Key Issues in Solidification Modeling— Vaughan Voller, University of Minnesota, Aditya Birla Chair ~0.5 m Process Scales: ~10 mm Micro-strcture Validation and Verification: Do governing equations model the correct physics? Is approximate solution a solution of governing Equations ? Ferreira et al How can we deal with this problem Micro-macro models How feasible is Direct Modeling of Microstructure? What can it tell us about the process Scale? Prediction Of microstructure

solid crystals + liquid Scales: An Example Problem: Macrosegregation—Ingot alloy solidification Csolid Cliquid In mushy region solute is partitioned at solid-liquid interface Result after full solidification is macro-scale areas with concentration above or Below the nominal concentration see Flemings (Solidification Processing) and Beckermann (Ency. Mat) solid crystals + liquid “mushy region” This solute is redistributed at process scale by fluid and solid motions shrinkage grain motion solid liquid alloy convection 1m

Computational grid size Key Scales in Macro-segregation ~0.5 m Process ~5 mm REV ~ 50 mm solid representative ½ arm space sub-grid model g Computational grid size Solute value in liquid phase controlled by local diffusion in solid “microsegregation” Solute transport controlled by advection

Scales in General Solidification Processes chill A Casting The REV Nucleation Sites columnar equi-axed The Grain Envelope The Secondary Arm Space The Tip Radius f = 1 f = -1 The Diffusive Interface ~ 0.1 m ~10 mm ~ mm ~100 mm ~1 nm Scales in General Solidification Processes 103 101 10-1 10-3 10-5 10-7 10-9 10-9 10 10-3 10-1 Length Scale (m) interface kinetics nucleation solute diffusion growth grain formation casting heat and mass tran. Time Scale (s) (after Dantzig) Question for later: Can we build a direct-simulation of a Casting Process that resolves to all scales?

Meso Solid-liquid interface NANO-Meter A solidification model has three components: The Domain: The Grid: The sub-Grid: Examples Problem Domains The Grid Sub-grid --Constitutive -- Controlled by averaged Properties in REV Realizations--Of multi-phase regions Element in numerical Calculation ---REV State described by averaged mixture values Macro Process Effect of morphology on flow REV METER Meso Microstructure The Grain Envelope Solid-liquid interface A representative Arm spacing— Form of Constitutive model T=F(g) f = 1 f = -1 The Diffusive Interface, e.g. NANO-Meter

Computational grid size Key Scales in Macro-segregation ~0.5 m Process ~5 mm REV ~ 50 mm solid representative ½ arm space sub-grid model g Computational grid size Solute value in liquid phase controlled by local diffusion in solid “microsegregation” Solute transport controlled by advection Develop a “Macro-Micro Model” (Rappaz) Solve transport equations at macroscopic scale (MACRO) Use sub-grid model to account for microsegregation (MICRO) a “constitutive model”

Macro (Process) Scale Equations Equations of Motion (Flows) mm REV Heat: Solute Concentrations: Assumptions for shown Eq.s: -- No solid motion --U is inter-dendritic volume flow If a time explicit scheme is used to advance to the next time step we need find REV values for T temperature Cl liquid concentration gs solid fraction Cs(x) distribution of solid concentration

Computational grid size The Micro-Macro Model MACRO: ~ 50 mm solid representative ½ arm space sub-grid model g microsegregation and solute diffusion in arm space MICRO: ~0.5 m ~5 mm Computational grid size Process REV from computation Of these Mixture values need to extract --

Primary Solidification Solver g Transient mass balance g model of micro-segregation Iterative loop Cl T (will need under-relaxation) Gives Liquid Concentrations A C equilibrium

before solidification Micro-segregation Model liquid concentration due to macro-segregation alone Solute Fourier No. ½ Arm space of length l takes tf seconds to solidify In a small time step new solid forms with lever rule of concentration transient mass balance gives liquid concentration Solute mass density before solidification of new solid (lever) after solidification Q -– back-diffusion Need an easy to use approximation For back-diffusion

The Ohnaka approximation The parameter Model --- Clyne and Kurz, Ohnaka For special case Of Parabolic Solid Growth And ad-hoc fit sets the factor and In Most other cases The Ohnaka approximation Works very well

The Profile Model Wang and Beckermann Need to lag calculation one time step and ensure Q >0 m is sometimes take as a constant ~ 2 BUT In the time step model a variable value can be use Due to steeper profile at low liquid fraction ----- Propose

