs. bond distance Valency (sum bond orders) 8
  9. 9. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 9 Extended Shell Terms: IDEA: Atomic Core: Single Gaussian with Fixed Charge (+4 for Ti) Electron Shell: Single Gaussian with Variable Charge equalized by QEq ρ η ρ η η π η π i c i c i c i c i s i s i s i s r Q r r r Q r r i c i s ( ) ( ) exp( | | ) ( ) ( ) exp( | | ) v v v v v v = − ⋅ − = − ⋅ − 3 2 3 2 2 2 9
  10. 10. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 10 Electrostatics Energy: E E Q E Q Q E Q E Q Q J Q Q E Q Q SH i self i N i s ij kl ij kl bal i j l c sk N j c si N k l i self i s io io s i c i s io s i c i s ij kl bal i j k l Erf = + = + ⋅ + + ⋅ + = = ≠ ==== ∑ ∑∑∑∑1 11 1 2 ( ) ( ) ( , ) ( ) ( ) ( ) ( , ) , ,, , ( χ ηη η η η ij kl ij kl ij kl ij kl r r i j k l Q Q ⋅ + ⋅ ⋅ ⋅ , , ) QEq Method (3 universal parameter per atom: χ, J, η0) ∂ ∂ ∂ ∂ ∂ ∂ = = = = = = −∑ ∑ Q SH Q SH Q SH i s i N total i c i N s s N sE E E Q Q Q 1 2 1 1 ... ⇒ Q Q Qs s N s 1 2, ... 10
  11. 11. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 11 Energy (3 parameters per ij: R0, D0, α) E E r E r D MS ij MS i j ij ij MS ij MS MS r r MS r r ij o ij o = = − − ⋅ −       ≠ ∑ ( ) ( ) exp[ ( )] exp[ ( )]α α 1 2 2 1 Distance energy Pauli Principle and Dispersion Terms 11
  12. 12. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 12 d dr Q dr Q r Q dr Q dr r Q dEi c i c i N i s i s i s i N r i s j c j N r i s j s i i N E i s j c j s v v v v v v v v v v vµ = ⋅ + ⋅ + ⋅ ∇ ⋅ + ∇ ⋅ + ⋅ ∇ ⋅ = = = = ∑ ∑ ∑ ∑( ) ( ) ( ) 1 1 1 1 Dipole Moment: α µβγ β β γ β β γ β γ γ γ = = ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ∂ ∂ − == ∂ ∂ − === = ∂ ∂∑∑ ∑∑∑ ∑ E i j ij kl j l c si k N k l i s r i s is kl l c sk N k l i N i s i N E i s Q H Q r Q H Q r Q k l v 1 1 1 11 1 , , ,, , , Polarizability: ε δ αβγ βγ π βγ= + ⋅4 V Dielectric Constant: Polarization in Ferroelectrics 12
  13. 13. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 13 T=0 K BaO: pressure induced phase transitions Pressure induced phase transitions B1 B8 B8 dist. B2 dist. B2 B2 P = 11.13 GPa (exp 9.2 GPa) P = 21.5 GPa (exp 14-18 GPa) P = 62 GPa 13
  14. 14. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 14 T=0 K Barium Oxide EOS: DFT-GGA Volume (Å3) Energy (eV) Pressure (GPa) Volume (Å3) B1 NaCl B8 NiAs Dist. B2 B2 (CsCl) B1 (this work) 42.898 11.01 71.22 44.85 39.78 B1 (exp) 42.141 10.15 75.8 45.5 36 V0 (Å3) Ec (eV) BT (GPa) Cs C44 14
  15. 15. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 15 T=0 K BaO B1 to B8 to dist B2 transitions Pressure (GPa) enthalpy(eV) Pressure (GPa) enthalpy(eV) B1 → B8 @ 11.13 GPa B8 → dist. B2 @ 21.5 GPa B1 B8 B8 Dist B2 15
  16. 16. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 16 T=0 K BaO dist-B2 to cubic B2 phase transition Dist. B2 → B2 @ 62 Gpa (continuous Phase transition) Pressure (GPa) enthalpy(eV) B1 → B8 3.6% vol. change B8 → dist B2 8.7% vol. change Dist. B2 → B2 0% vol. change (continuous Phase transition) Pressure (GPa) Volume(Å3) 16
  17. 17. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 17 T=0 K TiO2 EOS 3 phases: DFT-GGA Rutile(P 42/mnm) Exp(Ref1,2) DFT-GGA Error(%) a (Ang) 4.594 4.621289 0.59 b (Ang) 4.594 4.621289 0.59 c (Ang) 2.959 2.962367 0.11 K (Gpa) 210 192 -8.57 E(kcal/mol) -57389 Anatase (I 41/amd) Exp(Ref3) DFT-GGA Error(%) a (Ang) 3.785 3.771 -0.37 b (Ang) 3.785 3.771 -0.37 c (Ang) 9.512 9.765 2.66 K (Gpa) 194 E(kcal/mol) -57387.55 Brookite (P bca) Exp(Ref4) DFT-GGA Error(%) a (Ang) 9.174 9.216 0.46 b (Ang) 5.449 5.478 0.53 c (Ang) 5.138 5.161 0.45 K (Gpa) 205 E(kcal/mol) -57391.36 Rutile phase Anatase phase Brookite phase 17
