Nucleation rate isothermal crystallization

Literature has revealed limited kinetic data on secondary nucleation of alumina trihydrate in the precipitator of the Bayer Process for alumina production. A batch agitated, isothermal, three litre crystallizer was used in the study. A Coulter-Counter was utilized as the particle sizing equipment. The effects of seed density, supersaturation and temperature on secondary nucleation were investigated. Maximum nucleation rates were found to occur at about 70 C and for any crystallization temperature, the nucleation rate passed through a maximum. The correlated equation for the effective secondary nucleation rate of alumina trihydrate is... 

The above is for isothermal crystal growth. In nature, crystallization occurs in a continuously cooled magma. The cooling rate plays a main role in controlling crystallization, and the nucleation and interface-reaction rates shown in Figure 1-14 are instructive in understanding crystallization under various cooling rates. 

The isothermal crystallization of PEO in a PEO-PMMA diblock was monitored by observation of the increase in radius of spherulites or the enthalpy of fusion as a function of time by Richardson etal. (1995). Comparative experiments were also made on blends of the two homopolymers. The block copolymer was observed to have a lower melting point and lower spherulitic growth rate compared to the blend with the same composition. The growth rates extracted from optical microscopy were interpreted in terms of the kinetic nucleation theory of Hoffman and co-workers (Hoffman and Miller 1989 Lauritzen and Hoffman 1960) (Section 5.3.3). The fold surface free energy obtained using this model (ere 2.5-3 kJ mol"1) was close to that obtained for PEO/PPO copolymers by Booth and co-workers (Ashman and Booth 1975 Ashman et al. 1975) using the Flory-Vrij theory. 

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-... 

In conditions of high viscosity of the metastable phase the time of establishment of a stationary concentration of nuclei becomes longer. The process of non-stationary nucleation may be characterized by the stationary nucleation rate J and the lag time. Non-stationary nucleation shows up in the crystallization of amorphous layers of water. The crystallization of such layers proceeds during continuous heating or an isothermal allowance after a stepwise rise in the temperature. 

Figure 9. Temperature ranges of states of stable, superheated and supercooled water at atmospheric pressure. Stationary homogeneous nucleation rate during crystallization (1) and boiling-up (2). Inverse isothermal compressibility for stable and metastable states of water (3) in the absence of the spinodal in a supercooled liquid (3 ) and in the case of its presence according to (3 % T - the temperature of the spinodal of a superheated liquid.

Indicated in Fig. 9 are temperature ranges of supercooled, stable and superheated water at atmospherie pressure. Ibidem one can see curves representing the temperature dependenee of the logarithm of the homogeneous nucleation rate for crystallization (curve 1) and boiling-up (curve 2). The maximum rate of formation of vapor nuclei is attained at the approach of the spinodal determined by condition (3). Fig. 9 also shows how the inverse isothermal eompressibility =-v(5p/5v) changes with temperature (curve 3). An arrow shows the temperature of the spinodal of superheated water. 

The procedure of Zhdanov and Samulevich enables the calculation of isothermal nucleation rate profiles from determinations of growth rate and crystal size distribution [16,82]. Originally implemented in analyses of zeolite Na-A [83] and Na-X [82] crystallisation, the method has subsequently been applied to other zeolite systems, including silicalite [84,85]. If it is supposed that all the crystals in a batch have the same (known) growth rate behaviour, the total growth time of each crystal can be calculated. Assuming also that the nuclcation point for each crystal can be obtained by linear extrapolation to zero time, the nucleation profile for the whole batch can be determined from their final sizes. 

Plaris et al. [1993] investigated also the same blend system and reported that blending had a pronounced effect on the lamellar morphology. Furthermore, the isothermal crystallization experiments indicated that the spherulite growth rate, G, and the nucleation density of the PP phase were enhanced. The authors suggested that these observations could be related to the formation of additional nucleation sites, which arise from the polymer-polymer interfaces created by the blending. 

More extensive investigations have been performed on HDPE/PP blends by Martuscelli et al. [1980] and Bartczak and Galeski [1986]. From the isothermal crystallization experiments, it was found that the rate of crystallization of the HDPE matrix was markedly reduced upon addition of small amounts of PP (10 wt%). The authors attributed this phenomenon to the increased melt-viscosity of the sample caused by the presence of solidified PP domains. Moreover, Plesek and Malac [1986] have calculated from the surface tensions of the homopolymers at T, that PP crystallization will not cause the nucleation of the HDPE phase, while in the reverse case HDPE crystals will induce the nucleation of PP. 

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. 

Parameter Estimation. The kinetic parameters of the model given above that must be estimated for model identification include kg, g, Eg, kb, b, Eb,j. Parameter estimation for this type of model is quite difficult because the parameters appear nonlinearly, the nucleation rate parameters enter only in the boundary condition, and availability of accurate data is limited. Certainly a model that describes the behavior of a nonisothermally operated crystallizer is needed if the temperature is to be manipulated, but there have been only a few studies of the effect of temperature on crystallization processes (Kelt and Larson 1977 Randolph and Cise 1972 Rousseau and Woo 1980). For isothermal crystallization, the terms involving Eg and Eh are absorbed into kg and kb, and only kg, g, kb, b, and i need to be estimated. 

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... 

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