Heterogeneous Nucleation Mechanisms in Systems with Large Lattice Misfit Demonstrated by the Pb(l)/Cu(s) System
Round 1
Reviewer 1 Report
The authors present a numerical study investigating the heterogeneous nucleation of the Pb(l)/Cu(s) system characterized by a large lattice misfit. The authors specifically use molecular dynamics simulations to evaluate nucleation on the (111) Cu plane. They confirm the work previously presented by Palafox-Hernandez et al. [16] showing an initial prenucleation of Pb at temperatures above its melting temperature followed by a 26K undercooling to nucleate a crystal plane. The authors show the formation of the Pb prenucleation solid has a coincident site lattice relationship with the (111) Cu plane that absorbs a large fraction of the lattice misfit. The authors further discuss the implications for understanding the non-monotonic relationship between the required undercooling for nucleation and lattice misfit. The article is suitable for publication as-is.
Author Response
We would like to thank Reviewer 1 for his/her effort to review our manuscript, and for his/her constructive and positive comments.
Reviewer 2 Report
The author investigated the atomistic mechanisms of both prenucleation and heterogeneous nucleation in the system with large lattice misfit using MD simulations. The approach and the calculation results are fine, but the reviewer is confused at the discussion part.
The unit structure of CSL is defined to minimize the sum of elastic energy and dislocation network. So, it is influence by the atomic ratio and the interatomic potentials. Also, the structure is defined to minimize the energy of the CSL + L1Pb + L2Pb + .... If we don't consider the energy, misfit f can be the sum of small f_csl and positive f_r (vacancy type) or large f_csl and negative f_r (dislocation type). Reviewer recommends to add the discussions from the viewpoint of energy.
Author Response
We would like to thank all the Reviewers for their effort to review our manuscript, and for their constructive and positive comments. Here is a summary of our response to the reviewers’ comments, which are presented in italic.
Reviewer 2
“but the reviewer is confused at the discussion part. The unit structure of CSL is defined to minimize the sum of elastic energy and dislocation network. So, it is influence by the atomic ratio and the interatomic potentials. Also, the structure is defined to minimize the energy of the CSL + L1Pb + L2Pb + .... If we don't consider the energy, misfit f can be the sum of small f_csl and positive f_r (vacancy type) or large f_csl and negative f_r (dislocation type). Reviewer recommends to add the discussions from the viewpoint of energy.“
Response: We agree with the reviewer 2’s suggestions to add the discussions from the viewpoint of energy. We didn’t explicitly calculate the interfacial energy of Pb/Cu interface, which should include the contributions from the chemical interaction and strain energy of the misorientation. Here, for simplicity we assume that the chemical interaction does not change with variation in the interfacial structure with increasing misfit (this change is expected to be insignificant). The strain energy due to misorientation at the interface is dominant, particularly for large misfit. Thus, the introduction of CSL at the interface will reduce the interfacial energy significantly due to the good matching. The interfacial energy with large misfit includes the strain energy of the CSL and that of a small residual misfit (either negative or positive), plus a constant from the chemical interaction. We have added further explanation to include the contribution from the chemical interaction in the revised manuscript.
Reviewer 3 Report
Review of “Heterogeneous Nucleation Mechanisms in Systems with Large Lattice Misfit Demonstrated by the Pb(l)/Cu(s) System”
The paper is devoted to the computational study of crystallization of a supercooled liquid Pb in contact with a Cu solid. A thorough analysis of the atomistic simulation Pb/Cu data was performed. It allowed to reliably reveal the complex mechanism of heterogeneous crystal Pb nucleation and its further crystal growth. The obtained fundamental results are an important step in our understanding of heterogeneous crystallization in nature. Overall, in my opinion, the manuscript can be published in Metals after revision.
Comments and questions
1. Page 2. “discrepancies exist between the simulation results and the theoretical predictions by the CNT”. Note that the discrepancies between the CNT predictions and experimental nucleation rates in different liquids have been also widely reported, e.g. (i) Low-temperature nucleation anomaly in silicate glasses shown to be artifact in a 5BaO8SiO2 glass, Nat. Commun. 12 (2021) 2026; (ii) Critical assessment of the alleged failure of the classical nucleation theory at low temperatures, J. Non Cryst. Solids 547 (2020) 120297; (iii) Homogeneous crystal nucleation in silicate glasses: a 40 years perspective, J. Non Cryst. Solids 352 (2006) 2681–2714. These papers should be cited and discussed. Note also that some recent computer simulation findings favor the validity of the CNT, for example, (iv) Crystal nucleation kinetics in supercooled germanium: MD simulations versus experimental data, J. Phys. Chem. B 124 (2020) 7979–7988; (v) Nucleation kinetics in supercooled Ni50Ti50: computer simulation data corroborate the validity of the classical nucleation theory, Chem. Phys. Lett. 735 (2019) 136749; (vi) Successful test of the classical nucleation theory by molecular dynamic simulations of BaS, Comput. Mater. Sci. 161 (2019) 99–106; (vii) Diffusivity, interfacial free energy, and crystal nucleation in a supercooled Lennard-Jones liquid, J. Phys. Chem. C 122 (2018) 28884–28894; etc. These papers should be also cited and discussed.
