Coupling of multicomponent thermodynamic databases to a phase field model: application to solidification and solid state transformations of superalloys

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The phase field method has successfully been applied to predict microstructure evolution in metallic alloys such as dendritic solidification [1,2,3] as well as the precipitation of coherent ordered phases from a disordered matrix taking into account the effect of elastic strain on the morphology of the precipitates [4,5,6]. Recently, phase field methods employing multiple phase field parameters have been applied to eutectic and peritectic systems [7,8,9] in dilute model systems as well as to evaluate the kinetics of solid state grain growth [10]. In order to apply the method quantitatively to these phenomena occurring in technical alloys the method needs to be able to treat multicomponent multi-phase systems. Recently a simplified multicomponent approach using linearized phase diagrams has been applied to steels [11]. The ternary system Fe-Al-Co has been addressed by [5] who use real thermodynamic data within a phase field model. The main target of this work is to simulate solidification and heat treatment of technical single crystal superalloys. The present paper proposes a multicomponent extension to a multi-phase-field model (PFM) described in [12,13] employing a general method of obtaining thermodynamic data from databases being assessed according to the CALPHAD method [14]. This method provides realistic thermodynamic descriptions for all phases present in a given material. The present model can be applied to any system if a thermodynamic database is available and is not restricted to a special formulation of the Gibbs energy. To evaluate thermodynamic quantities the FORTRAN interface of the software ThermoCalc [15] is used for the calculation of molar Gibbs energies and chemical potentials to calculate the driving force at the diffuse interface. Furthermore, a subroutine of the software Dictra [16], which is also interfaced to Thermo-Calc in order to calculate thermodynamic factors of diffusion, has been coupled to the phase field code. The subroutine calculates the diffusion matrix for given multicomponent phase from a standardized kinetic database containing data on atomic mobilities. In order to validate the growth kinetics of this model 1D benchmark tests have been performed by comparison with a sharp interface calculation by Dictra using the same thermodynamic [17] and kinetic database [18] of the ternary Ni-Al-Cr system. Moreover, a 2D simulation of Ostwald-ripening of spherical g9 precipitates in a ternary Ni-Al-Cr alloy with a small g-g9 lattice mismatch is presented. The results are compared to experimental data by M. Doi [19], who determined the coarsening rate of the g9 precipitates in a Ni-6.2 at.-% Al-18.2 at.-% Cr alloy. Scripta mater. 42 (2000) 1179–1186