# The Diminishing Role of the Nucleation Rate as Crystallization Develops in Avrami-Type Models

## Abstract

**:**

## 1. Introduction

_{R}, and a constant linear growth rate, k, Equation (1) reduces to what we will simply call the Avrami exponential expression:

_{R}k

^{3}. This suggests that increasing the growth rate, k

^{3}, by a factor of 10 is the same as increasing the nucleation rate, N

_{R}, by a factor of 10. Notice that the product N

_{R}k

^{3}can be achieved in innumerable ways: we could have a nucleation rate of 1000 N

_{R}times a growth rate of k

^{3}/1000 or N

_{R}/1000 times 1000 k

^{3}. In Equation (2), this is irrelevant due to the assumption that both rates remain constant along the transformation process. Nevertheless, we know that in reality, these two rates vary as crystallization progresses. Consider the process in which the domain growth rate is slowed by the proximity of other domains (phenomenon known as impingement). To some extent, it matters if for a given value of X, we have a few large crystalline domains or a large number of small ones. When morphological details affect these two rates to a significant extent, Equation (1) would be better suited to deal with a crystallization kinetics involving time-dependent rates. We will follow a slightly different approach in order to study the roles of variations in these two rates. We will make use of an integral equation describing the isothermal crystallization kinetics that was introduced previously by the author to assess the relevance of phantom nuclei [18]. In it, nucleation and growth rates explicitly depend on the degree of crystallization. It will be shown that varying the expression describing the growth rate as a function of X has a much greater effect than varying the expression describing the nucleation rate as a function of X, in clear contrast to the situation described for Equation (2) above in this paragraph. The reason for this is of a geometric nature, and we explain it in Section 4.

## 2. Crystallization Kinetics Integral Equation

- (1).
- The time elapsed since the overall crystallization process started, beginning from a state of zero crystallization, is indicated by t.
- (2).
- The nucleation time, τ, indicates the time at which a particular spherical crystalline domain, originating from a nucleus, begins its growth process. This way, we say that, at a time t, the growth interval for a spherical crystalline domain born at a nucleation time τ is (t–τ).

_{tot}be the total volume of a sample and V

_{x}(t) be the volume of material transformed at time t. Thus, we can define the crystallized fraction at time t as $X\left(t\right)=\raisebox{1ex}{${V}_{x}\left(t\right)$}\!\left/ \!\raisebox{-1ex}{${V}_{tot}$}\right.$, with X(t = 0) = 0 and X(t = ∞) = 1. For simplicity, assume that the densities of non-crystallized (amorphous) and crystallized phases are the same.

^{3}G[X] is an instantaneous (volumetric) growth rate (G[X = 0] = 1 and G[X = 1] = 0). Thus, the volume of a crystalline domain born at a time τ is now given by

_{R}, such that the number of nuclei born between time t and t + dt is given by N

_{R}dt. However, as in the case of the growth rate for later stages, we assume that the nucleation rate decreases as crystallinity increases. Thus, we propose an instantaneous nucleation rate given by a function N

_{R}I[X], with 0 ≤ $I$[X] ≤ 1, $I$[X = 0] = 1, and $I$[X = 1] = 0. Thus, the number of crystalline domains born up to time t is given by

## 3. Dependence of the Evolution of X on γ and ν

^{1/4}τ = 0.001) is significantly larger than that of late comers (κ

^{1/4}τ = 1). This point will turn out to be important for the discussions given below. Figure 3 shows the rates dx(t, τ)/dt corresponding to the same τ values. Again, we see the predominance of domains born in the early stages.

^{1/4}τ = 0.1, 0.5, and 1 are now larger than for γ = 1 (see Figure 2, Figure 3, Figure 4 and Figure 5).

^{1/4}τ = 1.0 have a negligible contribution. This is not the case for γ = 3 and 5, in which case, at time κ

^{1/4}τ = 1.0, the old domains are smaller than for γ = 1, leaving more room for new domains to grow.

## 4. A geometric Explanation

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

- (1).
- If γ = 1 (that is, the growth rate factor G is proportional to the untransformed fraction (1 − X)) and if I(τ) is not an explicit function of X, then we obtain the KJMA model, Equation (1).
- (2).
- For γ = 1 and a constant nucleation rate (ν = 0), we obtain Equation (2).
- (3).
- If γ ≠ 1 and ν = 0, a simple mathematical expression proposed by Lee and Kim to study the transformation kinetics of Cu-Zn-Al shape memory alloys is obtained [11].

