Polymers, kinetic modeling

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

The thermally-initiated styrene system is considerably simpler than most industrial applications. Though these experiments provided useful guidelines, it was difficult to develop broadly applicable design criteria without carefully evaluating a broad range of monomer, polymer and initiator systems. Hence we extended our kinetic model to some other monomer systems such as styrene and methyl methacrylate using common initiators such as benzoyl peroxide (BPO) and... 

In conclusion, we have reviewed how our kinetic model did simulate the experiments for the thermally-initiated styrene polymerization. The results of our kinetic model compared closely with some published isothermal experiments on thermally-initiated styrene and on styrene and MMA using initiators. These experiments and other modeling efforts have provided us with useful guidelines in analyzing more complex systems. With such modeling efforts, we can assess the hazards of a polymer reaction system at various tempera-atures and initiator concentrations by knowing certain physical, chemical and kinetic parameters. 

TERMONU SMITH Kinetic Model for TensUe D ornuUioti of Polymers 117... 

Minimizing the cycle time in filament wound composites can be critical to the economic success of the process. The process parameters that influence the cycle time are winding speed, molding temperature and polymer formulation. To optimize the process, a finite element analysis (FEA) was used to characterize the effect of each process parameter on the cycle time. The FEA simultaneously solved equations of mass and energy which were coupled through the temperature and conversion dependent reaction rate. The rate expression accounting for polymer cure rate was derived from a mechanistic kinetic model. 

Advanced computational models are also developed to understand the formation of polymer microstructure and polymer morphology. Nonuniform compositional distribution in olefin copolymers can affect the chain solubility of highly crystalline polymers. When such compositional nonuniformity is present, hydrodynamic volume distribution measured by size exclusion chromatography does not match the exact copolymer molecular weight distribution. Therefore, it is necessary to calculate the hydrodynamic volume distribution from a copolymer kinetic model and to relate it to the copolymer molecular weight distribution. The finite molecular weight moment techniques that were developed for free radical homo- and co-polymerization processes can be used for such calculations [1,14,15]. 

Fig. 13 Amount of polymer as measured by the IR absorption at 2852 cm with respect to time. The kinetics observed on a rough catalyst is represented by crosses. The line through these points is a fit to the kinetic model given in the text. Kinetics observed for smooth catalysts is given hy full circles. The line is a guide for the eye...

Gel Permeation Chromatography (CPC) is often the source of molecular wei t averages used in polymerization kinetic modelling Q.,2). Kinetic models also r uire measurement of molecular weight distribution, conversion to polymer, composition of monomers in a copolymerization rea tion mixture, copolymer composition distribution, and sequence length distribution. The GPC chromatogram often reflects these properties (3,. ... 

A simple algorithm [17] makes it possible to find the probability of any fragment of macromolecules of Gordonian polymers. Comparison of these probabilities with the data obtained by NMR spectroscopy provides the possibility to evaluate the adequacy of a chosen kinetic model of a synthesis process of a particular polymer specimen. The above-mentioned probabilities are also involved in the expressions for the glass transition temperature and some structure-additive properties of branched polymers [18,19]. 

The Flory principle is one of two assumptions underlying an ideal kinetic model of any process of the synthesis or chemical modification of polymers. The second assumption is associated with ignoring any reactions between reactive centers belonging to one and the same molecule. Clearly, in the absence of such intramolecular reactions, molecular graphs of all the components of a reaction system will contain no cycles. The last affirmation concerns sol molecules only. As for the gel the cyclization reaction between reactive centers of a polymer network is quite admissible in the framework of an ideal model. 

When there is a need to calculate only the statistical moments of the distribution of molecules for size and composition, rather than to find the very distribution, the task becomes essentially easier. The fact is that for the processes of polymer synthesis which may be described by the ideal kinetic model the set of equations for the statistical moments is always closed. 

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

In subsequent sections some results will be reported relevant to the theoretical consideration of several principle processes of the synthesis of polymers described by various kinetic models. This information may be useful in gaining a better understanding of the potentialities of the statistical chemistry of polymers. 

A kinetic model based on the Flory principle is referred to as the ideal model. Up to now this model by virtue of its simplicity, has been widely used to treat experimental data and to carry out engineering calculations when designing advanced polymer materials. However, strong experimental evidence for the violation of the Flory principle is currently available from the study of a number of processes of the synthesis and chemical modification of polymers. Possible reasons for such a violation may be connected with either chemical or physical factors. The first has been scrutinized both theoretically and experimentally, but this is not the case for the second among which are thermodynamic and diffusion factors. In this review we by no means pretend to cover all theoretical works in which these factors have been taken into account at the stage of formulating physicochemical models of the process... 

This assumption is implicitly present not only in the traditional theory of the free-radical copolymerization [41,43,44], but in its subsequent extensions based on more complicated models than the ideal one. The best known are two types of such models. To the first of them the models belong wherein the reactivity of the active center of a macroradical is controlled not only by the type of its ultimate unit but also by the types of penultimate [45] and even penpenultimate [46] monomeric units. The kinetic models of the second type describe systems in which the formation of complexes occurs between the components of a reaction system that results in the alteration of their reactivity [47-50]. Essentially, all the refinements of the theory of radical copolymerization connected with the models mentioned above are used to reduce exclusively to a more sophisticated account of the kinetics and mechanism of a macroradical propagation, leaving out of consideration accompanying physical factors. The most important among them is the phenomenon of preferential sorption of monomers to the active center of a growing polymer chain. A quantitative theory taking into consideration this physical factor was advanced in paper [51]. 

The two major theories of flocculation, the bridging model (1) and the electrostatic patch model (2, 3 ), provide the conceptual framework for the understanding of polymer-aided flocculation, but they do not directly address the kinetics of the process. Smellie and La Mer (4) incorporated the bridging concept into a kinetic model of flocculation. They proposed that the collision efficiency in the flocculation process should be a function of the fractional surface coverage, 0. Using a modified Smoluchowski equation, they wrote for the initial flocculation rate... 

As shown in Sect. 2, the fracture envelope of polymer fibres can be explained not only by assuming a critical shear stress as a failure criterion, but also by a critical shear strain. In this section, a simple model for the creep failure is presented that is based on the logarithmic creep curve and on a critical shear strain as the failure criterion. In order to investigate the temperature dependence of the strength, a kinetic model for the formation and rupture of secondary bonds during the extension of the fibre is proposed. This so-called Eyring reduced time (ERT) model yields a relationship between the strength and the load rate as well as an improved lifetime equation. 

The opposite effects of the initiator on the structure of 1,4-DVB polymers in a range of low (1-16 mol %) and a high (17-200 mol %) concentration of 1,4-DVB were explained by a kinetic model of anionic polymerization of 1,4-DVB [239]. Calculations indicated that, at low concentrations of the initiator, the poly-... 

For step-change polymerization at steady-state in a CSTR represented by the kinetics model in Section 7.3.2, derive equations giving the weight fraction of r-mer, wr, on a monomer-free basis, as a function of r, the number of monomer units in polymer Pr at the reactor outlet. This is a measure of the distribution of polymers in the product leaving the reactor. 

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