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

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

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

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

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

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 ) ～ nm In animal cells, only composed of 2 types lamins

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

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

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

Basal Plane Crystal growth – Solid-Liquid interface

Anisotropy vs. Crystal structures fcc bcc WHY?

FF 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

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

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

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

Fe  100  110  111  4 (%) MD (MH(SA) 2 )177.0(11)173.5(11)173.4(11)1.0(0.6) GL theory 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!

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?

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)

(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

PFC Model – Phase Diagram Phase diagram Conserved Dynamics

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

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

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

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

What about Other Crystal Structures? Phase diagram

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

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

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

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

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

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

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

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

Intermediate Misorientations – cont.

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

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

Three-Grain System Grain Rotation? GB wiggles

Grain Rotation

Grain Translation

5.2º -5.2º 0º GB Wriggles

Dihedral angle follows Frank’s formula not the Herring relation

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

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

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 High stress concentration at the tip

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

Nonlinear Steady State for a Smaller k  k

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

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

PFC modeling of nonlinear elasticity Solid Liquid Inhomogeneous materials nonlinear elasticity

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

Nonlinear Evolution for k ~ k m  k

3D Island – BCC Systems

And More … VLS nanowires Nano-particles with defects

And More …

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

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