Thermal force induced by the presence of a particle near a solidifying interface

The presence of a foreign particle in the melt, ahead of a solid-liquid interface, leads to the onset of interfacial deformations if the thermal conductivity of the particle, k_p, differs from that of the melt, k_l. In this paper, the influence of the thermal conductivity contrast on the interaction between the solidifying interface and the particle is quantified. We show that the interface distortion gives rise to a thermal force whose expression is given by F_th=2πLGa³(1-α)/(2+α)T_m, where L is the latent heat of fusion per unit volume, T_m is the melting point, a is the particle’s radius, G the thermal gradient in the liquid phase and α=k_p/k_l. The derivation makes use of the following assumptions: (i) the particle is small compared to the horizontal extent of the interface, (ii) the particle is placed in the near proximity of the deformable solid-liquid interface, and (iii) the interface is practically immobile in the calculation of the thermal field, i.e., V≪k_lG/L, where V is the interface growth velocity. An order of magnitude analysis shows that the magnitude of the thermal force may be greater than that of the forces that are usually accounted for in studies of particle-interface interaction, namely, the drag force and the force due to the disjoining pressure. The inclusion of the thermal force in the overall force balance is found to modify the value of the growth rate at which equilibrium is attained. The analysis leads to the prediction that the growth rate at equilibrium is increased for particles with a low thermal conductivity and is decreased for particles with a high thermal conductivity. The derivation of the thermal force is preceded by a detailed analysis of the influence of the thermal conductance contrast on the interface profile.