Phase-field modeling of micropore formation in a solidifying alloy

This thesis is focused on the study of the morphology of micropores formed during solidification of metallic alloys. Micropores constrained to form in well-developed dendritic solid network adopt complex non-spherical shapes. Previous studies using X-ray tomography (XRT) have shown that the local mean curvature of micropores can be as large as 0.2μm−1. Such a high curvature induces an overpressure of 400 kPa in the pore with respect to the surrounding liquid and thus highly affects its volume fraction. While trying to predict pore formation at the macro-scale using average equations, the effect of this curvature is usually introduced using simple mathematical relationships, i.e., pinching model, describing the pore curvature as a function of the volume fraction and a typical length scale (e.g., the secondary dendrite arm spacing or DAS) of the primary phase. Such relationships, however, are based on simplifications of the pore morphology that are not generally backed up with an extensive study of the pore shape and its evolution during solidification. On the other hand, direct observations using XRT offer valuable information about micropore morphologies after solidification, but unfortunately their limited spatial resolution does not allow yet for a detailed study of the curvature of micropores during their formation. In this work, a multiphase-field model has been developed in order to study and better understand the formation of micropores constrained to grow in a solid network (i.e., pinching effect). The model accounts for the pressure difference due to capillarity forces between liquid and gas and the mechanical equilibrium condition at triple (solid-liquid-pore) lines. The partitioning and diffusion of dissolved gases such as hydrogen in aluminum alloys are also incorporated into the model by solving together Sievert’s law, the perfect gas law and Fick’s equation. The model was first implemented in 2-D, and then was extended to 3-D by developing a program for parallel Distributed Memory Processor (DMP) machines. The model was used to study the influence of the DAS, primary phase solid fraction and gas content on the morphology of micropores. After validating the multiphase-field approach for a spherical micropore growing freely in a supersaturated liquid, the calculations show that a pore constrained to grow in a narrow liquid channel exhibits a substantially higher mean curvature, a larger pressure and a smaller volume than an unconstrained pore. The morphology of pores at steady state, obtained with the model for different solid morphologies and initial gas concentrations, was also analyzed. From their predicted 3-D morphologies, entities such as the Interfacial Shape Distribution (ISD) were plotted and analyzed. As expected, it was verified that the mean curvature of the pore-liquid interface, and thus also the pressure inside the pore, is uniform. The local morphology of the pore, however, varies depending on the position of the pore-liquid interface with respect to the primary solid: In between two parallel dendrite arms, the pore adopts a cylindrical-type shape with one principal curvature being almost nil and the other being about twice the mean curvature of the pore-liquid interface growing with a spherical-type tip in between four parallel dendrite arms. The results were then compared with analytical pinching models. While predicting a similar trend, analytical models tend to underestimate the pore curvature at high solid fraction and gas concentration. For pores spanning over distances larger than the average DAS, the simulations showed that the mean curvature varies between two limits: a minimum curvature given by the largest sphere that can be fitted in the interdendritic liquid, and a maximum curvature given by the size of the narrowest section that the pore needs to pass in order to expand. The pore curvature is therefore a complex non-monotonic function of the DAS, solid fraction, gas content and statistical variations of the liquid channel widths. Based on this and considering the complex morphology of pores reconstructed using high-resolution XRT, the present phase-field results suggest that a simple pinching model based on a spherical tip growing in between remaining liquid channels is a fairly good approximation. This model was further validated by performing phase-field calculations for a pore growing in a representative volume element taken from XRT. For such condition, it was observed that as the pore grows and penetrates thin liquid channels, the fraction of cylindrical-type pore-liquid interfaces increases and becomes dominant over spherical-type ones, a feature already observed in XRT observations of as-solidified micropores. Finally, in-situ XRT observations were also performed on Al-Cu samples directionally viii solidified in a Bridgman furnace and then quenched. Macroscopic calculations of porosity formation using the software ProCast showed that a high fraction of pores can form during the quench itself, and not so much during directional solidification. After solidification, small specimens were analyzed by XRT on the TomCat beamline of the Paul Scherrer Institute in Villigen, during isothermal holding at a temperature slightly above the eutectic temperature. It was shown that the volume fraction of primary solid increases during holding time, as a result of solid state diffusion of copper, while coalescence of secondary dendrite arms simultaneously modifies the topology of the remaining liquid from continuous films to isolated droplets. This topology change is shown to modify substantially the average hydrogen diffusion coefficient in the mushy zone. In parallel to the evolution of the solid-liquid interface, the number of micropores and their volume fraction change over time. This evolution is analyzed in terms of a local mass balance of hydrogen and of diffusion of hydrogen toward the ambient atmosphere.