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From Pattern Formation 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 3/23/2011

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Pattern Formation in Crystal Growth by Wilson Bentley (The snowflake man), 1885

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Pattern Formation in Crystal Growth Al-Cu dendrite, Voorhees Lab Northwestern University At the nanoscale (atomistic scale) Liquid-Solid interfaces Anisotropy ↔ Morphology Atomistic details ↔ Anisotropy? Solid-Solid interfaces Grain boundary Atomistic details ↔ growth? Atomistic details ↔ Continuum theory at the nanoscale Hoyt, McMaster Schuh, MIT

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Pattern Formation in Macromolecules Polyelectrolyte Gels Hex (-) Hexagonal phase in solvent rich region Hex (+) Hexagonal phase in polymer rich region Competition between Enthalpy, Entropy, Elastic Network Energy, Electrostatic energy, … etc

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Pattern Formation in Biology Lincoln Park Zoo Chicago Rural Area, Wisconsin

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Pattern Formation in Biology Bleb Formation in Breast Cancer Cell Nucleus Goldman Lab, Northwestern University Confocal Immunofluorescence of a normal cell nucleus Goldman Lab, Northwestern University Lamin ( 核纖層蛋白 ) A/C Lamin B1, B2 Nuclear Lamina ( 核纖層 ) ～ 30-100nm In animal cells, only composed of 2 types lamins

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Crystal Growth at the Nanoscale Solid-Solid interface Grain boundaries Schuh/MIT Solid-Liquid interface Crystal growth from its melt with interfacial anisotropy Solid-Fluid interface under stress Quantum dots InAs/GaAs Ng et al., Univ. of Manchester, UK

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Crystal growth – Solid-Liquid interface n Motivation – anisotropy vs. crystal structures n Order-parameter models of equilibrium bcc-liquid interfaces –Ginzburg-Landau theory Density functional theory (DFT) of freezing Comparison with MD simulations –Phase-field crystal model Multi-scale analysis Determination of phase-field crystal model parameters Comparison with GL theory and MD simulations

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Gibbs-Thomson condition 1/TrS (Max ΔT) Phase-field simulations of solidification Morphology vs. Anisotropy Anisotropy of What causes the anisotropy?

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Basal Plane Crystal growth – Solid-Liquid interface

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Anisotropy vs. Crystal structures fcc bcc WHY?

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FF u 110 Liquid Solid GL Theory for bcc-liquid interface S(K) K (Å -1 ) K0K0 a 3 and a 4 are determined by equilibrium conditions Liquid structure factor Density Functional Theory of Freezing Free energy functional for a planar solid-liquid interface with normal

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For the crystal face {110} is separated into three subsets Bcc-liquid interface profile

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Anisotropic Density Profiles Symmetry breaks at interfaces → Anisotropy 2D Square Lattices

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n x 10 -23 (cm -3 ) Comparison with MD results BCC Iron

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Fe 100 110 111 4 (%) MD (MH(SA) 2 )177.0(11)173.5(11)173.4(11)1.0(0.6) GL theory144.26145.59137.571.02 Predict the correct ordering of and weak anisotropy 1% for bcc crystals Anisotropy (erg/cm 2 ) Comparison with MD results Atomistic details (Crystal structures) matter!

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Methodology for atomistic simulations Molecular Dynamics (MD) Mean field theory Ginzburg-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?

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Methodology for atomistic simulations Molecular Dynamics (MD) Mean field theory Phase field crystal (PFC) Average out vibration modes (ms) Atomistic details – elasticity, crystalline planes, dislocations, … etc. Realistic physics Resolve vibration modes (ps)

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(001) plane of bcc crystals (100) (110) Formulation - Phase Field Crystal Capillary Anisotropy? Elasticity? Swift & Hohenberg, PRA (1977) 2D Patterns – Rolls, Hexagons Elder et al., PRL (2002) Propose a conserved SH equation The Free Energy Functional Equation of Motion

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PFC Model – Phase Diagram Phase diagram Conserved Dynamics

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Maxwell construction Seek the perturbative solution The solid-liquid coexistence region A weak first-order freezing transition (The multi-scale analysis of bcc-liquid interfaces will be carried out around c ) Multi-scale Analysis Assumption – interface width is much larger than lattice parameter

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Small limit – diffuse interface Multi-scale analysis Equal chemical potential in both phases One of twelve stationary amplitude equations Multi-scale Analysis – Amplitude equation

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u 110 Order Parameter Profile Comparison For the crystal face Determination of the PFC model Parameter from density functional theory of freezing

