dorsal/arxiv
View SchemaBreakdown of Burton-Prim-Slichter approach and lateral solute segregation in radially converging flows
| Authors | J. Priede, G. Gerbeth |
|---|---|
| Categories | |
| ArXiv ID | physics/0605139 |
| URL | https://arxiv.org/abs/physics/0605139 |
| DOI | 10.1016/j.jcrysgro.2005.08.026 |
| Journal | J. Cryst. Growth 285 (2005) 261--269 |
Abstract
A theoretical study is presented of the effect of a radially converging melt flow, which is directed away from the solidification front, on the radial solute segregation in simple solidification models. We show that the classical Burton-Prim-Slichter (BPS) solution describing the effect of a diverging flow on the solute incorporation into the solidifying material breaks down for the flows converging along the solidification front. The breakdown is caused by a divergence of the integral defining the effective boundary layer thickness which is the basic concept of the BPS theory. Although such a divergence can formally be avoided by restricting the axial extension of the melt to a layer of finite height, radially uniform solute distributions are possible only for weak melt flows with an axial velocity away from the solidification front comparable to the growth rate. There is a critical melt velocity for each growth rate at which the solution passes through a singularity and becomes physically inconsistent for stronger melt flows. To resolve these inconsistencies we consider a solidification front presented by a disk of finite radius $R_0$ subject to a strong converging melt flow and obtain an analytic solution showing that the radial solute concentration depends on the radius $r$ as $\sim\ln^{1/3}(R_0/r)$ and $\sim\ln(R_0/r)$ close to the rim and at large distances from it. The logarithmic increase of concentration is limited in the vicinity of the symmetry axis by the diffusion becoming effective at a distance comparable to the characteristic thickness of the solute boundary layer. The converging flow causes a solute pile-up forming a logarithmic concentration peak at the symmetry axis which might be an undesirable feature for crystal growth processes.
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"abstract": "A theoretical study is presented of the effect of a radially converging melt\nflow, which is directed away from the solidification front, on the radial\nsolute segregation in simple solidification models. We show that the classical\nBurton-Prim-Slichter (BPS) solution describing the effect of a diverging flow\non the solute incorporation into the solidifying material breaks down for the\nflows converging along the solidification front. The breakdown is caused by a\ndivergence of the integral defining the effective boundary layer thickness\nwhich is the basic concept of the BPS theory. Although such a divergence can\nformally be avoided by restricting the axial extension of the melt to a layer\nof finite height, radially uniform solute distributions are possible only for\nweak melt flows with an axial velocity away from the solidification front\ncomparable to the growth rate. There is a critical melt velocity for each\ngrowth rate at which the solution passes through a singularity and becomes\nphysically inconsistent for stronger melt flows. To resolve these\ninconsistencies we consider a solidification front presented by a disk of\nfinite radius $R_0$ subject to a strong converging melt flow and obtain an\nanalytic solution showing that the radial solute concentration depends on the\nradius $r$ as $\\sim\\ln^{1/3}(R_0/r)$ and $\\sim\\ln(R_0/r)$ close to the rim and\nat large distances from it. The logarithmic increase of concentration is\nlimited in the vicinity of the symmetry axis by the diffusion becoming\neffective at a distance comparable to the characteristic thickness of the\nsolute boundary layer. The converging flow causes a solute pile-up forming a\nlogarithmic concentration peak at the symmetry axis which might be an\nundesirable feature for crystal growth processes.",
"arxiv_id": "physics/0605139",
"authors": [
"J. Priede",
"G. Gerbeth"
],
"categories": [
"physics.flu-dyn"
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"doi": "10.1016/j.jcrysgro.2005.08.026",
"journal_ref": "J. Cryst. Growth 285 (2005) 261--269",
"title": "Breakdown of Burton-Prim-Slichter approach and lateral solute segregation in radially converging flows",
"url": "https://arxiv.org/abs/physics/0605139"
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