A comprehensive model is developed for solving the heat and solute diffusion equations during solidification that avoids tracking the liquid—solid interface. The bulk liquid and solid phases are treated as regular solutions and an order parameter (the phase field) is introduced to describe the interfacial region between them. Two-dimensional computations are performed for ideal solutions and for dendritic growth into an isothermal and highly supersaturated liquid phase. The dependence upon various material and computational parameters, including the approach to conventional sharp interface theories, is investigated. Realistic growth patterns are obtained that include the development, coarsening, and coalescence of secondary and tertiary dendrite arms. Microsegregation patterns are examined and compared for different values of the solid diffusion coefficient.

Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method

A simple analytical model of microsegregation for the solidification of multicomponent steel alloys is presented. This model is based on the Clyne-Kurz model and is extended to take into account the effects of multiple components, a columnar dendrite microstructure, coarsening, and the δ/γ transformation. A new empirical equation to predict secondary dendrite arm spacing as a function of cooling rate and carbon content is presented, based on experimental data measured by several different researchers. The simple microsegregation model is applied to predict phase fractions during solidification, microsegregation of solute elements, and the solidus temperature. The predictions agree well with a range of measured data and the results of a complete finite-difference model. The solidus temperature decreases with either increasing cooling rate or increasing secondary dendrite arm spacing. However, the secondary dendrite arm spacing during solidification decreases with increasing cooling rate. These two opposite effects partly cancel each other, so the solidus temperature does not change much during solidification of a real casting.

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Simple Model of Microsegregation during Solidification of Steels

Mathematical modeling of mass, momentum, heat, and species transport phenomena occurring during solidification of metal alloys is reviewed. Emphasis is placed on the incorporation of the effects of the solid structure and the interactions between the solid and liquid phases on a microscopic scale into a (macroscopic) model of the transport phenomena occurring at the system scale. Both columnar and equiaxed growth structures, as well as laminar convection of liquid and solid crystals are considered. The macroscopic conservation equations are introduced via a volume averaging approach and commonly made simplifications are examined. Basic constitutive relations for the phase interactions occurring in alloy solidification are presented. Recent progress in including nucleation, microsegregation, undercooling and other microscopic phenomena in the macroscopic equations is reviewed. The specific areas where future theoretical and experimental research is needed are identified.

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Mathematical Modeling of Transport Phenomena During Alloy Solidification

A model is presented to predict the columnar to equiaxed transition (CET) in alloy castings. The model is based on a multiphase approach and accounts for heat and solute diffusion, as well as for grain nucleation, growth, and morphology. The model equations are applicable to both columnar and equiaxed dendritic solidification, thus offering an efficient single-domain formulation. A fixed grid, fully implicit finite-difference procedure is employed in the numerical solution, and a novel front tracking technique is incorporated that is also implicit in nature and readily applies to multidimensional situations. Calculations are performed for one-dimensional (1-D) and two-dimensional (2-D) castings of Al-Cu and Sn-Pb alloys. The calculated CET positions are compared with previous measurements in a (1-D) ingot cast under well-controlled conditions, and good agreement is found. The effects of various casting parameters on the CET are numerically explored.

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Prediction of Columnar to Equiaxed Transition during Diffusion-Controlled Dendritic Alloy Solidification

A unified solute diffusion model is proposed for columnar and equiaxed dendritic alloy solidification, in which nucleation, growth kinetics and dendrite morphology are taken into account. Various applications to a uniformly solidified system are demonstrated, with emphasis on three special cases: complete solute mixing in the liquid, columnar growth with significant dendrite tip undercooling, and equiaxed dendritic growth. Theoretical predictions of microsegregation, eutectic fractions and cooling curves are compared with a number of previous theoretical and experimental results, and good agreement is found.

A unified solute diffusion model for columnar and equiaxed dendritic alloy solidification

A mathematical model was developed to calculate microsegregation in binary metallic alloys. This model utilized the mathematical techniques of the method of lines combined with invariant imbedding (MOL/II) to solve the problem of combined heat and mass transfer during and after solidification. Model predictions were compared to experimental measurements in the Al-Cu system and to other microsegregation models. The MOL/II model predicted nonequilibrium second-phase contents within ±3 pct at low and intermediate cooling rates, when dendrite-arm coarsening was included in the model. It also was able to reproduce concentration profiles reasonably well. The analytical models commonly used (equilibrium cooling, Scheil equation, Brody/ Flemings model, Clyne/Kurz model, Solari/Biloni model, and Basaran equation) in micro-segregation calculations were shown to be considerably less accurate than the numerical models (MOL/II and the Ogilvy/Kirkwood model).

Mathematical Modeling of Microsegregation in Binary Metallic Alloys

Accurate relationships between equilibrium partition coefficients and solute concentration are required for the prediction of solute redistribution during solidification. Thermodynamic analyses are presented to relate these coefficients to fundamental thermodynamic quantities. Using the most accurate data available, partition coefficients are calculated for ten Fe−X (X=Al, C, Cr, Mn, Ni, N, P, Si, S, Ti) binary systems and compared with literature values. Equations are presented to allow for prediction of these partition coefficients as a function of temperature, as well as liquidus temperature as a function of composition. In addition, partition coefficient values are examined for the ternary systems Fe−Cr−C, Fe−Mn−Ni, and Fe−Ni−S.

Equilibrium partition coefficients in iron-based alloys

Viscosity measurements at high temperature were made using a concentric cylinder viscometer. These measurements were conducted on a typical lead blast furnace slag, and this master slag doped with various amounts of SiO2, A12O3, CaO, MgO, and/or ZnO. It was found that additions of A12O2 and/or SiO2 increased the viscosity of the master slag over the entire temperature range of interest (1150 °C to 1350 °C). Additions of the basic oxides CaO, MgO, and/or ZnO decreased the viscosity at high temperatures but raised the slag liquidus temperature. The majority of the measured viscosities are accurate within ± 10 pct. The viscosity data of this study, along with that of several similar studies from the literature, were correlated with the weight parameter (WP), a composition-dependent function similar to a basicity ratio. Analytical expressions were developed relating viscosity and the WP at several temperatures between 1150 °C and 1350 °C. Viscosities calculated using these expressions have estimated accuracies of ±1 poise (within certain temperature and composition limits). PbO and ZnO activity coefficient data from the literature were collected and correlated with composition-dependent functions. The calculated activity coefficients are considered accurate within ±0.3 for γpbo and ±0.5 for γzn within certain composition and temperature limits.

Viscosities and activities in lead-smelting slags

The importance of microsegregation in solidifying metals is discussed, with particular emphasis on the mathematical models which have been developed to help in the determination of solute concentration profiles and second phase contents after solidification. The predictions of a recently-developed numerical model are compared with other models and experimental data in the Al-Cu system. Results indicate that only the two numerical models were able to reproduce the amount of non-equilibrium second phase accurately. A plot is presented by which the amount of second phase can be estimated for hypoeutectic alloys as a function of cooling rate.

Thomas P. Battle

Papers

Mathematical Modeling of Microsegregation in Binary Metallic Alloys

Equilibrium partition coefficients in iron-based alloys

Viscosities and activities in lead-smelting slags

Microsegregation in cast alloys