The Avrami Equation

The kinetic approach relies on the establishment of a relation between the densities of the crystalline and melt phases and the time. This provides a measure of the overall crystallization rate. It is assumed that the spherulites grow from nuclei whose 

Relation between the Avrami Exponent and the Morphological Unit Formed for Sporadic Nucleation 

Sporadic nucleation is assumed to be a first-order mechanism, and if we consider that a two-dimensional disk is formed, then n = 2 -i- 1=3. Rapid nucleation is a zeroth-order process in which the growth centers are formed at the same time, and for each growth imit listed in Table 11.3, the corresponding values of the exponent would be (n — 1). Thus, the Avrami exponent is the sum of the order of the rate process and the number of dimensions the morphological unit possesses. 

The time dependence of crystallization, particularly primary crystallization, has often been interpreted or modeled in terms of 

FIGURE 10-23 Schematic graph of the degree of crystallinity versus time showing the induction period, and primary and secondary crystallization. 

FIGURE 10-24 Spherulidc growth. On the left is a snapshot during primary crystallization, while the snapshot on the right is during the secondary crystallization stage (Courtesy Prof. James Runt, Penn State). 

FIGURE 10-25 Schematic diagram showing the thickness of a lamellar. 

Excess free energy of surface per unit area = oe 

Crystallization theories have been developed for low-molecular-weight substances and later adapted to polymers. One of the first theories to study such phenomena was the free growth theory formulated by Goler and Sachs [5,8,32,33]. They established that once a given nuclei or center is initiated, it grows unrestrained or without the influence of others that may have also been nucleated and could be growing within the same time scale. If N is the steady-state nucleation rate per unit of untransformed mass (where all the material is in the liquid state or yet to be converted to the solid state), is the mass of a given center at time t, that was initiated at time T (t t), then  

Assuming that the polymer structure can be described by a two-phase model and considering the densities of the crystalline (Pc) and liquid phases (p ). Equation 11.2 can be alternatively written as 

The Avrami equation, also referred to as the Kolmogorov-Johnson-Mehl-Avrami equation [34-37,40], can be considered to be one of the possible solutions of Equation 11.3, and in its simplest form it can be expressed as [31,34—39,41] 

The Avrami theory usually provides a good fit of the experimental data at least in the conversion range up to the end of the primary crystallization, that is, up to the impingement of spherulites at approximately 50% conversion to the solid 

In Equation 11.4, the Avrami index can be considered as a first approximation to be composed of two terms [42]  

Many phase transformations such as nucleation and crystal growth from a melt, solid-state recrystallization during annealing, austenitic to ferritic transformation in steels, etc. are stochastic events because they involve either liquid- or solid-phase nucleation and therefore must be treated in terms of probabilities as done in Chapter 11. Let Vq be the total volume of the material, Vj be the volume of the transformed material, and Vu be the volume of material that has not yet been transformed. Recall from Poisson statistics that the probability of no events occurring in a given time is exp (expected number of events in that time). In this case the expected volume to be transformed is kt where k and n may be specified by models of the transformation mechanism or may be determined empirically. Therefore, the fraction of trarrsformed matter at time t can be written as 

This relationship is known as the Avrami equation (also known as the Kolmogorov-Johnson-Mehl-Avrami or KJMA equation). Half of the material will be transformed when fcf = 0.693. Only 1% of the transformation has occurred when fcf = 0.01 and the transformation is 99% complete when fcf = 4.6. The transformation time is generally specified as the time required for half of the material to be transformed, which is given by 

Fraction transformed versus time on a log scale as described by the Avrami equation. 

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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 ... 

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... 

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...

Figure 4-14 fitting bubble growth model results by the Avrami equation 367... 

Kinetics of crystallization. Trick (79) has reported a dilatometric study of the bulk crystallization of PTHF. The rates he observed for a polymer of Mw = 130,000 (Polymer A) are shown in Fig. 27. He also found that a lower molecular weight polymer (Polymer B, Mn = 6760) crystallized to a higher degree of crystallinity, whereas the introduction of comonomer units (Polymer C) decreased the degree of crystallinity (Fig. 28). From attempts to fit the Avrami Equation to the experimental data in the early stages of crystallization, a tentative value of n = 3 was... 

The key to modelling the crystallization process is the derivation a kinetic equation for a(t,T). It is possible to find different versions of this equation, including the classical Avrami equation, which allows adequate fitting of the experimental data. However, this equation is not convenient for solving processing problems. This is explained by the need to use a kinetic equation for non-isothermal conditions, which leads to a cumbersome system of interrelated differential and integral equations. The problem with the Avrami equation is that it was derived for isothermal conditions and... 

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