1. Ingot Casting Continuous Casting Welding & Laser Remelting Directional Casting Shaped Casting Solidification Processing

2.  R – Tip Radius   2 – Secondary Arm Spacing  1 – Primary Arm Spacing Dendritic Array Growth  Temperature Gradient, G   Growth Velocity, V   Diffusion + Convection exist in the Melt

3. Modeling Dendritic Array Growth  Experimental modeling: TGS + Transparent Materials Controlled G and V Minimum Convection  Numerical modeling: Self-consistent model G/V   Dendrites G/V   Cells A.Single Cell/Dendrite B.Cellular/Dendritic Array

4. Numerical Modeling of Cellular/Dendritic Array Growth (Diffusion Controlled Growth + No Convection concerned) Basic Parameters  Given : Materials Properties: C 0, m L, k, D L,  (  /  S), E 4, Solidification Condition: G and V  Unknown: R, 1,  T (T i )

5. Numerical Modeling of Cellular/Dendritic Array Growth (Diffusion Controlled Growth + No Convection concerned)

6. Numerical Method  :  Solute Flow:  i+1 C i+1 -  i C i = A N (V N C + DdC/dr) N dt – A S (V S C + DdC/dr) S dt + A E (V E C + DdC/dx) E dt – A w (V W C + DdC/dx) w dt

7. Spacing Adjustment of Array Growth Spacing,  1  as Velocity, V  Mechanism of Spacing Adjustment Lower Limit Upper Limit V

8. Array Stability Criterion Unstable Stable Solute

9. Result I: Shapes of Single Cell/Dendrite

10. Result I: Single Cell Growth in fine capillary tubes 200  m Stable CellPerturbed Cell

11. Result II: Primary Spacing

12. Result II: Primary Spacing – SCN – 5.6 wt.% H 2 O System

13. Result II: Primary Spacing – NH 4 Cl - 70 wt.% H 2 O System

14. Result III: Tip Radius 20  m The relation, R 2 V = Constant, is confirmed for all the cases examined in both experimental modeling and numerical modeling.

16. Result IV: Growth Undercooling TT TLTL TiTi

17. Result V: The Effect of Temperature Gradient

18. Modeling Rapid Solidification  Diffusion Coefficient – Temperature Dependent: D  as T  D = D 0 exp[-Q/(RT)]  Distribution Coefficient – Velocity Dependent: k  as V , Aziz (1988) where  Non-equilibrium vs. Equilibrium: Boettinger etc. (1986) G  , V  ,  T   Laser Remelting

19. Result VI: Rapid Solidification

20. Result VII: Global Structure Planar  Cellular  Dendritic  Cellular  Planar V

21. Development of Semi-analytical Expressions (Hunt/Lu Model) 1. Variables: Composition, C 0, Liquidus Slop, m, Distribution Coefficient, k, Diffusion Coefficient, D, Gibbs-Thompson Coefficient, , Surface Energy Anisotropy Coefficient, E 4, Growth Velocity, V, Temperature Gradient, G, Primary Spacing,, and Tip Undercooling,  T. 2.Dimensionless Parameters:  Temperature Gradient: G’ = G  k/  T 0 2  Growth Velocity: V’ = V  k/(D  T 0 )  Primary Spacing: ’ =  DT 0 /(k  )  Tip Undercooling:  T’ =  T/  T 0 where  T 0 = mC 0 (1-1/k) 3.Properties of the Non-dimensionalization:  G’ = V’: Constitutional Undercooling Limit --- V = GD/  T 0  V’ = 1: Absolute Stability Limit --- V =  T 0 D/(k  )   T’ = 1: The undercooling with a planar front growth ---  T =  T 0 = mC 0 (1-1/k)

22. Result VIII: Semi-analytical Expressions (Hunt/Lu Model) 1.Cellular Growth (Derived from the Array Stability Criterion):  Undercooling:  T’ =  T’ s +  T’ r  T’ s = G’/V’ + a +(1-a)V’ 0.45 – G’/V’[a + (1-a)V’ 0.45 ] where a = 5.273 x10 -3 + 0.5519k – 0.1865k 2  Tr’ = b(V’ – G’) 0.55 (1-V’) 1.5 where b = 0.5582 – 0.2267log(k) + 0.2034{log(k)] 2  Cell Spacing: ’ 1 = 8.18k -0.485 V’ -0.29 (V’ – G’) -0.3  T’ s -0.3 (1-V’) -1.4 2.Dendritic Growth:  Undercooling:  T’ =  T’ s +  T’ r  T’ s = G’/V’ + V’ 1/3  T’ r = 0.41(V’ – G’) 0.51  Primary Dendrite Spacing (Derived from the Array Stability Criterion): ’ 1 = 0.156V’ (c-0.75)( V’ – G’)0.75G’ –0.6028 where c = -1.131 – 0.1555log(G’) – 0.7598 x 10 -2 [log(G’)] 2 * Expressions are developed with the Array Stability Criterion

23. Experimental Modeling of Grain Formation in Casting

24. Tip Radius, R , Spacing,  1  as Velocity, V  Deceleration Experimental Modeling: Effect of Deceleration on the Dendritic Array Growth (SCN - 5.5 wt.% H 2 O System) R 1 Tip Radius, R: Rapid response to velocity change. Every individual dendrite follows the Marginal Stability criterion approximately during deceleration. Primary Spacing, 1 : Slow response to velocity change.The array is unstable and is in transient condition during deceleration.

25. Experimental Modeling: Effect of Deceleration on the Dendritic Array Growth – Fragmentation (SCN - 5.5 wt.% H 2 O System) Continuous Deceleration, a = -1.0  ms -2 High Velocity Low Velocity Secondary Arm, 2, Detached due to deceleration – Accelerated ripening process. The fragmentation rate is proportional to the deceleration.