Typical Shapes of Kinetic Curves

4 Kinetics of Ziegler-Natta Polymerization 9.4.1 Typical Shapes of Kinetic Curves 

In Ziegler-Natta polymerizations, the reaction systems are more often heterogeneous than homogeneous. While the relatively few polymerizations that are homogeneous behave in a marmer generally similar to ionic polymerizations, described in Chapter 8, the heterogeneous systems 

Type (b) behavior, in which the rate of polymerization increases in an acceleration period to reach a maximum and then decreases, may be observed for a more active but less stable catalyst, while Type (c) or Type (d) behavior in which the rate starts at a maximum value or rises very rapidly to a maximum value and then decrease rapidly with time is exhibited by many supported high-activity catalyst systems, e.g., MgQ2/ethylbenzoate/TiCl4-AlEt3 in ethylene or propylene polymerization. Type (c) behavior is also shown by many homogeneous catalyst systems, e.g., Cp2TiEtCl-AlEtCl2 in ethylene polymerization (Cp = cyclopentadiene). 

Type (e) behavior may be observed when there is an almost instantaneous breakdown of porous catalyst particles on treatment with the cocatalyst so that an acceleration or settlement period is nearly nonexistent. This behavior is shown by ether-treated highly porous catalysts in propylene polymerization (Tail and Watkins, 1989). The polymerization rate decreases very slowly with time due to good stability of the catalyst system. 

Type (f) behavior, featuring constant polymerization rate, is not often found in practice. One example of this type is the polymerization of 4-methyl-1-pentene on MgC -supported catalysts containing phthalate esters. The polymerization rate for this system is found to be almost constant with time (Kissin, 1985). 

In Ziegler-Natta polymerizations, the reaction systems are more often heterogeneous than homogeneous. While the relatively few polymerizations that are homogeneous behave in a manner generally similar to ionic polymerizations, described in Chapter 8, the heterogeneous systems usually exhibit complicated behavior, as can be seen from some typical kinetic rate-time profiles of Ziegler-Natta poymerizations. Types (a)-(f) in Fig. 9.5. The shapes of these profiles may be characteristic of particular catalysts or catalyst-monomer systems and may be considered to consist of three periods, viz., an acceleration period, a stationary period, and a decay period. Some catalyst systems, however, show all three types. 

Type (a) behavior is observed for many first generation catalyst systems, e.g., or-TiCls, VQ3, etc. with diaUcylaluminum hahdes as cocatalysts in the polymerization of propylene in hydrocarbon media. During an initial acceleration period, which is of 20-60 minutes duration for many propylene polymerizations at 1 atm pressure in the temperature range 50-70°C, the rate increases from the beginning to reach a more or less steady value. Natta and Pasquon (1959) attributed this behavior to the breakdown of the or-TiCls matrix to smaller crystallites due to the pressure of the growing polymer chains in the initial stages, leading to exposure of fresh Ti atoms and creation of new active centers with consequent increase in 

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Figure 1. Typical shape of kinetic curves of O2 absorption. Determination of oxidation induction time (tj and maximum oxidation rate (r. ...

The above expressions allow us to describe the shape of kinetic curves gi(t) in general terms. In particular, as it follows fromEqs. [6.1.39] - [6.1.41] atq< 1, the kinetic curves in the coordinates (t,gi) are upward convex and have the shape typical for pseudo-normal sorption. This takes place at slinear mode of liquid absorption. At s>Se/2-l the lower part of the kinetic curve is convex in a downward direction and the whole curve becomes S-shaped. Note that in terms of coordinates (t ,gi) at s > 0 all the kinetic curves are S-shaped. Hence, the obtained solutions enable one to describe different anomalies of sorption kinetics observed in the experiments on elastomer swelling in low-molecular liquids. 

Two typical cases are illustrated in Fig. 2.24 the first scheme (Fig. 2.24 a) is related to high-temperature polymerization, in which newly formed polymer is molten and the processes of polymerization (part Ob of the full curve) and crystallization (part bK of the full curve) are separated in time. The second case (Fig. 2.24 b) illustrates low-temperature polymerization in this situation crystallization starts before the full process of polymerization is completed. This is typical superposition of two kinetic processes, and the shape of the curve in Fig. 2.24 b does not allow the separation of these processes without additional information and assumptions.The net heat effect is the same in... 

Somewhat more critical are interpretations in which the inner layer is modelled according to some site-binding model. Figure 3.63 gives an example, which is typical in that the o°(pH) curves are well recovered but the electro-kinetic potentials are not. The former feature follows simply from the fact that the shapes of the curves are relatively simple and that at least seven parameters can be adjusted C, C. pK. pK. and (sec. 3.6e.g). In the... 

Figure 1.4 shows typical energy spectnims for the neutron population in the three principal reactor types. The ordinate in these curves gives the relative density of neutrons as a function of their kinetic energy. The intent here is to present some illustrative sketches which will give some idea of the nature of the distribution function. Other than indicating the general shape of these curves, the only feature of interest is the relative sharpness of the spectrum in the thermal reactors. 

A typical profile for melt viscosity as a function of time for a polymerization reaction was shown in Figure 2. The shape of the curve reflects the kinetic processes which are occuring during the polymerization. All of the reactions were carried out neat, at a. 200 C. The viscosity/time profiles that were observed with both anionic and coordination type catalysts were essentially the same. The data not fit a normal "living" mechanism, with no side reactions. There is... 

Various mechanisms and kinetics of coal liquefaction have been proposed and examined by many investiga tors(l,2,4-8). As a general kinetic model of coal lique-action, scheme 1 was assumed. The reaction rate of every reaction step in the scheme assumed to be first order with respect to reacting species and dissolved hydrogen. A few typical cases of a general kinetic model and the general characteristics for their cases are illustrated on Table 3. When compared these typical figures, the curves are apparently different in shape. 

The authors [1] studied kinetics of poly (amic acid) (PAA) solid phase imidization in the presence of nanofiller (Na+-montmorillonite) and in its absence. It was found out, that the kinetic curves conversion (imidization) degree Q versus reaction duration t were have typical for polymerization reactions shape with autodeceleration showing imidization rate reduction as time is passing. As it is known [2], such curves Q(t) are specific for reaction passing in heterogeneous medium and are described by the simple relationship ... 

Figure 8.9 shows that the concentration of intermediate in reversible series reactions need not pass through a maximum, while Fig. 8.10 shows that a product may pass through a maximum concentration typical of an intermediate in the irreversible series reaction however, the reactions may be of a different kind. A comparison of these figures shows that many of the curves are similar in shape, making it difficult to select a mechanism of reaction by experiment, especially if the kinetic data are somewhat scattered. Probably the best clue to distinguishing between parallel and series reactions is to examine initial rate data—data obtained for very small conversion of reactant. For series reactions the time-concentration curve for S has a zero initial slope, whereas for parallel reactions this is not so. 

The typical form of the kinetic curves is presented in Fig. 9 14,149 15U. The curves are distinctly S-shaped. The inductive period becomes shorter and the maximum reaction rate increases upon addition of alcohols. At a given alcohol concentration, the inductive... 

Useful insights into the kinetics of a phase transformation that proceeds by nucle-ation and growth can be obtained by observing the fraction transformed, , under isothermal conditions at a series of different temperatures. This is usually done by undercooling rapidly to a fixed temperature and then observing the resulting isothermal transformation. The kinetics generally follows the typical C-shaped behavior described in Exercise 18.4. If a series of such curves is obtained at different temperatures, the time required to achieve, for example, ( = 0.01, 0.50, and... 

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