Overall crystallization rate

Nuclei form by the addition of molecules this is a process of diffusion for which the following relation holds  

Because the growth of nuclei proceeds by the process of heterogeneous nucleation, the growth rate v can also be represented by an equation of the form 

Around the glass transition temperature, the following holds [10]  

An examination of Eq. (11.3.9) shows that the overall growth rate is independent of time under isothermal conditions. Also, the plot of G as a function of temperature is beU shaped (see Fig. 11.9). The rate is zero in the vicinity of the glass transition temperature because the rate of diffusion is small and the first exponential in Eq. (11.3.9) tends to zero, and it is also zero close to the melting point because the second exponential is driven to zero due to AT tending to zero. The rate is a maximum approximately midway between these two limits. 

Solution Taking the natural logarithm of both sides of Eq. (11.3.9) and rearranging gives 

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Tanimoto, A.K., Kobayashi, K. and Fujita, S., 1964. Overall crystallization rate of copper sulfate pentahydrate in an agitated vessel. International Chemical Engineering, 4(1), 153. 

It has been reported that the overall rate of crystallization of pure PHB is relatively low compared to that of common synthetic polymers, showing a maximum in the temperature range of 55-60°C [23]. The spherulite growth rate kinetics have been evaluated [59] in terms of the theory by Hoffmann et al. [63], At about 90 °C, the spherulite growth rate displayed a maximum, which is not excessively low compared to that of common synthetic polymers. Therefore it was stated that the low overall crystallization rate of PHB centers on the nuclea-tion process rather than the subsequent crystal growth. Indeed, it has been shown that PHB has an exceptionally low level of heterogeneous nuclei [18]. 

The overall rate of crystallization is determined by both the rate of nuclei formation and by the crystal growth rate. The maximum crystal growth rate lies at temperatures of between 170 and 190 °C [71, 72], as does the overall crystallization rate [51, 61, 75], The former is measured using hot stage optical microscopy while the latter is quantified by the half-time of crystallization. Both are influenced by the rate of nucleation on the crystal surface and the rate of diffusion of polymer chains to this surface. It has been shown that the spherulite growth rate decreases with increasing molecular weight due to the decrease in the rate of diffusion of molecules to this surface [46, 50, 55, 71, 74],... 

Phenomenology of the crystallization. The conversion versus time curves obtained at three different temperatures are shown in Figure 1. With the synthesis procedure used, the sigmoid curves were characterized by shorter induction periods than the traditional method (11,12). As expected, temperature had a strong effect on the rate of crystallization. The overall crystallization rates may be approximated by the reciprocal of the times of half conversion. From these values an apparent activation energy of 22 1 kcal/mol was obtained. With respect to literature data, this value exceeds that reported, for instance, for zeolite Na-X (1,4) but compares well with the 19.8 kcal/mol found for ZSM-11 (13). 

The overall crystallization behavior was studied by depolarized light intensity (DLI) and DSC (15,16,17). We observed a decrease in overall crystallization rate in the PBT-rich blends when the PET percentage... 

To circumvent such seemingly insurmountable difficulties, approximations and simplifications can often be made. For example, even though the formation of a solid from a saturated solution can involve several successive reactions, one may judiciously select for analysis or prediction the one that presents the slowest rate. The so-called Oswald s phase rule states that a supersaturated solution that undeigoes a sudden alteration that takes it out of such a state will produce a metastable solid (instead of the expected thermodynamically stable solid). It is a very useful rule, although it is not always obeyed. Three limiting cases are shown in Figure 5.8 for different values of the overall crystallization rate (nucleation -1- crystal growth), v. 

Agitation. Often, crystallization from solution occurs while the mixture is stirred or otherwise agitated. This generally increases the overall crystallization rate, and it may also affect crystal size. The following effects can be involved ... 

Assume a solution supersaturated with one crystallizable solute. To part of the solution, a crystal growth inhibitor is added. It is established that in both cases the relation Lc = a(ln fi)2 holds, only the constant a being different. Assume now that both solutions have the same initial, rather weak, supersaturation and that crystallization is started by adding tiny seed crystals. Would the ratio of the overall crystallization rates d(p/dt now be the same as the ratio of both a values ... 

The overall crystallization rate also depends on (a) the total crystal surface area, which will be larger if more crystals are formed and hence if nucleation is faster (b) stirring, which increases the rate if diffusion of molecules to the crystal surface is growth limiting and (c) removal of the released heat of fusion, since an increase in temperature causes a decrease in supersaturation. Naturally, the growth rate will decrease when most of the crystallizable material has crystallized. 

The effect of blending on the overall crystallization rate is the net combined effect of the nucleation and sphemlite growth. Martuscelli [1984] observed that in blends of PP with LDPE, crystallized at a T high enough to prevent any LDPE crystallization, the overall rate of crystallization of the PP matrix... 

