Mathematical modelling of microsegregation in binary metallic alloys.

Abstract

The process of microsegregation has been modelled as a combined heat and mass transfer problem in one dimension, utilizing the Method of Lines/Invariant Imbedding (MOL/II) mathematical treatment. Dendrite-arm coarsening, the eutectic reaction, and post-solidification heat and mass transfer were modelled as part of this process. The predictions of the model were compared to simpler models, as well as the experimental measurements in the Fe-Ni (25 mole % Ni) and Al-Cu (1-3 mole % Cu) systems. Qualitative results from the model were physically reasonable. Quantitatively, the model was capable of predicting the amount of non-equilibrium second phase in Al-Cu within $\\pm$3% for sample cooling rates from 0.1 to 200 K/s. Concentration profile predictions in Al-Cu were considerably more sample-dependent. Predictions of maximum and minimum concentrations in the Fe-Ni system did not match experiment. It is possible that a more general microsegregation model would be required for accurate calculations in this system. The MOL/II model was slightly more accurate than the simpler numerical model developed by Ogilvy and Kirkwood. Both numerical models were considerably more accurate than analytical microsegregation models (Scheil equation, Brody/Flemings and Clyne/Kurz models). Run times were significantly longer when systems with low solid-state diffusivity were modelled, or when dendrite-arm coarsening was included in the algorithm. As an example, a simulation that would run in a few hours on an Apollo workstation without considering coarsening might take several days with coarsening included. A two-dimensional model in cylindrical coordinates was also developed, and preliminary tests indicated that reasonable results could be obtained in the Al-Cu system. Predicted concentrations were very close to the one-dimensional case. This model has not been extensively tested and its utility for general dendritic solidification is unknown. Finally, in order to assist in accurate use of the model, a literature review and analysis of values for the equilibrium partition coefficient was conducted. This parameter was examined in eleven binary systems. Ten of these alloys were iron-based, with the solutes Al, C, Cr, Mn, N, Ni, P, S, Si, and Ti. The Al-Cu system was also investigated. Results indicated significant errors in literature values often used in microsegregation calculations.