My first suggestion is to do a proper uncertainty budget analysis. The values of n and Ea can only be correlated with confidence when you have their uncertainties. Values of n = 0.85 ± 0.10 and n = 0.70 ± 0.10 have a significantly lower confidence of being different than do values n = 0.85 ± 0.01 and n = 0.70 ± 0.01.

My second suggestion is to recognize that Avrami is a semi-empirical formulation, not a first principles formulation. When you want to deconstruct both the nucleation and the growth terms with a reasonable degree of confidence, you would have to move to a first-principles consideration of the overall rate as a proper combination of the two fundamental rates. Avrami is not this proper fundamental combination.

My third suggestion is that changes in Ea are changes in the kinetic reaction barrier while changes in n are changes in the kinetic mechanism. The latter is correlated with the former, but the former may happen independently of the latter. In particular, Avrami "fudges" the mechanism as a one-step process, while the true process is multiple elemental steps. When something fundamental changes in any one of the elemental steps, both n and Ea will likely change. One only needs to have a two-step nucleation become a single-step nucleation to alter the entire landscape in n and Ea. On the other hand, the mechanism could remain the same, yet the activation energy for a certain step may be dependent on temperature. In this case, n might stay only mildly dependent on temperature yet Ea change with temperature.

Your overall question I cannot answer directly. The significance of n < 1 to interpretation as favoring nucleation or growth is an area that others might be better suited to address with clarity.

Dear Dr. Jeffrey J. Weimer,

Thank you very much sir for your insightful suggestions. It helps me to overcome some of confusions. Your suggestions develops some enquires in my mind and they are as follows :- 

I had already  done uncertainty budget analysis (sorry for not mentioning in the details section) and they are as follows 0.85+-0.08; 0.7+-0.05 and 0.6 +-0.09. Now kindly suggest me how to correlate  values of n and Ea .

Secondly, you have suggested that " Avrami is a semi-empirical formulation, not a first principles formulation and  to move to a  first-principles consideration of the overall rate as a proper combination of the two fundamental rates."  Will you kindly suggest some of them.

Your third suggestion is very interesting and it actually helps me to understand relationship between n and Ea better.

 Sir, I am very novice in this field, suggestions/comments form experts like you will help me to improve myself in this field. Thank you once again for your valuable time. 

Before discussing the meaning of the apparent kinetic parameters, could you tell us some more details about the reaction and the procedure of kinetic analysis?

(1) There are three compound. Each compound decomposes in three reaction steps. Right? Are these reaction steps overlapping partially?

(2) How did you obtain the kinetic rate data? Thermal Analysis? Isothermal condition or dynamic nonisothermal condition? How about sample mass and reaction atmosphere?

(3) How did you determine the apparent kinetic parameters? Classical isothermal kinetic analysis?

(4) Why did you assume nucleation-growth mechanism for the thermal decomposition. The shape of the master plot for JMA(n) with n < 1 is similar to diffusion controlled model with contracting geometry. Pure theoretically, JMA(n) with n=0.5 describes the one dimensional growth of nuclei controlled by diffusion and with n = 1 can also be interpreted as that of two dimensional growth. So with n = 1.5 for three dimensional growth.

I am interesting in your question and answers from specialists. If you could provide us above information, our discussion becomes more meaningful and many specialists can attend this discussion.

Thank you very much in advance.

Response to Dr. Nobuyoshi Koga sir,

Thank you very much sir for your interest and enquires. I am trying to provide the informations point by point-

(1) Yes sir, all the three compounds shows a three stage thermal decomposition pattern. The last two stages (Stage II and Stage III) for each compound are overlapping in nature and quite difficult to separate.  

(2) I have performed TG ( model- NETZSCH STA 449F1) analysis at three different heating rates (2.5, 5 and 10 oC/min) in nitrogen atmosphere. The sample mass was around ~20 mg. Precautions were taken to keep the sample mass more or less constant for each experiment. 

(3)   (a) TG, dTG and fractional mass loss data (derived from TG data) were used to have an initial idea about the decomposition reaction and progress of the reaction with time and/or temperature.