An Important wrinkle ---Coarsening Due to dissolution processes some arms will melt and arm-space will coarsen Time 1 Time 2 > Time 1

Coarsening Arm-space will increase in dimension with time This will dilute the concentration in the liquid fraction—can model be enhancing the back diffusion  A model by Voller and Beckermann suggests If we assume that solid growth is close to parabolic m =2.33 in Parameter model In profile model

Verification: of Micro Models: Constant Cooling of Binary-Eutectic Alloy With Initial Concentration C0 = 1 and Eutectic Concentration Ceut = 5, No Macro segregation , k = 0.1 T Cl As solidification proceeds the concentration in liquid increases. When the eutectic composition is reached remaining liquid solidifies isothermally, Eutectic Fraction In model calculate the transient value of g from Use 200 time steps and equally increment 1 < Cl < 5 Parameter or Profile

Verification of Micro Models: Verify approximate model for back-diffusion by comparing solution with FD solution of Fick’s equation in arm space. Parameter or Profile Remaining Liquid when C =5 is Eutectic Fraction

Validation of Micro Model: Predictions of Eutectic Fraction With constant cooling Co = 4.9 Ceut = 33.2 k = 0.16 Al-4.9% Cu Comparison with Experiments Sarreal-Abbaschian Met Trans 1986 X-ray analysis determines average eutectic fraction

Macro-Micro Model of Solidification Predict g predict Cl predict T Calculate Transient solute balance in arm space Macro-Micro Model of Solidification Two MICRO Models For Back Diffusion My Method of Choice Parameter Robust Easy to Use Poor Performance at very low liquid fraction— can be corrected Profile A little more difficult to use With this Ad-hoc correction Excellent performance at all ranges Account for coarsening

Modeling the fluid flow could require a Two Phase model, that may need to account for: Both Solid and Liquid Velocities at low solid fractions A switch-off of the solid velocity in a columnar region A switch-off of velocity as solid fraction g  o. An EXAMPLE 2-D form of the momentum equations in terms of the interdentrtic fluid flow U, are: Extra Terms Magically vanish Buoyancy Friction between solid and liquid Accounts for mushy region morphology Can requires a solid-momentum equation

Verification: of Macro-Micro Model—Inverse Segregation in a Binary Alloy chill solid mushy liquid riser y Shrinkage sucks solute rich fluid toward chill – results in a region of +ve segregation at chill 100 mm Flow by simple app. of continuity Fixed temp chill results in a similarity solution Parameter Current estimate empirical

Validation: Comparison with Experiments chill solid mushy liquid riser y 100 mm Ferreira et al Met Trans 2004

Direct Microstructure Modeling chill A Casting The REV Nucleation Sites columnar equi-axed The Grain Envelope The Secondary Arm Space The Tip Radius f = 1 f = -1 The Diffusive Interface ~ 0.1 m ~10 mm ~ mm ~100 mm ~1 nm Macro-Micro models at process scale grid sub-grid domain domain grid constitutive Micro-Nano model for micro-structure

Example: Growth of dendritic crystal in an under-cooled melt (seminar on July 14) Solved in ¼ Domain with A 200x200 grid Growth of solid seed in a liquid melt Initial dimensionless undercooling T = -0.8 Resulting crystal has an 8 fold symmetry

Grid independent results with correct dynamics can be readily obtained Tip Velocity Interacting grains grid anisotropy prediction of concentration field Scale of calculations shown 1 mm

Computational grid size So can we use DMS to predict microstructure at the process level? ~0.5 m ~5 mm Computational grid size Process REV ~ 1 mm Sub grid scale For 2-D calc at this scale Will need 1018 grids

For 2-D calc at this scale Will need 1018 grids Voller and Porte-Agel, JCP 179, 698-703 (2002 1000 20.6667 Year “Moore’s Law” 2055

CONCLUSIONS Validated and Verified Models that Conclusion: Can Currently Build Validated and Verified Models that Can successfully model across ~ 4 decades Of length scales chill A Casting The REV Nucleation Sites columnar equi-axed The Grain Envelope The Secondary Arm Space The Tip Radius f = 1 f = -1 The Diffusive Interface ~ 0.1 m ~10 mm ~ mm ~100 mm ~1 nm Able to use Macro-Micro Approach To model all scales of Heat and Mass Transport Able to build Local Microstructure models But a long way from DMS Direct microstructure simulation at the process scale In the meantime what Value Added can we get from Local microstructure models Use as generator for constitutive models Use in volume averaging approaches