The Diminishing Role of the Nucleation Rate as Crystallization Develops in Avrami-Type Models
Round 1
Reviewer 1 Report
Avrami equation is recently successfully applied for a very broad spectrum of applications far beyond its initial use for quantitative crystallization kinetics description. Particularly it is reported to span the range from alloys [1] and polymers [2] crystallization to transformation of human cells into pathogenic ones [3] and even to COVID-19 wave kinetics [4]. Hundreds of reports are published annually on Avrami equation applications. However, the need of Avrami equation refinements is periodically stated [5-13].
Thus, any critical publication on this equation peculiarities, restrictions, limitations and validity surely deserves attention. In this respect, the refinement of Avrami equation by consideration of crystallization as a process consisting of nucleation and growth of spherical crystalline domains, and by assumption that the nucleation and domain growth might be a function of the level of crystallization is actual, and is reasonable. The manuscript may be published with minor revision.
1.Line 30. The phrase “Though these two expressions were introduced about 80 years ago, they are still widely used to fit a great variety of experimental data [here some references would be good to add to illustrate wide spectrum of Avrami equation successful applications ]” has to be supplied by corresponding references. Then, here it is reasonable to add some phrases and corresponding references to mention papers on Avrami equation criticism, e.g. [5-13] and/or some others. Some comments of author on these works are highly desirable. In fact, ref.[12] gives an example of the primary process alone responsible for kinetics.
2.The reference list has to be extended, see comment 1.
References
[1] Kurkin, A. S. Mathematical Research of the Phase Transformation Kinetics of Alloyed Steel. Inorganic Materials (2020), 56(15), 1471-1477. DOI:10.1134/s0020168520150091
[2] Huang, R. et al, Crystallization behavior of polyvinyl alcohol with inorganic nucleating agent talc and regulation mechanism analysis. J.Polymers Environ. (2022), Ahead of Print., DOI:10.1007/s10924-022-02408-2
[3] Fornalski, K. W.; Dobrzynski, L. Modeling of single cell cancer transformation using phase transition theory: application of the Avrami equation, Radiat. Environ. Biophys. (2022), 61(1), 169-175. DOI:10.1007/s00411-021-00948-0
[4] Takase, Y. Analysis of COVID-19 infection waves in Tokyo by Avrami equation, arXiv.org, e-Print Archive, Physics (2021), 1-10.
[5] Oladoja, N. A. A critical review of the applicability of Avrami fractional kinetic equation in adsorption-based water treatment studies. Desalination and Water Treatment (2016), 57(34), 15813-15825, DOI:10.1080/19443994.2015.1076355
[6] Guimaraesa, J. R. C.; Riosa, P. R.; Alves, A. L. M. An alternative to Avrami equation. Materials Research (Sao Carlos, Brazil) (2019), 22(5), e20190369/1-e20190369/8.DOI:10.1590/1980-5373-mr-2019-0369
[7] Malek, J. The applicability of Johnson-Mehl- Avrami model in the thermal analysis of the crystallization kinetics of glasses.Thermochimica Acta (1995), 267, 61-73. DOI:10.1016/0040-6031(95)02466-
[8] Wang, Jun; et al Limitation of the Johnson-Mehl-Avrami equation for the kinetic analysis of crystallization in a Ti-based amorphous alloy. International Journal of Minerals, Metallurgy and Materials (2010), 17(3), 307-311. DOI:10.1007/s12613-010-0309-5
[9] Todinov, M. T. On some limitations of the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation, Acta Materialia (2000), 48(17), 4217-4224. DOI:10.1016/S1359-6454(00)00280-9
[10] Almansour, A.; Matsugi, K.; Hatayama, T.; Yanagisawa, O. Modeling of growth and impingement of spherical grains. Materials Transactions, JIM (1996), 37(10), 1595-1601., DOI:10.2320/matertrans1989.37.1595
[11] Chiang, Ch.-Ya.; Starov, V. M.; Lloyd, D. R. Crystallization kinetics of a polymer-solvent system: 1. Derivation of model equations. Colloid Journal (Translation of Kolloidnyi Zhurnal) (1995), 57(5), 715-24.
[12] Aziz, A. A.; et al. The effect of a secondary process on polymer crystallization kinetics - 3. Co-poly(lactic acid), European Polymer Journal (2017), 94, 311-321.DOI:10.1016/j.eurpolymj.2017.07.006
[13] Albano, C.; Papa, J.; Bare, W.; Gonzalezx, J. Avrami equation inadequacy for modeling the HDPE isothermal crystallization rates. Revista de la Facultad de Ingenieria, Universidad Central de Venezuela (2007), 22(1), 71-77.
