Modelling of microsegregation in ternary alloys: Application to the solidification of Al–Mg–Si

Abstract

A fast numerical model has been developed for the quantitative prediction of microsegregation during solidification of ternary alloys. Considering a small volume of uniform temperature, the back-diffusion equations in the primary solid phase are solved in a 1-dimensional configuration using an implicit finite difference formulation with a Landau transformation onto a fixed [0,1] interval. The other phases which may precipitate during solidification are supposed to be stoichiometric and at equilibrium while the liquid is in a state of complete mixing. These calculations are coupled with phase diagram data through the use of mapping files: the liquidus surface, the monovariant lines and all the pertaining information are mapped through calls to Thermo-Calc [B. Sundman, B. Jansson and J. O. Andersson, CALPHAD, 9, 153 (1985)], prior to starting the microsegregation calculation itself. This very efficient microsegregation model can thus be coupled directly to macrosegregation computations performed at the scale of a whole casting: from the average enthalpy and concentrations variations computed at each mesh point of a casting during one time step, this microsegregation model is capable of predicting the variations of temperature, of the volume fractions of the various phases, of the liquid concentrations and of the average density. The efficiency of this coupling between microsegregation calculation and thermodynamic mapping files is demonstrated in the particular case of the Al–Mg–Si system.

Introduction

Since the first analytical approach of Mehrabian and Flemings for a ternary alloy [2], numerous models based on phase diagrams have been developed for the simulation of microsegregation in multicomponent systems [3]. Simple calculations for the limiting cases of equilibrium (i.e., Lever-rule) and of no diffusion in the solid phase [4] (i.e., Scheil–Gulliver) are even offered in commercial programs such as Thermo-Calc [1] and Chemsage [5]. These programs are based on the CALPHAD method, in which phenomenological thermodynamic models are given under the form of Gibbs free energy for the various phases which may appear in a given alloy [6].

By using Thermo-Calc and the Scheil's approximation, Saunders predicted the nature and amount of phases present in a variety of commercial aluminium alloys [7]. He obtained a good agreement between measured and simulated solid fraction–temperature relationships, except near the very end of solidification. Under Scheil's assumption (no back-diffusion), it is clear that microsegregation calculations always predict the formation of some eutectic phases [8], which are not always observed experimentally. Chen et al. [9] have simulated the solidification of nine-component alloys, for the case of slightly modified Scheil's. Coupling the analytical model of Clyne and Kurz [10] for back-diffusion with the calculated phase diagram, Yamada et al. [11] computed the solidification path of a ternary Fe–Cr–Ni alloy.

In order to give a more accurate description of the kinetic effects that take place during solidification (e.g., solid state diffusion and coarsening), many numerical models of microsegregation have been developed. Usually based upon a one-dimensional (1D) plate approximation to describe the geometry of a dendrite arm, the first numerical models approximated the phase diagrams with constant partition coefficients [12], [13], [14]. However, in ternary and higher-order systems, the partition coefficients can have an appreciable temperature dependence and change dramatically when the solidification path intersects a line of two-fold saturation. This is why, over the past twelve years, several authors have coupled phase diagram calculations with microsegregation computations, first for low-alloyed steels [15], but also for aluminium alloys [16], [17], [18] and even for eight-element superalloys [19]. In most cases, direct access to thermodynamic calculations was performed within the microsegregation model. It should be also mentioned that Sigli et al. [20] have developed a new method, based on the solubility concept, in order to calculate phase diagram equilibrium in aluminium alloys. This requires less adjustable parameters than the standard Gibbs' free energy method and therefore allows a faster and simpler assessment of thermodynamic equilibrium in multicomponent alloys.

The calculation of thermodynamic equilibrium in multicomponent alloys can be coupled with a back-diffusion model in the case of a small specimen solidified under given conditions, but certainly not to a whole casting in which mass-, heat- and solute-transport occurs (i.e., macrosegregation). In order to alleviate this difficulty, we have developed recently a new approach to microsegregation modelling in which access to thermodynamic data is made through mapping files. Although the back-diffusion calculation was initially limited to the solidification of a primary phase in a closed system [21], the present paper describes a consistent model in which back diffusion is calculated during the entire solidification process, i.e., even during precipitation of intermetallics, for a closed or open system. It is an extension of the model proposed by Combeau et al. [22] for binary alloys.