Theoretical treatment of network formation by a multistage process

Dušek, Karel; Scholtens, Boudewijn J. R.; Tiemersma-Thoone, G. P. J. M.

doi:10.1007/bf00285356

Cited by 26 publications

(11 citation statements)

References 10 publications

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“…The branching theory gives a positive answer [14] and predicts the sol fraction and concentration of elastically active network chains. The dependence of the sol fraction on the transesterification conversion has been predicted to have a peculiar character: it never reaches zero and remains relatively high.…”

Section: ) May Serve As Examplesmentioning

confidence: 98%

Polymer Networks: A Challenge to Theorist and Technologist

Dušek

1991

Journal of Macromolecular Science: Part A - Chemistry

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Polymer networks challenge a theorist to develop methods of their generation, describe their topology, and understand critical phenomena (gelation) during their formation. They challenge a technologist because of their unique properties based on their largescale three-dimensional connectivity and their easy transformation of relatively low-viscosity liquids into crosslinked solids. Polymer networks challenge both, in that they are structurally very complex for the technologist who needs the theorist for understanding and controlling their structure and properties. Also, the theorist is challenged by problems appearing in technology which may initiate a basic reconsideration of the theories' assumptions. In this contribution, methods for modeling polymer network structures and their recent applications to relatively complex systems of technological importance are briefly discussed.

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Section: ) May Serve As Examplesmentioning

confidence: 98%

Polymer Networks: A Challenge to Theorist and Technologist

Dušek

1991

Journal of Macromolecular Science: Part A - Chemistry

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“…The cascade theory has been developed to deal with problems concerning polymer network formation [37][38][39][40][41][42][43][44]. Consider the polymer network made by polycondensation of f -functional monomers, represented as a rooted tree in Figure 9.13.…”

Section: Probability Generating Functions and Cascade Theorymentioning

confidence: 99%

“…We will discuss pgfs as used in a transformation procedure to solve population balance equations [3], and as employed in the cascade theory of polymer networks [37][38][39][40][41][42][43][44].…”

Section: Introductionmentioning

confidence: 99%

Mathematical Methods

Iedema¹,

Kolhapure²

2005

Handbook of Polymer Reaction Engineering

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IntroductionIn this chapter some mathematical methods to solve kinetic modeling problems are explained. A very sound basis for this was already laid many years ago by Flory [1]. Here, we want to present modern mathematical tools that have recently been developed through the use of computers. The focus is on the link between kinetic rate data and reactor type on one hand, and distributive properties -in one or more dimensions -on the other. These distributive or microstructural properties are concerned not only with countable quantities, such as the number of monomer units in a polymer molecule, but also with structure in the case of branched polymer molecules. With structure, we discuss the connectivity of branch points and the lengths of the segments between them. In the greater part of the text, reactors are treated in a simplified manner. We consider continuous and batch reactors, but all of them ideally mixed. The effect of incomplete mixing (segregation, macroand micromixing) is addressed in classical textbooks such as Biesenberger [2] and Dotson et al. [3]. Some recent attempts to include the impact of micromixing on distributions are available in the literature [57][58][59], but this field is still in its infancy. Nevertheless, to still provide a sound basis for issues of mixing, we devote one section to the use of computational fluid dynamics in polymer reaction engineering problems.The microstructural properties that we address are chain length, number of monomer units of one kind (copolymer), number of branch points, number of unsaturated bonds, and number of reactive monomer units (end groups in polycondensation). Problems may require solution of one or more of these properties simultaneously. Here, we will denote this as the dimensionality of the problem at hand. For instance, growth in addition polymerization can be described by a simple 1D (chain length) reaction equation and population balance.In contrast, growth in polycondensation of a trifunctional monomer A with a bifunctional monomer B requires a 3D description. The 3D distribution R n; i; k , where subscripts denote chain length, number of A end groups, number of B end groups, respectively, obeys:Note that this describes a reaction between single end groups of two different molecules; end group combinations within one (longer) molecule require a similar approach. These examples illustrate the importance of the dimensionality of the problem at hand. In general, low dimensionality can be dealt with using analytical or differential methods, while higher dimensionality soon requires a Monte Carlo sampling approach. Note that all of these methods, except MC, start with a population balance. Here we will mainly discuss ways to solve such balances of lower or higher dimensionality. This chapter will start with the description of lower-dimensional problems and show to what extent these can be successful. This often involves the reduction of the problem to lower dimensionality, inevitably leading to averaging over one or more dimensions. The most well...

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“…Recently, procedures for treating multistage processes have been developed (44,45). In a multistage process, branched prepolymers are formed first, they can be further modified by reaction with some other monomers and eventually crosslinked, this is a theoretically rather difficult problem especially if the substitution effect on reactivity of groups is operative even in a ring-free system.…”

Section: Application Of Branching Theories To More Complex Systemsmentioning

confidence: 99%

Cross-Linking and Structure of Polymer Networks

Dušek

MacKnight

1988

ACS Symposium Series

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The article is an overview on network formation theories, their experimental verification and application to more complex systems of industrial importance. Network formation controlled by specific or overall diffusion is also discussed.The development of an understanding of formation-structure-properties relations for polyfunctional systems undergoing crosslinking and branching is much more difficult than for bifunctional systems leading to linear polymers. The degree-of-polymerization distribution becoming very broad when the gel point is approached, the presence of an "infinite" molecule (the gel) beyond the gel point and the limited number of experimental methods for examining the gel structure are among the main reasons. Therefore, realistic network formation theories verified by experiments are of major importance in understanding and predicting these relations. Experiments on model low-functionality systems as well as polyfunctional systems serve as input information for the formulation of a correct and concrete theoretical approach, they also give values of initial parameters controlling the crosslinking reaction. A combination of organic and physical chemistry with branching theory appears to be necessary in order to understand the network formation and structure of real systems.A knowledge of the factors controlling the crosslinking reaction is one of the necessary prerequisites for selection and application of the branching theory. The crosslinking reaction can be controlled by:-Chemical equilibrium or chemical kinetics: The reactivity of functional groups is independent of the size and topology of the

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Theoretical treatment of network formation by a multistage process

Cited by 26 publications

References 10 publications

Polymer Networks: A Challenge to Theorist and Technologist

Polymer Networks: A Challenge to Theorist and Technologist

Mathematical Methods

Cross-Linking and Structure of Polymer Networks

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