Avrami equation cases

The gradient of this graph therefore permits the determination of n, and the intercept allows k to be calculated. The advantage of using a Sharp-Hancock plot rather than a least squares fitting process with the Avrami equation is that if Avrami kinetics are not applicable, this can be seen in the former plot, and hence other kinetic models may be investigated. Purely diffusion controlled processes can be identified using a Sharp-Hancock plot n is foimd to be 0.5 in such cases. 

Isothermal curves derived from this equation are shown in Fig. 2.19. It is clear that this equation fits the experimental data. A comparison of the kinetic equation (2.48) and the Avrami equation shows93 that any experimental data described by the Avrami equation can be approximated by Eq. (2.48) for any arbitrary set of constants. The divergence of the curves does not exceed 1% in the range 0.2 experimental data (in the isothermal case) can be analyzed by both equations with practically the same reliability. Thus the choice of approximating equation depends on the goal of this procedure if we are interested in physi-... 

The surfaces of real substrates are inhomogeneous and exhibit surface defects such as steps, kinks, pits etc. These defects do significantly influence not only the energetics of 2D nucleation [3.249], but also the overlapping of growing 2D islands. Thus, the assumptions of the Avrami equation (3.65) are not fulfilled in this case. In the following, the influence of surface inhomogeneities such as monatomic steps on the kinetics of 2D Meads phase formation is briefly discussed. 

In the case of a sufficiently large crystal face intersected by a large number A/disi of randomly distributed single screw dislocations, the overlapping of the growing new pyramids can be taken into account using the Avrami equation (5.13). 

For the case of (hypothetical) spherical crystals and reaction-limited crystallization, the Avrami equation takes the form... 

For the case of heterogeneous nucleation, the exponent n of the Avrami equation is equal to the dimensionality of growth. 

Avrami equation hold, Eq. (8) agrees well with the data. Figure 3.86 was computed from the crystal-growth rates listed in Fig. 3.83 and the constant numbers of nuclei of Table 3.2. With these data it was possible to compute the overall curves shown in Fig. 3.22 for V , using n = 3, and thus, complete the discussion of the complicated case of simultaneous LiPOj polymerization, a chemical reaction, with overlapping crystallization, a physical phase transition. 

The kinetic evolution is usually represented by a sigmoid-type curve. Such a typical curve is given in Figure 9 for the case of displacement reaction, Ni + CuO Cu -F NiO realized in planetary ball mill [58]. Author of this overview analyzed these experimental results by one of the most frequently used kinetic model applied to various solid-state reactions, namely Johnson-Mehl-Avrami equation ... 

The overall crystallization rate is used to follow the course of solidification of iPP. Differential scanning calorimetry (DSC), dilatometry, dynamic X-ray diffraction and light depolarization microscopy are then the most useful methods. The overall crystallization rate depends on the nucleation rate, 1(0 and the growth rate of spherulites, G(0. The probabilistic approach to the description of spherulite patterns provides a convenient tool for the description of the conversion of melt to spherulites. The conversion of melt to spherulites in the most general case of nonisothermal crystallization is described by the Avrami equation ... 

For quiescent crystallization under a constant temperature, in the case of instantaneous nucleation with a constant number density No, one obtains Nq t) = NoH t) for the activated quiescent nuclei number density, where H(t) is the Heaviside unit step function, zero for f < 0 and unity for t > 0. Then the rate of the nuclei number density is Ng = NoS t), with (5(f) being the Dirac delta function concentrated at f = 0. Equations 4.1 and 4.3 lead to the familiar Avrami equation ... 

The plots at the bottom of Fig. 2.12 illustrate the Avrami expressions for the experimental data for the UPO3 case of Figs. 2.10 and 2.11. Equation (19), finally, gives the general form of the Avrami equation. The constant K collects all crystal-geometry- and nucleation-dependent terms that arise from crystal growth. Crystal growths other than athermally nucleated spherical... 

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... 

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] ... 

Hyperbolic curves can also be described by the modified Avrami equation, but in this case n should set to 1. The resulting equation is an exponential growth curve. There also was a good fit of the model to the data over the same range of fractional crystallizations. Tables 5 and 6 show the values of and the correlation coefficients for samples 16, 15, and 14 (second hydrogenation process) and 8, 7, and 6 (first hydrogenation process), respectively. 

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 ... 

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. 

The Avrami exponent n is evaluated by the curve fitting procedure that has been described. It only applies to that portion of the isotherm that fits Eq. (9.3 la). However, there are situations where a subjective decision has to be made. These are cases where, although a significant portion of the transformation can be fitted by n = 3, about half the transformation is also satisfied by n = 4. The problem is whether n = 4 represents the actual mechanism with deviations from the Avrami equation ensuing. 

A similar plot for poly(butylene naphthalene 2,6-dicarboxylate) is given in Fig. 9.25.(96) In contrast to the previous figure, these plots are only linear at the lower portion and curvature is observed at the higher levels of crystallinity. These results indicate that in this case the crystallization has not been limited to Region I. Curvature in the Ozawa type plot has also been observed with poly(aryl ether ether ketone) (40), poly(aryl ether ether ketone ketone) (97) and poly(aryl ether ether sulfide).(97a) Curvature and deviation from the theory will be observed, if crystallization occurs beyond Region I, because the derived Avrami equation is no longer vaUd. 

M = 2 X 10" to 2 X 10 , from dilute solutions of either /7-xylene or ethanol showed good adherence to the derived Avrami equation with n = 4, over an appreciable extent of the transformation.(33,34) An unfractionated isotactic poly(propylene), 3.6 X 10 , yields a set of superposable isotherms, when crystallized from a 0.3% solution of either ether tetrahn or decaUn.(35) A plot of the data according to Eq. (9.32) that is given in Fig. 13.14 makes clear both the superposition and adherence to a derived Avrami expression. In this case the exponent n is again found to be 4. An exponent of 4 has also been found for poly(ethylene terephthalate) crystallizing from several different solvents.(36)... 

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