2. Page 2. “12.5%, which is the theoretical upper limit for dislocation mechanism”. The reference should be given.
3. The expression for a lattice misfit should be given.
4. Page 2. “there exists a ‘clean’ interface between liquid Pb and (111) Cu substrate”. The liquid Pb / solid Al system demonstrates the solid-liquid interfacial premelting layer, see, Yang, Y., Asta, M., & Laird, B. B. (2013). Solid-liquid interfacial premelting. Physical review letters, 110(9), 096102. Is the same phenomenon observed for the Pb/Cu system in the current simulation?
5. Page 3. Only one experimental work on Pb/Cu study (ref 33) was once mentioned. Ref 33 is outdated (1993), are there other researches?
6. Page 3. It should be explained why <111> Cu plane was chosen.
7. Page 3. “The Cu(s) is ... fully relaxed”. How was the full relaxation confirmed?
8. Page 3. “a vacuum region is inserted”. The schematic figure of the whole Pb/Cu/vacuum system should be given.
9. Page 4. “components of the pressure tensor”. Which expression for the pressure tensor (Irving–Kirkwood, Kirkwood-Buff, etc.) was employed and why?
10. Page 4. “An atom is identified as solid if the number of the connections that this atom has with its neighbours reaches a threshold of 6”. Why six? The physical arguments should be provided.
11. Page 4. “z = 0 marks the Gibbs dividing interface”. It is not clear. The Gibbs dividing interface surface is determined via the surface excess properties. How did the authors determine it?
12. Figure 3d. Why are the Pb/Cu contact layers stretched, i.e. under negative pressures?
13. Page 7. How does the computational atomic spacing agree with the experimental values?
14. Page 8. “Theoretically, the CSL should have accommodated 25% of the lattice misfit”. The reference should be given.
15. Page 10. “Here we define L3Pb as the 2D nucleus”. Can the observed Pb crystallization be treated as two crystal-growth processes with different velocities without involving the 3rd layer nucleation?
Author Response
We would like to thank all the Reviewers for their effort to review our manuscript, and for their constructive and positive comments. Here is a summary of our response to the reviewers’ comments, which are presented in italic.
Reviewer 3
(1). Page 2. “discrepancies exist between the simulation results and the theoretical predictions by the CNT”. Note that the discrepancies between the CNT predictions and experimental nucleation rates in different liquids have been also widely reported, e.g. (i) Low-temperature nucleation anomaly in silicate glasses shown to be artifact in a 5BaO8SiO2 glass, Nat. Commun. 12 (2021) 2026; (ii) Critical assessment of the alleged failure of the classical nucleation theory at low temperatures, J. Non Cryst. Solids 547 (2020) 120297; (iii) Homogeneous crystal nucleation in silicate glasses: a 40 years perspective, J. Non Cryst. Solids 352 (2006) 2681–2714. These papers should be cited and discussed. Note also that some recent computer simulation findings favor the validity of the CNT, for example, (iv) Crystal nucleation kinetics in supercooled germanium: MD simulations versus experimental data, J. Phys. Chem. B 124 (2020) 7979–7988; (v) Nucleation kinetics in supercooled Ni50Ti50: computer simulation data corroborate the validity of the classical nucleation theory, Chem. Phys. Lett. 735 (2019) 136749; (vi) Successful test of the classical nucleation theory by molecular dynamic simulations of BaS, Comput. Mater. Sci. 161 (2019) 99–106; (vii) Diffusivity, interfacial free energy, and crystal nucleation in a supercooled Lennard-Jones liquid, J. Phys. Chem. C 122 (2018) 28884–28894; etc. These papers should be also cited and discussed.
Response: We agree with reviewer 1, and have included these references in the revised manuscript.
(2). “Page 2. “12.5%, which is the theoretical upper limit for dislocation mechanism”. The reference should be given.”
Response: We have added the reference and provided further explanation in the revised manuscript.
(3) “The expression for a lattice misfit should be given”
Response: We have included an expression for misfit in the revised manuscript and the corresponding reference.
(4) “Page 2. “there exists a ‘clean’ interface between liquid Pb and (111) Cu substrate”. The liquid Pb / solid Al system demonstrates the solid-liquid interfacial premelting layer, see, Yang, Y., Asta, M., & Laird, B. B. (2013). Solid-liquid interfacial premelting. Physical review letters, 110(9), 096102. Is the same phenomenon observed for the Pb/Cu system in the current simulation?”
