Branching theory

Throughout this chapter, predictions are made concerning the effect of external variables on the network structure evolution and these predictions are compared with experimental results. The predictions are based on branching theories. The purpose of this section is to outline the basis of present network formation theories and the underlying assumptions. 

One can observe positive deviations in the region of rH < 1 (excess of isocyanate groups) which are due to side reactions (allophanate, urea and biuret groups). In the region of rH > 1 the agreement of wg values is good. In the case of i e, the predicted curves depend not only on the results of the branching theory but also... 

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 ... 

It may be of Interest to list the most important structural parameters which can be calculated by applying the branching theory. 

Newly developing experimental methods for looking into the structure of crossllnked systems may require additional information which very likely can be obtained by branching theories. 

The measurement of the equilibrium modulus offers another possibility to compare the branching theory and experiment. 

Application of Branching Theories to More Complex Systems... 

The branching theories have been successfully applied to a number of systems of industrial importance which are usually rather complex. The examples of these applications are listed below ... 

The application of the branching theories to acid curing was rather successful the discovery of the relative Importance of the transesterification following the epoxy-carboxyl addition initiated the theoretical treatment of branching. The prediction was that... 

Chain crosslinking (co)polymerlzatlon involving a polyvinyl monomer represents a process where the application of the branching theory based on a (perturbed) tree-like model falls due to strong cycllza-tion. Strong cyclization is characteristic for these systems not because of a special configuration of the monomers but because of... 

The branching theories In their present state can treat a number of complex branching reactions of Industrial importance. It is to be stressed, however, that there does not exist any universal approach to all systems. The understanding of the reaction mechanism and kinetics is a necessary prerequisite for adaptation of the proper theory to give relations for structural parameters. Further progress in the network formation theory seems highly desirable particularly in the field of cycllzatlon and diffusion control and in understanding the network structure-properties relations. 

The cascade theory is probably the oldest branching theory. It was developed by the English chaplain, the Reverend Watson16,181 and the biometrician Galton17,181 in 1873 who were evidently stimulated by Darwin s famous book on The Origin of Species . Nowadays cascade theory is widely used in evolution theory19,201, in actuarial mathematics (birth and death processes), in the physics of cosmic ray showers and in the chemistry of combustion due to branched chain reactions21-241. 

This article shows how successfully the cascade branching theory works for systems of practical interest. It is a main feature of the Flory-Stockmayer and the cascade theory that all mentioned properties of the branched system are exhaustively described by the probabilities which describe how many links of defined type have been formed on some repeating unit. These link probabilities are very directly related to the extent of reaction which can be obtained either by titration (e.g. of the phenolic OH and the epoxide groups in epoxide resins based on bisphenol A206,207)), or from kinetic quantities (e.g. the chain transfer constant and monomer conversion106,107,116)). The time dependence is fully included in these link probabilities and does not appear explicitly in the final equations for the measurable quantities. 

Ill) The Flory-Stockmayer and the equivalent cascade theory are not the only branching theories, and a few words have to be said about the others. 

Chapter D gives details on the common evaluation and interpretation of scattering experiments. Many experimental results are discussed in comparison with the behavior predicted by theory. This will show how much of this behavior can already be described by the cascade branching theory in spite of its obvious limitations. Furthermore, the great advantage of a combined measurement of both the static and dynamic LS is shown. 

In this section some details of the static and dynamic structure factors and on the first cumulant of the time correlation function are given. Hie quoted equations are needed before the cascade theory can be applied. This section may be skipped on a first reading if the reader is concerned only with the application of the branching theory. 

Test results, plotted in Figures 5, 6, and 7, show that impact strengths vary with difference in sonic speed in the way predicted by the crack/ craze branching theory. 

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