1. An Enthalpy Model for Dendrite Growth Vaughan Voller University of Minnesota

2. Growing Numerical Crystals Objective: Simulate the growth (solidification) of crystals from a solid seed placed in an under-cooled liquid melt Some Physical Examples snow-flakes-ice crystals Germs Dendrite grains in material systems (IACS), EPFL science.nasa.gov Voorhees, Northwestern

3. ~0.5 m ~5 mm Computational grid size ProcessREV ~ 1  m In terms of the process Sub grid scale

4. Simulation can be achieved using modest models and computer power Growth of solid seed in a liquid melt Initial dimensionless undercooling T = -0.8 Resulting crystal has an 8 fold symmetry Solved in ¼ Domain with A 200x200 grid Box boundaries are insulated Since the thermal boundary layer is thin Boundaries do not affect growth until seed approaches edges

17. By changing conditions can generate any number of realistic shapes in modest times PC CPU ~5mins BUT—WHY do we get these shapes ?—WHAT is Physical Bases for Model ? Solution is with a FIXED grid -HOW does the Numerical solution work ? IS the solution “correct” ? Complete garbage

18. First recognize that there are two under-coolings The bulk liquid is under-cooled, i.e., in a liquid state below the equilibrium liquidus temperature T M The temperature of the solid-liquid interface is undercooled Conc. of Solute, m < 0 slope of liquidus Gibbs Thomson  curvature,  surface tension Kinetic v n normal interface velocity

19. Can describe process with the Sharp Interface Model—for pure material With dimensionless numbers Assumed constant properties Insulated domain Initiated with small solid seed n vnvn Capillary length ~10 -9 for metal On interface Angle between normal and x-axis The heat flows from warm solid into the undercooeld liquid drives solidification Preferred growth direction and interplay between curvature and liquid temperature gradient determine Growth rate and shape.

20. Smear out interface n Use smooth interpolation for Liquid fraction f across interface f=1 f=0 A Diffusive interface model: Usually implies Phase Field—Here we mean ENTHALPY (see Tacke 1988, Dutta 2006) The Enthalpy -sum of sensible and latent heats-change continuously and smoothly throughout domain—Hence can write Single Field Eq. For Heat transport undercooling Can be solved on a Fixed grid

21. Growth of Equiaxed Crystal In under-cooled melt A microstructure model Phase change temperature depends on interface curvature, speed and concentration Sub-grid models Account for Crystal anisotropy and “smoothing” of interface jumps

22. Four fold symmetry Sub grid constitutive If f= 0 or f = 1If 0 < f < 1 curvature Local direction Capillary length 10 -9 m in Al alloys ENTHALPY Numerical Solution Very Simple—Calculations can be done on regular PC

23. seed Typical grid Size 200x200 ¼ geometry At end of time step if solidification Completes in cell i Force solidification in ALL fully liquid neighboring cells. Physical domain ~ 2-10 microns Initially insulated cavity contains liquid metal with bulk undercooling T 0 < 0. Solidification induced by placing solid seed at center. Some Results

24. Problem: range of cells with 0 < f < 1 restricted to width of one cell Accuracy in curvature calc? Remedial scheme: smear out f value, e.g., Remedial Scheme: Use nine volume stencil to calculate derivatives Tricks and Devices

25. Produces nice answers BUT are they correct

26. 4d o (blue) 3.25d o (black) 2.5d o (red) Dendrite shape with 3 grid sizes shows reasonable independence  = 0.05, T 0 = -0.65 Dimensionless time  = 6000  = 0.25,  = 0.75

27. Solvabillity (kim et al) Long term tip dynamics approaches theory BUT results begin to deteriorate if grid is made smaller !!

28. Verify solution coupling by Comparing with one-d solidification of an under-cooled melt T 0 = -.5 Compare with Analytical Similarity Solution Carslaw and Jaeger Temperature at dimensionless time t =250 Front Movement

29. Verification 1 Looks Right!! k = 0 (pure),  = 0.05, T 0 = -0.65,  x = 3.333d 0 Enthalpy Calculation Dimensionless time  = 0 (1000) 6000 k = 0 (pure),  = 0.05, T 0 = -0.55,  x = d 0 Level Set Kim, Goldenfeld and Dantzig Dimensionless time  = 37,600 Red my calculation for these parameters With grid size

30. The Solid color is solved with a 45 deg twist on the anisotropy and then twisted back —the white line is with the normal anisotropy Note: Different “smear” parameters are usedin 0 0 and 45 0 case Tip position with time Dimensionless time  = 6000  = 0.05, T 0 = -0.65 Not perfect: In 45 0 case the tip velocity at time 6000 (slope of line) is below the theoretical limit. Low Grid Anisotropy

31. m For binary system need to consider – solute transport, discontinuous diffusivities and solutes Use smoothly interpolated Variables across the interface Sharp Interface Model Single Domain Eq.

32. Comparison with one-d Analytical Solution Constant T i, C i k = 0.1, Mc = 0.1, T 0 = -.5, Le = 1.0 Concentration and Temperature at dimensionless time t =100 Front Movement

33. Effect of Lewis Number: small Le  interface concentration close to C 0 k = 0.15, Mc = 0.1, T 0 = -.65  = 0.05,  x =3.333d 0 All predictions at time  =6000

34. k = 0.15, Mc = 0.1, T 0 = -.55, Le = 20.0  = 0.02,  x = 2.5d 0 Concentration field at time  = 30,000 Profile along dashed line Concentration

35.  = 0.05, T 0 = -0.65 time  = 6000  = 0.25,  = 0.75,  x =4d 0 FAST-CPU This On This In 60 seconds !

36. Conclusion –Score card for Dendritic Growth Enthalpy Method (extension of original work by Tacke) Ease of CodingExcellent CPUExcellent (runs shown here took between 1 and 2 hours on a regular PC) Convergence to known analytical sol. Excellent Convergence to known operating state Very Good (if grid is not too fine and remedial parameters well chozen Grid AnisotropyGood (see comment above) AlloyPromising

37. Playing Around A Problem with Noise Multiple Grains-multiple orientations Grains in A Flow Field Thses calculations were performed by Andrew Kao, University of Greenwich, London Under supervision of Prof Koulis Pericleous and Dr. Georgi Djambazov.

38. Can this work be related to other physical cases ? Extensions: Grain Growth Couple with porosity formation ? 100  m seconds 100 k m 1000’s of years e.g., shoreline in sedimentary basin

39. NCED’s purpose: to catalyze development of an integrated, predictive science of the processes shaping the surface of the Earth, in order to transform management of ecosystems, resources, and land use The surface is the environment!

40. Research fields Geomorphology Hydrology Sedimentary geology Ecology Civil engineering Environmental economics Biogeochemistry Who we are: 19 Principal Investigators at 9 institutions across the U.S. Lead institution: University of Minnesota

41. Fans Toes Shoreline Two Sedimentary Moving Boundary Problems of Interest Moving Boundaries in Sediment Transport

42. 1km Examples of Sediment Fans Moving Boundary How does sediment- basement interface evolve Badwater Deathvalley

43. sediment h(x,t) x = u(t) bed-rock ocean x shoreline x = s(t) land surface   A Sedimentary Ocean Basin

44. An Ocean Basin Melting vs. Shoreline movement

45. Experimental validation of shoreline boundary model ~3m

46. Base level Measured and Numerical results ( calculated from 1 st principles) 1-D finite difference deforming grid vs. experiment +Shoreline balance

47. Is there a connection Grain Growth in Metal Solidification From W.J. Boettinger  m  10km “growth” of sediment delta into ocean Ganges-Brahmaputra Delta