Equation Avrami

Knowing from the microscopic analysis the number of athermally growing nuclei and the linear crystal growth rate, it is easy to calculate the initial volume fraction, v, crystallized at time t. This volume fraction is needed for the link of the crystallization 

V = volume at time t 9, 9a, and 9c represent the densities of the overall sample, the amorphous and crystalline phase, all at time t P = chance of an event not occurring 

the equation for the evaluation of the crystallized volume-fraction is derived in Eqs. (3-5) of Fig. 3.84. Equation (5) is also used to compute the volume fraction crystallinity of semicrystaUine polymers from dUatometric experiments (see Sect. 4.1). The more common mass-fraction crystallinity is derived in Sect. 5.3.1. Simple density determinations as a function of time can, according to Eq. (5), establish the overall progress of crystallization. 

K = geometry and nucleation dependent constant n = Avrami exponent 

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Table 4.3 Summary of Exponents in the Avrami Equation for Different Crystallization Mechanisms...

The decrease in amorphous content follows an S-shaped curve. The corresponding curve for the growth of crystallinity would show a complementary but increasing plot. This aspect of the Avrami equation was noted in connection with the discussion of Eq. (4.24). 

Next let us examine an experimental test of the Avrami equation and the assortment of predictions from its various forms as summarized in Table 4.3. Figure 4.9 is a plot of ln[ln(l - 0)" ] versus In t for poly (ethylene terephtha-late) at three different temperatures. According to Eq. (4.35), this type of... 

The testing of the Avrami equation reveals several additional considerations of note ... 

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... 

The crystallization of poly(ethylene terephthalate) at different temperatures after prior fusion at 294 C has been observed to follow the Avrami equation with the following parameters applying at the indicated temperatures ... 

The Avrami—Erofe ev equation, eqn. (6), has been successfully used in kinetic analyses of many solid phase decomposition reactions examples are given in Chaps. 4 and 5. For no substance, however, has this expression been more comprehensively applied than in the decomposition of ammonium perchlorate. The value of n for the low temperature reaction of large crystals [268] is reduced at a 0.2 from 4 to 3, corresponding to the completion of nucleation. More recently, the same rate process has been the subject of a particularly detailed and rigorous re-analysis by Jacobs and Ng [452] who used a computer to optimize curve fitting. The main reaction (0.01 < a < 1.0) was well described by the exact Avrami equation, eqn. (4), and kinetic interpretation also included an examination of the rates of development and of multiplication of nuclei during the induction period (a < 0.01). The complete kinetic expressions required to describe quantitatively the overall reaction required a total of ten parameters. 

Assuming the formation of N0 nuclei at the first stages of oxidation, the effective relaxed area (taking into account the overlap between neighboring expanding conductive regions) at every overpotential tj can be estimated by means of the Avrami equation.177 We arrive at... 

The overall isothermal crystallization kinetics of polymers can be described by the Avrami equation [88-90] ... 

Solid PET feedstock for the SSP process is semicrystalline, and the crystalline fraction increases during the course of the SSP reaction. The crystallinity of the polymer influences the reaction rates, as well as the diffusivity of the low-molecular-weight compounds. The crystallization rate is often described by the Avrami equation for auto-accelerating reactions (1 — Xc) = cxp(—kc/"), with xc being the mass fraction crystallinity, kc the crystallization rate constant and n a function of nucleation growth and type. 

Crystallization of PET proceeds in two distinct steps [97], i.e. (1) a fast primary crystallization which can be described by the Avrami equation, and (2) a slow secondary crystallization which can be described by a rate being proportional to the crystallizable amorphous fraction dXc/dt = (Xmax — tc)kc, with Xmax being the maximum crystallinity (mass fraction) [98], Under SSP conditions, the primary crystallization lasts for a few minutes before it is replaced by secondary crystallization. The residence time of the polymer in the reactor is of the order of hours to days and therefore the second rate equation can be applied for modelling the SSP process. 

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. 

Note 3 The Avrami equation addresses the problem that crystals growing from different nuclei can overlap. Accordingly, the equation is sometimes called the overlap equation . 

One approach is to roughly estimate how the degree of crystallization would vary with time by making main simplifications in treating solidification, leading to the Avrami equation. 

Figure 4-13 Degree of crystallization based on the Avrami equation.

Figure 4-14 Calculated F versus for bubble growth using the program of Proussevitch and Sahagian (1998) modified by Liu and Zhang (2000). F is the volume of the bubble versus the final equilibrium volume of the bubble. The calculated trend may he fit by the Avrami equation with an n value of 0.551.

The above set of equations can be solved numerically given input parameters, including surface tension a, temperature, solubility relation, D and p as a function of total H2O content (and pressure and temperature), initial bubble radius ao, initial outer shell radius Sq, initial total H2O content in the melt, and ambient pressure Pf. For example. Figure 4-14 shows the calculated bubble radius versus time, recast in terms of P versus t/tc to compare with the Avrami equation (Equation 4-70). 

Figure 4-13 Calculated results using the Avrami equation 366...

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