Response: In terms of prenucleation, the Pb(l)/Cu(s) and Pb(l)/Al(s) systems are similar since both systems are not only immiscible, but also have a large misfit between the substrate and the corresponding solid. However, there are some differences in these two systems. The MD simulation has demonstrated that in the Pb(l)/Cu(s) system, there is neither solute Cu in liquid Pb nor solute Pb in solid Cu at the liquid Pb/solid Cu interface, and hence a ‘clean’ interface. In contrast, mutual solubility has been observed in the Pb(l)/Al(s) system.
(5) “Page 3. Only one experimental work on Pb/Cu study (ref 33) was once mentioned. Ref 33 is outdated (1993), are there other researches?”
Response: Ref. [33] is one of best classical works for the study on heterogeneous nucleation of Pb droplet in Cu matrix, and widely cited in the literature to date. We are not aware of any other experimental work on this system.
(6) “Page 3. It should be explained why <111> Cu plane was chosen”.
Response: The <111> Cu is the densely packed atomic plane, most stable, and in Ref. [33] it is reported that the heterogeneous nucleation of droplet Pb occurred on <111> Cu of Cu matrix. Further explanation has been included in ten revised manuscript.
(7) “Page 3. “The Cu(s) is ... fully relaxed”. How was the full relaxation confirmed?”
Response: The movements of the atoms in Cu(s) has no any restriction during the simulations with enough equilibration time, except for the periodical boundary conditions. We have measured the atomic spacing of the atoms in Cu(s), which is very close to the theoretical value of Cu at the corresponding temperature. It could be indicative of a full relaxed simulation system.
(8) “Page 3. “a vacuum region is inserted”. The schematic figure of the whole Pb/Cu/vacuum system should be given.”
Response: Inserting a vacuum region in the simulation system according to periodical boundary condition is a common method used for atomistic simulations. We agree with the good suggestion of the reviewer 3, but believe that the description in the main text is enough for the readers.
(9) “Page 4. “components of the pressure tensor”. Which expression for the pressure tensor (Irving–Kirkwood, Kirkwood-Buff, etc.) was employed and why?”
Response: For MD simulations, it is the simple and straightforward way to calculate an average stress based on the Virial stress, with equation (1) in this manuscript from Ref. [16]. We may adopt Irving–Kirkwood formulation to calculate the local stress in the future.
(10) Page 4. “An atom is identified as solid if the number of the connections that this atom has with its neighbours reaches a threshold of 6”. Why six? The physical arguments should be provided.
Response: It is tricky to identify an atom in a solid or liquid state at the liquid/solid interface. For a fcc (solid) structure, its nearest neighbor is 12, and it is a general method to consider an atom with 6 nearest solid neighbors at the interface as in a solid state. This has been proved to be one of the most reasonable descriptions about the status of atoms for heterogeneous nucleation at the liquid/substrate, homogeneous nucleation and crystal growth at the liquid/solid interface in the community of atomistic simulations.
(11) “Page 4. “z = 0 marks the Gibbs dividing interface”. It is not clear. The Gibbs dividing interface surface is determined via the surface excess properties. How did the authors determine it?”
Response: As we stated previously, the liquid Pb/solid Cu interface is ‘clean’, and therefore the Gibbs dividing interface simply means the interface of liquid Pb and solid Cu in terms of the chemical compositions
(12) “Figure 3d. Why are the Pb/Cu contact layers stretched, i.e. under negative pressures?”
Response: The Pb atoms has a much larger atomic size than Cu atoms, but the 1st Pb interfacial layer is epitaxial to the 1st Cu interfacial layer. Thus, the 1st layer of Pb atoms must be subject to the compressive (negative) stress to squeeze Pb atoms into the epitaxial positions templated by the lattice of 1st Cu interfacial layer.
(13) “Page 7. How does the computational atomic spacing agree with the experimental values?”
Response: As we previously stated, the Cu and Pb in the simulation system are well equilibrated. Thus, the atomic spacing of Cu in the solid substrate will be very closed to the equilibrium value (the experimental values at corresponding temperature) since the atomic spacing is usually a fitting parameter to the experimental value during the construction of the potentials.
(14) “Page 8. “Theoretically, the CSL should have accommodated 25% of the lattice misfit”. The reference should be given.”
Response: This is obtained according to equation (5), which give a misfit of 25%, as we stated in Line 364-366, Page 8. Further clarification has been provided in the revised text.
(15) “Page 10. “Here we define L3Pb as the 2D nucleus”. Can the observed Pb crystallization be treated as two crystal-growth processes with different velocities without involving the 3rd layer nucleation?”
Response: We agree with this interesting suggestion. Overall, the crystallisation of Pb on a Cu substrate can be treated as a growth process, that can be devised into two stages with different growth velocity: a slow growth stage involving only 3 atomic layers, which as define as nucleation; and a faster growing stage which may need further undercooling. This is presented in more details in a recent publication (Fan, Z.; Men, H. Heterogeneous nucleation and grain initiation on a single substrate. Metals 2022, 12, 1454. https:// doi.org/10.3390/met12091454).
Round 2
Reviewer 3 Report
The revised version can be published as is.
