Crystallization kinetic models

Density data and crystallization kinetics model parameters for various iPPs... 

Figs. 17.a and b and 18.a and b show a comparison of the density dependence upon cooling rate for the iPP grades studied, whereas Table 2 reports the crystallization kinetics model parameters calculated by a best fitting procedure not only on the basis of the final monoclinic and mesomorphic content of the quenched samples, taken from the deconvolution of the WAXD patterns, but also accoimting for results which provide the time and the temperature at the maxima of the crystallization rate (isothermal tests and DSC measurements) respectively. For this purpose a mulfiobjecfive optimization code was adopted. 

Sorrentino, A., Pantani, R., Titomanlio, G. Two-phase crystallization kinetic model of syndiotactic polystyrene. Submitted. 

Qualitative examples abound. Perfect crystals of sodium carbonate, sulfate, or phosphate may be kept for years without efflorescing, although if scratched, they begin to do so immediately. Too strongly heated or burned lime or plaster of Paris takes up the first traces of water only with difficulty. Reactions of this type tend to be autocat-alytic. The initial rate is slow, due to the absence of the necessary linear interface, but the rate accelerates as more and more product is formed. See Refs. 147-153 for other examples. Ruckenstein [154] has discussed a kinetic model based on nucleation theory. There is certainly evidence that patches of product may be present, as in the oxidation of Mo(lOO) surfaces [155], and that surface defects are important [156]. There may be catalysis thus reaction VII-27 is catalyzed by water vapor [157]. A topotactic reaction is one where the product or products retain the external crystalline shape of the reactant crystal [158]. More often, however, there is a complicated morphology with pitting, cracking, and pore formation, as with calcium carbonate [159]. 

Rielly and Marquis (2001) present a review of crystallizer fluid mechanics and draw attention to the inconsistency between the dependence of crystallization kinetic rates on local mean and turbulent velocity fields and the averaging assumptions of conventional well-mixed crystallizer models. 

Hartel, R.W. and Randolph, A.D., 1986. Mechanisms and kinetic modelling of calcium oxalate crystals in urine-like liquor Part II kinetic modelling. American Institution of Chemical Engineers Journal, 32, 1186-1195. 

Figure 8.10. Methanol synthesis rate over a Cu(lOO) single crystal in the zero conversion limit as a function of the H2 mole fraction. The full line corresponds to the kinetic model in Eqs. (23-35) with reaction (33),...

A full analysis of the rate expression reveals that all data on the Cu(lOO) single crystal are modeled very well, as shown in Fig. 8.10. Even more important is that the model also describes data obtained on a real catalyst measured under considerably different conditions reasonably well, indicating that the micro-kinetic model captures the most important features of the methanol synthesis (Fig. 8.11). 

The contribution of different crystal planes to the overall surface area of the particle can thus be calculated and is shown in Fig. 8.12(b). The results have been included in a dynamical micro-kinetic model of the methanol synthesis, yielding a better description of kinetic measurements on working catalysts [C.V. Ovesen, B.S. Clausen, J. Schiotz, P. Stoltze, H. Topsoe and J.K. Norskov, J. Catal. 168 (1997) 133]. 

In this chapter, we have summarized (recent) progress in the mechanistic understanding of the oxidation of carbon monoxide, formic acid, methanol, and ethanol on transition metal (primarily Pt) electrodes. We have emphasized the surface science approach employing well-defined electrode surfaces, i.e., single crystals, in combination with surface-sensitive techniques (FTIR and online OEMS), kinetic modeling and first-principles DFT calculations. 

Verhoyen, 0., Dupret, F. and Legras, R., Isothermal and non-isothermal crystallization kinetics of polyethylene terephthalate mathematical modeling and experimental measurement, Polym. Eng. Sci., 38, 1592-1610 (1998). 

Clearly this is a very interesting problem and of great practical relevance, very well suited to Monte Carlo simulation. At the same time, simulations of such problems have just only begun. In the context of crystal growth kinetics, models where evaporation-condensation processes compete with surface diffusion processes have occasionally been considered before . But many related processes can be envisaged which have not yet been studied at all. 

A kinetic model for single-phase polymerizations— that is, reactions where because of the similarity of structure the polymer grows as a solid-state solution in the monomer crystal without phase separation—has been proposed by Baughman [294] to explain the experimental behavior observed in the temperature- or light-induced polymerization of substimted diacetylenes R—C=C—C=C—R. The basic feature of the model is that the rate constant for nucleation is assumed to depend on the fraction of converted monomer x(f) and is not constant like it is assumed in the Avrami model discussed above. The rate of the solid-state polymerization is given by... 

The purity of amino acids recovered by batch crystallization has been examined using L-isoleucine as a model system. The concentration of impurities in the feed solution were shown to affect crystal purity, as were variables that affect crystallization kinetics (e.g., agitation, precipitant addition rate, and cooling rate). 

Before addressing these fundamental questions, we present a brief review on phenomenology, classical thermodynamics, and kinetic models of polymer crystallization. Advances made recently (as of 2003) using molecular modeling are reviewed next. 

Sanchez and DiMarzio identified [41] a crossover regime 11 [Fig. 1.15(c)], where g is more rapid than in 1 and less than in 111. On the basis of the LH model, crystallization kinetics in these three regimes are obtained as follows. 

The link between the microscopic description of the reaction dynamics and the macroscopic kinetics that can be measured in a catalytic reactor is a micro-kinetic model. Such a model will start from binding energies and reaction rate constants deduced from surface science experiments on well defined single crystal surfaces and relate this to the macroscopic kinetics of the reaction. 

A kinetic model consists of a description of the elementary steps at the atomic level is presented. Input data for elementary steps are taken from available single crystal studies. The model is successfully tested against kinetic data for a working catalyst. 

The purpose of our studies is not to present a kinetic model, which will reproduce one or a few of the aspects of water-gas shift reaction very accurately. The point we make is that a physically reasonable treatment of the proposed reaction mechanism with kinetic and thermodynamic data measured for Cu single crystals leads to a reasonably accurate description of most aspects of the observed kinetics. 

In conclusion, we observe that the crossing of crystal phase boundaries by matter means the transfer of SE s from the sublattices of one phase (a) into the sublattices of another phase (/ ). Since this process disturbs the equilibrium distribution of the SE s, at least near the interface, it therefore triggers local SE relaxation processes. In more elaborated kinetic models of non-equilibrium interfaces, these relaxations have to be analyzed in order to obtain the pertinent kinetic equations and transfer rates. This will be done in Chapter 10. 

In many non-equilibrium situations, this local equilibrium assumption holds for the crystal bulk. However, its verification at the phase boundaries and interfaces (internal and external surfaces) is often difficult. This urges us to pay particular attention to the appropriate kinetic modeling of interfaces, an endeavour which is still in its infancy. 

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