  18. 18. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 18 Rutile:Homogeneous Expansion/Compression -200 0 200 400 600 800 0 50 100 150 V/Vo (%) Stress(GPa) Stress[1,1] Stress[3,3] Rutile:Uniaxial Expansion/Compression in z direction (keep a=b, Vol=const) -40 -20 0 20 40 60 80 70 80 90 100 110 120 130 c/a Stress Stress[1,1] Stress[3,3] Rutile:Uniaxial Expansion/Compression in y direction (keep c=const, Vol=const) -30 -20 -10 0 10 20 0 50 100 150 b/a (%) Stress(GPa) Stress[1,1] Stress[2,2] Stress[3,3] Rutile:Shear in z-plane (Keep a/c=b/c, Vol=const) 0 20 40 60 90 95 100 105 110 Gamma (Degree) Stress[1][2](GPa) QM Deformation of TiO2 Rutile Phase Rutile:Shear in x-plane (Keep b/a=c/a, Vol=const) -10 0 10 20 30 40 90 95 100 105 110 115 120 Alpha (Degree) Stress[2][3](GPa) 18
  19. 19. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 19 Anatase:Homogeneous Expansion/Compression -20 -10 0 10 20 30 40 90 95 100 105 110 V/Vo (%) Stress(GPa) Stress[1,1] Stress[2,2] Stress[3,3] Brookite:Homogeneous Expansion/Compression -10 -5 0 5 10 15 94 96 98 100 102 104 106 V/Vo (%) Stress(GPa) Stress[1,1] Stress[2,2] Stress[3,3] QM Deformation of TiO2 Anatase/Brookite Phases 19
  20. 20. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 21 BaTiO3 phasesRhombohedral a=b=c; α=β=γ≠90 Ti: (0.5+ ∆Ti, 0.5+ ∆Ti, 0.5+∆Ti) O: (0.5+∆O1, 0.5+∆O1, 0.0+∆O2) O: (0.5+∆O1, 0.0+∆O2, 0.5+∆O1) O: (0.0+∆O2, 0.5+∆O1, 0.5+∆O1) Orthorhombic a≠b≠c; α=β=γ=90 Ti: (0.5, 0.5 + ∆Ti, 0.5+∆Ti) O: (0.5, 0.5+∆O1, 0.0+∆O2) O: (0.5, 0.0+∆O2, 0.5+∆O1) O: (0.0, 0.5+∆O3, 0.5+∆O3) Tetragonal a=b≠c; α=β=γ=90 Ti: (0.5, 0.5, 0.5+∆Ti) O: (0.5, 0.5, 0.0+∆O1) O: (0.5, 0.0, 0.5+∆O2) O: (0.0, 0.5, 0.5+∆O2) cubic a=b=c; α=β=γ=90 Ti: (0.5, 0.5, 0.5) O: (0.5, 0.5, 0.0) O: (0.5, 0.0, 0.5) O: (0.0, 0.5, 0.5) 21
  21. 21. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 22 BaTiO3 pressure induced phase transitions Pressure induced phase transitions rhombohedral orthorhombic orthorhombic tetragonal tetragonal cubic P ~ 5 GPa P ~ 6 GPa P ~ 7.5 GPa 22
  22. 22. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 23 T=0 K BaTiO3 EOS: DFT-GGA Volume (Å3) energy(eV) orthorhombic tetragonal cubic rhombohedral Energy-volume curves for the experimentally observed phases 23
  23. 23. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 24 DFT-GGA (zero stress) 65.9118 1.049 DFT-GGA (exp. volume) 64.2555 1.03 Experiment (T=300 K) 64.2555 1.011 BaTiO3-Tetragonal Structure Ti: (0.5, 0.5, 0.5-∆Ti) O: (1.0, 0.5, 0.5+∆O1) O: (0.5, 0.5, 1.0+∆O2) ∆Ti = 0.01653 (exp. 0.0224) ∆O2 = 0.02721 (exp. 0.0244) ∆O1 = 0.01561 (exp. 0.0105) Thermodynamically stable phase at room conditions V0 (A3) c/a Kwei et al. J. Phys. Chem. 97, 2368 (1993) 24 Using experimental lattice parameters
  24. 24. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 25 BaTiO3-Tetragonal Structure 25 Pressure (GPa) Fractional z position (z/c) Titanium Oxygens Variation of the internal parameters of the tetragonal structure with pressure DFT-GGA
  25. 25. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 26 MD simulations of domain walls in BaTiO3 Ba-centered 180º wall Domain wall energy: qMS-FF: 0.009 J/m2 (using preliminary qMS-Q FF) DFT-LDA: 0.006 J/m2 [J. Padilla et al. Phys. Rev. B (1996)] Work in progress 26
  26. 26. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 27 MD simulations of domain walls in BaTiO3 Ba-centered 90º walls Domain wall energy: qMS-FF ~ 0.06 J/m2 (preliminary FF) Work in progress Future Work (using ReaxFF) Mobility of domain walls Geometrically necessary dislocations in 90º walls 27
  27. 27. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 28 Mesoscale Simulations (Cagin) Quantum Mechanics Molecular Dynamics Quasi-continuum Mesoscale parameters Domain walls Defects Domain patterns Grains Ferroelectric film Connect from First Principles (QM) to Materials Applications (Films)