A different case has also been explored by Martuscelli [1984] for PA-6 blended with an EPR-rubber. As shown in Figure 3.40, the PA-6/EPR blend decreased (faster overall crystallization rate) as the content of the rubbery phase increased, especially at lower concentrations of the EPR phase. 

Shingankuli [1990] studied the crystallization behavior of PP in the presence of solidified PVDF domains. A higher crystallization temperature of the PP matrix phase was observed, indicating an enhanced nucleation in the blends. The degree of crystallinity of PP was found to increase by about 30 to 40% with increasing PVDF content. Isothermal crystallization studies also confirmed the acceleration of the overall crystallization rate in terms of shorter crystallization half-times for PP. 

The DSC crystallization isotherms of pure iPP, pure PB-1, and blends compared at the same demonstrate that the overall crystallization rate constant progressively decreases with increase in the amount of the diluent component in the sample. 

The reduction in the overall crystallization rate of iPP in the iPP/HOCP blends was more drastic to the one engendered by the addition of PB-1 as the second component. 

The addition of HOCP to PB-1 causes a striking reduction in the overall crystallization rate. The overall kinetic rate constant K was calculated by using the Avrami equation (38) ... 

The isotherms of crystallization of pure iPP and iPP in the ternary blends showed that the overall crystallization rate constant in most cases decreases with the decrease in the amount of iPP in the blend and, with a constant fraction of iPP, increases when the fraction of HOCP decreases (Table 6.15). For all the examined samples, values of Avrami exponent, n, close to 3 have been obtained, suggesting a three-dimensional growth of crystalline unit, developed by heterogeneous nucleation. 

The investigations have shown that the spherulitic growth rate, the overall crystallization rate, and the melting temperature of iPP are depressed by the presence of HOCP. Together with the detection of a single glass transition, the results suggest that the two polymers are miscible in the melt. 

The sphemlitic growth rate, the overall crystallization rate, and the melting temperature of PB-1 are depressed by the presence of HOCP. The verified HOCP interferences on the kinetics of PB-1 crystal transformation from form II to form I indicate that in the crystallized mixtures the HOC molecules are rejected in inter-lamellar and/or interfibriUar regions of PB-1 spherulites where, dependening on the blend composition, they can form a homogeneous mixture with uncrystaUized PB-1 molecules or a conjugated amorphous phase, one rich in PB-1 component and the other rich in HOCP component. 

The miscibility in the melt of iPP, PB-1, and HOCP has been evidenced by the presence of a single glass transition and by the interference of PB-1 and HOCP on sphemlite growth rate, overall crystallization rate, and melting equilibrium temperature of iPP. The addition of the HOCP component to iPP and PB-1 has been found to increase the stability of the blends. The demixing phenomena in the ternary blends occur during and/or after the crystallization of the PB-1 component. The possibility of preparing ternary blends where the PB-1 component directly crystallizes in form I, could offer new opportunities about their application. 

The crystallization behavior and kinetics under isothermal conditions of iPP/SBH and HDPE/SBH blends, compatibilized with PP-g-SBH and PE-g-SBH copolymers, respectively, have been investigated (71). It has been established that the LCP dispersed phase in the blends plays a nucleation role for the polyolefin matrix crystallization. This effect is more pronounced in the polypropylene matrix than in the polyethylene matrix, due to the lower crystallization rate of the former. The addition of PP-g-SBH copolymers (2.5-10 wt%) to 90/10 and 80/20 iPP/SBH blends provokes a drastic increase of the overall crystallization rate of the iPP matrix and of the degree of crystallinity. Table 17.4 collects the isothermal crystallization parameters for uncompatibilized and compatibilized iPP/SBH blends (71). On the contrary, the addition of PE-g-SBH copolymers (COP or COP 120) (2.5-8 wt%) to 80/20 HDPE/SBH blends almost does not change or only slightly decreases the PE overall crystallization rate (71). This is due to some difference in the compatibilization mechanism and efficiency of both types of graft copolymers (PP-g-SBH and PE-g-SBH). The two polyolefin-g-SBH copolymers migrate to blend interfaces and... 

Overall crystallization rate (from DRS) and spherulitic growth rate (from HSOM) for PDS homopolymer as a function of temperature. 

Numerous experimental data by Toda et al. [45,46] indicate that the logarithmic derivative of the microscopically measured growth rate is equivalent to the logarithmic derivative of the overall crystallization rate obtained from DSC. For this reason, the temperature dependence of the effective activation energy... 

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... 

The variation of the overall crystallization rate has thus been related to the nucleation density in the melt by using the relation ... 

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 crystallization rate was further investigated by the measurement of ti/2 in DSC experiments. The ti/2 represents the overall crystallization rate and is determined by the rates of nucleation and linear growth. A plot of the half-life times against the crystallization temperature is shown in Figure 11.1. 

Spherulites are produced because the overall crystallization rate is the same for all directions in space. However, the growth rates of crystals in the spherulites may be directionally dependent. Conversely, if the overall... 

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