Mats Hillert

On the theory of normal and abnormal grain growth

Abstract A growth equation for individual grains in single-phase materials is suggested. It is used to calculate a rate equation for normal grain growth and the size distribution in the material. It predicts a maximum size of twice the average size. The theory is modified to take into account the effect of second-phase particles. In an alternative treatment the array of grains is described in terms of a kind of defects introduced into a perfect array. The defects move through the array during grain growth. The rate of grain growth is calculated from the number of defects and their mobility. The defect concentration is predicted by comparing the two treatments. The defect-model predicts two grain size limits due to second-phase particles. Normal grain growth takes place below the lower limit. Abnormal grain growth can take place between the two limits if the material contains at least one very large grain. No grain growth can take place above the higher limit. Several possible mechanisms for the development of abnormal grain growth are examined. An explanation is offered for the observation that most of the well-known cases occur as the second-phase is dissolving.

A model for alloying in ferromagnetic metals

Abstract A mathematical representation of the magnetic specific heat, recently suggested by Inden, was applled to iron in an evaluation of the difference in Gibbs energy between the fcc and bcc states. The resulting equations were then used for a treatment of alloying effects in ferromagnetic metals due to the change of the Curie temperature. The result was approximated in order to conform to the subregular solution model. A strong asymmetric term was obtained.

The compound energy formalism

The compound energy formalism for solution phases with sublattices is very flexible and thermodynamic models for a large variety of phases have been constructed within this formalism. The range of applications is reviewed and the methods of handling various problems are examined. Recent developments including treatments of short range order within the compound energy formalism are reviewed.

Empirical methods of predicting and representing thermodynamic properties of ternary solution phases

Abstract Various empirical methods of predicting the properties of a ternary system from its binary components are reviewed and their generalization to higher systems is discussed, The methods are compared by applying the subregular solution model and expressions are derived which allow a direct comparison. For symmetric treatments, a numerical method and equivalent analytical methods are recommended for universal usage. For asymmetric treatments, a numerical method is recommended but the possibility of treating such cases by transforming an asymmetric ternary system into a reciprocal system and using a formally symmetric description is emphasized.

A two-sublattice model for molten solutions with different tendency for ionization

Different models accounting for the introduction of an excess of cations or anions in an ionic melt are considered. A new model which describes the variation in the thermodynamic properties in a melt as it gradually changes its composition from purely metallic state to complete ionization is developed. The model is based upon Temkin’s expression for the configurational entropy of mixing. Neutral species and hypothetical vacancies with an induced charge are introduced on the anion sublattice. The induced charge is equal to the average valency of the species occupying the cation sublattice. When the tendency of ionization is weak, the model approaches the substitutional model. For binary systems the new model is formally identical to the associate solution model if the associates are defined in such a way that they contain one atom of the electronegative element. For higher-order systems the new model works with a lower number of composition variables and parameters than the associate solution model.

Chemically induced grain boundary migration

Abstract Experimental evidence is presented, showing that gram boundary motion can be induced by changing the composition by means of grain boundary diffusion. The experiments were carried out by treating specimens of pure iron in an atmosphere of zinc or iron-zinc specimens in an atmosphere with a lower zinc potential. The reaction resembles the grain boundary migration in discontinuous precipitation and provides an experimental method of isolating grain boundary migration for study. The experimental results can be used to evaluate rather directly the grain boundary diffusivity and mobility. The diffusivities thus obtained are several orders of magnitude larzer than the values reported for stationary boundaries. It is concluded that the boundary diffusivity is greatly enhanced by grain boundary motion.

Phase equilibria, phase diagrams and phase transformations : their thermodynamic basis

Computational tools allow material scientists to model and analyze increasingly complicated systems to appreciate material behavior. Accurate use and interpretation however, requires a strong understanding of the thermodynamic principles that underpin phase equilibrium, transformation and state. This fully revised and updated edition covers the fundamentals of thermodynamics, with a view to modern computer applications. The theoretical basis of chemical equilibria and chemical changes is covered with an emphasis on the properties of phase diagrams. Starting with the basic principles, discussion moves to systems involving multiple phases. New chapters cover irreversible thermodynamics, extremum principles, and the thermodynamics of surfaces and interfaces. Theoretical descriptions of equilibrium conditions, the state of systems at equilibrium and the changes as equilibrium is reached, are all demonstrated graphically. With illustrative examples - many computer calculated - and worked examples, this textbook is an valuable resource for advanced undergraduates and graduate students in materials science and engineering.

A treatment of the solute drag on moving grain boundaries and phase interfaces in binary alloys

Abstract A previous treatment of the solute drag effect on the movement of grain boundaries and phase interfaces, which was based upon the evaluation of the free-energy dissipation, is developed further. The new treatment becomes identical to the treatments by Cahn and Lucke and Stuwe for grain boundaries in dilute solutions but is not limited to low solute contents, to ideal solutions or to single-phase systems. Some numerical calculations for grain boundaries are presented and the dissipation of free energy in different zones of the interface is discussed. The maximum solute drag, calculated with a variable diffusivity. is always lower than the value found by using any average value of the diffusivity. The simple expression for the estimation of the solute drag, previously given, is approximately correct at constant diffusivity. only. A number of calcultions for diffusionless phase transformations are presented and allow some conclusions regarding the spontaneous start of such a transformation.

Inhibition of grain growth by second-phase particles

The difficulty for a second-phase particle to pinch off when the grain boundary migrates, affects the final grain size. Some theoretical models for this effect are discussed. The predictions are quite different for three-dimensional and two-dimensional systems. In the first case one finds that Dr should be proportional to ƒ−0.93 for reasonably small ƒ That is close to the Smith-Zener prediction of ƒ−1. For large ƒ it is predicted that Dr should be proportional to ƒ−1/3. In two-dimensional systems it is predicted that one should always come close toƒ−0.5, in agreement with results from computer simulations. There is no region where one should expect Dr to be proportional to ƒ−1 in two dimensions.

A compound-energy model of ordering in a phase with sites of different coordination numbers

Abstract When applied to a phase where all the sites do not have the same coordination number, the bond-energy model gives a term in u AA − u BB , which cannot be assigned a numerical value because the energy of different elements cannot be compared. As an alternative a compound-energy model is now developed from the well-known model for reciprocal systems. It gives a similar result but the u AA − u BB term does not appear. In addition, the new model takes into account two kinds of factors resulting in ordering, those which make different elements prefer different kinds of lattice sites and those which give a preference for unlike neighbors. The ordering in the sigma phase is considered in some detail.