       (b) Model free analyses (Friedman and Ozawa-Flynn-Wall method) were exploited to study the variation in Ea with progress of the reaction. 

      (c) Then a multivariate non-linear regression analysis on the dynamic thermal mass loss data at different heating rate  was performed to determine the most probable kinetic model based on statistical fit using Thermokinetic software (Netzsch). 

       In this software, a direct fit of a model to the experimental data is achieved by non-linear regression analysis. This direct fit of the model without any transformation does not affect the error structure and offers no limitations with respect to complexity of the model. In this iterative process, a 5th-degree Runge-Kutta embedding method with automatic optimization of the interpolation nodes for numerical solution of the equations by Dormand–Prince method is employed to estimate the kinetic parameters . This involves Gauss-Newton algorithm (Marquardt procedure)  with a step-length optimization.

 (4)  Avrami model for clay de-hydroxylation reaction is reported earlier. The selection of Avrami model for stage-I ( for my sample -Layered double hydroxide) was  based on the following consideration- i) Literature ii) the TG analysis indicated the existence of three stages of mass loss. Each of these stages may be modelled with 18 standard kinetic models (literature). Thus for a three- stage consecutive decomposition reaction (A →B→C→D) a total of 5832 (=18x18x18)  models were possible (along with other parameters). Although there was hardly a comprehensive report about the kinetics for all the three stages of decomposition but those informations help me to reduce the number of possible combinations.

Now one a question automatically arises how one could justify the validity of the assigned kinetic model in terms of practical application ? I am sharing my findings-

Based on the kinetic parameters I have construct a TTT (Time-Temperature-Transformation) diagram and  by exploiting this i successfully engineered the decomposition reaction to any extent viz. partial removal of hydroxyl groups. 

One more interesting thing sir, when i plot Avrami coefficient (n) and Ea against the layer charge - they shows a linear relationship. With increasing interaction (layer charge) the Avrami coefficient and Ea increases. 

Thank you very much sir for your patience. Hope I have not confused you. Please let me know anything I miss. 

(1) So, which reaction step within paertially overlapping three reaction steps?

(2) I recommend you to measure at least four kinetic curves. mo=20 mg means that you are measuring the reaction behavior of sample bed.

(3) I know the software and its algorithm. So, you are assuming each reaction steps are independent. Right?

(4) How do you correlate the structural characterisitcs of layered double hydroxide with phenomenological nucleation and growth model?

(5) For the discussion, (4) should be clearly explained before examining the empirical linear relationships, I think.

We specialists are always seeking phycial and chemical correlation between the structural factor of the sample with the apprent kinetic results obtained by the thermoanalytical study. So, we are very interested in your discussion. This is why we are asking about measurement conditions and procedure of kinetic calulation. In my opinion, if you can explain (4) with supplimental experimental evidences, it would be nice to carry out more systematic kinetic measurements under differnt measurement conditions and confirm  the reliablity of the kinetic rate data before kinetic calulation. The following article may be helpful to consider about reliable kinetic measurements for your system:   http://dx.doi.org/10.1016/j.tca.2014.05.036

I am looking forward to seeing your successful work in a scientific journal. Thank you for the interesting discussion.

Response to Dr. Nobuyoshi Koga sir,

Thank you very much sir for your nice suggestions. This work already been re- communicated to Thermochimica Acta with minor revisions. Hope I shall be able to answer your questions but it may take some time as I am very novice to this field. One more request sir, the paper you have recommended is not downloadable from my institute. If you kindly send me the paper it will a great help for me. 

Thank you once again for your valuable suggestions. 

You can find it in my profile in ReasrchGate. I do not know, but someone uploaded.

Kind regards.

Thank you sir

Response to Dr. Nobuyoshi Koga sir,

1) It is the 2nd and 3rd steps that are overlapping in nature

2) I will try to follow your recommendation. 

