The Scheil Equation

Many materials are manufactured by solidification, either as a final product by casting, or as an intermediate ingot or bar. The Scheil equation describes the partitioning that takes place during solidification and the resulting spatial redistribution of solute, which makes it difficult to maintain a homogeneous material composition, and which leads to unwanted concentrations of harmful impurities. This chapter explains nucleation and growth processes during solidification, the resulting dendritic, faceted, equiaxed and columnar structures depending on thermal conditions and material type, coupled solidification of two-phase eutectic materials, and typical casting methods and associated structures and defects. Very little is known about Erich Scheil, who worked at the Max Planck Institute in Stuttgart in the mid-20th century.

The Equations of Materials

This book describes some of the important equations of materials and the scientists who derived them. It is aimed at anyone interested in the manufacture, structure, properties and engineering application of materials such as metals, polymers, ceramics, semiconductors and composites. It is meant to be readable and enjoyable, a primer rather than a textbook, covering only a limited number of topics and not trying to be comprehensive. It is pitched at the level of a final year school student or a first year undergraduate who has been studying the physical sciences and is thinking of specialising into materials science and/or materials engineering, but it should also appeal to many other scientists at other stages of their career. It requires a working knowledge of school maths, mainly algebra and simple calculus, but nothing more complex. It is dedicated to a number of propositions, as follows: 1. The most important equations are often simple and easily explained; 2. The most important equations are often experimental, confirmed time and again; 3. The most important equations have been derived by remarkable scientists who lived interesting lives. Each chapter covers a single equation and materials subject. Each chapter is structured in three sections: first, a description of the equation itself; second, a short biography of the scientist after whom it is named; and third, a discussion of some of the ramifications and applications of the equation. The biographical sections intertwine the personal and professional life of the scientist with contemporary political and scientific developments. The topics included are: Bravais lattices and crystals; Bragg’s law and diffraction; the Gibbs phase rule and phases; Boltzmann’s equation and thermodynamics; the Arrhenius equation and reactions; the Gibbs-Thomson equation and surfaces; Fick’s laws and diffusion; the Scheil equation and solidification; the Avrami equation and phase transformations; Hooke’s law and elasticity; the Burgers vector and plasticity; Griffith’s equation and fracture; and the Fermi level and electrical properties.

Relationship between cooling rate and microsegregation in bottom-chilled directionally solidified ductile irons

This study explores the relationship between cooling rate and microsegregation of directionally solidified ductile iron. The unidirectional heat transfer system used in this research is made up of a copper mold kept chilled by circulating water and embedded in the bottom of Furan sand mold. Thermocouples are connected to the computer measuring system to record the cooling curves of the castings at a distance of 0, 30, 60 and 90 mm from the chilled copper mold surface. Alloys including Mn, Cr, Cu, Ni and Ti were added to the specimens. Electron microprobe analysis (EPMA) was employed to examine distribution of elements between the dendrite arms and nodular graphite. Results show that unidirectional heat transfer affects directly the solidification mode and microstructure of the casting. The cooling curves reveal that local solidification time increases with increasing distance from the chilled copper mold surface. Different solidification rates with corresponding microstructure and element segregation were observed in the same unidirectionally solidified casting. Local solidification time was closely related to element segregation. The effective segregation coefficient (Keff) calculated using the Scheil equation was found to vary, according to the stage of solidification. The actual segregation characteristics of complex alloys generally follow the Scheil equation.

Segregation and grain refinement in cast titanium alloys

The growth restriction factor is a parameter derived from binary phase diagrams and is a useful predictor for the grain refining response when a solute is added to a base alloy. This work investigates the relevance of growth restriction theory to titanium alloys where solidification rates are an order of magnitude faster than previous studies in aluminum- and magnesium-based systems. In particular, the segregation of Fe and Cr in titanium is investigated and the effects on grain size studied. It was found that the Scheil equation reasonably modeled solidification of titanium where cooling rates approach 120 °C/s, and the growth restriction factors for Fe and Cr were useful in predicting prior-β grain refinement. However, it was found that caution must be used when calculating growth restriction factors from binary phase diagrams.