"Each single plant has a single animating principle of its own, since each instance of a plant exists separately, and there is no cause to wonder that each should be equipped with its own peculiar shape. But to imagine an individual soul for each and any starlet of snow is utterly absurd, and therefore the shapes of snowflakes are by no means to be deduced from the operation of soul in the same way as with plants." Johannes Kepler, 1611
"I only had difficulty to imagine what could have formed and made so exactly symmetrical these six teeth around each grain in the midst of free air and during the agitation of a very strong wind, until I finally considered that this wind had easily been able to carry some of these grains to the bottom or to the top of some cloud, and hold them there, because they were rather small; and that there they were obliged to arrange themselves in such a way that each was surrounded by six others in the same plane, following the ordinary order of nature." Rene Descartes, 1635
Listen to what Data will say about this stuff!
Boettinger, W.J., Warren, J.A., Beckermann, C., and Karma, A., "Phase-Field Simulation of Solidification," Annual Review of Materials Research, Vol. 32, pp. 163-194, 2002.
Images are for increasing imposed temperature gradient from top to bottom.
View of entire dendrite.
Close-up of center region.
The above simulations are described in detail in the following publication: Ramirez, J.C., and Beckermann, C., "Examination of Binary Alloy Free Dendritic Growth Theories with a Phase-Field Model," Acta Materialia, Vol. 53, pp. 1721-1736, 2005.
Parameters: dimensionless supercooling = 0.55, dimensionless composition = 0.04, anisotropy strength = 0.02, Lewis number = 100, partition coefficient = 0.15, no solute diffusion in solid, no kinetics, no solute trapping.
The left movie shows how the grid is adapted in the simulation: a fine grid is used only in the inner quadrant where the dendrite grows and where there are strong species concentration gradients (U field); a grid that is four times coarser is used for the outer quadrants in order to accomodate the long tails of the thermal boundary layer (Theta field) in front of the growing dendrite; the total domain size is constant; as the dendrite grows, the grid is adapted periodically (five times in the movie) by enlarging the fine grid region at the expense of the course grid region. This saves a lot of computer time.
The simulation was performed using the phase-field model published in: J.C. Ramirez, C. Beckermann, A. Karma, and H.J. Diepers, "Phase-field modeling of binary alloy solidification with coupled heat and solute diffusion," Physical Review E, Vol. 69, 051607 (16 pages), 2004.
Click on the picture for the full movie (animated gif)!
Parameters: dimensionless supercooling = 0.55; anisotropy strength: 0.03, Prandtl number = 23.1; D = 2; uniform inlet velocity = 1.0 (in dimensionless form); others.
Computational details: Grid: 1024x2048; the entire transient simulation took about 200 CPU hours on an alpha workstation - corresponding simulations without flow take about 2 hours (the no-flow version of the code is from A. Karma).
We would like to offer this and other simulations as a computational benchmark. We can provide quantitative information on the tip speeds, curvatures, velocities, etc. Please contact C. Beckermann at becker@engineering.uiowa.edu.
... more on phase-field with convection
Sun, Y., and Beckermann, C., "Sharp Interface Tracking Using the Phase-Field Equation," J. Computational Physics, Vol. 220, 2007, pp. 626-653.
Badillo, A., and Beckermann, C., "Phase-Field Simulation of the Columnar-to-Equiaxed Transition in Alloy Solidification," Acta Materialia, Vol. 54, 2006, pp. 2015-2026.
Sun, Y., and Beckermann, C., "Phase-Field Simulation of Two-Phase Micro-Flows in a Hele-Shaw Cell," in Computational Methods in Multiphase Flow III, eds. A.A. Mammoli and C.A. Brebbia, WIT Press, Southampton, UK, 2005, pp. 147-157.
Lu, Y., Beckermann, C., and Ramirez, J.C., "Three-Dimensional Phase-Field Simulations of the Effect of Convection on Free Dendritic Growth," J. Crystal Growth, Vol. 280, pp. 320-334, 2005.
