Coarsening of solidβ-Sn particles in liquid Pb-Sn alloys: Reinterpretation of experimental data in the framework of trans-interface-diffusion-controlled coarsening

Abstract

Previously published data, not ours, on the coarsening of solid \ensuremath{\beta}-Sn particles in a liquid Pb-Sn matrix of near-eutectic composition are reanalyzed within the framework of the trans-interface-diffusion-controlled (TIDC) theory of coarsening. The data were obtained under conditions of microgravity from specimens heat-treated at 458 K and containing four equilibrium volume fractions ${f}_{e}$ equaling 0.10, 0.15, 0.20, and 0.30. We show that the rate constants $k({f}_{e})$ in the traditional coarsening equation ${\ensuremath{\langle}r\ensuremath{\rangle}}^{3}\ensuremath{\approx}k({f}_{e})t$ for the kinetics of growth of the average particle radius $\ensuremath{\langle}r\ensuremath{\rangle}$ are nearly independent of ${f}_{e}$, in disagreement with numerous theories wherein coarsening is controlled by diffusion in the host matrix phase. Atom transport in TIDC coarsening is instead controlled by slow diffusion through the diffuse interface, of width \ensuremath{\delta}, separating the dispersed particles from the matrix; the kinetics of this process is independent of ${f}_{e}$. Atomistic simulations were performed to estimate the properties of the solid-liquid (S-L) interface at 458 K, 2 K above the Pb-Sn eutectic temperature. The S-L interfaces normal to (001) and (010) of tetragonal \ensuremath{\beta}-Sn were examined and found to have nearly identical properties, including interface widths of \ensuremath{\sim}1.7 nm. In conjunction with the diffusivities in solid \ensuremath{\beta}-Sn and liquid hypereutectic Pb-Sn at 458 K, we estimate that TIDC coarsening should prevail for solid Sn particles \ensuremath{\sim}1700 \ensuremath{\mu}m in radius, far exceeding the maximum radius of \ensuremath{\sim}100 \ensuremath{\mu}m measured experimentally. The TIDC theory also predicts that the kinetics of growth obeys the equation ${\ensuremath{\langle}r\ensuremath{\rangle}}^{n}\ensuremath{\propto}t$. The temporal exponent $n$ was evaluated to be \ensuremath{\sim}2.5, as ascertained by analyzing data on the particle size distributions (PSDs; histograms) for the alloys with ${f}_{e}=0.15$, 0.20, and 0.30. The histograms were converted to experimental cumulative distribution functions (ECDFs) and analyzed using the Kolmogorov-Smirnov (K-S) test applied to the theoretical CDFs predicted by the TIDC theory. We report the first successful application of the K-S test to experimental PSDs concomitant with particle coarsening. From every aspect of the experimental data amenable to analysis, we conclude that the coarsening behavior of solid Sn particles in liquid hypereutectic Pb-Sn alloys is fully consistent with the predictions of the TIDC theory of coarsening.