Progress in modelling solidification microstructures in metals and alloys: dendrites and cells from 1700 to 2000

ABSTRACT

This is the first account of the history of our understanding of, and ability to model, solidification microstructures. Its objective is to retrace the scientific steps made, from the earliest observations of the eighteenth century to our present-day understanding of dendrites and eutectics. Because of the abundance of information, especially that added during the present century, sub-division was essential: this being the first of three articles. They cover dendrites and cells from 1700 to 2000, and then from 2001 to 2015 and finally eutectics and peritectics from 1700 to 2015. The authors have striven always to identify the genesis of every advance made, being aware that such a compact history must leave many worthy contributions by the wayside; others will doubtless complete the history. This review shows how cross-fertilisation between theory and experiment, and basic and applied research led to both the posing and answering of challenging fundamental questions, thus rewarding society with beneficial results.

KEYWORDS:

• Dendrites
• marginal stability
• solvability
• phase field
• rapid solidification
• dendrite spacing
• fragmentation
• columnar-equiaxed transition
• cell growth

Acknowledgements

The authors gratefully acknowledge private communication of precious information and helpful contributions from many colleagues; especially Bernard Billia, Jon Dantzig, Peter Galenko, Alain Karma, James Langer, Heiner Müller-Krumbhaar, Nikolas Provatas, Michel Rappaz and Ingo Steinbach. The authors also appreciate the contributions to various sections by Christoph Beckermann, Richard Grugel, Dieter Herlach, Hugh Kerr, Gérard Lesoult and Aurele Mariaux. The authors are indebted to Hélène Rappaz for meticulously verifying and organising the many references, and the librarians of Ecole Polytechnique Fédérale de Lausanne (EPFL) especially D. Gueritault, J. Yerly, J. Tanari, and of Eidgenössische Technische Hochschule Zürich (ETHZ) for delivering numerous copies of older papers. Last but not least the authors appreciate the constructive contributions of the reviewers which helped improving the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The acceptance of Zener’s conclusions was not immediate, especially by established experts in the field, as the long and animated discussion – filling 13 of the 46 pages – at the end of the article [30] shows. The reader is encouraged to read the highly interesting analysis of Zener’s impact upon metallurgy by Hillert [32].

2 Chalmers states in his book that J.C. Fisher in 1950 was the first to predict the steady-state dendrite. The priority, however, belongs clearly to Zener who, in 1946 (to the best knowledge of the present authors), was the first to combine an approximate transport equation for plate-like precipitates with capillarity. Zener’s contribution was probably not recognised for dendrite growth since his theory was proposed for solid-state transformation where plate-like precipitates are more common than needles. Zener also developed an exact solution for diffusional growth of a sphere that can be used for modelling of spherical precipitates or dendrite tips [34]. Once more we mention in this context the informative overview on Zener’s impact on phase transformations by Hillert [32].

3 A ‘cell’ is a well-defined morphology of the solid–liquid interface. In its fully developed form, it is a ‘finger’ without side arms and a non-paraboloidal tip.