Application of an Enthalpy Method for Dendritic Solidification Vaughan R. Voller and N. Murfield

1. Application of an Enthalpy Method for Dendritic Solidification Vaughan R. Voller and N. Murfield Voller, University of Minnesota volle001@umn,edu

2. The problem—simulate the growth of a crystal into an undercooled melt contained in an insulated cavity How does solidification proceed? Why do we get a dendritic shape? Voller, University of Minnesota volle001@umn,edu

3. How does solidification proceed? seed Initial liquid at a temperature below equilibrium solidification temperature T = 0 seeded with solid at solidification temperature T0 < 0 H = cT + fL Liquid layer adjacent to seed uses latent heat to heat up to T = 0 H = 0 + fL, 0 < f < 1 Negative gradient into liquid removes residual latent heat and drives solidification T = T0 < 0 If a solute is present the equilibrium temp and gradient slope will be lower—resulting in a slower advance for the solidification Voller, University of Minnesota volle001@umn,edu

4. Why do we get a dendritic shape? anisotropic surface energy liquidus slope Surface of seed is under-cooled due to curvature (Gibbs-Thompson) and solute (Not Shown kinetic) To < Tm With dimensionless numbers Capillary length Angle between normal and x-axis As crystal grows the sharper temp grad at tip drives sol harder BUT the increased tip curvature holds it back A steady tip operating Velocity is reached Anisotropic term makes under cooling less in preferred growth directions 0.25 1 0.25 Initial seed with radius r = 1/k Voller, University of Minnesota volle001@umn,edu

5. Current Approaches (Pure Melt) Level Set Interface Tracking H. S. Udaykumar, R. Mittal,y and Wei Shyy Also see Juric Tryggvason and Zhao and Heinrich Solve for level set Using speed function from Stefan Cond. Maintain distance function properties by re-in Enthalpy Method-Proposed by K H Tacke Solve heat con. Use level set to mod FD at interface A function of f if 0 < f < 1 (f determines curvature) Kim, Gldenfeld, Dantzig And Chen, Merriman, Osher,andSmereka In this work: use iterative sol. Include anisotropy and solute -1 < phase marker < 1 Phase Filed Thermodynamic equation –minimizing free energy across a diffusive interface Heat equation with source Alain Karma and Wouter-Jan Rappel Voller, University of Minnesota volle001@umn,edu

6. Governing Equations With additional dimensional numbers Governing equations are If 0 < g < 1 Chemical potential Continuous at interface concentration Voller, University of Minnesota volle001@umn,edu

7. Numerical Solution Very Simple—Calculations can be done on regular PC Use square finite difference grid, set length scale to In a time step Solve for H and C (explicit time integration) Calculate curvature and orientation from current nodal g field Calculate interface undercooling If 0 < g < 1 then Update f from enthalpy as Check that calculated liquid fraction is in [0,1] Update Iterate until At end of time step—in cells that have just become all solid introduce very small solid seed in ALL neighboring cells. Required to advance the solidification ON A FIXED UNIFORM GRID Initial condition— Circle r = 15do Typical grid Size 200x200 ¼ geometry Voller, University of Minnesota volle001@umn,edu

8. Level Set Kim, Goldenfeld and Dantzig k = 0 (pure), e = 0.05, T0 = -0.55, Dx = d0 Dimensionless time t = 37,600 Looks Right!! Verification 1 Enthalpy Calculation k = 0 (pure), e = 0.05, T0 = -0.65, Dx = 3.333d0 Dimensionless time t = 0 (1000) 6000 Voller, University of Minnesota volle001@umn,edu

9. Symbol-numeric sol. Red-line Numeric sol. Covers analytical Concentration and Temperature at dimensionless time t =100 Front Movement Verification 2 Verify solution coupling by Comparing with one-d solidification of an under-cooled binary alloy Constant Ti, Ci k = 0.1, Mc = 0.1, T0 = -.5, Le = 1.0 Compare with Analytical Similarity Solution—Rubinstein Carslaw and Jaeger combo. Voller, University of Minnesota volle001@umn,edu

10. Verification 3 Compare calculated dimensionless tip velocity with Steady state operating sate calculated from the microscopic solvability theory Voller, University of Minnesota volle001@umn,edu

11. Verification 4 Check for grid anisotropy Solve with 4 fold symmetry twisted 45o Then Twist solution back Dx = .36do Dx = .38do Dimensionless time t = 6000 Voller, University of Minnesota volle001@umn,edu

12. Result: Effect of Lewis Number k = 0.15, Mc = 0.1, T0 = -.65 e = 0.05, Dx = 3.333d0 All predictions at Dimensionless time t =6000 Voller, University of Minnesota volle001@umn,edu

13. Result: Prediction of Concentration k = 0.15, Mc = 0.1, T0 = -.55, Le = 20.0 e = 0.02, Dx = 2.5d0 Dimensionless time t = 30,000 Voller, University of Minnesota volle001@umn,edu

14. Conclusion –Score card for Dendritic Growth Enthalpy Method (extension of original work byTacke) Voller, University of Minnesota volle001@umn,edu