Nucleation Avrami equation

A-i [—In (1 — a)]1 3 =s kt Random nucleation Avrami equation 11 Phase boundary reaction, cylindrical symmetry... 

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 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. 

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. 

This is the general solution to the problem of multistep nucleation followed by a constant rate of growth in three dimensions. Equation (113) is of the same form as Eq. (107) and therefore is called the generalized Avrami equation. The limit for short times is... 

Nucleation and Growth Johnson-Mehl-Avrami Equation... 

The primary crystallization process is characterized by three parameters. These are the rate of radial growth of the spherulite, G, the time constant for nucleation, t , and the time constant for the primary crystallization process, Tc, which is determined from the Avrami equation. All three parameters seem to depend on the stereoregularity of the polymer, but the nucleation rate seems to depend most strongly. 

Wu and Woo [26] compared the isothermal kinetics of sPS/aPS or sPS/PPE melt crystallized blends (T x = 320°C, tmax = 5 min, Tcj = 238-252°C) with those of neat sPS. Crystallization enthalpies, measured by DSC and fitted to the Avrami equation, provided the kinetic rate constant k and the exponent n. The n value found in pure sPS (2.8) points to a homogeneous nucleation and a three-dimensional pattern of the spherulite growth. In sPS/aPS (75 25 wt%) n is similar (2.7), but it decreases with increase in sPS content, whereas in sPS/PPE n is much lower (2.2) and independent of composition. As the shape of spherul-ites does not change with composition, the decrease in n suggests that the addition of aPS or PPE to sPS makes the nucleation mechanism of the latter more heterogeneous. 

Crystallization data have typically been treated theoretically using either the Fisher-Tumbull model or the Avrami equation. These analyses not only allow lipid crystallization to be modeled but may also shed some hght on the mechanisms of nucleation and growth. However, there is some recent debate about the validity of such models, especially the application of the Avrami equation (42) to accurately depict crystallization of lipids. 

Avrami Analysis The Avrami equation, a general approach for description of isothermal phase transformation kinetics originally developed for polymers (46), is often used for describing nucleation and crystal growth in fats. The Avrami equation is given as... 

The crystallization rate constant k) is a combination of nucleation and growth rate constants, and is a strong function of temperature (47). The numerical value of k is directly related to the half time of crystallization, ti/2, and therefore, the overall rate of crystallization (50). For example, Herrera et al. (21) analyzed crystallization of milkfat, pure TAG fraction of milkfat, and blends of high- and low-melting milk-fat fractions at temperatures from 10°C to 30°C using the Avrami equation. The n values were found to fall between 2.8 and 3. 0 regardless of the temperature and type of fat used. For temperatures above 25°C, a finite induction time for crystallization was observed, whereas for temperatures below 25°C, no induction time was... 

The caution by Marangoni (8) on the misuse of a modified form of the Avrami equation is rightly justified within the context of the warning. However, if the equation is to be applied to the crystallization of oils and fats in order to elucidate information on the mechanisms of their nucleation and growth, the equation will have to be modified to suit the conditions prevailing in the system. 

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. 

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