Cascade theory

Polyoxypropylene triol-based urethane networks are simulated. Simulations show that cyclic molecules are present in substantial amounts in the sol fraction when the vhnmotr-. tin rss1.0. Simulations also show that the fractions of loops are much higher than those obtained from modified cascade theory. 01 lc r... 

Finally, the applicability of the cascade theory to rather complicated systems with unequal functional groups, substitution effect, vulcanization of chains and long rang correlation as a result of directed chain reactions is shown. The limitation of the theory to essentially tree-like molecules and their unperturbed dimensions is outlined and the consequence of this error for the prediction of reed systems is discussed. 

This article deals with one of the above mentioned subjects already treated in the 1940 s branched polymers. We present a survey of a number of scattering functions for special branched polymer structures. Hie basis of these model calculations is still the Flory-Stockmayer (FS) theory1,14,15) but now endowed with the more powerful technique of cascade theory which greatly simplifies the calculations. 

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. 

For example, in recent years Macosko and Miller (MM)37-40 have developed an attractively simple method which at first sight appears to be basically new. However, a closer inspection reveals the MM approach as being a degenerate case of the more general cascade theory. The simplicity is unfortunately gained at the expense of generality, and up-to-date conformation properties are not derivable by the MM-technique. 

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. 

To complete the theoretical framework, we treat now characteristic examples which also demonstrate the full power of the cascade theory. This case is once again the polyconden-sation of trifunctional monomers with unlike functional groups1055. Another example is the random co-condensation of an f-functional monomer RAf and a bifunctional RB2 monomer, which is treated in the next chapter. All other cases can in principle be reduced to these general examples. 

Moreover, the overcrowding effect can be avoided in the cascade theory by introducing a second shell substitution effect. This was done by Gordon and Parker177. In the... 

In Chap. C we have discussed in some detail the application of the cascade theory to polycondensates in their unperturbed state. In Chap. D some experimental results were already given for cross-linked or vulcanisated linear chains. In this chapter we shall now outline in brief how cross-linking chain reactions or the vulcanization of preformed chains of an arbitrary length distribution can be treated by cascade theory. Second, we shall discuss how heterogeneities in branching or a rigidity of a certain domain can be taken into account. 

The principles of the calculation by means of the cascade theory is sketched in Fig. 58 and compared with the random polycondensation. Instead of selecting a single monomeric unit as root of a tree, a whole primary chain is placed on the zero-th generation, and the same is done for all the other primary chains from the cross-linked polymer. 

The present model may impose too strong an obstruction on a real system, and it will be of great interest to know whether the idea of a higher shell substitution effect can be extended and adjusted to the excluded volume problem. Certainly, nobody can characterize the cascade theory with substitution effects as a mean field theory. 

In 1963 the remarkable phenomenon of gas breakdown by laser radiation was discovered in experiments, which laid the ground for broad new directions in plasma physics and the physics of the interaction of laser radiation with a substance. Soon Ya.B. and Yu. P. Raizer developed the cascade theory of laser breakdown [23]. Hardly a single article of the great number devoted to optical breakdown manages without a reference to this work. 

Fig. 1. The weight average degree of polymerization, PW) in a linear polymerization with monomer reacting with a substitution effect. Solid line - calculated according to Eq. (33), broken line - calculated by using cascade theory for the specified relative rate constant k...

The reader familiar with the cascade theory will notice that the root 0 < < 1 is related to the extinction probability, v, i.e., the probability of a unit chosen at random to belong to a sol molecule [ 14,55]. This probability is [53]... 

Two theoretical approaches are mentioned in Section 4.1 (Refs. but the first one is based on incorrect statistical treatment. A comparison with experiments is not available, so that the seriousness of deviations cannot be estimated. Also, a cascade theory has been developed to cover polyetherification released by the reaction of diepoxides with dicarboxylic acids but an analysis of experimental data is also lacking. An extensive pregel study of polymerization of diepoxides released by diphenols is analyzed by W. Burchard in another chapter of this volume... 

In the literature, two additional reactions following addition esterification have been treated using the cascade theory the addition esterification followed by polyetherification with epoxide groups in excess (a reaction used for crosslinking of carboxyl terminated polydienes) and addition esterification followed by transesterification. Transesterification often interferes wherever hydroxyester groups are formed, for example, in synthesis of linear oligomeric polyesters from diepoxide and acids. As has been explained before, polyetherification is an initiated reaction and, therefore, the statistical treatment offerend in Refs. should be revised. Below we show the treatment of transesterification for a system composed of a diepoxide and a dicar boxylic acid. 

The cascade theory, developed by Gordon, is an extension of the classical Flory-Stockmayer concepts of gelation. An expression for the shear modulus, G, can be written as ... 

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