Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-16T10:55:29.246Z Has data issue: false hasContentIssue false
  • English
  • Français

Particle capture in binary solidification

Published online by Cambridge University Press:  14 April 2009

JUSTIN C. T. KAO ,
ALEXANDER A. GOLOVIN  and
STEPHEN H. DAVIS
JUSTIN C. T. KAO*
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL, USA
ALEXANDER A. GOLOVIN
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL, USA
STEPHEN H. DAVIS
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL, USA
*
†Present address: Massachusetts Institute of Technology, Cambridge, MA, USA. E-mail address for correspondence: kaoj@mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the interaction of a spherical foreign particle with a propagating solidification front in a binary alloy. Depending on the material properties and the speed of the front, the particle may be pushed ahead of the front, or engulfed and incorporated into the solid phase. We apply numerical boundary integral and continuation methods to determine the critical speed for particle capture, as a function of the system parameters. We reconcile the differing predictions of previous theoretical works, and show that many typical systems may obey a new scaling of the critical speed, as obtained here. We show that due to constitutional undercooling, the presence of solute decreases particle speeds by an order of magnitude below those for a single-component system. We briefly consider the case of spherical bubbles, where thermocapillary and solutocapillary effects play a large role.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 625 , 25 April 2009 , pp. 299 - 320
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

Professor Alexander A. Golovin passed away on 10 September 2008.

