Avrami law

Fig. 16. Profile analysis of an elementary mass step recorded at a 10 Hz sampling frequency, characteristic of a 2D nucleation and growth process. Conditions R = 2.4 > Rc, deposition temperature 50 C the dashed line represents a fit according to the Avrami law (Eq. 6) (from [147]). With permission of Electrochem. Soc.

Fig. 18. Dependence of the step time constant on deposition temperature for various bath compositions determined by a step profile analysis according to the Avrami law (Eq. 37).

The exponents a and f) which determine the kinetics of the structural change of oriented PET are shown in Fig. 6.6. The values of a and /3 are obtained by the fitting procedure mentioned above. In Fig. 6.6, it is foimd that the exponent a for the kinetics from the nematic structure to the smectic structure decreases from 2.3 to 1 with decreasing temperature within the temperature range investigated here. In the case of the Avrami law [17], the crystalline fraction (j)c is given by... 

The growth process of the large micellar structures, which are strongly aligned, has been studied in more detail by transient SANS experiments. In these experiments the shear rate for the samples was raised stepwise from zero to a certain finite value. These experiments showed that the large micelles grow according to the Avrami law... 

Here, i/gph is the hardness value of the spherulites, is the hardness of the amorphous interspherulitic regions, and is the volume fraction of crystallized spherulites. During primary crystallization, Hgph remains constant and hardness is directly proportional to the volume occupied by the spherulites (12). The hardness variation in the course of isothermal crystallization of PET and poly(ethylene naphthalene-2,6-dicarboxylate) (PEN) has been shown to follow Avrami law (13,14). 

Decompositions may be exothermic or endothermic. Solids that decompose without melting upon heating are mostly such that can give rise to gaseous products. When a gas is made, the rate can be affected by the diffusional resistance of the product zone. Particle size is a factor. Aging of a solid can result in crystallization of the surface that has been found to affect the rate of reaction. Annealing reduces strains and slows any decomposition rates. The decompositions of some fine powders follow a first-order law. In other cases, the decomposed fraction x is in accordance with the Avrami-Erofeyev equation (cited by Galwey, Chemistry of Solids, Chapman Hall, 1967)... 

Following a common practice in the literature, significant rate expressions will be referred to by names which have become accepted through common usage these may be descriptive (power law, etc.) or recall the names of workers who contributed towards their development (the Avrami—Erofe ev equation, etc.). Examples of systems obeying each expression are restricted in the present section since the applications are exemplified more generally in the literature surveys which constitute Chaps. 4 and 5. 

The strongly acceleratory character of the exponential law cannot be maintained indefinitely during any real reaction. Sooner or later the consumption of reactant must result in a diminution in reaction rate. (This behaviour is analogous to the change from power law to Avrami—Erofe ev equation obedience as a consequence of overlap of compact nuclei.) To incorporate due allowance for this effect, the nucleation law may be expanded to include an initiation term (kKN0), a branching term (k N) and a termination term [ftT(a)], in which the designation is intended to emphasize that the rate of termination is a function of a, viz. 

Fig. 6. Two reaction models which result in obedience to the power law [eqn. (2), n = 2 ] at low a, or the Avrami—Erofe ev equation [eqn. (6), n = 2 ] over a more extensive range of a. In (a), there is growth of semi-circular nuclei in a thin plate of reactant in (b), there is cylindrical growth of linear internal nuclei. In both examples, rapid nucleation (0 = 0) is followed by two-dimensional growth (X = 2).

Table 8.1 Values of a as a Function of Time for a Reaction Following an Avrami-Erofeev Rate Law with n = 3 and k = 0.020 min-1.

Methods of data analysis for reactions in solids are somewhat different from those used in other types of kinetic studies. Therefore, the analysis of data for an Avrami type rate law will be illustrated by an numerical example. The data to be used are shown in Table 8.1, and they consist of (a,t) pairs that were calculated assuming the A3 rate law and k = 0.025 min-1. 

FIGURE8.2 A graph of a versus time for an Avrami rate law with n = 3and = O.MOmirT1. 

The first of these reactions takes place as the sample is heated in the range 47 to 63 °C and the second in the range 70.5 to 86 °C. When the data were analyzed to determine the rate law for the processes, it was found that both reactions followed an Avrami rate law with an index of 2 as the extent of reaction varied from a = 0.1 to a = 0.9 (Ng et al, 1978). Another reaction for which most data provide the best fit with an Avrami rate law is... 

An enormous number of phase transitions are known to occur in common solid compounds. For example, silver nitrate undergoes a displacive phase transition from an orthorhombic form to a hexagonal form at a temperature of approximately 162°C that has a enthalpy of 1.85 kj/mol. In many cases, the nature of these transitions are known, but in other cases there is some uncertainty. Moreover, there is frequently disagreement among the values reported for the transition temperatures and enthalpies. Even fewer phase transitions have been studied from the standpoint of kinetics, although it is known that a large number of these transformations follow an Avrami rate law. There is another complicating feature of phase transitions that we will now consider. 

Prepare rate plots of the data shown in Table 8. lusing an Avrami rate law with n values of 2, 3, and 4. 

In a real system there will be several clusters growing simultaneously. At first the clusters are separated, but as they grow, they meet and begin to coalesce (see Fig. 10.5), which complicates the growth law. For the case of circular growth considered here, the Avrami theorem [4]... 

The most commonly employed approach is that of Avrami and Erofe ev. This rate law has been derived in a munber of ways, indicating its general apphcabiUty and vaUdity. This equation takes the form... 

We can apply classical germination laws to this supersaturated system thus, the Avrami-Mempel laws confirm the unidimensional growth of the solid-like gel network. Induction times can also be studied in this framework 11). Here, we are interested first by the different kinetic behaviors which are dependent upon the location in the phase diagram of the initial solution defined by its supersaturation degree. 

Table 12.2 lists a number of rate equations that are commonly applied to dissolution processes.Two equations that are used quite frequently are the cube rate law (Hixon Crowell, 1931) which takes the geometry of the dissolving particle into account, and the Avrami-Erofejev law which applies to sigmoidal dissolution curves. 

Extensive measurements of the kinetics to determine rate constants for the nanocrystal transition have been made only on the CdSe system (Chen et al. 1997, Jacobs et al. 2001). Both the forward and reverse transition directions have been studied in spherically shaped crystallites as a function of pressure and temperature. The time-dependence of the transition yields simple transition kinetics that is well described with simple exponential decays (see Fig. 5). This simple rate law describes the transformation process in the nanocrystals even after multiple transformation cycles, and is evidence of the single-domain behavior of the nanocrystal transition. Rate constants for the nanocrystal transition are obtained from the slope of the exponential fits. This is in contrast to the kinetics in the extended solid, which even in the first transformation exhibits complicated time-dependent decays that are usually fit to rate laws such as the Avrami equation. 

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 ... 

Care should be taken in defining the procedure for calculating values of k fi om the experimental data. There is always the possibihty that the apparent is a compound term containing several individual rate coefficients for separable processes (such as nucleation and growth). It is important that the dimensions of k (and hence of A) should be (time). For example, the power law (Table 3.3.) should be written as = kt and not as ar = k t. Similarly the Avrami-Erofeev equation (An) is [-ln(l - a)Y = kt. The use of k in place of A in the Arrhenius equation will produce an apparent activation energy /i, which is n times the conventional activation energy obtained using k. 

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