Author Response
Thirteen references have been added to illustrate the widespread use of Avrami-like models, as well as critical reviews and alternative approaches. Reflecting this, the second paragraph of the Introduction now reads: ” Though these two expressions were introduced about 80 years ago, they are still widely used to fit a great variety of experimental data, ranging from metals and alloys [1, 4], polymers [2, 5], growth of cancerous cells [6], etc. A few examples of critical reviews and/or alternatives to Avrami-like formulations are given in Refs. 7 – 17”.
Reviewer 2 Report
The manuscript is an original theoretical work and deals with role of nucleation rate in Avrami-type models of isothermal crystallization. The structure of the paper is easy to understand and follow. However a few improvements, corrections should be implemented. In particular, the following observations should be taken into account by the authors.
The Authors might want to consider mentioning papers regarding alternative, with respect to Avrami-type, isothermal crystallization models (Tobin, Malkin). It seems that introduction would be a right place for additional references.
Another point of concern is figure 1. In particular it is very hard to observe presence of two curves since they practically look superimposed. Using symbols instead would probably help the readers.
The authors are encouraged to address these points.
Author Response
1) Thirteen references have been added to illustrate the widespread use of Avrami-like models, as well as critical reviews and alternative approaches. (These include Tobin's and Malkin's work.) Reflecting this, the second paragraph of the Introduction now reads: ” Though these two expressions were introduced about 80 years ago, they are still widely used to fit a great variety of experimental data, ranging from metals and alloys [1, 4], polymers [2, 5], growth of cancerous cells [6], etc. A few examples of critical reviews and/or alternatives to Avrami-like formulations are given in Refs. 7 – 17”.
2) One of the curves in Fig. 1 was substituted by the symbol ‘+’ for clarity.
3) Font text within the figures has been enlarged.
4) Different colors for different curves are now used in all figures.
Reviewer 3 Report
The author of this manuscript uses an old model to rethink the role of both domain and nucleation rate in the crystal growth process. Although the model is very simple and many ideal assumptions are imposed by the author, the overall content is still helpful to understand crystalline growth. So I agree it is published after the author makes some minor revisions.
1. Line 77, the integration equation is quite redundant to me. From the author's hypothesis of infinite small nucleation which is not mentioned in the context, it's just the volume of the sphere in an unimpeded growing process in the short time window. The author can eliminate it.
2. The author lists too less reference papers.
3. The figure fonts are not in a good uniform. For example, the font size of the x and y-axis labels of all figures is too small.
4. Some expressions are very colloquial. For example:
Line 178, "Let us summarize."
Line 184, "Next we explain why."
Typo:
Line 177, for what?
Author Response
1) Thirteen references have been added to illustrate the widespread use of Avrami-like models, as well as critical reviews and alternative approaches. Reflecting this, the second paragraph of the Introduction now reads: ” Though these two expressions were introduced about 80 years ago, they are still widely used to fit a great variety of experimental data, ranging from metals and alloys [1, 4], polymers [2, 5], growth of cancerous cells [6], etc. A few examples of critical reviews and/or alternatives to Avrami-like formulations are given in Refs. 7 – 17”.
2) One of the curves in Fig. 1 was substituted by the symbol ‘+’ for clarity.
3) Font text within the figures has been enlarged.
4) Different colors for different curves are now used in all figures.
5) ‘Obvious’ integral in Eq. (5) has been omitted.
6) Colloquial expressions “Let us summarize…” (Line 178) and “Next we explain why…” (Line 184) have been modified into more formal language.
7) Legend of Fig. 9 (Line 177) has been completed.
Reviewer 4 Report
This is a good manuscript about the role of nucleation in crystallization, which provides novel knowledge different from classical theories.
Author Response
1) Thirteen references have been added to illustrate the widespread use of Avrami-like models, as well as critical reviews and alternative approaches. Reflecting this, the second paragraph of the Introduction now reads: ” Though these two expressions were introduced about 80 years ago, they are still widely used to fit a great variety of experimental data, ranging from metals and alloys [1, 4], polymers [2, 5], growth of cancerous cells [6], etc. A few examples of critical reviews and/or alternatives to Avrami-like formulations are given in Refs. 7 – 17”.
2) One of the curves in Fig. 1 was substituted by the symbol ‘+’ for clarity.
3) Font text within the figures has been enlarged.
4) Different colors for different curves are now used in all figures.
5) ‘Obvious’ integral in Eq. (5) has been omitted.
6) Colloquial expressions “Let us summarize…” (Line 178) and “Next we explain why…” (Line 184) have been modified into more formal language.
7) Legend of Fig. 9 (Line 177) has been completed.
