Metallurgical and Materials Transactions A

, Volume 50, Issue 2, pp 1094–1094 | Cite as

Authors’ Reply to Comments on “A New Analytical Approach to Predict Spacing Selection in Lamellar and Rod Eutectic Systems”

  • Adrian V. CatalinaEmail author
  • Subhayu Sen
  • Doru M. Stefanescu
The authors, Song et al., of the article[1] claim that a “fatal mistake” was found in our paper[2] that discusses the spacing selection in binary eutectic systems. The issue that Song et al. bring into question is the correctness of Eq. [9a] of Reference 2, which is also presented in an explicit form as Eq. [4] in Reference 1 and reproduced here for clarity from our original paper:
$$ \begin{aligned} B_{\text{o}} & = \left[ {\left( {f_{\alpha } \nu_{\alpha } + f_{\beta } \nu_{\beta } } \right) - \left( {f_{\alpha } \nu_{\alpha } k_{\alpha } + f_{\beta } \nu_{\beta } k_{\beta } } \right)} \right] \cdot \left( {C_{\text{E}} + B_{\text{o}} } \right) \\ & \quad + \left[ {\nu_{\alpha } \left( {1 - k_{\alpha } } \right) - \nu_{\beta } \left( {1 - k_{\beta } } \right)} \right] \cdot \sum\limits_{n = 1}^{\infty } {B_{n} \frac{{\sin (n\pi f_{\alpha } )}}{n\pi }} . \\ \end{aligned} $$
At this point, we would like to invite the authors of Reference 1 to carefully consider the boundary condition given by Eq. [3] of their paper, which is identical to Eq. [8] of our paper, and integrate it within the appropriate limits, also by accounting that CI(x) is the liquid concentration at the solid/liquid interface as given by Eq. [1] of Reference 1 (i.e., for z = 0). We trust that they will obtain the result shown in Eq. [4] of Reference 1. Certainly, this is different from Eq. [10] of Song et al. The reason for these different results resides in the fact that Song et al. substituted CI(x) for the eutectic concentration, CE, in the boundary condition (see Eqs. [7-1] and [7-2] of Reference 1). This substitution is a simplification proposed in the original treatment of Jackson and Hunt (JH)[3] which, as explained by JH in their note, can be used “with quite good accuracy for most cases.” Therefore, when setting να = 1 and νβ = 1, the result obtained by Song et al. for the Bo term (Eq. [10] of Reference 1) becomes identical to that of JH, except for being expressed in a different form. Consequently, the results of Song et al. are also correct, as they are exactly the JH solution.

The reason for the simplification proposed in the JH treatment was to make the mathematics tractable when calculating the Fourier coefficients of the general solution for liquid concentration. This was used by JH when calculating the Bo term as well as the coefficients Bn. Catalina et al.[2] also used this simplification when calculating Bn, but, in an attempt to improving the solution accuracy, chose to use the actual CI(x) for the calculation of Bo. Unfortunately, this detail was missed by Song et al., although it is clearly stated when the JH approximation was used in Reference 2. It must be pointed out that there is still an ongoing effort aimed to refining the solution of eutectic growth and extend it from binary to multicomponent systems. Examples of such effort can be found in References 4 through 6, just to name a few.

Most of the time, the solidification processes can be described mathematically as boundary value problems, with the solid/liquid interface boundary being also the solution of the problem. Consequently, many different solutions can be obtained for the same process, depending on the assumptions used in the mathematical formulation. It is exactly the case discussed in this article. Therefore, the statement of Song et al. that a “fatal mistake” made its way in our formulation has no validity as it is the consequence of their failure to recognize the difference in the boundary conditions used in our approach compared to the JH treatment.


  1. 1.
    K. Song, H. Zhao, W. Zhao, B. Su, and Z. Wang: Metall. Mater. Trans. A, 2018,
  2. 2.
    A.V. Catalina, S. Sen, and D.M. Stefanescu: Metall. Mater. Trans. A, 2003, vol. 34A, pp. 383-394 CrossRefGoogle Scholar
  3. 3.
    K.A Jackson and J.D Hunt: Trans. Metall. Soc. of AIME, 1966, vol. 236, pp. 1129-42.Google Scholar
  4. 4.
    A. V. Catalina and D.M. Stefanescu: in Proceedings of the 8th Pacific Rim Confereence on Modeling of Casting and Solidification Processes (MCSP8-2010), J.K. Choi, H.Y. Hwang and J.T. Kim, eds., April 12–15, 2010 pp. 125–32.Google Scholar
  5. 5.
    A.V. Catalina, P.W. Voorhees, R.K. Huff, and A.L. Genau: IOP Conf. Series: Mat. Sci. Engr., 2015, vol. 84, 012085CrossRefGoogle Scholar
  6. 6.
    O. Senninger and P.W. Voorhees: Acta Mat., 2016, vol. 116, pp. 308-320CrossRefGoogle Scholar

Copyright information

© The Minerals, Metals & Materials Society and ASM International 2018

Authors and Affiliations

  • Adrian V. Catalina
    • 1
    Email author
  • Subhayu Sen
    • 2
  • Doru M. Stefanescu
    • 3
    • 4
  1. 1.Flow Science, Inc.Santa FeUSA
  2. 2.Geocent, LLCHuntsvilleUSA
  3. 3.The University of AlabamaTuscaloosaUSA
  4. 4.The Ohio State UniversityColumbusUSA

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