to Phase Field Crystal Model 吳國安 Kuo An Wu 清華大學物理系 Department of Physics National Tsing Hua University 3232011 Pattern Formation in Crystal Growth by Wilson Bentley The snowflake man 1885 ID: 1040762
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1. From Pattern Formation to Phase Field Crystal Model吳國安 (Kuo-An Wu)清華大學物理系Department of PhysicsNational Tsing Hua University3/23/2011
2. Pattern Formation in Crystal Growthby Wilson Bentley (The snowflake man), 1885
3. Pattern Formation in Crystal GrowthAl-Cu dendrite, Voorhees Lab Northwestern UniversityAt the nanoscale (atomistic scale)Liquid-Solid interfacesAnisotropy ↔ MorphologyAtomistic details ↔ Anisotropy?Solid-Solid interfacesGrain boundaryAtomistic details ↔ growth?Atomistic details ↔ Continuum theory at the nanoscaleHoyt, McMasterSchuh, MIT
4. Pattern Formation in MacromoleculesPolyelectrolyte GelsHex (-)Hexagonal phase in solvent rich regionHex (+)Hexagonal phase in polymer rich regionCompetition between Enthalpy, Entropy, Elastic Network Energy,Electrostatic energy, … etc
5. Pattern Formation in BiologyLincoln Park ZooChicagoRural Area, Wisconsin
6. Pattern Formation in BiologyBleb Formation in Breast Cancer Cell NucleusGoldman Lab, Northwestern UniversityConfocal Immunofluorescence of a normal cell nucleusGoldman Lab, Northwestern UniversityLamin (核纖層蛋白) A/CLamin B1, B2Nuclear Lamina (核纖層)～30-100nmIn animal cells, only composed of 2 types lamins
7. Crystal Growth at the NanoscaleSolid-Solid interfaceGrain boundariesSchuh/MIT Solid-Liquid interfaceCrystal growth from its melt with interfacial anisotropySolid-Fluid interface under stressQuantum dots InAs/GaAsNg et al., Univ. of Manchester, UK
8. Crystal growth – Solid-Liquid interfaceMotivation – anisotropy vs. crystal structuresOrder-parameter models of equilibrium bcc-liquid interfacesGinzburg-Landau theoryDensity functional theory (DFT) of freezingComparison with MD simulationsPhase-field crystal modelMulti-scale analysisDetermination of phase-field crystal model parametersComparison with GL theory and MD simulations
9. Gibbs-Thomson condition1/TrS(Max ΔT)Phase-field simulationsof solidificationMorphology vs. AnisotropyAnisotropy of gWhat causes the anisotropy?
10. Basal PlaneCrystal growth – Solid-Liquid interface
11. Anisotropy vs. Crystal structuresfccbccWHY?
12. DFu110gLiquidSolidGL Theory for bcc-liquid interfaceS(K)K (Å-1)K0a3 and a4 are determinedby equilibrium conditionsLiquid structure factorDensity Functional Theory of FreezingFree energy functional for a planarsolid-liquid interface with normal
13. For the crystal face{110} is separated into three subsetsBcc-liquid interface profile
14. Anisotropic Density ProfilesSymmetry breaks at interfaces → Anisotropy2D Square Lattices
15. n x 10-23 (cm-3)Comparison with MD resultsBCC Iron
16. Fe1001101114 (%)MD (MH(SA)2)177.0(11)173.5(11)173.4(11)1.0(0.6)GL theory144.26145.59137.571.02Predict the correct ordering of and weak anisotropy 1% for bcc crystalsAnisotropy (erg/cm2)Comparison with MD resultsAtomistic details (Crystal structures) matter!
17. Methodology for atomistic simulationsMolecular Dynamics (MD)Mean field theoryGinzburg-Landau theory Realistic physics Resolve vibration modes (ps) Rely on MD inputs Average out atomistic details Diffusive dynamics (ms) Larger length scale (m) Elasticity, defect structure, … etc?
