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Metallurgical and Materials Transactions A

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01-09-2015 | Discussion

Discussion of “On the Boltzmann–Matano Analysis of Diffusion in a Semi-infinite Medium”

Authors: Masayuki Okugawa, Hiroshi Numakura

Published in: Metallurgical and Materials Transactions A | Issue 9/2015

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Abstract

Ernst et al. analyzed the concentration-dependent diffusion of carbon in austenitic steel in the semi-infinite geometry by the Boltzmann–Matano method. They assumed that the origin of the position coordinate be at the surface, and tried to justify it from the particular nature of interstitial diffusion. We point out that their ‘assumption’ is prerequisite to applying the method to the problem, irrespective of the mechanism.

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Here we ignore the formal mathematical problem that the value of \(\eta \) for Eq. [1] is indefinite. That for Eq. [2] is evaluated to be \(+\infty \) by taking the limit \(t\rightarrow 0+\), since we consider only \(t \ge 0\).

For example, Crank[4] states ‘It is only when the initial and boundary conditions are expressible in terms of \(\eta \) alone, and x and t are not involved separately, that the transformations ... can be used.’

X. Gu, G.M. Michael, F. Ernst, H. Kahn, and A.H. Heuer: Metall. Mater. Trans. A, 2014, vol. 45A, pp. 3790–99.CrossRef

F. Ernst, A. Avishai, H. Kahn, X. Gu, G.M. Michael, and A.H. Heuer: Metall. Mater. Trans. A, 2009, vol. 40A, pp. 1768–80.CrossRef

H.S. Carslaw, and J.C. Jaeger: Conduction of Heat in Solids, 2^nd edn., Oxford University Press, Oxford, 1959, pp. 89–90.

J. Crank: The Mathematics of Diffusion, 2^nd edn., Oxford University Press, Oxford, 1975, p. 106.

J. Philibert: Atom Movements—Diffusion and Mass Transport in Solids, Les Editions de Physique, Les Ulis, 1991

L.F. Shampine: Q. Appl. Math., 1973, vol. 30, pp. 441–52

Title: Discussion of “On the Boltzmann–Matano Analysis of Diffusion in a Semi-infinite Medium”
Authors: Masayuki Okugawa
Hiroshi Numakura
Publication date: 01-09-2015
Publisher: Springer US
Published in: Metallurgical and Materials Transactions A / Issue 9/2015
Print ISSN: 1073-5623
Electronic ISSN: 1543-1940
DOI: https://doi.org/10.1007/s11661-015-3022-1

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