Growth of lamellar eutectic structures: The axisymmetric state.

Abstract

We study steady symmetric lamellar eutectic growth in directional solidification by an extensive analysis of the pertinent boundary integral equation. We find a discrete set of solutions that differ in their average undercoolings. As the wavelength \ensuremath{\lambda} increases, the branches coalesce by pairs to form fold singularities above which axisymmetric solutions cease to exist. Front shapes are computed in a wide range of wavelengths, and a systematic comparison with an improved Jackson and Hunt theory [Trans. Metall. Soc. AIME 236, 1129 (1966)] is made. The last one turns out to be accurate in general, for the lowest branch, but does not provide any hint at other branches. In the experimentally relevant parameter range, the front equation reduces to a similarity equation containing two dimensionless parameters \ensuremath{\sigma}==${\mathit{d}}_{0}$l/${\ensuremath{\lambda}}^{2}$ and \ensuremath{\chi}==l/${\mathit{l}}_{\mathit{T}}$, where ${\mathit{d}}_{0}$,l,${\mathit{l}}_{\mathit{T}}$ are the capillary, diffusion, and thermal lengths. We explicitly demonstrate the similarity properties of the pattern. The selected wavelength scales as \ensuremath{\lambda}\ensuremath{\simeq} \ensuremath{\surd}${\mathit{d}}_{0}$l f(l/${\mathit{l}}_{\mathit{T}}$). At the minimum undercooling, \ensuremath{\lambda} varies with the growth velocity V as ${\mathit{V}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$, where \ensuremath{\beta} is an increasing function of V that saturates to about 1/2 at large V. This feature is in agreement with experiments [G. Lesoult, Ann. Chim. Fr. 5, 154 (1980)]. The general scaling of the wavelength allows the exact prediction of a new scaling exponent that is independent of the nature of the selection criterion. This prediction inspires another experimental test of the theory.