1. Modeling of Micro segregation in Metal Alloys Vaughan R. Voller University of Minnesota

2. After Flemings (Solidification Processing) and Beckermann (Ency. Mat) 1m What is Macrosegregation C solid C liquid Partitioned solute at solid-liquid interface Redistributed by Fluid and solid motion convection grain motion shrinkage

3. 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 To advance to the next time step we need find REV values for T temperature C l liquid concentration g s solid fraction C s  distribution of solid concentration

4. Modeling the fluid flow requires 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: turbulence Buoyancy Friction between solid and liquid Accounts for mushy region morphology Requires a solid-momentum equation Extra Terms

5. 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 To advance to the next time step we need find REV values for T temperature C l liquid concentration g s solid fraction C s  distribution of solid concentration

6. Need four relationships which can be obtained from a micro-scale model Under the assumptions of: 1.Equilibrium at solid-liquid interface 2.Perfect solute mixing in the liquid 3.Identification of a solid-liquid interface length scale (e.g., a ½ secondary arm space) Possible Relationships are X s (t) X l (t)~  1/3 ~ 10’s  m coarsening Definitions of mixture terms in arm space T=G(C l 1,C l 2 …….) 4. Account of local scale redistribution of solute during solidification of the arm space 3. 1 1. 2. Thermodynamics Primary + Secondary C l k C s k (interface) C l k =F(C l 1,C l 2 …….) g s = X s /X l

7. solid Liquid The Micro-Scale Model ~ 1  m Macro Inputs X s (t) X l (t)~  1/3 Define: macro coarsening back Average Solute Concentration in X l -X s old During time step  Treat like initial state for a new problem Iteration: Guess of T in [H]  X s –X s old  Assume Lever on C *  C l --, T new solid thermo

8. 3 ***. At each time step approx. solid solute profile as Choose to satisfy Mass Balance 1. Numerical Solution in solid Requires a Micro-Segregation Model that to estimate “back diffusion” of solute into the solid at the solid-liquid interface solid Liquid ~ 1  m X s (t) X l (t)~  1/3 ( Three Approaches) 2. Approximate with “average” parameter Function of  can be corrected for coarsening

9. solid Testing: Binary-Eutectic Alloy. Cooling at a constant rate Predictions of Eutectic Fraction at end of solidification coarsening Numerical back diff model Approx profile model 

10. Calculate kC l /C 0 with time. Calculations terminate when g s = 1. Parabolic growth The segregation of phosphorus in  -Fe Testing: solid fraction is calculated from g s =  0.5 Profile Model pow = 2

11. z Z eut Z liq lever Gulliver -Scheil Effect of Microsegregation On Macrosegregation  = 0.2 Coarsening g A uni-directional solidification of a Of binary alloy cooled from a fixed chill. Microsegregation (back diffusion into solid) modeled in terms off rate of change of solute in liquid No Coarsening   ,  = 1  =0,  = 0 Solute concentration in mushy region

12. ~ 1 mm 1m Summary T,g, C s and C l ~ 1  m solid Microsegregation And Themodynamics From macro variables Find REV variables Accounting for at solid-liquid interface Key features -- Simple Equilibrium Thermodynamics -- External variables consistent with macro-scale conservation statements -- Accurate approximate accounting of BD and coarsening at each step based on current conditions

13. 1m I Have a BIG Computer Why DO I need an REV and a sub grid model ~ 1  m solid ~mm (about 10 9 ) Model Directly (about 10 18 ) Tip-interface scale

14. Well As it happened not currently Possible 1000 2 0.6667 Year “Moore’s Law” Voller and Porte-Agel, JCP 179, 698-703 (2002) Plotted The three largest MacWasp Grids (number of nodes) in each volume 2010 for REV of 1mm 2055 for tip