  28. 28. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 29 Multi-scale modeling hierarchy
  29. 29. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 30 Ferroelectric materials: Hundreds of ferroelectric compounds and solid solutions are known Many of the most important are based on the perovskite structure (our current focus) Perovskite structures exhibit an enormous range of compositions (only partially explored) Ferroelectric properties are very sensitive to composition allowing a seemingly unlimited range of tunable properties. This diversity provides an enormous opportunity to search for new ferroelectrics with desired properties (including high piezoelectric and dielectric constants). Atomistic Simulations Provides an excellent tool to search for novel systems with novel properities Many other families of ferroelectrics exist (oxides, hydrogen-bonded, semiconductor, organics).
  30. 30. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 31 Issues to be Discussed • Polarization • Electric fields • Solid solutions • Piezoelectricity • Temperature • Formation, dynamics of microstructures • Thin films, interfaces and hetero-structures • Loss mechanisms, fatigue, and aging
  31. 31. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 32 Polarization Short Term: We will use the ReaxFF with Molecular Dynamics to describe charge transfer and polarization. This allows the prediction of dielectric, pyroelectric, and piezoelectric properties as a function of Temperature and stress. Long term Problems: Only recently was a fundamental Quantum Mechanics (QM) based theory developed for describing polarization in solids. (Previously it was thought that bulk polarization could be determined from the charge density, an approach that is correct only for finite crystals where it is dominated by surface effects). There are questions concerning the polarization dependence of the exchange- correlation in Kohn-Sham theory. We will be exploring these issues with First Principles QM
  32. 32. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 33 Important Issues Electric fields: There are significant conceptual and technical problems problems in understanding crystal behavior under finite external macroscopic electric fields. We will explore these issues using MD with ReaxFF. As the issues are understood we may need to readdress the QM Solid Solutions: The single-crystal piezoelectrics Pb(Mg,Nb)O3--PbTiO3 (and most other ferroelectrics) are complex solid solutions. The nature of the medium- and short-range order in these materials and the impact on properties is not known. We will determine the relation between order and properties using MD and mesoscale modeling combined with spectroscopic experiments.
  33. 33. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 34 Temperature It is very important to determine the temperature dependence of the equilibrium state and dynamical behavior of bulk crystals and interfacial structures in ferroelectric and ferroelastic materials. We will be particularly concerned with the transport and pinning of oxygen vacancies along twin boundaries and the effect of domain switching. Here the simulation methods will focus on Kinetic Monte Carlo using the ReaxFF.
  34. 34. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 35 Piezoelectricity. Piezoelectricity is a central useful properties of ferroelectrics. We will use first-principles computations of piezoelectricity to understand the intrinsic response of structure and polarization to fields and stress. The goal will be to understand the relative roles of crystal structure and microstructure in producing the measured values. This should allow improvements in be designed into the materials based on the microscopic understanding. This may lead to improved devices.
