Avrami equations examples

The Avrami—Erofe ev equation, eqn. (6), has been successfully used in kinetic analyses of many solid phase decomposition reactions examples are given in Chaps. 4 and 5. For no substance, however, has this expression been more comprehensively applied than in the decomposition of ammonium perchlorate. The value of n for the low temperature reaction of large crystals [268] is reduced at a 0.2 from 4 to 3, corresponding to the completion of nucleation. More recently, the same rate process has been the subject of a particularly detailed and rigorous re-analysis by Jacobs and Ng [452] who used a computer to optimize curve fitting. The main reaction (0.01 < a < 1.0) was well described by the exact Avrami equation, eqn. (4), and kinetic interpretation also included an examination of the rates of development and of multiplication of nuclei during the induction period (a < 0.01). The complete kinetic expressions required to describe quantitatively the overall reaction required a total of ten parameters. 

The above set of equations can be solved numerically given input parameters, including surface tension a, temperature, solubility relation, D and p as a function of total H2O content (and pressure and temperature), initial bubble radius ao, initial outer shell radius Sq, initial total H2O content in the melt, and ambient pressure Pf. For example. Figure 4-14 shows the calculated bubble radius versus time, recast in terms of P versus t/tc to compare with the Avrami equation (Equation 4-70). 

The function ((t) in Eq. 21.12 has a characteristic sigmoidal shape with a maximum rate of transformation at intermediate times. Examples are shown in Fig. 21.2. The d = 3 form of Eq. 21.12 is commonly known as the Johnson-Mehl-Avrami equation. 

EXAMPLE 11.2 A reaction N is 20% complete after 45 s and 85% complete after 1.25 min. Determine the value of n in the Avrami equation. 

The crystallization rate constant k) is a combination of nucleation and growth rate constants, and is a strong function of temperature (47). The numerical value of k is directly related to the half time of crystallization, ti/2, and therefore, the overall rate of crystallization (50). For example, Herrera et al. (21) analyzed crystallization of milkfat, pure TAG fraction of milkfat, and blends of high- and low-melting milk-fat fractions at temperatures from 10°C to 30°C using the Avrami equation. The n values were found to fall between 2.8 and 3. 0 regardless of the temperature and type of fat used. For temperatures above 25°C, a finite induction time for crystallization was observed, whereas for temperatures below 25°C, no induction time was... 

In reality, polymer crystaHizatiOTi is too complex to be described by a simple expression such as the Avrami equation. For example, the assumption in Avrami s expression that the volume does not change is inaccurate because the specimen tends to shrink during crystallization. In addition, secondary crystallization and crystal perfecting processes are not taken into account. 

In this chapter, the possibilities of developing new ideas about self-organization processes and about nanostmctures and nanosystems are discussed on the example of metal/carbon nanocomposites. It is proposed to consider the obtaining of metal/carbon nanocomposites in nanoreactors of polymeric matrixes as self-organization process similar to the formation of ordered phases, which can be described with Avrami equation. The application of Avrami equations during the synthesis of nanofilm stmctures containing copper clusters has been tested. The influence of nanostmctures... 

In the past few years, we have been employing the Avrami equation [31,42,45-76] to analyze DSC overall isothermal crystallization kinetics in many different types of polymeric materials and we have encountered many practical problems [31] that are sometimes apparent in published isothermal crystallization calorimetry data. Some of those problems are treated here as examples. 

We use differential scanning calorimetry (DSC) thermograms to monitor the crystallization process, for example, isothermal crystallization from the melt. We pack the sample in a DSC pan, heat it, and maintain it at the temperature of the melt for a short time in order to erase the sample preparation history. Then we reduce the temperature to the preset crystallization temperature Tc, and trace the thermal energy change. Figure 5.12a is a schematic illustration of this curve. In many cases the curve follows the Avrami equation ... 

Avrami theory is derived by presuming random nucleation, a constant rate of nucleation (or a constant nucleation density). However these assumptions may not always hold true. The linear growth rate, for example, is not always constant with time. In addition, the number of nuclei may not increase continuously but may instead reach a limiting level after exhaustion of heterogeneous nuclei. Ihe use of the Avrami equation is further complicated [12] by additional factors such as ... 

The Avrami equation is frequently applied directly to isothermal DSC data where the heat output of the crystallisation process is thought to describe exactly the crystallisation. It can also be applied in modified form to non-isothermal data such as the crystallisation of PET on cooling. It should be noted that there is a certain ambiguity in the exponent value. For example, a value of 3 can mean either 2D nucleation from nuclei appearing randomly in time, or 3D nucleation from pre-existing nuclei. 

This type of approach does not appear to have been used extensively in food research. The decomposition observed in TGA measurements is invariably in the solid state and consequently reactions can be extremely complex, leading to many different equations being put forward to describe the loss of weight (see [170] for a discussion on this point). For example, a reaction can proceed by a mechanism similar to crystallisation, that is points of initiation followed by circular growth fronts. As might be expected, a variant of the Avrami equation (see Section 1.5.9) has been proposed to describe such reactions. 

Another example that demonstrates the influence of chain entanglements is found in the melt crystallization kinetics of pure poly(dimethyl siloxane) (Mn = 740 000).(50b) It is found that in this case the derived Avrami equation with n = 3 can account for 95% of the transformation. This relatively rare event can be explained by the fact the molecular weight between entanglements of this polymer is 12 000 as compared, for example, with 830 for linear polyethylene.(50c) Thus, the number of entanglements per chain is relatively low as compared to other polymers. It will also be shown in Chapter 13 that a derived Avrami expression explains more than 90% of the crystallization of linear polyethylene from dilute solution. 

Fig. 13.13 Examples of superposition of isotherms, (a) Data from Fig. 13.10, M = 3.0 X 10. (b) Data from Fig. 13.12, M = 3.1 x 10 . Solid curves derived Avrami equation with n = 4.

Those exponents which we have discussed expUcitly are identified by equation number in Table 4.3. Other tabulated results are readily rationalized from these. For example, according to Eq. (4.24) for disk (two-dimensional) growth on contact from simultaneous nucleations, the Avrami exponent is 2. If the dimensionality of the growth is increased to spherical (three dimensional), the exponent becomes 3. If, on top of this, the mechanism is controlled by diffusion, the... 

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. 

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).

Analyses of rate measurements for the decomposition of a large number of basic halides of Cd, Cu and Zn did not always identify obedience to a single kinetic expression [623—625], though in many instances a satisfactory fit to the first-order equation was found. Observations for the pyrolysis of lead salts were interpreted as indications of diffusion control. More recent work [625] has been concerned with the double salts jcM(OH)2 yMeCl2 where M is Cd or Cu and Me is Ca, Cd, Co, Cu, Mg, Mn, Ni or Zn. In the M = Cd series, with the single exception of the zinc salt, reaction was dehydroxylation with concomitant metathesis and the first-order equation was obeyed. Copper (=M) salts underwent a similar change but kinetic characteristics were more diverse and examples of obedience to the first order, the phase boundary and the Avrami—Erofe ev equations [eqns. (7) and (6)] were found for salts containing the various cations (=Me). 

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