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

Excerpt

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: 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 C_I(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 C_I(x) for the eutectic concentration, C_E, 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 ν_α = 1 and ν_β = 1, the result obtained by Song et al. for the B_o 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. …