Avrami theory equation

The isothermal crystallization study on binary blends was performed with P(HB80-ET20)/PET (weight ratio, 4 3) and P(HB80-ET20)/PEN (weight ratio, 4 3). The Avrami theory was applied, as shown in the following equation [23, 46] ... 

Transformation kinetics according to Nakamura and Ziabicki Nakamura (3) extended Avrami theory to non-isothermal transformations and proposed the following equation ... 

The kinetics of crumb firming have been described by an equation derived from the Avrami theory (Avrami 1939). This theory describes the rate of change of a supercooled amorphous material to an ordered crystalline structure when the process is governed by random production of stable nuclei ... 

On the other hand, kinetic data on first-order transformations are often obtained by isothermal analysis. The isothermal crystallization kinetics of the amorphous phase can be usually analyzed in terms of the generalized theory of the well-known Kolmogorov-Johnson-Mehl-Avrami (JMA) equation (Christian, 2002) for a phase transition ... 

The Isothermal dilatometric growth rate data were fitted to the equation predicted by the Avrami theory (25,26,27) ... 

Non-isothermal crystallization kinetics can be analyzed by using the extension of the Avrami theory [68-70] and proposed by Ozawa [71, 72], This analysis accounts for the effect of cooling rate on crystallization from the melt by replacirtg the time variable in the Avrami equation with a variable cooling rate term, that is, by replacing f in the equation (3) with TIa as shown in the equation (4) ... 

In this chapter, we take a practical approach to briefly explain how to experimentally determine both spherulitic growth rates by polarized light optical Microscopy (PLOM) and overall isothermal crystallization kinetics by differential scanning calorimetry (DSC). We give examples on how to fit the data using both the Avrami theory and the Lauritzen and Hoffman theory. Both theories provide useful analytical equations that when properly handled represent valuable tools to understand crystallization kinetics and its relationship with morphology. They also have several shortcomings that are pointed out. 

Isothermal crystallization of a polymer is frequently characterized by the induction time and the crystallization half-time. The crystallization induction time is the time that elapses from the moment when the desired crystallization temperature is reached to the onset of crystallization, characterized by the formation of the first nuclei. The crystallization half-time is the time when relative crystallinity reaches 0.5. More detailed analyses of the isothermal crystallization are usually based on the Avrami-Evans theory. Equation (7.10) yields ... 

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 ... 

There arc other criticisms of the Avrami method here. Negahban points out that modeling of crystallization kinetics using Avrami-type equations is incompatible with the entropy production inequahty (i.e., Clausius Duhem inequahty) that is basic to die termination of crystallization in real situations. Much of his work has concentrated on the crystallization of rubber (hterature results) with all its ancillary effects/property-wise material functions that arc amenable to simulation, hr the same marmer, the kinetic theory of crystallization, based upon concepts of molecular chain-folding and it s related parameters,at the same time recognizing some of the shortcomings involved. An attempt has been made to keep... 

Vanrolleghem 2006), worked on a differential equation derived from the original theory of crystallization described by Avrami (1939 1940 1941). The main hypothesis of this work is that crystallization could be interpreted as an nth order reaction and melting as a first order reaction,... 

The Avrami model (19,20) states that in a given system under isothermal conditions at a temperature lower than V. the degree of crystallinity or fractional crystallization (70 as a liinction of time (t) (Fig. 11) is described by Equation 5. Although the theory behind this model was developed for perfect crystalline bodies like most polymers, the Avrami model has been used to describe TAG crystallization in simple and complex models (5,9,13,21,22). Thus, the classical Avrami sigmoidal behavior from an F and crystallization time plot is also observed in TAG crystallization in vegetable oils. This crystallization behavior consists of an induction period for crystallization, followed by an increase of the F value associated with the acceleration in the rate of volume or mass production of crystals, and finally a metastable crystallization plateau is reached (Fig. 11). 

In calculating the Avrami exponent from DSC data, it was observed that fractional Avrami exponents are obtained, and not whole numbers, as is observed for the application of the equation to metal melts for which the theory was developed or in the majority of cases for polymeric crystallization from melts. In certain instances the practice of rounding off the Avrami exponent value could lead to ambiguity, especially where rounding off intermediate values between two integers, e.g., where exponents of 2.45 or 2.48 are obtained. 

The crystalline phase typically grows as spherical aggregates called spherulites. However, other geometries such as disks or rods may be found with, as shown below, a consequent modification of the rate equation. M. Avrami [26] first derived these rate equations in the form used for polymer kinetics for the solidification of metals. The weight of the crystalline phase is calculated as a function of time at constant temperature. As will be described below, the temperature dependence of crystallization can be derived from classical nucleation theory. 

---