Avrami-Erofeev models

Figure 10.7 Arrhenius plot for solid-state proton transfer in dihydrogen-bonded complex LiBFLj triethanolamine. The data are obtained in the framework of the Avrami-Erofeev model. (Reproduced with permission from ref. 14.)...

An example of an intensively studied set of polymorphs whose decompositions are of great theoretical and practical importance (see Chapter 12) is CaCOj which may exist (in order of decreasing thermodynamic stability) as calcite, aragonite or vaterite [18]. Vaterite can be prepared by precipitation from aqueous solutions under carefully controlled conditions. A DTA curve for the vaterite calcite transition is shown in Figure 2.3. The transition is exothermic AH = -34.3 J g ) with onset at 704 K. Isothermal extent of conversion against time curves were described [18] by the Johnson, Mehl, Avrami, Erofeev model (see Chapter 3) with n = 2. The measured Arrhenius parameters were F, = 210 kJ mol and A = 1.15x10 min. The decomposition of vaterite and its concurrent transformation to calcite under various conditions were compared [18] with the decomposition of calcite xmder the same conditions (see Chapter 12). 

Elder [45] has modelled several multiple reaction schemes, including mutually independent concurrent first-order reactions, competitive first-order reactions, mutually independent n-th order reactions, and mutually independent Avrami-Erofeev models with n = 2 or 3. The criteria identified for recognizing the occurrence of multiple reactions were (i) the apparent order of reaction, n, varies with the method of calculation, and (ii) the kinetic parameters, A and vary with the extent of reaction, a. 

As can be seen, with the exception of the R3, D3 and D4 models, the exclusion of the In (j)m (o max term in the generalized Kissinger equation (9) has less than a 3% effect, and for the simple n order and Avrami-Erofeev models, A2, A3, less than a 1% effect. Similar figures result if one sets m - h or m = 1 in the rate equation (1). As the Arrhenius exponent E/RTjjjqx increases, there is a gradual small increase in Umax models. For all n order and diffusion controlled... 

Figure 1 Solid-state degradation models, a—fraction decomposed. A = Prout-Tompkins, kt = ln(-p ), B = 2 dimensional phase boundary kl = l-(l-a)1 2, C=Avrami-Erofeev, kt — [—ln(l — a)] , w=l, D = Avrami-Erofeev, = 0.5.

For a solid-state reaction, one of the solutions of Equation 3.1 is the Avrami-Erofeev equation [3], The phase transition model that derives this equation supposes that the germ nuclei of the new phase are distributed randomly within the solid following a nucleation event, grains grow throughout the old phase until the transformation is complete. Then, the Avrami-Erofeev equation is [3]... 

Fig. 11.13. Calculated Fe3C>4 —>Fe metal TPR peaks for six reduction models using E = 111 kj/mol (TPR on dry H2/ Ar) 0.2 K/min (a) three-dimensional nucleation according to Avrami-Erofeev, (b) two-dimensional nucleation according to Avrami-Erofeev, (c) two-dimensional phase boundary, (d) three-dimensional phase boundary, (e) unimolecular decay, (f) three-dimensional...

Fig. 11.14. Comparison between measured TPR patterns for Fe2C>3 and calculated TPR peaks for Fe304 —>Fe metal reduction step, using the three-dimensional nucleation model according to Avrami-Erofeev at 0.2 K/min. Calculated dotted line (a) dry series, (b) wet series, E = 172...

Figure 11.14a compares the model based on Avrami-Erofeev kinetics with the... 

Figure 12.14a compares the model based on Avrami-Erofeev kinetics with the experimental curve. The agreement is satisfactory in light of the fact that no adjustable parameter has been used to allow curve-fitting procedures. The difference at the leading edge is attributed to the first reduction step which is not included in the model. 

Incorporation of a diffusion term in nucleation and growth reaction models has been proposed by Hulbert [68]. Interface advance is assiuned to fit the parabolic law and is proportional to but the nucleation step is uninhibited. The overall rate expressions have the same form as the Avrami-Erofeev equation ... 

Because (1 - is a constant for a given value of n, a value for 7i, may be obtained from the slope of a plot of ln(/ /r , against l/r ,ax for a series of experiments at different heating rates, ft,. Augis and Bennett [65] modified the Kissinger treatment for use with the Avrami - Erofeev (or JMAEK) model (An). They plotted - TJ) against l/7] ax where is the initial temperature at the... 

A recent kinetic study [108] of the overall dehydration rates of KA1(S04)2.12H20 and KCr(S04)2.12H20 showed that measured ur-time data were well described by the Avrami-Erofeev equation with = 2. This was not consistent with expectation for the growth of three dimensional nuclei for which A = 3 and = 0 or 1, respectively. In accordance with the reaction models described above, there must be limited water losses from all surfaces together with an overall reaction controlled by product recrystallization that confers the apparent topotacticity on the overall... 

Kinetic runs in step b in Fig. 8c started with a very fast reduction of approximately e per molecule, after which a slow reductioh took place, yielding sigmoidal reduction curves. This, indicates that reduction of Co2+ to Co° is controlled by the formation and slow growth of reduction nuclei of metallic cobalt on. the surface of the reduced phase in step a (nucleation model). Initially, the reduction rate increases because of the growth of nuclei already formed and the appearance of new ones. At a certain point the reduction nuclei start to overlap at the inflection point, the interface of. the oxidized and reduced phases and the reduction rate both begin to decrease. Reduction of this type is described by the Avrami-Erofeev equation (118)... 

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