References

REFERENCES

Ahuja, S., Stefanescu, D. M. & Dhindaw, B. K. 1994 Role of processing/system parameters on the interactions between liquid/solid interfaces and insoluble particles. In Proceedings of the Second International Conference on Cast Metal Matrix Composites (ed. Stefanescu, D. M. & Sen, S.), pp. 44–56. American Foundrymen's Society.Google Scholar
Allgower, E. L. & Georg, K. 1979 Introduction to Numerical Continuation Methods, Classics in Applied Mathematics, vol. 45. Society for Industrial and Applied Mathematics.Google Scholar
Asthana, R. 1998 Solidification Processing of Reinforced Metals, Key Engineering Materials, vol. 151–152. Trans Tech Publications.CrossRefGoogle Scholar
Asthana, R. & Tewari, S. N. 1993 The engulfment of foreign particles by a freezing interface. J. Mater. Sci. 28, 5414–5425.CrossRefGoogle Scholar
Aubourg, P. 1978 Interaction of second-phase particles with a crystal growing from the melt. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
Azouni, M. A. & Casses, P. 1998 Thermophysical properties effects on segregation during solidification. Adv. Colloid Interface Sci. 75, 83–106.CrossRefGoogle Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242–251.CrossRefGoogle Scholar
Bronstein, V. L., Itkin, Y. A. & Ishkov, G. S. 1981 Rejection and capture of cells by ice crystals on freezing aqueous solutions. J. Crystal Growth 52, 345–349.CrossRefGoogle Scholar
Catalina, A. V., Mukherjee, S. & Stefanescu, D. M. 2000 A dynamic model for the interaction between a solid particle and an advancing solid/liquid interface. Metall. Mater. Trans. A 31A, 2559–2568.CrossRefGoogle Scholar
Chang, A., Dantzig, J. A., Darr, B. T. & Hubel, A. 2007 Modeling the interaction of biological cells with a solidifying interface. J. Comput. Phys. 226 (2), 1808–1829.CrossRefGoogle Scholar
Chernov, A. A., Temkin, D. E. & Mel'nikova, A. M. 1976 Theory of the capture of solid inclusions during the growth of crystals from the melt. Soviet Phys.: Crystallogr. 21, 369–373.Google Scholar
Chernov, A. A., Temkin, D. E. & Mel'nikova, A. M. 1977 The influence of the thermal conductivity of a macroparticle on its capture by a crystal growing from a melt. Soviet Phys.: Crystallogr. 22, 656–658.Google Scholar
Dash, J. G., Rempel, A. W. & Wettlaufer, J. S. 2006 The physics of premelted ice and its geophysical consequences. Rev. Mod. Phys. 78 (3), 695–741.CrossRefGoogle Scholar
Davis, S. H. 2001 Theory of Solidification. Cambridge University Press.CrossRefGoogle Scholar
Deville, S., Saiz, E., Nalla, R. K. & Tomsia, A. P. 2006 Freezing as a path to build complex composites. Science 311 (5760), 515–518.CrossRefGoogle ScholarPubMed
Garvin, J. W. & Udaykumar, H. S. 2003 a Particle-solidification front dynamics using a fully coupled appraoch. Part I: Methodology. J. Crystal Growth 252, 451–466.CrossRefGoogle Scholar
Garvin, J. W. & Udaykumar, H. S. 2003 b Particle-solidification front dynamics using a fully coupled appraoch. Part II: Comparison of drag expressions. J. Crystal Growth 252, 467–479.CrossRefGoogle Scholar
Garvin, J. W. & Udaykumar, H. S. 2005 Drag on a ceramic particle being pushed by a metallic solidification front. J. Crystal Growth 276, 275–280.CrossRefGoogle Scholar
Garvin, J. W. & Udaykumar, H. S. 2006 Effect of a premelted film on the dynamics of particle-solidification front interactions. J. Crystal Growth 290, 602–614.CrossRefGoogle Scholar
Garvin, J. W., Yang, Y. & Udaykumar, H. S. 2007 a Multiscale modeling of particle-solidification front dynamics, Part I: Methodology. Int J. Heat Mass Transfer 50, 2952–2968.CrossRefGoogle Scholar
Garvin, J. W., Yang, Y. & Udaykumar, H. S. 2007 b Multiscale modeling of particle-solidification front dynamics. Part II: Pushing-engulfment transition. Int J. Heat Mass Transfer 50 (15–16), 2969–2980.CrossRefGoogle Scholar
Hadji, L. 1999 Asymptotic analysis of particle engulfment. Phys. Rev. E 60 (5), 6180–6183.CrossRefGoogle Scholar
Hadji, L. 2002 Modelling and asymptotic analysis of particle-interface interaction. Math. Comput. Model. 36, 147–156.CrossRefGoogle Scholar
Hadji, L. 2003 Morphological instability prior to particle engulfment by a solidifying interface. Script Materialia 48, 665–669.CrossRefGoogle Scholar
Hadji, L. 2007 Migration of a bubble in front of a directionally solidified interface. Phys. Rev. E 75 (4), 2602.CrossRefGoogle ScholarPubMed
Hecht, U. & Rex, S. 1997 On the transition from pushing to engulfment during directional solidification of the particle-reinforced aluminum-based metal-matrix composite 2014+ 10 vol pct Al2O3. Metall. Mater. Trans. A 28 (3), 867–874.Google Scholar
Hoyt, J. J. & Asta, M. 2002 Atomistic computation of liquid diffusivity, solid–liquid interfacial free energy, and kinetic coefficient in Au and Ag. Phys. Rev. B 65, 214106.CrossRefGoogle Scholar
Ishiguro, H. & Rubinsky, B. 1994 Mechanical interactions between ice crystals and red blood cells during directional solidification. Cryobiology 31 (5), 483–500.CrossRefGoogle ScholarPubMed
Israelachvili, J. N. 1991 Intermolecular and Surface Forces. Academic Press.Google Scholar
Juretzko, F. R., Stefanescu, D. M., Dhindaw, B. K., Sen, S. & Curreri, P. A. 1998 Particle engulfment and pushing by solidifying interfaces: Part I. Ground experiments. Metall. Mater. Trans. A 29 (6), 1691–1696.CrossRefGoogle Scholar
Karlsson, J. O. M. & Toner, M. 1996 Long-term storage of tissues by cryopreservation: critical issues. Biomaterials 17 (3), 243–256.CrossRefGoogle ScholarPubMed