18. Methodology for atomistic simulationsMolecular Dynamics (MD)Mean field theoryPhase field crystal (PFC) Average out vibration modes (ms) Atomistic details – elasticity, crystalline planes, dislocations, … etc. Realistic physics Resolve vibration modes (ps)
19. (001) plane of bcc crystals(100)(110)Formulation - Phase Field CrystalCapillary Anisotropy?Elasticity?Swift & Hohenberg, PRA (1977)2D Patterns – Rolls, HexagonsElder et al., PRL (2002)Propose a conserved SH equationThe Free Energy FunctionalEquation of Motion
20. PFC Model – Phase DiagramPhase diagramConserved Dynamics
21. Maxwell constructionSeek the perturbative solutionThe solid-liquid coexistence regionA weak first-order freezing transition(The multi-scale analysis of bcc-liquid interfaces will be carried out around c)Multi-scale AnalysisAssumption – interface width is much larger than lattice parameter
22. Small limit – diffuse interfaceMulti-scale analysisEqual chemical potential in both phasesOne of twelve stationary amplitude equationsMulti-scale Analysis – Amplitude equation
23. u110Order Parameter Profile ComparisonFor the crystal faceDetermination of the PFC model Parameter from densityfunctional theory of freezing
24. Fe1001101114 (%)MD (MH(SA)2)177.0(11)173.5(11)173.4(11)1.0(0.6)GL theory144.26141.35137.571.02PFC144.14140.67135.761.22Predict the correct ordering of 100 > 110 > 111and weak anisotropy 1% for bcc crystalsAnisotropy (erg/cm2)Comparison with MD results
25. What about Other Crystal Structures?Phase diagram
26. Fu110xyzBCC-LiquidFFCC-LiquidGL theory of fcc-liquid interfaces
27. The Two-mode fcc modelThe PFC modelFCC ModelPhase DiagramTwin BoundaryFCC Polycrystal
28. Design Desired LatticesExample: Square Lattices Single-mode modelMulti-mode modelDictate interaction angle(lattice symmtry)Elasticity
29. Grain BoundaryGrain boundary is composed of dislocationsGeometric arrangement of crystals determines dislocation distributionDistinct evolution for low and high angle grain boundary Symmetric tilt planar grain boundary in goldby STEMD
30. GB sliding and couplingGB Coupling – Low Angle GBGB Sliding – High Angle GBSutton & Balluffi, Interfaces in Crystalline Materials, 1995Well described bycontinuum theory
31. Large MisorientationsCurvature driven motionG.B. sliding (fixed misorientation)g remains constantWell described by classical continuum theory
32. Small OrientationsAtoms at the center of the circular grainTheory that only considers gMisorientation decreases?Misorientation increases!
33. Small MisorientationsG.B. couplingMisorientation-dependent mobility:For symmetric tilt boundaries(Taylor & Cahn)Misorientation increasesGB energy increases
34. Intermediate Misorientations – cont.
35. Intermediate MisorientationsFaceted–Defaceted Transition Frank-Bilby formula Tangential motion of dislocations Annihilation of dislocations
36. Intermediate Misorientations – cont.Instability of tangential motion occurs when0p/3FSpacing d1 is a function of GB normal
37. Three-Grain SystemGrain Rotation?GB wiggles
38. Grain Rotation
39. Grain Translation
40. 5.2º-5.2º0ºGBWriggles
41. Dihedral anglefollows Frank’s formulanot the Herring relation
42. Self-Assembled Quantum DotsLee et al., Lawrence Livermore National LaboratoryQuantum-dot LEDsOther Applications Tunable QD Laser Quantum Computing Telecommunication and moreQuantum dots InAs/GaAsNg et al., Univ. of Manchester, UK
43. Linear perturbationcalculationFilmSubstrateStress Induced Instability – Asaro-Tiller-Grinfeld InstabilitySchematic plot from Voorhees and JohnsonSolid State Physics, 59Cullis et al. (1992): 40 nm thick Si0.79Ge0.21 on (001) Si substrate - Grown at 1023 K (Defect-free growth)Misfit Parameterasaf
44. Later Stage Evolution - Cusp Formation - DislocationsSi0.5Ge0.5/Si(001)Jesson et al., Z-Contrast, Oak Ridge Natl. Lab., Phys. Rev. Lett. 1993High stress concentration at the tip
45. Simulation ParametersThe PFC modelSimulation parametersVarious sizesHexagonalPhaseConstantPhaseConstantPhase(1+xx)Lx
46. Nonlinear Steady State for a Smaller kk
47. Quantitative Comparison of Strain Fields Correct elastic fields Elastic fields relax much faster than the density field
48. Critical Wavenumber vs StrainLinear perturbation theory Sharp Interface Homogeneous MaterialsPFC simulationsPFC simulationsClassical Elasticity TheoryXie et al., Si0.5Ge0.5 films, PRLLinear Elasticity kc ~ xx2 for small strains Nonlinear elasticity modifies length scale
49. PFC modeling of nonlinear elasticitySolidLiquid Inhomogeneous materials nonlinear elasticity
50. Finite Interface Thickness EffectSolid, E=EoLiquid, E=0E(x,y)c~1/2·xx-2W~-1/2Finite interface thickness WElastic constants vary smoothlyacross the Interface region Upper bounds Interface thickness is no longer negligible at the nanoscale
51. Nonlinear Evolution for k ~ kmk
52. 3D Island – BCC Systems
53. And More …VLS nanowiresNano-particles with defects
54. And More …
55. Pattern Formation - ExamplesGrapheneNorth Pole Hexagon on SaturnIce CrystalAgular et al, Oxford UniversityHoneycombRock Formation in Ireland
56. CollaboratorsMathis PlappLaboratoire de Physique de la Matière Condensée Ecole PolytechniqueAlain KarmaNortheatsern UniversityPeter W. VoorheesNorthwestsern University