Publon Academy: Second Project

Post-publication review of paper: H. Neumann-Heyme et al. (2015): Dendrite fragmentation in alloy solidification due to sidearm pinch-off, PHYSICAL REVIEW E 92, 060401(R) (2015).

Mahdi Torabi Rad

Summary

This paper addresses a very challenging and open question in the field of solidification. The authors present a mathematical model to predict development and morphological transitions of a side-branch of a dendrite. They investigate cases with zero and non-zero net solidification and for the case of zero net solidification they verify their predictions against a capillary pinching theory available in the literature. They show that the pinch-off transition is bounded between retraction and coalescence transitions and they propose scaling relations to predict these bounds.

Strengths of the Article

This is a very well-written which presents a mathematical model to predict the challenging phenomena of dendrite fragmentation. Results are presented very clearly. The scaling relations discovered in this paper can guide future designs of fragmentation experiments.

Weaknesses of the paper

The main weakness of the paper is that it is not clear how the main equations, i.e. equations (5-7), are derived.

Specific Comments and Questions

Comments about literature review

The literature review of the paper clearly discusses the importance of dendrite fragmentation during solidification and the difficulties in understanding it (the first paragraph). It also provides the very fundamental knowledge necessary to understand the rest of the paper (the second paragraph) and available knowledge about Capillary-driven pinching in related fields (the third paragraph). The limitation in the previous studies of pinching during dendritic solidification, as correctly mentioned in the literature review, is that all the previous capillary pinching theories are limited to the situation where solidification takes place at a constant temperature, i.e., isothermal conditions. However, in the paper, the authors are investigating capillary in the presence of cooling. The major consequence of considering cooling is that now the capillary and solidification will have competing effects: capillary wants to decrease the neck radius during pinching while solidification wants to increase this radius. It would have been better if the authors had included the following two papers in their literature review:

1. Galenko et al. Metallurgical Research and TechnologyVolume 111, Issue 5, 2014, Pages 295-303

2. Liotti et al. Materials Science ForumVolume 765, 2013, Pages 210-214

The authors mention that they have derived equation (5-7) from equation (1-3) using the scaling in equation (4). They need to provide more details about their mathematical derivation. I tried to use the information in the paper and derive equation (5-7) myself; but, unfortunately, what I ended up with is different from what is in the paper.

Note: Equations that are taken from the paper are listed with their number as they appear in the paper with suffix “-paper” added to the end. Equations that I derived are numbered in a successively.

Equation (1) in the paper reads:

(1-paper)

Equation 4 in the paper reads

(4-paper)

which can be rearranged as

(1)

If I substitute equation (1) into the right and left hand sides of equation (1-paper) I get

(2)

(3)

If I substitute equations (2) and (3) into equation (1-paper) I get:

Now in the limit this equation reduces to

(4)

which is clearly different from equation (5) in the paper which reads:

(5-paper)

To clarify this confusion, the authors need to provide details on how they derive equation (5-paper) from equation (1-paper) using the scaling in equation (4-paper)

Similarly, when I start from equation (2) in the paper:

(2-paper)

and substitute equation (1) into (2-paper) I get

In the limit of this equation reduces to

(2)

which is again clearly different from equation (6) in the paper which reads:

(6-paper)

Similarly, starting from equation (3) in the paper

(3-paper)

If I substitute equation (1) into (3-paper) I get

(3)

Which is different from equation (7) in the paper which reads

(7-paper)

Again, the above discrepancies is because the authors are not providing enough details about how they have derived equations (5-7) from equations (1-3).

Other comments

1. Equations (1-3) are not referenced.

2. In Fig.1 it is clear how the region enclosed by the red eclipse is undergoing coalescence. However, it is not clear, or at least I don’t understand, how the region enclosed by the yellow eclipse corresponds to fragmentation.

3. Fig.1 shows that initially the radius of the arm, R, is equal to the radius of the curved surface connecting the arm to the primary stem. Now from the relations listed inside Fig. 2c this should result in . However, in the plot, at t = 0, the magnitude of and don’t seem to be same. What am I missing here?

4. My next comment is about the verification of the results. For the case with zero net solidification, the authors are verifying their results against the theory in [17]. This verification is clear and I understand it. However there is no verification for the results with non-zero net solidification. The authors mention that there is no theory for capillary pinching in the presence of solidification; but they don’t mention whether there are any experimental results or not! If yes, they should use it to verify their results with non-zero net solidification. If not, they should mention it in the introduction.

5. They mention that close to the pinch-off time, the shape of the neck becomes a double cone with angle 80 degrees. It is not clear for me how can they define or calculate meridional curvature for this shape.

6. Since the theory in [17] doesn’t consider net solidification, I was expecting that, in the inset of Fig. 3(a), the blue line (their result with zero net solidification) would collapse with the black line (the theory in [17]). But it doesn’t! Why?

7. In Fig. 3(a), the other point that is not clear for me is that with net solidification, the neck radius should be wider, which is true as I look at the inset of the figure. Therefore, in the plot, I was expecting that the radius of the case with the highest net solidification (the red curve) to be higher than the radius of the case with no net solidification (the blue curve). But what I see is exactly the opposite. Why?

8. In Fig. 3(b), it is not clear what the cooling rates for the three shapes shown in the plot are! Are they the cooling rate lying in the middle of the shape?

9. In Fig. 3(b), why is there a discontinuity from the dashed blue curve to the dashed red curve? If I chose to be equal to the value at this discontinuity, then can their model predict whether the pinched-off fragment will re-melt or coalesce after pinch-off?

10. Are their results completely independent of the time step they are using? The reason I am raising this issue is because the final stage of pinch-off is very fast so it is important to show that results obtained for tp are time step independent. I do understand that the relation they use for adaptive time stepping, , will reduce the time step close to the pinch off time and therefore help getting results that are completely time step independent. But I think it is important to show that.

11. In Fig. 4 why some of the data (the cross symbols) are excluded?

12. In section 3 of the supplementary part, equation (A.6) is solved. However, the equation for , on the right-hand-side of (A.6), is not listed.

13. It is not clear why the authors are scaling the length with R and not d0?