Spherulitic growth rate equation

For miscible blends, the spherulitic growth rate equation can be employed to predict the crystallization kinetics. This equation, commonly employed for unblended crystalline polymers, is ... 

This equation has been widely used although the validity of applying the WLF equation to spherulitic growth rate is merely a repetitive assertion (Hoffman et al., 1959 Hoffman and Weeks, 1962), not involving any direct proof of substantiation, as Mandelkern has stated. 

The spherulite growth rate depression is proportional to the type of energy barrier that has to be overcome, and can be quantitatively expressed by a modified equation of the spherulite growth rate [Martuscelli, 1984] ... 

For the crystalline component, the parameters from the WLF equation, Ci and Cl, can be found from literature (Icenogle 1985). Tg and can be measured for pure crystaUizable component. The parameters G° and C3 can be calculated from the experiments that give the spherulite growth rate G as... 

Gi Undisturbed spherulite growth rate of the homopolymer described by the Tumbull-Fisher equation... 

Keywords entanglement, disentanglement, cross-hatching, lamellae, crystallization, nucleation, reptation, nucleation (crystallization) regimes, nucleation agents, nucleation rate, spherulitic growth rate, Avrami-equation, Ozawa-equation, isothermal crystallization, nonisothermal crystallization, secondary nucleation, supercooling. 

The spherulite growth rate in regime I, previously described in equation (6.18), can be expressed as a function of temperature with the relationship... 

In this chapter, we take a practical approach to briefly explain how to experimentally determine both spherulitic growth rates by polarized light optical Microscopy (PLOM) and overall isothermal crystallization kinetics by differential scanning calorimetry (DSC). We give examples on how to fit the data using both the Avrami theory and the Lauritzen and Hoffman theory. Both theories provide useful analytical equations that when properly handled represent valuable tools to understand crystallization kinetics and its relationship with morphology. They also have several shortcomings that are pointed out. 

For homopolymers, the temperature dependence of the isothermal spherulite growth rate, G, is described by the equation below ... 

Equation (7.31) can also be derived assuming that the ratio of spontaneous nucleation rate to spherulite growth rate remains constant during cooling, that is ... 

Similar effects were found by analyzing the overall crystallization kinetics from the melt at the same the crystallization half-time ty of the blend increased exponentially with increasing concentration of the amorphous component. The temperature dependence of the spherulite growth rate (G) and kinetic constant was then examined according to Equation (10.13)... 

In order to examine the role of composition on the spherulite growth rate of the crystallizing component in a miscible blend, and indirectly on the overall crystallization rate, the effect of dilution on each of the terms in the growth rate equation for a one-component system needs to be modified. Consequently, starting with Eq. (9.205) it is found that (23-25)... 

One of the more important uses of OM is the study of crystallization growth rates. K. Cermak constructed an interference microscope with which measurements can be taken to 50° (Ref 31). This app allows for study of the decompn of the solution concentrated in close proximity to the growing crystal of material such as Amm nitrate or K chlorate. In connection with this technique, Stein and Powers (Ref 30) derived equations for growth rate data which allow for correct prediction of the effects of surface nucleation, surface truncation in thin films, and truncation by neighboring spherulites... 

The isotherms obtained in dilatometric measurements of the crystallization rate could be fitted with an Avrami (3) type equation only by assuming the existence of a secondary crystallization process much slower than the rate of spherulitic growth observed microscopically, and by taking into account the experimentally determined form of the nucleation rate. The nucleation rate was found to be a first-order process. Assuming that the secondary crystalliza-... 

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. 

According to this theory, the equation describing the growth rate G of spherulites of a crystallizable polymer in a one-phase melt containing a second polymer acting as a diluent assumes the form... 

The influence of PMMA content on the kinetic and thermodynamic parameters controlling the isothermal spherulitic growth and the overall crystallization rate of PEG from the molten blends has been analyzed on the basis of the modified Tumbul 1-Fisher equation ... 

The spheruhte growth may be initiated by either heterogeneous nucleation by foreign particles or homogeneous spontaneous nucleation. The growth rate of spherulites from the polymer melt increases as the temperature decreases. It reaches a maximum in the temperature range of 140-150°C, and then decreases upon further decrease of temperature. The time dependency of the primary crystallization can be described by the known Avrami equation [282]. 

Figure 4. Variation of spherulite radius growth rate with temperature. Notice that the data can be fitted to equation (5) using tlie appropriate characteristic constants.

If the crystallization takes place over an extended temperature range, most if not all homopolymers display a maxima in rates of spherulitic growth and in the overall crystallization. The equation for spherulitic growth is written as follows [42] ... 

Examples of the change in spherulite radius with time for selected temperatures are shown in Fig. 39.6, where it can be seen that linear growth rates result. Plots of growth rate versus temperature for iPP can be seen in Fig. 39.7. When the data are analyzed using the Hoffman-Lauritzen equation, Fig. 39.8, it is seen that iPP shows the Regime II-Regime III transition, previously identified by several groups of workers [7-10]. In these analyses the values of and U were 186.1 °C and 1,500 cal/mol, respectively. 

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 some industrial polymers, crystallization rates are too high so that the observation of spherulite growth in the interesting temperature range is not experimentally possible. An approximate method is therefore useful. Van Krevelen (1976) proposed a semi-empirical equation for the growth rate of variety of common polymers as follows ... 

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