The general Avrami equation

The general Avrami equation is applicable to any type of crystallization. It is not restricted to polymers. It describes the time evolution of the overall crystallinity. The pioneer work was conducted during the 1930s and 1940s by Evans, Kolmogoroff, Johnson and Mehl, and Avrami. Wunderlich (1978) concludes that without the parallel knowledge of the microscopic, independently proven mechanism, the macroscopic, experimentally derived Avrami equation and the Avrami parameters are only a convenient means to represent empirical data of crystallization. However, interest in the Avrami equation has been 

To illustrate the general Avrami equation, a particularly simple case is selected athermal nucleation followed by a spherical free growth in three dimensions. All nuclei are formed and start to grow at time t = 0. The spherical crystals grow at a constant rate f. It is an established fact that crystallization from a relatively pure melt occurs at a constant linear growth rate. All nuclei within the radius rt from 

The probability p(0) is equivalent to the volume fraction (1 — 1 ) o( the polymer which is still in the molten state  

Crystallization based on different nucleation and growth mechanisms can be described by the same general formula, the general Avrami equation  

A slightly more complex case involves thermal nucleation. The nuclei are here formed at a constant rate both in space and time, similar to normal rain. Let us select the case of three-dimensional growth at a constant linear rate. The number of waves (d ) which pass the arbitrary point (P) for nuclei within the spherical shell confined between the radii r and r + dr is given by  

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This is the general solution to the problem of multistep nucleation followed by a constant rate of growth in three dimensions. Equation (113) is of the same form as Eq. (107) and therefore is called the generalized Avrami equation. The limit for short times is... 

The Avrami Constant. The calorimetric data were analyzed according to the general Avrami equation ... 

The above-mentioned expressions of polynucleation and growth processes, as triggered by single potential step experiments El El, may be represented by the general Avrami equation [15,16,188]... 

Inserting the equation above into (10.36), one derives the general Avrami equation as given by... 

It should be noted that now the time is raised to the fourth power. In general it is found that crystallization based on diflferent nucleation and growth mechanisms can be described by the same general formula, the general Avrami equation ... 

Figure 2. Kinetic curves calculated from the general Avrami equation for different values of the e3cponent n.

This chapter presents first some fundamental aspects of nucleation, and second the general Avrami equation, which is frequently used to describe overall crystallization. The growth theories of Lauritzen and Hoffman and Sadler and Gilmer are discussed in sections 8.4.2 and 8.4.3. Molecular fractionation and orientation-induced crystallization are dealt with in sections 8.5 and 8.6. 

Expressions are derived in this section for a few selected cases and it is shown that the derived equations have a certain common mathematical form. This is expressed in the general Avrami equation. Figure 8.7 illustrates the fundamentals of the model. It is assumed that crystallization starts randomly at different locations and propagates outwards from the nucleation sites. The problem which is dealt with can be stated as follows. If raindrops fall randomly on a surface of water and each creates one leading expanding circular wave, what is the probability that the number of waves which pass a representative point P up to time t is exactly cl The problem was first solved by Poisson in 1837 and the resulting equation is referred to as the Poisson distribution ... 

Fit the general Avrami equation to the following data obtained by DSC at two different temperatures for a highly branched polyethylene sample ... 

The kinetics of crystallization, which is of interest for both academic and industrial reasons, is preferably studied under isothermal conditions by DSC, dilatometry or TOA. These methods reveal the overall crystallinity, volume (vf) or mass wf) crystallinity as a function of time (f), and the general Avrami equation can be applied ... 

One way to incorporate the effect of stress on crystallization is described here (Katayama and Yoon, 1985). One starts with the generalized Avrami equation ... 

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... 

A similar derivation, which assumes that the transformation rate is controlled by the nucleation rate, gives an equation where t is raised to the fourth power. The classical Avrami equation generalizes these results by assuming that a is related to t by an equation with an arbitrary value for n. 

A plot of log V against log t should give a straight line with the intercept of log F T) = log[H(T)/k] and the slope of equal to -n/M. In fact, the Ozawa equation and the isothermal Avrami equation (Eq. 7.10) are derived from the same general Equation (7.6a) and Equation (7.6b) but with different assumptions in the Ozawa equation a constant cooling rate is assumed, while in the Avrami Equation (7.10) the growth rate G is a constant. Therefore, the combination of the equations proposed in References [68] and used by many authors has no justification and is erroneous. 

As has been indicated earlier, both Eq. (9.31) and Eq. (9.31a) have been commonly termed the Avrami relation. Either one is a very simple and convenient expression to use and hence their widespread adoption. However, it should be recognized that they are restricted in scope because of the limited nucleation and growth processes that have been considered. Thus, caution must be exercised when interpreting results using these equations. Theoretical isotherms based on the derived Avrami equations are plotted in Fig. 9.10 in accordance with Eq. (9.31 a) for n = 1, 2,3 and 4. The curves in Fig. 9.10 have several important features in common. Their general shapes are in qualitative accord with the experimental observations that were described previously. When examined in detail, differences exist that distinguish one curve from another. As n increases from 1 to 4, the time interval at which crystallinity becomes apparent becomes greater. However, once the transformation... 

The isotherms illustrated in Fig. 9.12 only represent one type of nuclei activation. If, for example, all of the potential nuclei are activated at r = 0, equations of the form of Eqs. (9.28) to (9.31) result. There is obviously a variety of activation processes that can be postulated, each of which will result in its own unique isotherm. Hence, for a given growth geometry there can be many differently shaped isotherms that have n values that not only differ but apparently vary with the time course of the transformation. Hence, it should not be unexpected when considering the complete transformation to observe decreasing values of the Avrami exponent with time. This observation is not caused by a deficiency in the general Avrami formulation (Eqs. 9.26 and 9.27) but rather by the varying nucleation rate with time. 

Deviations from the Avrami equation are frequently encountered in the long time limit of the data. This is generally attributed to secondary nucleation occurring at irregularities on the surface of crystals formed earlier. 

Experimental results are in general conformity with the Avrami equation, but the interpretation of various observations is still complicated in many instances. One intriguing observation is that the induction period for nucleation is inversely proportional to the length of time the liquid is held in the liquid state after previous melting. This dependence on prior history may be qualitatively understood... 

Avrami Analysis The Avrami equation, a general approach for description of isothermal phase transformation kinetics originally developed for polymers (46), is often used for describing nucleation and crystal growth in fats. The Avrami equation is given as... 

The Avrami equation [Eq. (2.15)], which was originally proposed in the general context of phase changes, has provided the starting point for many studies of polymer crystallization and spherulitic growth. It relates the fraction of a sample still molten, 9, to the time, t, which has elapsed since crystallization began. The temperature must be held constant. 

Avrami equation, Eq. (7.1), is generally performed to follow the rate constant of the included AITC released from the CD-based inclusion complexes under various relative humidities [14,15]. 

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