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Fe 100 110 111 4 (%) MD (MH(SA) 2 )177.0(11)173.5(11)173.4(11)1.0(0.6) GL theory144.26141.35137.571.02 PFC144.14140.67135.761.22 Predict the correct ordering of 100 > 110 > 111 and weak anisotropy 1% for bcc crystals Anisotropy (erg/cm 2 ) Comparison with MD results

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What about Other Crystal Structures? Phase diagram

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F u 110 x y z BCC-Liquid F FCC-Liquid GL theory of fcc-liquid interfaces

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The Two-mode fcc modelThe PFC model FCC Model Phase Diagram Twin Boundary FCC Polycrystal

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Design Desired Lattices Example: Square Lattices Single-mode model Multi-mode model Dictate interaction angle (lattice symmtry) Elasticity

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Grain Boundary n Grain boundary is composed of dislocations n Geometric arrangement of crystals determines dislocation distribution n Distinct evolution for low and high angle grain boundary Symmetric tilt planar grain boundary in gold by STEM D

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GB sliding and coupling GB Coupling – Low Angle GBGB Sliding – High Angle GB Sutton & Balluffi, Interfaces in Crystalline Materials, 1995 Well described by continuum theory

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Large Misorientations Curvature driven motion G.B. sliding (fixed misorientation) remains constant Well described by classical continuum theory

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Small Orientations Atoms at the center of the circular grain Theory that only considers Misorientation decreases? Misorientation increases!

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Small Misorientations G.B. coupling Misorientation-dependent mobility: For symmetric tilt boundaries (Taylor & Cahn) Misorientation increases GB energy increases

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Intermediate Misorientations – cont.

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Intermediate Misorientations Faceted–Defaceted Transition Frank-Bilby formula Tangential motion of dislocations Annihilation of dislocations

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Intermediate Misorientations – cont. Instability of tangential motion occurs when 0 Spacing d 1 is a function of GB normal

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Three-Grain System Grain Rotation? GB wiggles

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Grain Rotation

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Grain Translation

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5.2º -5.2º 0º GB Wriggles

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Dihedral angle follows Frank’s formula not the Herring relation

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Self-Assembled Quantum Dots Lee et al., Lawrence Livermore National Laboratory Quantum-dot LEDs Other Applications - Tunable QD Laser - Quantum Computing - Telecommunication - and more Quantum dots InAs/GaAs Ng et al., Univ. of Manchester, UK

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Linear perturbation calculation Film Substrate Stress Induced Instability – Asaro-Tiller-Grinfeld Instability Schematic plot from Voorhees and Johnson Solid State Physics, 59 Cullis et al. (1992): 40 nm thick Si 0.79 Ge 0.21 on (001) Si substrate - Grown at 1023 K (Defect-free growth) Misfit Parameter asas afaf

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Later Stage Evolution - Cusp Formation - Dislocations Si 0.5 Ge 0.5 /Si(001) Jesson et al., Z-Contrast, Oak Ridge Natl. Lab., Phys. Rev. Lett. 1993 High stress concentration at the tip

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Simulation Parameters The PFC model Simulation parameters Various sizes Hexagonal Phase Constant Phase Constant Phase (1+ xx )L x

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Nonlinear Steady State for a Smaller k k

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Quantitative Comparison of Strain Fields Correct elastic fields Elastic fields relax much faster than the density field

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Critical Wavenumber vs Strain Linear perturbation theory - Sharp Interface - Homogeneous Materials PFC simulations Classical Elasticity Theory Xie et al., Si 0.5 Ge 0.5 films, PRL Linear Elasticity k c ~ xx 2 for small strains Nonlinear elasticity modifies length scale

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PFC modeling of nonlinear elasticity Solid Liquid Inhomogeneous materials nonlinear elasticity

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Finite Interface Thickness Effect Solid, E=E o Liquid, E=0 E(x,y) c ~ 1/2 · xx -2 W~ -1/2 Finite interface thickness W Elastic constants vary smoothly across the Interface region Upper bounds Interface thickness is no longer negligible at the nanoscale

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Nonlinear Evolution for k ~ k m k

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3D Island – BCC Systems

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And More … VLS nanowires Nano-particles with defects

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And More …

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Pattern Formation - Examples Graphene North Pole Hexagon on Saturn Ice Crystal Agular et al, Oxford UniversityHoneycomb Rock Formation in Ireland

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Collaborators Mathis Plapp Laboratoire de Physique de la Matière Condensée Ecole Polytechnique Alain Karma Northeatsern University Peter W. Voorhees Northwestsern University

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