Crystallisation Avrami equation

An analysis of crystallisation rates is conveniently performed in terms of the so-called time-temperature-transformation (TTT) curves, which relate the time taken to crystallise a given fraction of the undercooled liquid or the supersaturated solution to the temperature. Experimentally, the crystallisation rates are measured by quenching the liquid phase to some predetermined temperature T and measuring the time taken for the solid to crystallise at that temperature, either by monitoring the latent heat of crystallisation or by microscopic observation. The volume fraction 4>(T) that crystallises out in time 1 is given by one form of the Avrami equation ... 

Kinetic data from scattering experiments could be analysed in terms of the Avrami equation (Eq. 26). This equation describes the development of crystallinity (or other separate phase) with time, at a constant temperature, in terms of a rate coefficient k and an exponent n is the fraction of crystalHsable material which has crystallised at time t. The value of n reflects a combination of the kinetics of nucleation of the second phase and the number of dimensions in which crystallinity (or second phase) develops. A value of n=3 is consistent with a constant niunber of growing nuclei and growth in three dimensions, i.e. the kinetics of a constant number of expanding spheres. The Avrami equation allows for impinging growing phases and departures from the Avrami equation, at long times, are often explained in terms of secondary crystallisation within the already-formed spheruHtes ... 

Rates of radial growth of spherulites, constant with time, were highest for pure PCL and decreased with increasing SAN content and with increasing temperature in the range 34-50 C [1061 he. the presence of SAN decreased the rate of crystallisation. The variation in extent of crystallinity with time was sigmoidal and the kinetics of crystallisation were consistent with the Avrami equation (Eq. 26) with an exponent of 3 0.02, consistent with three-dimensional growth... 

The kinetics of isothermal crystallisation were studied in quenched (at -120 ° C) amorphous samples after heating to 127 °C and rapid cooling to the crystallisation temperature [137]. Crystallisation kinetics, for PCL crystallisation, were consistent with the Avrami equation (Eq. 26) with values of n close to 3, consistent with spontaneous heterogeneous nucleation and three-dimensional growth of spherulites, at crystallisation temperatures from 30 °C to 40 °C. Rates of crys-... 

The Avrami equation (Eq. 26), appHcable to non-isothermal crystallisation, can be written as... 

As it is known [3, 33], crystallisation process kinetics can be described with the aid of the Kolmogorov-Avrami equation ... 

The Kolmogorov-Avrami equation for the case when the crystallisation parameter is the stress in a uniaxially stretched sample can be presented as follows [39] ... 

Eet us consider crystallisation mechanism changes for HDPE/EP nanocomposites in comparison with the initial HDPE, which define the degree of reduction in crystallinity with an increase in c p. As it is known [8], the crystallisation kinetics of polymers is often described with the aid of the Kolmogorov-Avrami equation, obtained for low-molecular substances (Equation 4.19), in which the exponent n value can be changed within the range of 1-4 [9]. 

The crystallisation behaviour of blow moulded PETP bottles, which helps determine the product s transparency, was investigated by DSC dynamic cooling experiments that simulated the cooling that occurs in the injection blow moulding manufacturing process. DSC measurements were used to obtain information on related aspects, such as the ease of crystallisation from glassy and molten states and crystallinity in the products. An Avrami equation was used for calculation of the crystallisation kinetic parameters. 40 refs. 

The crystallisation kinetics of polymeric materials xmder isothermal conditions for various modes of nucleation and growth can be well approximated by the Avrami equation[9,10]. 

The end results is an over simplification of crystallisation process. In general however, the Avrami equation is usually found to represent a good fit with the data. This has shown to be a consequence of the inherent strong correlation between tp the induction time of nucleation, and k and n, from Equation (2.4) [12,13]. 

The Avrami equation is crucial in the description of crystallisation and other processes. The equation has been applied to areas as diverse as corrosion, reaction kinetics and the growth of micro-organisms. If applied correctly, it can give information on the type of nucleation (homogeneous or heterogeneous) as well as the geometry of crystallisation, for... 

Table 1.3 The different exponents and expressions for the rate constant in the Avrami equation for different crystallisation morphology...

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. 

Similarly, the value of the rate constant is frequently put inside the brackets in order to have constant dimensions. The Avrami equation has found increasing application in the foods area, for instance, in the crystallisation of fats as well as more traditional polymer areas. 

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. 

According to Avrami (1939-1941) the progress of the isothermal crystallisation can be expressed by the equation ... 

Lasocka parameter, activation energy of crystallisation, frequency factor, and Avrami parameter can be evaluated for different systems. Along with these parameters, crystallisation constants as a function of different temperatures as well as different heating rates (p) should be studied. Tg and its dependence on heating rates can be given by the following original empirical Lasocka equation. 

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