An ICME Method for Predicting Phase Dissolution During Solution Treatment in Advanced Super Vacuum Die Cast Magnesium Alloys

Abstract

An integrated computational materials engineering (ICME) methodology was applied in this study to systematically and quantitatively study the second phase dissolution kinetics during the solution treatment process of a high pressure die cast magnesium sample. The study was conducted on Mg–9 wt% Al, Mg–5 wt% Al, and Mg–11 wt% Al binary alloys after isothermal solution treatments ranging from 380 to 420 °C. The experimental measurements revealed an exponential decrease of the volume fraction of the second β-phase (Mg17Al12) during the solution treatment. A CALPHAD-based computational tool (Thermocalc DICTRA Diffusion Module) was used to simulate the dissolution process. In this study, measurements from as-cast samples were used as input parameters to improve the accuracy of the predicted results. An analytical, physics-based micro-model, based on the Johnson–Mehl–Avrami type equation, was also applied to study the dissolution kinetics. Both the simulation and micro-model quantitatively agree with the experimental results at all solution treatment temperatures. Calibration and verification of a few input parameters help to understand the assumptions made in the modeling, improve the accuracy of the prediction, and can be applicable in different Mg–Al binary alloys. Adopting such an ICME methodology for modeling development and verification in predicting the dissolution kinetics during solution treatment will allow for more rapid optimization of solution treatments procedures to be used in industrial applications.

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Acknowledgements

The authors acknowledge the support of U.S. Department of Energy under Contract DE-EE0006434. We also gratefully acknowledge M. Li, J. Zindel and L. Godlewski of Ford Motor Company who provided the SVDC cast plates.

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Correspondence to Zhenjie Yao.

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Data Availability

The experimental data supporting this publication and the MATLAB code for β phase characterization are available in the Materials Commons at http://doi.org/10.13011/m3-8xg4-dq75.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Appendices

Appendices

Appendix 1: Relationship Between Volume Fraction and Surface Area

In the multi-particle assumption, each particle now has an initial average radius of \( R_{\beta ,n} \), and n represents the number of particles in the actual samples. In the DICTRA simulation, the initial radius of the secondary phase is \( R_{\beta ,n} \).

In DICTRA simulation setup, the initial volume fraction can be expressed as: \( \frac{{R_{\beta }^{3} }}{{R_{\text{tot}}^{3} }} = \frac{{V_{\beta } }}{{V_{\text{tot}} }} = V_{\beta }^{f} \). For the n particles scenario, the initial volume fraction is: \( \frac{{n \times R_{\beta ,n}^{3} }}{{R_{\text{tot}}^{3} }} = \frac{{V_{\beta ,n} }}{{V_{\text{tot}} }} = V_{\beta ,n}^{f} \). The initial volume fraction should be satisfied that \( V_{\beta }^{f} = V_{\beta ,n}^{f} \), and therefore, \( R_{\beta }^{3} = nR_{\beta ,n}^{3} \Rightarrow R_{\beta } = \sqrt[3]{n}R_{\beta ,n} \)

The surface area of the sphere for n particle scenario is \( A_{n} = 4\pi R_{\beta ,n}^{2} \) for each individual particle. The surface area of the sphere for DICTRA simulation is \( A = 4\pi R_{\beta }^{2} \). Hence, the ratio of the two surface area in these two case is \( \frac{A}{{A_{n} }} = \frac{{R_{\beta }^{2} }}{{n \times R_{\beta ,n}^{2} }} = \frac{{\sqrt[3]{{n^{2} }}R_{\beta ,n}^{2} }}{{n \times R_{\beta ,n}^{2} }} = \sqrt[3]{{\frac{1}{n}}} < 1 \).

Appendix 2: The Volume Fraction Change with Respect to Time

During a diffusion-controlled dissolution process, after time t, the shrinkage scale (solute diffusion length) \( l = a\sqrt {Dt} \), where a is a prefactor constant.

The ratio of the volume fraction of two model can be express as

$$ \frac{{V_{\beta }^{f} \left( t \right)}}{{V_{\beta ,n}^{f} \left( t \right)}} = \frac{{V_{\beta } \left( t \right)}}{{V_{\beta ,n} \left( t \right)}} = \frac{{R_{\beta }^{3} \left( t \right)}}{{n \times R_{\beta ,n}^{3} \left( t \right)}} = \frac{{\left( {R_{\beta } - l} \right)^{3} }}{{n \times \left( {R_{\beta ,n} - l} \right)^{3} }} = \frac{{\left( {\sqrt[3]{n}R_{\beta ,n} - l} \right)^{3} }}{{n \times \left( {R_{\beta ,n} - l} \right)^{3} }} $$
$$ \sqrt[3]{{\frac{{V_{\beta }^{f} \left( t \right)}}{{V_{\beta ,n}^{f} \left( t \right)}}}} = \frac{{\sqrt[3]{n}R_{\beta ,n} - l}}{{\sqrt[3]{n} \times \left( {R_{\beta ,n} - l} \right)}} = \frac{{\sqrt[3]{n}R_{\beta ,n} - \sqrt[3]{n}l + \sqrt[3]{n}l - l}}{{\sqrt[3]{n} \times \left( {R_{\beta ,n} - l} \right)}} = 1 + \frac{{\left( {\sqrt[3]{n} - 1} \right)l}}{{\sqrt[3]{n}\left( {R_{\beta ,n} - l} \right)}} > 1 $$

The volume fraction is now \( V_{\beta }^{f} \left( t \right) = \left( {1 + \frac{{\left( {\sqrt[3]{n} - 1} \right)l}}{{\sqrt[3]{n}\left( {R_{\beta ,n} - l} \right)}}} \right)^{3} V_{\beta ,n}^{f} \left( t \right). \)

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Yao, Z., Berman, T. & Allison, J. An ICME Method for Predicting Phase Dissolution During Solution Treatment in Advanced Super Vacuum Die Cast Magnesium Alloys. Integr Mater Manuf Innov 9, 301–313 (2020). https://doi.org/10.1007/s40192-020-00186-0

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Keywords

  • Dissolution kinetics
  • Solution treatment
  • SVDC
  • Magnesium alloys
  • ICME
  • DICTRA