Axisymmetric shapes and forces resulting from the interaction of a particle with a solidifying interface

A steady state model of the interaction of a foreign particle with a solidifying interface in a microgravity environment is presented in order to examine the phenomenon of particle engulfment. The model considers the interception of a spherical impurity particle by a deformable solid-liquid interface. Three forces are present whenever the particle is in near contact with the solid front, namely, the disjoining pressure force, the thermal force, and the hydrodynamic pressure force. These forces arise due to (i) the disjoining pressure gradients resulting from the deformation of the interface behind the particle, (ii) the interface distortion caused by the difference between the coefficients of thermal conductance of the particle and melt, and (iii) the hydrodynamic pressure resulting from the melt flow driven by the pressure gradients in the gap between the particle and the interface. The model accounts for the modification of the melt's freezing point by these pressure terms. The dependence of the interface morphology on the various physical and processing parameters is revealed. These include the rate of solidification, the thermal gradient, the gap width, and the coefficients of thermal conductivity of the particle and the melt. The gap profile is calculated and used in the evaluation of the three forces that characterize the interaction. The analysis shows that the outcome of the interaction, i.e., particle engulfment or rejection, depends primarily on the competition between the hydrodynamic and thermal forces.