Polymerization Kinetics and Mathematical Modeling

Because of the nature of the active species, coordination polymerization has been classified as ionic polymerization, which follows the polyaddition mechanism s characteristic steps, in the growing of the polymeric chain initiation, propagation, and termination. As for the initiation step, the ionic active species is produced by the reaction between the catalyst and cocatalyst. Usually, the catalysts are actually precursor catalysts or precatalysts, which become the real cationic active species after the activation or reaction with the cocatalyst (Fig. 5.8). 

As for the termination step, different reactions have been detected, according to the structure of the polymers, such as terminal vinylic groups, which are evidence of -hydride elimination from the polymeric chains. 

Mathematical models for olefin polymerization with coordination catalysts are usually classified into microscale. 

This division is very useful during model development and implementation. Differentiated emphasis should be placed on these modeling scales depending on the model objectives. For instance, detailed polymerization kinetics is important if the precise prediction of polyolefin microstructure is required. On the other hand, apparent kinetics suffices if the model s objective is to follow the evolution of particle fragmentation or to describe reactor residence time effects on polymerization. These ideas are detailed in the following sections. 

Polyethylene microstructure is defined by its distributions of chain length (CLD) or molecular weight (MWD), chemical composition (CCD), comonomer sequence length (CSLD), and LCB. In addition, polypropylene microstructure is further characterized by its distribution of regio- and stereoregularity [53, 54]. 

---

Engineering of polymerization reactions requires a detailed knowledge of the phenomena that take place in the polymer reactor. This entails a model of the polymerization kinetics and the heat and mass transfer features of the particular polymerization and process. Polymerization reactions are usually complex and a certain degree of mathematical sophistication is required for effective modeling. Excursions into the details of particular reactions or modeling techniques are beyond the scope of this introductory text and this chapter is therefore limited to a review of the special considerations that apply in the case of various polymerizations and processes. Most of the following discussions are necessarily qualitative, for space reasons. 

The mathematical modeling of polymerization reactions can be classified into three levels microscale, mesoscale, and macroscale. In microscale modeling, polymerization kinetics and mechanisms are modeled on a molecular scale. The microscale model is represented by component population balances or rate equations and molecular weight moment equations. In mesoscale modeling, interfacial mass and heat transfer... 

The last part of this chapter deals with coordination polymerization kinetics and mechanism, mathematical models at different scales, as well as some analyses on the supported catalyst particle breakup and growth. 

Strategies for controlling the copolymer composition and MWD of latices based on linear and non-linear copolymers, such as styrene/butyl acrylate copolymers and methyl methacrylate/n-butyl acrylate copolymers, are described. These strategies involve on-line procedures based on calorimetric measurements and open-loop processes employing a mathematical model for determining the trajectories of the manipulated variables, such as monomer feed flow rates and chain transfer agent. 35 refs. (3rd lUPAC-Sponsored International Symposium on Free-Radical Polymerization Kinetics and Mechanism, II Ciocco (Lucca), Tuscany, Italy, 3rd-9th June, 2001)... 

A main distinction has been made between deterministic and stochastic modeling techniques. A further distinction has been proposed based on the scale for which the mathematical model must be derived (eg, micro-, meso-, and/or macroscale). Notably, the complexity of the model approach depends on the desired model output. Detailed microstractural information is only accessible using advanced modeling tools but these are associated with an increase high in computational cost. The advanced models allow one to directly relate macroscopic properties to the polymer synthesis procedure and, thus, to broaden the application market for polymer products, based on a fundamental understanding of the polymerization kinetics and their link with polymer processing. 

The polymerization system for which experiments were performed is represented by the mathematical model consisting of Equations 1 and 7. Their steady state solutions are utilized for kinetic evaluation of rate constants. Dynamic simulations incorporate viscosity dependency. 

Detailed studies on the lipase-catalyzed polymerization of divinyl adipate and 1,4-butanediol were performed [41-44]. Bulk polymerization increased the reaction rate and molecular weight of the polymer however, the hydrolysis of the terminal vinyl ester significantly limited the formation of the polyester with high molecular weight. A mathematical model describing the kinetics of this polymerization was proposed, which effectively predicts the composition (terminal structure) of the polyester. 

It should be emphasized that in all these cases, combined or superimposed phenomena must be dealt with, viz. for stage IV, fluiddynamics, kinetics of polymerization, and rheokinetic changes caused by chemical reactions for stage V, polymerization kinetics, crystallization kinetics and heat transfer effects a thermomechanical problem in combination with crystallization kinetics. Construction of a mathematical model requires simultaneous solution of a set of equations in order to describe these related phenomena. 

Values of both parameters, p and Mn, are generalized characteristics of any polymerization process therefore, it is reasonable to construct a mathematical model of the kinetics for the parameter P and then calculate Mn fromp with Eq. (2.3). 

Polymerization of lactams in reactive processing proceeds with the involvement of a catalyst and direct or indirect activators. A mathematical model of the process must be a kinetic equation relating the rate of conversion of a monomer to a polymer to the reagent concentrations and temperature. The general form of the model is... 

A quantitative kinetic model of the polymerization of a-pyrrolidine and cyclo(ethyl urea) showed,43 that two effects occur the existence of two stages in the initiation reaction and the absence of an induction period and self-acceleration in a-pyrrolidine polymerization. It was also apparent that to construct a satisfactory kinetic model of polymerization, it was necessary to introduce a proton exchange reaction and to take into consideration the ratio of direct and reverse reactions. As a result of these complications, a complete mathematical model appears to be rather difficult and the final relationships can be obtained only by computer methods. Therefore, in contrast to the kinetic equations for polymerization of e-caprolactam and o-dodecalactam discussed above, an expression... 

In theory, by feeding the MWD and experimental rate data into a mathematical model containing a variety of polymerization mechanisms, it should be possible to find the mechanism which explains all the experimental phenomena and to evaluate any unknown rate constants. As pointed out by Zeman (58), as long as there are more independent experimental observations than rate parameters, the solution should, in principle, be unique. This approach involves critical problems in choice of experiments and in experimental as well as computational techniques. We are not aware of its having yet been successfully employed. The converse— namely, predicting MWD from different reactor types on the basis of mathematical models and kinetic data—has been successfully demonstrated, however, as discussed above. The recent series of interesting papers by Hamielec et al. is a case in point. 

Based on these kinetic and microscopic observations, olefin polymerization by supported catalysts can be described by a shell by shell fragmentation, which progresses concentrically from the outside to the centre of the support particles, each of which can thus be considered as a discrete microreactor. A comprehensive mathematical model for this complex polymerization process, which includes rate constants for all relevant activation, propagation, transfer and termination steps, serves as the basis for an adequate control of large-scale industrial polymerizations with Si02-supported metallocene catalysts [A. Alex-iadis, C. Andes, D. Ferrari, F. Korber, K. Hauschild, M. Bochmann, G. Fink, Macromol. Mater. Eng. 2004, 289, 457]. 

---