For a uniform nucleation in 3 dimension, the value of n is given by D+1, where D is the number of dimensions. Hence for a crystal growing uniformly in 3-dimensions, the value of n will be 4. For your experiments where you obtained a value of n greater than 4, there is a possibility that the crystal was not growing uniformly in all the 3 dimensions. This might be the reason you obtained the Avrami exponent greater than 4. I hope my explanation is clear.

Dear Michael Emmanuel

Thank you for your response. Your explanation makes sense, but do you have any reference or similar observation to mention?.

Happy to hear from you.

https://link.springer.com/article/10.1134/S1063783414080241

Dear Dr. Fedor M. Shakhov

Thank you for sharing your paper. I read your paper, but there is no indication of anomalous Avrami exponents and also discussion about the reasons for large Avrami values (larger than 4). It would be great if you provide some further explanations.

Happy to hear from you.

The applicability of the Avrami equation is very limited. The equation describes the phase transition in a solid. It is assumed that the reaction begins from the formation of a finite number of the transformation centers which are arbitrarily distributed in a volume. The new phase forms homogeneously and isotropically. In addition, during formation of the new phase, new transformation centers form, but these centers are not combined. Does your reaction meet these conditions?

Dear Rashidi, I forgot where I saw similar explanation, but I will still search for it and furnish you with the reference in due time. I am currently working on a flow-driven crystallization system too.

The Avrami exponent gives the exponent of the time dependency of the crystallization process. This becomes apparent directly from the general JMAK-equation that can be found e.g. in J.W. Christian, "The Theory of Transformations in Metals and Alloys", 2002, in Chapter 12. In the argument of the exponential it is integrated over the nucleation rate I and the crystal growth velocity vg in order to get an expression about the changing crystallized volume ratio. In order to get the ratio, spherical grains are assumed (3d) and their size is found from integrating the crystal growth velocity three times and then multiplying by 4pi/3 to get the volume of that crystalline sphere of each grain. This volume expression is multiplied by the integral of the nucleation rate I. When the nucleation rate and the crystal growth velocity are both constant in time, the integral will be proportional to (vg*t)^3*I*t = vg^3*I*t^4. The Avrami exponent in this case is 4.

In two dimentions, the integral would be proportional to (vg*t)^2*I*t = vg^2*I*t^3. Here the Avrami exponent would be 3.

However, these results are only valid, when the crystal growth velocity and nucleation rate would be constant with time. For the nucleation rate, this is ususally not the case as the subcritical grain distribution has to evolve first to the steady-state distribution in order for the nucleation rate to become equal to the constant value of the steady-state nucleation rate, see Kelton and Greer "Nucleation in Condensed Matter". The resulting transient behavior is of course time dependent and the time dependence of the nucleation rate may be approximated by I(t) = I^m. Now, evaluating the argument of the expontial in the JMAK-equation will result in a different Avrami exponent than 4 (3d) or 3 (2d). If m > 1, the Avrami exponent will be larger than these values.

For the phase-change materials Ge2Sb2Te5 in our study, we measured an Avrami exponent of 5.8. More details on our measurements can be found here: Data Supporting Information - Switching between Crystalllization ...

Hope, this may help you!

Dear Julian Pries

Thank you very much for your comprehensive and helpful response. After a precise review of your paper, I will come back to you for further discussion.

Dear Michael Emmanuel

Thanks again for your kindness. This means a lot to me. Always happy to hear from you.

Take a look at the attached pdf document Reza Rashidi

How to interpret the value of Avrami exponent of 0.7 for the formation of epsilon-carbides and of 1.4 for the formation of cementites under tempering of low-alloy carbon medium spring steel

Dear Mr. Jassim Alhamid,

Thanks for your comment. I have looked through the paper of Reza Rashik.

The case when Avrami exponent great than four is considered in the document.

However my question about case when Avrami exponent of 0.7 and 1.4.

Best regards,

Aleksandr

Dear Reza,

This was the question raised by the reviewers of our submitted manuscript to J. Non-Cryst. Solids, and we described it as the following:

Based on JMA analysis and transformation theory, the value of n(α) should be lower than 4 and the values of n(α) exceeding 4 are anomalous. The extremely large values of n(α) at the beginning of crystallization (α < 0.02) are due to the large uncertainties caused in calculating n(α) based on JMA equation and the DSC data which rely on the choice of baseline. During crystallization, n(α) also exceeds 4 in our alloys (Zr-Cu-Al-Ni-(Nb)) and this behavior has been observed in several metallic glass systems. In a recent work (A.H. Cai, et al., Journal of Alloys and Compounds 846 (2020) 156250), the n(α) values as large as 32-37 have been reported in Zr-Al-Ni-Cu-Fe-C metallic glass system. The large values of n(α) may be attributable to the surface-induced abnormal grain growth having fractal dimensionality. Further, the large value of n(α) are characteristics of transient nucleation process during which the nucleation rate and the crystal growth velocity are not constant.

A detailed discussion can be found in our recent paper:

Article Effect of Nb minor addition on the crystallization kinetics ...