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Dynamical Selection of Interface Patterns SCR/RDR Alain Karma & Rohit Trivedi

Dynamical Selection of Interface Patterns SCR/RDR Alain Karma & Rohit Trivedi. Numerical Modeling Blas Echebarria & Mathis Plapp Supported by NASA. Charges # 1, 2. Why Phase-Field ?.

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Dynamical Selection of Interface Patterns SCR/RDR Alain Karma & Rohit Trivedi

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  1. Dynamical Selection of Interface Patterns SCR/RDRAlain Karma & Rohit Trivedi Numerical Modeling Blas Echebarria & Mathis Plapp Supported by NASA

  2. Charges # 1, 2 Why Phase-Field ? • Powerful and accurate computational approach to simulate the complete pattern evolution in a purely diffusive growth regime • Makes it possible to carry out quantitative comparisons between modeling predictions and experiments beyond the limitations of past uncontrolled approximations • Provides crucial guide to conceive new experiments and interpret their results

  3. Charges # 1, 2 Outline • Example of Recent Success • Classical Sharp-Interface Model • New Quantitative Phase-Field Model • Preliminary Quantitative Comparisons with Sharp-Interface Theories & Experiments • Fundamental Issues

  4. Charges # 1, 2 Tip Operating State (Plapp & Karma, 2000) Solvability (Langer, Kessler-Levine, Brener)

  5. Charges # 1, 2 z [001] [010] y f r [100] x Comparison with Microgravity Data (Karma, Lee, & Plapp, 2000) SCN LaCombe et al (1995) La Combe et al. 1995

  6. Charges # 1, 2 Fundamental Questions L Dendrites ??? Cells V • Steady-States • Stability • Dynamical Selection • Growth Conditions (V,G,C)

  7. Charges # 1, 2 Liquid V x G y Solid Sharp-Interface Model Hot Frozen T approximation: Cold

  8. Charges # 1, 2 Interface Kinetics Hoyt, Asta & Karma 2002 • Wilson-Frenkel • Turnbull (1981) • Broughton Gilmer-Jackson (1982) • Mikheev- Chernov (1992) • Shen-Oxtoby (1996)

  9. Charges # 1, 2 Phase-Field Approach Langer (78,86), Collins-Levine (85)… • Naturally circumvents front tracking by making the solid-liquid interface spatially diffuse • Can cope efficiently with a vast disparity of length and time scales • Permits the incorporation of thermal fluctuations that are crucially important for the generation of secondary and tertiary branches • Permits the exploration of parameter ranges that are not directly accessible to experiments

  10. Charges # 1, 2 PHASE FIELD MODEL Dilute Alloy Binary Alloy

  11. Charges # 1, 2 Alloy Phase-Field Model Wheeler, Boettinger, McFadden (92)

  12. Charges # 1, 2 W j g~ hW 10-4 Crystal Melt x f r L, l W~r/10 lT h j Bridging Micro-Macro Scales (m) 10-6 10-8 10-10

  13. Charges # 1, 2 binary alloy Thin-Interface Limit (W << r) Karma-Rappel (96); Almgren (99); Karma (01) Interface stretching Surface diffusion 3 constraints:

  14. Charges # 1, 2 New Alloy Phase-Field Formation A. Karma, PRL 87, 115701 (2001) j V Solute x Anti-trapping current • Anti-trapping current makes it possible to eliminate all corrections to the classical sharp-interface equations. • Makes experimentally relevant length and time scales computationally accessible. • Can simulate quantitatively the small V limit with local equilibrium at the interface.

  15. Charges # 1, 2 Numerical Example

  16. Charges # 1, 2 Composition Field around an Isothermal Dendrite Old

  17. Charges # 1, 2 Composition Field around an Isothermal Dendrite New

  18. Charges # 1, 2 Isothermal Solutal Dendrite

  19. Charges # 1, 2 Concentration in Solid along Dendrite Axis

  20. Charges # 1, 2 Comparison with Mullins-Sekerka Analysis Mullins-Sekerka (solid line) wl2/2Dl kl

  21. Charges # 1, 2 Comparison with Scheil Equation

  22. Charges # 1, 2 Comparison with Ground-based Experiments Trivedi et al. (SCN-salol)

  23. Charges # 1, 2 Role of Fluctuations V=32 mm/s, L=105 mm, G=105 K/cm Without Fluctuations With Fluctuations

  24. Charges # 1, 2 Steady States and Stability I V=32 mm/s & G=140 K/cm D “Cells” Cell Elimination Tertiary branching “Dendrites” L (mm)

  25. Charges # 1, 2 Steady States and Stability II V=32 mm/s & G=140 K/cm “Dendrites” “Cells” r Tertiary branching Cell Elimination L/r L (mm)

  26. Charges # 1, 2 Dynamically Induced Cell to Dendrite Transition

  27. Charges # 1, 2 Amplitude of Sidebranching x V=32 mm/s & G=140 K/cm Y(t) A (mm) Cells “Dendrites” L (mm)

  28. Charges # 1, 2 L Dendrites ??? Cells V Fundamental Questions • Steady-States • Stability • Dynamical Selection • Growth Conditions (V,G,C)

  29. Charges # 1, 2 Morphologies for Different G V=32 mm/s & L=105 mm G=35 K/cm G=70 K/cm G=140 K/cm

  30. Charges # 1, 2 y(t) A (mm) G (K/cm) Sidebranching Amplitude vs. G V=32 mm/s & L=105 mm x

  31. Charges # 1, 2 Noise Amplification Langer (86); Brener (95); Karma-Rappel (98) G=140 K/cm G=105 K/cm G=70 K/cm G=35 K/cm y (mm) r A y V x G (K/cm) x (mm) Sidebranch Formation

  32. Charges # 1, 2 Comparison of 2D and 3D-Axisymmetric Results Top view of cell Tip splitting

  33. Charges # 1, 2 Dendrites ??? Cells V Full 3D L Hexagons, squares ? (McFadden et al, 1985)

  34. Charges # 1, 2 Dynamics of Interface Motion Pattern Evolution in Bulk Samples Phase Field Calculations

  35. Charges # 1, 2 Dynamic Evolution of Interface (side-view)

  36. Charges # 1, 2 Dynamics of Interface Motion Pattern Evolution in Bulk Samples Phase Field Calculations

  37. Charges # 1, 2 Dynamic Evolution of Interface (top-view)

  38. Charges # 1, 2 Conclusions • Accurate phase-field simulations are now possible for experimentally relevant length/time scales • Good quantitative agreement between phase-field simulations and sharp-interface theories (Mullins-Sekerka, Scheil) • Good quantitative agreement with thin-sample ground based experiments • No quantitative agreement between 3D-axisymmetric phase-field simulations and thin-tube ground based experiments • 2D and full 3D fundamentally different  strong need for microgravity

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