A quantitative model with new scaling for silicon carbide particle engulfment during silicon crystal growth

We present rigorous numerical modeling and analytical arguments to describe data on the engulfment of silicon carbide particles during silicon crystal growth obtained via advanced terrestrial and microgravity experiments. For the first time in over a decade of research on SiC inclusions in silicon, our model is able to provide a quantitative correlation with experimental results, and we are able to unambiguously identify the underlying physical mechanisms that give rise to the observed behavior of this system. In particular, we identify a significant and previously unascertained interaction between particle-induced interface deflection (originating from the thermal conductivity of the SiC particle being larger than that of the surrounding silicon liquid) and curvature-induced changes in melting temperature arising from the Gibbs-Thomson effect. For a particular range of particle sizes, the Gibbs-Thomson effect flattens the deflected solidification interface, thereby reducing drag on the particle and increasing its critical velocity for engulfment. We show via numerical calculations and analytical reasoning that these effects give rise to a new scaling of the critical velocity to particle size as v_c∼R^-5/3, whereas all prior models have predicted either v_c∼R^-1 or v_c∼R^-4/3. This new scaling is needed to quantitatively describe the experimental observations for this system.