Keller, H. B. 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. Rabinowitz, P. H.). Academic Press.Google Scholar
Kim, J. K. & Rohatgi, P. K. 1998 The effect of the diffusion of solute between the particle and the interface on the particle pushing phenomena. Acta Metallurgica 46 (4), 1115–1123.Google Scholar
Körber, C., Rau, G., Cosman, M. D. & Cravalho, E. G. 1985 Interaction of particles and a moving ice-liquid interface. J. Crystal Growth 72 (3), 649–662.CrossRefGoogle Scholar
Kurz, W. & Fisher, D. J. 1992 Fundamentals of Solidification. Trans Tech Publications.Google Scholar
Lide, D. R. (ed.) 1991 Handbook of Chemistry and Physics. CRC Press.Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water. I. A numerical method of computation. Proc. Roy. Soc. Lond. A 350, 1–26.Google Scholar
Meyyappan, M., Wilcox, W. R. & Subramanian, R. S. 1981 Thermocapillary migration of a bubble normal to a plane surface. J. Colloid Interface Sci. 83 (1), 199–208.CrossRefGoogle Scholar
Miksis, M. J. 1981 Numerical solution of hydrodynamic free boundary problems. PhD thesis, Courant Institute, New York University.Google Scholar
Miksis, M. J., Vanden-Broeck, J.-M. & Keller, J. B. 1981 Axisymmetric bubble or drop in a uniform flow. J. Fluid Mech. 108, 89–100.CrossRefGoogle Scholar
Mortensen, A. & Jin, I. 1992 Solidification processing of metal matrix composites. Intl Mater. Rev. (USA) 37 (3), 101–128.CrossRefGoogle Scholar
Mullins, W. W. & Sekerka, R. F. 1964 Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 35, 444–451.CrossRefGoogle Scholar
Pang, H., Stefanescu, D. M. & Dhindaw, B. K. 1994 Influence of interface morphology on the pushing/engulfment transition of polystyrene particles in succinonitrile + water matrices. In Proceedings of the Second International Conferences on Cast Metal Matrix Composites (ed. Stefanescu, D. M. & Sen, S.), pp. 57–69. American Foundrymen's Society.Google Scholar
Park, M. S., Golovin, A. A. & Davis, S. H. 2006 The encapsulation of particles and bubbles by an advancing solidification front. J. Fluid Mech. 560, 415–436.CrossRefGoogle Scholar
Poirier, D. R. & Speiser, R. 1987 Surface tension of aluminum rich Al–Cu liquid alloys. Metall. Mater. Trans. A 18A, 1156–1160.CrossRefGoogle Scholar
Pötschke, J. & Rogge, V. 1989 On the behaviour of foreign particles at an advancing solid–liquid interface. J. Crystal Growth 94, 726–738.CrossRefGoogle Scholar
Ratke, L. & Voorhees, P. W. 2002 Growth and Coarsening: Ostwald Ripening in Materials Processing. Springer-Verlag.CrossRefGoogle Scholar
Rempel, A. W. & Worster, M. G. 1999 The interaction between a particle and an advancing solidification front. J. Crystal Growth 205, 427–440.CrossRefGoogle Scholar
Rempel, A. W. & Worster, M. G. 2001 Particle trapping at an advancing solidification front with interfacial-curvature effects. J. Crystal Growth 223, 420–432.CrossRefGoogle Scholar
Rogerson, M. A. & Cardoso, S. S. S. 2000 Patterns of bubble desorption during the solidification of a multicomponent melt. J. Fluid Mech. 419, 263–282.CrossRefGoogle Scholar
Sasikumar, R. & Ramamohan, T. R. 1991 Distortion of the temperature and solute concentration fields due to the presence of particles at the solidification front—effects on particle pushing. Acta Metallurgica et Meterialia 39 (4), 517–522.CrossRefGoogle Scholar
Sekhar, J. A. & Trivedi, R. 1991 Solidification microstructure evolution in the presence of inert particles. Mater. Sci. Engng A 147, 9–21.CrossRefGoogle Scholar
Sen, S., Kaukler, W. F., Curreri, P. & Stefanescu, D. M. 1997 Dynamics of solid/liquid interface shape evolution near an insoluble particle—an x-ray transmission microscopy investigation. Metall. Mater. Trans. A 28 (10), 2129–2135.CrossRefGoogle Scholar
Temkin, D. E., Chernov, A. A. & Mel'nikova, A. M. 1977 Capture of foreign particles by a crystal growing from a melt containing impurities. Soviet Phys.: Crystallogr. 22 (1), 13–17.Google Scholar
Uhlmann, D. R., Chalmers, B. & Jackson, K. A. 1964 Interaction between particles and a solid-liquid interface. J. Appl. Phys. 35, 2986–2993.CrossRefGoogle Scholar
Wang, Z., Mukai, K. & Lee, I. J. 1999 Behavior of fine bubbles in front of the solidifying interface. ISIJ Intl 39 (6), 553–562.CrossRefGoogle Scholar
Wei, P. S., Huang, C. C. & Lee, K. W. 2003 Nucleation of bubbles on a solidification front-experiment and analysis. Metall. Mater. Trans. B 34 (3), 321–332.CrossRefGoogle Scholar
Wettlaufer, J. S., Worster, M. G. & Huppert, H. E. 1997 Natural convection during solidification of an alloy from above with application to the evolution of sea ice. J. Fluid Mech. 344, 291–316.CrossRefGoogle Scholar
Wilde, G. & Perepezko, J. H. 2000 Experimental study of particle incorporation during dendritic solidification. Mater. Sci. Engng A 283 (1–2), 25–37.CrossRefGoogle Scholar
Wu, Y., Liu, H. & Lavernia, E. J. 1992 Solidification behavior of Al–Si/SiC MMCs during wedge-mold casting. In Microstructure Formation During Solidification of Metal Matrix Composites (ed. Rohatgi, P. K.), pp. 41–61. Minerals, Metals & Materials Society.Google Scholar
Yang, Y., Garvin, J. W. & Udaykumar, H. S. 2008 Sharp interface numerical simulation of directional solidification of binary alloy in the presence of a ceramic particle. Intl J. Heat Mass Transfer 51 (1–2), 155–168.CrossRefGoogle Scholar
Young, N. O., Goldstein, J. S. & Block, M. J. 1959 The motion of bubbles in a vertical temperature gradient. J. Fluid Mech. 6, 350–356.CrossRefGoogle Scholar