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Mechanism of Dendritic Branching

  • 2011 Edward DeMille Campbell Memorial Lecture ASM International
  • Published:
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Abstract

Theories of dendritic growth currently ascribe pattern details to extrinsic perturbations or other stochastic causalities, such as selective amplification of noise and marginal stability. These theories apply capillarity physics as a boundary condition on the transport fields in the melt that conduct the latent heat and/or move solute rejected during solidification. Predictions based on these theories conflict with the best quantitative experiments on model solidification systems. Moreover, neither the observed branching patterns nor other characteristics of dendrites formed in different molten materials are distinguished by these approaches, making their integration with casting and microstructure models of limited value. The case of solidification from a pure melt is reexamined, allowing instead the capillary temperature distribution along a prescribed sharp interface to act as a weak energy field. As such, the Gibbs-Thomson equilibrium temperature is shown to be much more than a boundary condition on the transport field; it acts, in fact, as an independent energy field during crystal growth and produces profound effects not recognized heretofore. Specifically, one may determine by energy conservation that weak normal fluxes are released along the interface, which either increase or decrease slightly the local rate of freezing. Those responses initiate rotation of the interface at specific locations determined by the surface energy and the shape. Interface rotations with proper chirality, or rotation sense, couple to the external transport field and amplify locally as side branches. A precision integral equation solver confirms through dynamic simulations that interface rotation occurs at the predicted locations. Rotations points repeat episodically as a pattern evolves until the dendrite assumes a dynamic shape allowing a synchronous limit cycle, from which the classic repeating dendritic pattern develops. Interface rotation is the fundamental mechanism responsible for dendritic branching.

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Notes

  1. The mathematical expressions for the transport fields surrounding dendrites in pure materials and alloys are identical; only the transport coefficients defining the Péclet number differ.

  2. The characteristic temperatures, \((\Updelta{H}_{f}/C_p)\) for body centered cubic (bcc) succinonitrile (SCN) and face centered cubic (fcc) pivalic acid anhydride (PVA) are 23 K, and 11 K, respectively.

  3. Interested readers may access NASA’s official archives for the IDGE-USMP series, available at http://pdlprod3.hosc.msfc.nasa.gov. To locate associated NASA Technical Reports for the IDGE-USMP series please go to http://naca.larc.nasa.gov/index.jsp?method=aboutntr.

  4. The gravitational acceleration in low-earth orbit, \(g_{{LEO}},\) affecting the IDGE experiments on NASA’s USMP missions was reduced to a quasi-static level of \(g_{{LEO}}\approx10^{-7}g_0,\) where \(g_0=9.807\,\hbox{m/s}^{2}\) is the average, or standard terrestrial value of the gravitational acceleration.

  5. The higher the Prandtl number of a melt, the more that hydrodynamic flow affects heat transfer. The Prandtl number of a fluid Pr is the ratio of its kinematic viscosity, or momentum diffusivity \(\nu [\hbox{m}^2/\hbox{s}],\) to its thermal diffusivity, \(\alpha [\hbox{m}^2/\hbox{s}].\) Plastic crystals, such as succinonitrile and pivalic acid anhydride, are stable and conveniently transparent for microphotography but have relatively large Prandtl numbers, \(Pr=\nu/\alpha>10,\) whereas molten metals, which suffer from opaqueness, reactivity, and much higher melting temperatures, exhibit small Prandtl numbers, \(Pr\ll 1.\)

  6. The capillary length, \(d_0=2.82\pm0.17\) nm, is defined from marginal stability as \(d_0\equiv(C_{p}T_{{m}}\Upomega\gamma_{s\ell})/\Updelta{H_{{f}}}^2.\) All the constituent thermo-physical constants for \(d_0\) for SCN are fully documented,[45] including, its equilibrium melting point, \(T_{{m}}=331.233\pm0.001\,\hbox{K};\) as well as the molar specific heat of the melt, \(C_p=160.91\pm1.6\,\hbox{J/mol-K};\) the molar volume of the melt, \(\Upomega=0.816\pm0.006\times10^{-4}\,\hbox{m}^3/\hbox{mol};\) the interfacial energy density, \(\gamma_{s\ell}=8.94\pm0.5\,\hbox{mJ/m}^{2};\) and the molar heat of fusion, \(\Updelta{H_{{f}}}=3.704\pm0.002\,\hbox{kJ/mol}.\)

  7. Conventional, i.e., bulk, thermal conductivities bear System International (SI) units of [watts/m-K]; however, surface, or interfacial, thermal conductivities must carry SI units of [watts/K] in order that the interface flux exhibits proper units of [watts/m].

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Acknowledgments

The author is honored by the ASM International for his selection as the 2011 Edward DeMille Campbell Lecturer at MS&T Columbus, OH, October 18, 2011, prompting preparation of this paper. Thanks are extended to colleagues Professor John Lowengrub, Mathematics Department, University of California, Irvine, CA, and Professor Shuwang Li, Department of Mathematics, Illinois Institute of Technology, Chicago, IL, for applying their integral equation solver that provided the initial independent dynamical checks on the local-response theory presented here; and to Professor Markus Rettenmayr and Mr. Klemens Reuther, Friedrich Schiller University, Jena, Germany, for their independent checks using their unstructured grid dynamic solver. The author acknowledges helpful discussions held on this subject with Dr. Geoffrey McFadden, NIST, Gaithersburg, MD; Dr. Alexander Chernov, Lawrence Livermore National Laboratory, Livermore, CA; Professor Bernard Billia, Faculté des Sciences et Techniques, University of Marseille, France. The author is grateful for the encouragement received from Professor Emerita Jean Taylor, Rutgers University, Cream Ridge, NJ, and the Courant Institute of Mathematical Sciences, New York University, New York City, and from Professor John W. Cahn, University of Washington, Seattle, WA.

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  1. Florida Institute of Technology, Melbourne, FL, 32901, USA

    Martin E. Glicksman (Allen S. Henry Chair and University Professor of Engineering)

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Correspondence to Martin E. Glicksman.

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Dr. Martin Eden Glicksman is a recognized expert on the solidification of metals and semiconductors, atomic diffusion processes, the energetics and kinetics of network structures, grain growth, phase coarsening, and microstructure evolution. He has co-authored over 300 technical papers, reviews, and monographs, and has authored two major textbooks: Diffusion in Solids (Wiley Interscience, 2000) and Principles of Solidification (Springer USA, 2011). He is a Fellow of the Metallurgical Society, the American Society for Materials International, American Association for the Advancement of Science, and the American Institute for Aeronautics and Astronautics, and a member of the American Physical Society. For his research accomplishments on solidification, he received the Rockwell medal, Case-Western University’s van Horn Award, the Metallurgical Society’s Chalmers Medal, ASM’s International Gold Medal and Honorary Membership, and the Alexander von Humboldt Senior Research Prize. Professor Glicksman’s experiments aboard Space Shuttle Columbia led to his receiving NASA’s Award for Technical Excellence, and AIAA’s 1998 National Space Processing Medal. In 2010 he was awarded the Sir Charles Frank Prize of the International Organization for Crystal Growth for his fundamental contributions to dendritic crystal growth. He is a member of the National Academy of Engineering, and serves as Chairman of the Materials Engineering Section of the National Academy of Engineering for 2011–2012.

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Glicksman, M.E. Mechanism of Dendritic Branching. Metall Mater Trans B 43, 207–220 (2012). https://doi.org/10.1007/s11663-011-9594-2

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