GROWING NUMERICAL CRYSTALS

1. GROWING NUMERICAL CRYSTALS Vaughan Voller, and Man Liang, University of Minnesota We revisit an analytical solution in Carslaw and Jaeger for the solidification of an an under-cooled melt in a cylindrical geometry. We show that when the one-d axi-symmetry is exploited a fixed grid enthalpy model Produces excellent results. BUT—when a 2-D Cartesian solution is sort—”exotic” numerical crystals grow. This IS NOT A numerical crystal

2. Consider a LIQUID melt infinite in extent At temperature T< 0 BELOW Freezing Temp At time t = 0 a solid seed at temperature T = 0 is placed in the center This sets up a temp gradient that favors the growth of the solid The Carslaw and Jaeger Solution for a cylindrical sold seed in an under-cooled melt

3. Exponential integral Assume radius grows as Then Found from With Similarity Solution Also develop similarity solutions for planar and spherical case

4. Enthalpy Solution in Cylindrical Cordiantes Assume an arbitrary thin diffuse interface where liquid fraction Define Throughout Domain a single governing Eq Numerical Solution Very Straight-forward

5. Initially seed Transition: When If Set

6. Excellent agreement with analytical when predicting growth R(t) R(t)

7. Concentration and Temperature Profiles for spherical seed at time Time 20, Le= 50 Time 250, Le= 2 Similarity and enthalpy solutions can be extended to account for a binary alloy and a spherical seed

8. A 2-D Cartesian application of enthalpy model If Start with a single solid cell When cell finishes freezing “infect” -- seed liquid cells in mane compass directions

9. Initial-Seed Infection This choice will grow a fairly nice four-fold symmetry dendritic crystal is a stable configuration Pleasing at first!!! But not physically reasonable • where is the anisotropy • Why is growth stable (no surface tension of kinetic surface under-cooling) • The initial seed, grid geometry and infection routine introduce artificial anisotropy • The grid size enforces a stable configuration—largest microstructure has to • be at grid size

10. Demonstration of artificial anisotropy induced by seed and infection routine Initial-Seed Infection Similarity Solution

11. Numerical Crystal With the Cartesian grid Hard to avoid non-cylindrical perturbations Which will always locate in a region favorable for growth If imposed anisotropy is weak this feature will swamp Physical effect and lead to a Numerical Crystal Similarity Solution Can The similarity solution be used to test the intrinsic grid anisotropy in numerical crystal growth simulators ? Serious codes impose anisotropy-and include surface tension and kinetic effects But the choice of seed shape and grid can (will) cause artificial anisotropic effects

12. 2. On Cartesian structured grid however the enthalpy method breaksdown and—due to artificial grid anisotropy grows NUMERICAL crystals stabilized by grid cell size. Conclusions: 1. Growth of a cylindrical solid seed in an undercooled binary alloy melt can (in the absence of imposed anisotropy, surface tension and kinetic effects) be resolved with a similarity solution and axisymmetric enthalpy code 3. Similarity solution stringent test of ability Of a given method to suppress grid anisotropy

13. We get a sea-weed pattern What about unstructured meshes ? Can we use this as a CA solver for channels in a delta