## References

- Christian, J.W. The Theory of Transformations in Metals and Alloys, Part I, 2nd ed.; Pergamon Press Ltd.: Oxford, UK, 1975. [Google Scholar]
- Mandelkern, L. Crystallization of Polymers, Vol. 2, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Avrami, M.J. Kinetics of Phase Change. I. J. Chem. Phys.
**1939**, 7, 1103–1112. [Google Scholar] [CrossRef] - Kurkin, A.S. Mathematical Research of the Phase Transformation Kinetics of Alloyed Steel. Inorg. Mater.
**2020**, 56, 1471–1477. [Google Scholar] [CrossRef] - Huang, R.; Zhang, Y.; Xiang, A.; Ma, S.; Tian, H.; Ouyang, Y.; Rajulu, A.V. Crystallization behavior of polyvinyl alcohol with inorganic nucleating agent talc and regulation mechanism analysis. J. Polym. Environ.
**2022**, 30, 3163–3173. [Google Scholar] [CrossRef] - 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, 169–175. [Google Scholar] [CrossRef] - Tobin, M.C. The Theory of Phase Transition Kinetics with Growth Site Impingement. II Heterogenous Nucleation. J. Polym. Sci
**1976**, 14, 2253–2257. [Google Scholar] - Malkin, A.Y. General Treatment of Polymer Crystallization Kinetics–Part 1. A New Macrokinetic Equation and its Experimental Verification. Polym. Eng. Sci.
**1984**, 24, 1396–1401. [Google Scholar] [CrossRef] - Hilliar, I.H. Modified Avrami Equation for the Bulk Crystallization Kinetics of Spherulitic Polymers. J. Polym. Sci. Part A
**1965**, 3, 3067–3078. [Google Scholar] [CrossRef] - Pérez-Cárdenas, F.C.; del Castillo, L.F.; Vera-Graziano, R. Modified Avrami expression for polymer crystallization kinetics. J. Appl. Polym. Sci.
**1991**, 43, 779–782. [Google Scholar] [CrossRef] - Lee, E.-S.; Kim, Y.G. A transformation kinetic model and its application to Cu-Zn-Al shape memory alloys–I. Isothermal conditions. Acta Metal. Mater.
**1990**, 38, 1669–1676. [Google Scholar] [CrossRef] - Guimaraesa, J.R.C.; Riosa, P.R.; Alves, A.L.M. An alternative to Avrami equation. Mater. Res. (Sao Carlos Braz.)
**2019**, 22, e20190369/1–e20190369/8. [Google Scholar] [CrossRef] - Malek, J. The applicability of Johnson-Mehl-Avrami model in the thermal analysis of the crystallization kinetics of glasses. Thermochim. Acta
**1995**, 267, 61–73. [Google Scholar] [CrossRef] - Wang, J.; Kou, H.C.; Chang, H.; Gu, X.F.; Li, J.S.; Zhong, H.; Zhou, L. Limitation of the Johnson-Mehl-Avrami equation for the kinetic analysis of crystallization in a Ti-based amorphous alloy. Int. J. Miner. Metall. Mater.
**2010**, 17, 307–311. [Google Scholar] [CrossRef] - Todinov, M.T. On some limitations of the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation. Acta Mater.
**2000**, 48, 4217–4224. [Google Scholar] [CrossRef] - Almansour, A.; Matsugi, K.; Hatayama, T.; Yanagisawa, O. Modeling of growth and impingement of spherical grains. Mater. Trans. JIM
**1996**, 37, 1595–1601. [Google Scholar] [CrossRef] - Aziz, A.A.; Samsudin, S.A.; Hay, J.N.; Jenkins, M.J. The effect of a secondary process on polymer crystallization kinetics—3. Co-poly(lactic acid). Eur. Polym. J.
**2017**, 94, 311–321. [Google Scholar] [CrossRef] - Pérez-Cárdenas, F.C. The irrelevance of phantom nuclei in crystallization kinetics: An integral equation approach. J. Non-Cryst. Solid X
**2019**, 1, 100002. [Google Scholar] [CrossRef]

**Figure 1.**X(t) as predicted by Equation (10) using two sets of data. When γ = 1 and ν = 0, Equation (10) transforms into Equation (2), as shown in Appendix A.

**Figure 6.**X(t) as predicted by Equation (10), when ν = 0. For γ = 1 (equivalent Avrami model, Equation (2)), for γ = 3, and for γ = 5. For these last two cases, Equation (10) is equivalent to Equation (A6) in Appendix A.

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**MDPI and ACS Style**

Pérez-Cárdenas, F.C.
The Diminishing Role of the Nucleation Rate as Crystallization Develops in Avrami-Type Models. *Solids* **2022**, *3*, 447-456.
https://doi.org/10.3390/solids3030031

**AMA Style**

Pérez-Cárdenas FC.
The Diminishing Role of the Nucleation Rate as Crystallization Develops in Avrami-Type Models. *Solids*. 2022; 3(3):447-456.
https://doi.org/10.3390/solids3030031

**Chicago/Turabian Style**

Pérez-Cárdenas, Fernando C.
2022. "The Diminishing Role of the Nucleation Rate as Crystallization Develops in Avrami-Type Models" *Solids* 3, no. 3: 447-456.
https://doi.org/10.3390/solids3030031