Field Method of Simulation of Phase Transformations in Materials. Alex Umantsev Fayetteville State University, Fayetteville, NC
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1 Field Method of Simulation of Phase Transformations in Materials Alex Umantsev Fayetteville State University, Fayetteville, NC
2 What do we need to account for? Multi-phase states: thermodynamic systems may have multiple stable phase coexisting at the same conditions. Heterogeneous states: phase transformations may go along very complicated paths. Dynamic structures: rate of transformation can make a difference for the final structure and properties Nucleation: phase transformations are initiated through the process of overcoming of some kind of a potential barrier.
3 Phase Field Theory 101: Six Pillars of the Field Theory of Phase Transformations Field Theory Landau theory Free Energy Order parameter Heterogeneity Dynamics Fluctuations Hydrody namics Ginzburg- Landau
4 X Pillar 1: Order Parameter: Symmetries of the System Order-disorder transformation in -brass (Cu-Zn alloy) Coarse-graining procedure --Zn atom --Cu atom concentration C N NCu N Cu Zn order N Cu/ red N parameter Cu N N Zn/ blue Zn C R R R
5 Pillar 2: Free Energy g( T, ) = g ( T) W2+ ( ) 3 4 Landau potential stable phase: crystal metastable phase: liquid reaction coordinate W potential barrier energy scale -chemical potential difference /W-driving force
6 Pillar 3: Heterogeneous Systems: Gradient Energy Length scale: G= [ g( W, ) + 1( ) 2 ] dv 2 Correlation length : W interface thickness 1 Interfacial energy : W Order- Disorder Fe-Al Allen, Cahn Circa 82
7 Pillar 4: Kinetics W The Gradient Flow Equation: d dt Time V int = G scale : 1 W M.I. Mendelev, J. Schmalian, C.Z. Wang, J.R. Morris, and K.M. Ho, Phys. Rev. B, 74, (2006).
8 Pillar 5: Internal fluctuations: Langevin force dη = γ( ) ζ(x, t) dt δg δη T,P ζ(x, t) 0 ζ(x, t)ζ(x',t') Γδ(x' x) δ(t' t) Γ 2γk T (x, t) noise: Gaussian, white, additive One more energy scale: thermal fluctuations-k B T B
9 Pillar 6: Hydrodynamic Modes : Momentum Flow (Pressure or Stress) Diffusion (Concentration of species) Electromagnetic Field Variation (Waves) Heat Flow (Temperature Variation): Thermodynamically consistent heat equation A. Umantsev and A. Roytburd. Sov. Phys. Solid State 30(4), , (1988) General Heat-Equation CdT= ( T) +Q( x, t) dt ( x, t) [( e) Heat Source Q 2 V, T E Latent Heat L d ] dt
10 l l l l I C T Length Scales W CTE 2 L L CV K 1 U n interfacial thickness capillary length kinetic length thermal length radius of curvature l C l I CT E 2 L W C T E L W dynamic R Time scales 1 l l W relaxation time C C L2 kinetic time C
11 Time scale Truly Multi-Scale Method D. Danilov temperature -----melting temp solid phase order parameter transition state liquid phase distance into the sample Å nm m Length scales mm l C l l T X
12 Real Material Modeling Problem A: Convert thermodynamic functions of two or more phases into a continuous Landau-Gibbs Free Energy Solution 1: LTPT: Symmetry expansion Solution 2: Speculate { i } G{ i ; T, P, C} CALPHAD {T, P, C} Problem B: Find the PFM parameters: {W i, i, i } Solution 1: Calculate from First Principles Solution 2: Calibrate from experiments or MD/MC simulations
13 Au-Sn Binary (Bulk) Phase Diagram Landau-Gibbs Free Energy of Au-Sn Alloy
14 Landau-Gibbs Free Energy of Carbon -740 molar Gibbs free energy [kj/mol] graphite coordination OP diamond liquid crystallization OP
15 Measurable Quantities Interface thickness: l I ~ W Interface energy: free energy unit area ~ W Kinetic coefficient: i nterface velocity 1K of supercooling W T L E Equilibrium Fluctuations of OP η V k 2 = Non-Equilibrium Fluctuations of OP k B T V 2 g η2 η + κ k 2 Dynamic structure factor time
16 Materials Genome Database of Interfacial Properties
17 Soldering: InterMetallic Phase Growth Soldering 101 Solder IMP Cross-section 100m Cu Sn-Solder IMP 5m Liquid-state solder Solid-state solder Where was the original interface between the tin and copper? Sn-Solder Cu-Plate Gagliano and Fine 00 Onishi and Fujibuchi 75 Cu-Plate
18 Experimental results Lord & Umantsev, J. Appl. Phys. 98, (2005) (Centennial Campus, 2004) PLC baseline IMP
19 free energy J/mol ordering Total Free Energy Umantsev, JAP 07 molecular liquid inter metallic F dr[ f(,, c) Aw( ) Bw( ) ( ) 2 ( ) 2 ] f(,, c) [ [ f f f solid liquid intermet Homogeneous part ( c) ( c) f ( c) liquid f ( c) solid ] s( ( c) ) ] s( ) normal liquid crystallization disordered solid Thermocalc G. Ghosh molecular liquid intermetallic compound atomic liquid solution fcc solid solution concentration Cu
20 Grain Boundary Contribution to Free Energy relaxation coefficient 1. f ori =f() not invariant against rotation of the reference frame 2. f ori =f( 1, 2, 3, ) orientation order parameter 3. f ori =f( )= s + q 2 Kobayashi, Warren, Carter f=+[1-s()]f liquid (c)+[s() -s()]f solid (c)+ s()f intermet (c) Dynamics 120 +(,) f ori t (, ) F Grain-boundary diffusion: Mobility: M=M(,, ) KWC G; this work liquid order parameter crystall
21 Crystallization Diffusion Evolution Equations Ordering GB orientation t t t F (, ) F F Parameters: 3 Diffusion coefficients: solder intermetallic substrate + 9 interfacial parameters (interface energies thicknesses, and mobilities) M c t M [ M( liq ( M sol,, M liq c) ) ( M F c int ] M Initial Conditions: slab, no nucleation sol) Scales: Length=0.25nm Time=0.25s
22 ordering 2D Modeling molecular liquid inter metallic C-concentration -grad angle orient disordered solder solid crystallization Liquid state soldering Copper Solid State Soldering Copper Problem: rotation of grain orientation Tin Tin
23 Phase Field Simulation of Nucleation at Large Driving Forces Lifetime: time for a supercritical nucleus to appear in the system. Advantages:1. more reliable theory 2. free-energy landscape 3D as opposed to 2D Correlation properties of the fluctuations are very different. Numerical simulations x, t are not just grid parameters. They are physical quantities the noise correlation length and time x x t t
24 Supercritical Nucleus X-Y plane Y-Z plane plane Z-X
25 Time Evolution of the System transition state V t x, tdx V equilibrium fluctuations perturbation theory escape rectification of fluctuations -phase
26 Free Energy Barrier
27 3D Structure of Supercritical Nucleus Shape characterization 1. Volumetric content 2. Eccentricity 3. Roughness 4. Probability distribution
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