3) I am not sure about that sir. During the optimization there is a parameter called extent of reaction. What i have understand is- the parameter defines the extent of reaction of each stage in the complete reaction. Say if extent of reaction is 0.4 for stage I- i have interpreted as after 40% completion of the reaction the significance change for  stage-II starts. As it is a three stage decomposition reaction so i got two such values and interpreted accordingly. I am not sure about my interpretation. Hope you can guide me.

4) It is the most difficult question sir and currently i am working in it. Truly speaking i have no idea about how to correlate the kinetic parameters and the structural change/ characteristics till now. I have gone through different kinetic literature but i have not found anything regarding this topic. 

I wish i could understand the kinetic models physically like a organic reaction models (viz., SN1 or SN2 ) . If a layman asked about the SN reactions i can tell them in a way so that they can understand. But if an expert ask me about the model i proposed for the LDH decomposition- then it is nearly impossible for me. i have answer indirectly or in a superficial way. 

> Truly speaking i have no idea about how to correlate the kinetic parameters and the structural change/ characteristics till now.

I suggest, you should use other techniques to examine your samples. In particular, you need tools that will give you direct information about the structure at different stages of decomposition. Have you explored any microscopy or diffraction methods?

> I wish i could understand the kinetic models physically like a organic reaction models (viz., SN1 or SN2 )

The SN1 and SN2 reactions are elemental steps. The 18 x 18 x 18 models you have are not. The system is more complex than elemental steps. Focus instead on what is supposedly proposed to be happening to the structure in each overall step A -> B then B -> C then C -> D. Confirm your hypotheses with other characterization methods.

Response to Dr. Jeffrey J. Weimer sir

In one of my paper i have used SEM to justify the proposed kinetic model and mechanistic pathway.  In recent paper (communicated) i have indirectly support my proposed kinetic model by the following way-

1. First assigned kinetic model for the three stage decomposition reaction

2. Based on the kinetic parameters a TTT diagram is developed

3. The TTT diagram predicts different extent of decomposition for a different set of time and temperature

4. We justified one of the predicted % decomposition with help of IR study with reasonable accuracy

Thus in an indirect way we justified our proposed model.

In addition, we have done the similar kind of research to study the  spinelization reaction of Mg-Al system (modelling, TTT and justification- communicated) and justified our proposed model by matching the % spinel formation from XRD analysis.  

Having said that, I am still unable to explain the mechanism based on the kinetic models (like SN1 and SN2 for example) . Hope you can guide me on how to handle such questions ( i am facing this type of questions very often). Thank you very much sir for the suggestions, comments and help. 

 Dear Mostofa Shamim,

 Given the above discussion I dare a small contribution.

1. Values of the Avrami-constant n only and only differ due to different suppositions made in deriving the corresponding versions of the Avrami-equation. 

2. In applying the Avrami-equation one has to study firstly whether or not the suppositions made correspond to the experimental configuration of study. This means that physical considerations resulting from an independent study should yield an insight into the nature of nucleation and growth/progress of the transformation concerned.

3. Given point 1 & 2, to treat n as an experimental parameter in order to elucidate point 2, seems to me an error.

4.In general the determination of the activation energy is based on the following assumptions: i. there is a rate-determining step in the transformation process under study, and ii. this rate-determining step is thermally activated. Whether or not these assumptions are met, needs an independent study. An example may illustrate this point: in precipitation solute diffusion can often be regarded as the rate determining step.

5. A link between n and values of activation energy, other than stated in the attachment, seems to me difficult to establish, as n reflects especially the mechanism of the reaction (nucleation/growth or decomposition) whereas the activation energy usuallly reflects the kinetical hindering of the progress of the transformation once started.

The attachment I found on the internet may be helpful to you. Good luck!

Have You tried the models like Sestak-Berggren (in Netzsch software called Bna) instead of Avrami during non-linear regression? it seems that at least some of these three reactions could have non-nucleation-growth mechanism.. I mean that Avrami-Erofeev model with non-digit reaction order can be a mark of some other reaction type and the reaction orders in Sestak-Berggren model can help You to find this real reaction type.

I agree with Dr. Koga suggestion - fourth nonisothermal run is highly recommended to perform since You have 3-staged model (=many adjustable parameters) and need to have more restrictions during regression.