  35. 35. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 36 Thin films, interfaces and hetero structures. The electronics applications of ferroelectrics will be based on thin films and related structures. There have been rapid advances in the synthesis of such structures via IBAD, CVD, MBE, sputtering, etc, yielding devices of high quality and reproducibility. For thin films the properties are largely determined by surface effects and finite size effects. These effects are determined by the atomic and electronic structure at interfaces between ferroelectrics and metals or ferroelectrics and other dielectrics. We will use MD based on ReaxFF to determine the nature of the structures and properties at these interfaces and then use the critical structures from the MD to examine the electronic properties with QM.
  36. 36. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 37 Loss mechanisms, fatigue, and aging. Crucial for device applications of ferroelectric materials are the loss mechanisms, fatigue, and aging. We expect that the electronic structure of defect and domain structures will play an important role, particularly the behavior of space-charge-limited currents and dielectric breakdown. Again the role of disorder is likely to be quite important. These complex properties are very difficult to characterize, either experimentally or computationally. We expect that computational and simulation methods will provide the basis for understanding these phenomena and for interpreting experiments
  37. 37. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 38 Formation and dynamics of microstructures. A critical problem is the characterization and control of microstructures, including twin boundaries, needle domains, and surface structures. The extension of the atomistic modeling into the mesoscale and then to the continuum is expected to be critical in explaining and characterizing the microstructure
  38. 38. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 39 Success Story in Bridging from QM to Macroscale Multiscale modeling of plasticity •Identify the unit processes that determine plasticity •Atomistic Modeling of the unit processes to obtain fundamental materials parameters Macroscopic driving force Macroscopic response Atomistic processes Atomistic data derived from First Principles Predictive power (do new materials) 39
  39. 39. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 40 Multiscale modeling of plasticity Mitchell and Spitzig. Acta Metallurgica1965. 99.97 % pure Ta single crystals Tensile load along [213] Temperature and strain rate dependence of stress-strain curves strain stress strain stress Experimental results 40
  40. 40. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 41 Core energy=0.827 eV/Å Eedge/Escrew = 1.77 Core energies of screw and edge dislocations in Ta Core energy=0.467 eV/Å 1/2a<111> screw dislocation 1/2a<111> edge dislocation 41
  41. 41. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 42 1/2<111> screw dislocation Single Kinks (<112> kink) b Molecular Mechanics simulations qEAM FF 40500 atoms 150 b long dislocations (quadrupolar system) Negative core (n) Positive core (p) 0.59 (eV) n n 0.60 (eV) n n 1.11 (eV) p n 0.57 (eV) p n 0.10 (eV) n p 0.62 (eV) n pKink Kink formation energies in Tantalum 42
  42. 42. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 43 Ortiz model of plasticity in single crystals • Dislocation mobility SLIP PLANE DISLOCATION FOREST • Dislocation interactions Energetic condition for bow-out process L.Stainer,A.Cuitino, M. Ortiz, J. Mech. Phys. Sol. (2001) 43
  43. 43. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 44 Additional parts for Ortiz model of plasticity • Dislocation density evolution DISLOCATION PRODUCTION by Frank-Reed sources DISLOCATION ANNIHILATION L.Stainer,A.Cuitino, M. Ortiz, J. Mech. Phys. Sol. (2001) 44
  44. 44. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 45 Multiscale modeling of plasticity •Dislocation mobility kink pair mechanism •Dislocation Interactions obstacle strength •Disloc. density evolution multiplication & attrition * not obtained from atomistics * atomistic value Parameter Fitted value Atomistic value 0.7 0.725 13 17 0.2 0.216 1.77 1.77 0.67 0.725 2.3 4.5 1250 500 Ekink Lkink / b Uedge / µb3 Eedge /Escrew Ejog λFR Kc/b * * * 45
  45. 45. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 46 Experiments Atomistic input Optimized parameters Strain-rate dependence of stress-strain curves strain strain strain stress stress stress L.Stainer,A.Cuitino, M. Ortiz, J. Mech. Phys. Sol. (2001) Mitchell and Spitzig. Acta Metallurgica1965. 46
  46. 46. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 47 Temperature dependence of stress-strain curves Experiment strain stress Fit Ortiz Model to experiment strain stress Mitchell and Spitzig. Acta Metallurgica1965. strain stress Use First Principles parameters Goddard-Ortiz Collaboration L.Stainer,A.Cuitin o,M. Ortiz J. Mech. Phys. Sol. (2001) 47
  47. 47. Engineering Microstructural Complexity in Ferroelectric Devices MURI Kick-off May 30, 2001 48 Time for Ortiz “Believe me, Forman, the problem is never on the molecular level” 48

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