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Mathematical modelling of Tyndall star initiation

  • A. A. LACEY (a1), M. G. HENNESSY (a2), P. HARVEY (a3) and R. F. KATZ (a4)

Abstract

The superheating that usually occurs when a solid is melted by volumetric heating can produce irregular solid–liquid interfaces. Such interfaces can be visualised in ice, where they are sometimes known as Tyndall stars. This paper describes some of the experimental observations of Tyndall stars and a mathematical model for the early stages of their evolution. The modelling is complicated by the strong crystalline anisotropy, which results in an anisotropic kinetic undercooling at the interface; it leads to an interesting class of free boundary problems that treat the melt region as infinitesimally thin.

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[1]Angenent, S. B. & Gurtin, M. E. (1989) Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface. Arch. Rat. Mech. Anal. 108, 323391.
[2]Atthey, D. R. (1974) A finite difference scheme for melting problems. IMA J. Appl. Math. 13 (3), 353366.
[3]Barles, G. & Souganidis, E. (1998) A new approach to front propagation problems: Theory and applications. Arch. Rat. Mech. Anal. 141, 237296.
[4]Boettinger, W. J., Warren, J. A., Beckermann, C. & Karma, A. (2002) Phase-field simulation of solidification. Ann. Rev. Mater. Res. 32, 163194.
[5]Burton, W. B., Cabrera, N. & Frank, F. C. (1951) The growth of crystals and the equilibrium structure of their surfaces. Philos. Trans. R. Soc. London, Ser. A 243 (866), 299358.
[6]Cahoon, A., Maruyama, M. & Wettlaufer, J. S. (2006) Growth-melt asymmetry in crystals and twelve-sided snowflakes. Phys. Rev. Lett. 96, 255502.
[7]Chadam, J., Howison, S. D. & Ortoleva, P. (1987) Existence and stability for spherical crystals growing in a supersaturated solution. IMA J. Appl. Math. 39, 115.
[8]Coriell, S. R., McFadden, G. B. & Sekerka, R. F. (1999) Selection mechanisms for multiple similarity solutions for solidification and melting. J. Cryst. Growth 200, 276286.
[9]Coriell, S. R., McFadden, G. B., Sekerka, R. F. & Boettinger, W. J. (1998) Multiple similarity solutions for solidification and melting. J. Cryst. Growth 191, 573585.
[10]Davis, S. H. (2001) Theory of Solidification, Cambridge University Press, Cambridge.
[11]Font, F., Mitchell, S. L. & Myers, T. G.One-dimensional solidification of supercooled melts. Int. J. Heat Mass Tran. 62, 411421.
[12]Gurtin, M. E. (1993) Thermomechanics of Evolving Phase Boundaries in the Plane, Clarendon, Oxford.
[13]Hu, H. & Argyropoulos, S. A. (1996) Mathematical modelling of solidification and melting: A review. Modelling Simul. Mater. Sci. Eng. 4, 371396.
[14]Harvey, P. (2013) An Experimental Analysis of Tyndall Figures. Technical Report, Department of Earth Science, University of Oxford, Oxford.
[15]Hennessy, M. G. (2010) Liquid Snowflake Formation in Superheated Ice, M.Sc. thesis, University of Oxford, Oxford.
[16]Howison, S. D., Ockendon, J. R. & Wilson, S. K. (1991) Incompressible water-entry problems at small deadrise angles. J. Fluid Mech. 222, 215230.
[17]Huppert, H. E. (1990) The fluid mechanics of solidification. J. Fluid Mech. 212, 209240.
[18]Lacey, A. A. & Herraiz, L. A. (2000) Macroscopic models for melting derived from averaging microscopic Stefan problems I: Simple geometries with kinetic undercooling or surface tension. Eu. J. Appl. Math. 11 (2), 153169.
[19]Lacey, A. A. & Herraiz, L. A. (2002) Macroscopic models for melting derived from averaging microscopic Stefan problems II: Effect of varying geometry and composition. Eu. J. Appl. Math. 13 (3), 261282.
[20]Lacey, A. A. & Shillor, M. (1983) The existence and stability of regions with superheating in the classical two-phase one-dimensional Stefan problem with heat sources. IMA J. Appl. Math. 30 (2), 215230.
[21]Lacey, A. A. & Tayler, A. B. (1983) A mushy region in a Stefan problem. IMA J. Appl. Math. 30 (3), 303313.
[22]Mae, S. (1975) Perturbations of disc-shaped internal melting figures in ice. J. Crystal Growth. 32 (1), 137138.
[23]Maruyama, M., Kuribayashi, N., Kawabata, K. & Wettlaufer, J. S. (2000) A test of global kinetic faceting in crystals. Phys. Rev. Lett. 85 (12), 25452548.
[24]Nakaya, U. (1956) Properties of Single Crystals of Ice, Revealed by Internal Melting, Technical report, Snow Ice and Permafrost Research Establishment, U.S. Army.
[25]Maruyama, M. (2011) Relation between growth and melt shapes of ice crystals. J. Cryst. Growth 318, 3639.
[26]Mullins, W. W. & Sekerka, R. F. (1963) Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Phys. 34, 323329.
[27]Ockendon, J., Howison, S., Lacey, A. & Movchan, A. 2003 Applied Partial Differential Equations, Oxford University Press, Oxford.
[28]Shimada, W. & Furukawa, Y (1997) Pattern formation of ice cystals during free growth in supercooled water. J. Phys. Chem. B 101, 61716173.
[29]Takeya, S. (2006) Growth of internal melt figures in superheated ice. Appl. Phys. Lett. 88, 074103.
[30]Tyndall, J. (1858) On some physical properties of ice. Phil. Trans. Roy. Soc. Lond. 148, 211229.
[31]Tsemekhman, V. & Wettlaufer, J. S. (2003) Singularities, shocks, and instabilities in interface growth. St. Appl. Math. 110, 221256.
[32]Uehara, T. & Sekerka, R.F. (2003) Phase field simulations of faceted growth for strong anisotropy of kinetic coefficient. J. Cryst. Growth 254, 251261.
[33]Wettlaufer, J. S. (2001) Dynamics of ice surfaces. Interface Sci. 9, 117129.
[34]Wettlaufer, J. S., Jackson, M. & Elbaum, M. (1994) A geometric model for anisotropic crystal growth. J. Phys. A 27, 59575967.
[35]Yokoyama, E. & Kuroda, T. (1990) Pattern formation in growth of snow crystals occurring in the surface kinetic process and the diffusion process. Phys. Rev. A 41, 20382050.
[36]Yokoyama, E. & Sekerka, R. F. (1992) A numerical study of the combined effects of anisotropic surface tension and interface kinetics on pattern formation during the growth of two-dimensional crystals. J. Cryst. Growth 125, 289403.
[37]Yokoyama, E., Sekerka, R. F. & Furukawa, Y. (2009) Growth of an ice disk: Dependence of critical thickness for disk instability on supercooling of water. J. Phys. Chem. B 113, 47334738.

Keywords

Mathematical modelling of Tyndall star initiation

  • A. A. LACEY (a1), M. G. HENNESSY (a2), P. HARVEY (a3) and R. F. KATZ (a4)

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