Avrami equation general

In the context of the crystallization process, p(0) is equivalent to the volume fraction (1 — vj of the polymer which is still in the molten state  

This equation indicates that in the case of spherical growth the growth rate depends on the cube of the time. 

If now the growth rate is assumed to be constant and linear in space and time then 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 

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  

Equation (6.42) can be expanded according to exp(-ift ) 1 - Kf +. .. and for the early stages of crystallization where there is little restriction of crystallization due to impingement  

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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... 

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. 

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... 

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 ... 

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... 

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 ... 

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