Ramirez, J.C., and Beckermann, C., "Examination of Binary Alloy Free Dendritic Growth Theories with a Phase-Field Model," Acta Materialia, Vol. 53, pp. 1721-1736, 2005.
Sun, Y., and Beckermann, C., "Diffuse Interface Modeling of Two-Phase Flows Based on Averaging: Mass and Momentum Equations," Physica D, Vol. 198, pp. 281-308, 2004.
Ramirez, J.C., Beckermann, C., Karma, A., and Diepers, H.-J., "Phase-Field Modeling of Binary Alloy Solidification with Coupled Heat and Solute Diffusion," Physical Review E, Vol. 69, 051607 (16 pages), 2004.
Sun, Y., and Beckermann, C., "A Diffuse Interface Model for Two-Phase Flows Based on Averaging," in Multiphase Phenomena and CFD Modeling and Simulation in Materials Processes, eds. L. Nastac and B.Q. Li, TMS, Warrendale, PA, 2004, pp. 109-118.
Ramirez, J.C., and Beckermann, C., "Examination of Binary Alloy Free Dendritic Growth Theories with a Phase-Field Model," in Solidification Processes and Microstructures - A Symposium in Honor of Wilfried Kurz, eds. M. Rappaz, C. Beckermann, and R. Trivedi, TMS, Warrendale, PA, 2004, pp. 373-378.
Lu, Y., Beckermann, C. and Karma, A., "Convection Effects in Three-Dimensional Dendritic Growth," Proceedings of ASME IMECE2002, Paper No. IMECE2002-32838, Nov. 2002.
Boettinger, W.J., Warren, J.A., Beckermann, C., and Karma, A., "Phase-Field Simulation of Solidification," Annual Review of Materials Research, Vol. 32, pp. 163-194, 2002.
Lu, Y., Beckermann, C. and Karma, A., "Convection Effects in Three-Dimensional Dendritic Growth," Proceedings of the 2001 Fall MRS Meeting in Boston, MA, 2001 (invited paper).
Tong, X., Beckermann, C., Karma, A. and Li, Q., "Phase-Field Simulations of Dendritic Crystal Growth in a Forced Flow," Physical Review E, Vol. 63, 061601 (16 pages), 2001.
Tong, X., Beckermann, C., and Karma, A., "Velocity and Shape Selection of Dendritic Crystals in a Forced Flow," Physical Review E, Vol. 61, pp. R49-R52, 2000.
Beckermann, C., Diepers, H.J., Steinbach, I., Karma, A., and Tong, X., "Modeling Melt Convection in Phase-Field Simulations of Solidification," J. Computational Physics, Vol. 154, pp. 468-496, 1999.
Diepers, H.J., Beckermann, C., and Steinbach, I., "Simulation of Convection and Ripening in a Binary Alloy Mush Using the Phase-Field Method," Acta Materialia, Vol. 47, pp. 3663-3678, 1999.
Diepers, H.J., Beckermann, C., and Steinbach, I., "A Phase-Field Method for Alloy Solidification with Convection," in Solidification Processing 1997, eds. J. Beech and H. Jones, Dept. Engineering Materials, The University of Sheffield, Sheffield, UK, 1997, pp. 426-430.
Tong, X., Beckermann, C., and Karma, A., "Phase-Field Simulation of Dendritic Growth with Convection," in Modeling of Casting, Welding and Advanced Solidification Processes VIII, eds. B.G. Thomas and C. Beckermann, TMS, Warrendale, PA, 1998, pp. 613-620.
Diepers, H.J., Beckermann, C., and Steinbach, I., "Modeling of Convection-Influenced Coarsening of a Binary Alloy Mush Using the Phase-Field Method," in Modeling of Casting, Welding and Advanced Solidification Processes VIII, eds. B.G. Thomas and C. Beckermann, TMS, Warrendale, PA, 1998, pp. 565-572.
