The cascade theory

Our starting point is the classical view of Kolmogorov (see Tennekes and Lumley, 1972) for homogeneous, isotropic, three-dimensional turbulence. 

At each spatial scale (r) there is a characteristic fluctuating velocity l/(r), related to (r) by the condition 

We must compare the characteristic frequencies l/(r)/r to the Zimm relaxation rate of one coil (Stockmayer, 1976 de Gennes, 

At large scales r, the hydrodynamic frequency U/r is smaller than 1/Tj. But, if we go down in scale, we may reach a value r=r where the two frequencies become equal. Thus 

Note that r (and U ) depend on molecular weight, but not on concentration. 

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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. 

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. 

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. 

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]... 

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 ... 

More advanced branching models are the so-called recursive theory by Miller and Macosko (Macosko and Miller 1976) and the cascade theory by Gordon (Gordon 1962). Both are able to include nonidealities such as cyclization and long-range substitution effects. All branching theories are mean-field theories and... 

The field of generating functions was actually opened to polymer science with the work of Burchard [1], DuSek [2], and Gordon [4]. Gordon introduced the cascade theory [5] to derive finally the degree of polymerization in various systems. In contrast to radical polymerization the concept of directionality [6] is not of importance for equilibrium condensation reactions. Extensive work on the subject has been done by Tang [26, 27]. 

Fig. 184. Young s modulus as a function of relative conversion a/a (finrni optical rotation) for aqueous gelatin solutions (in tripio) i eing temperature 26.9 °C. Solid lines theoretical predictions based on the cascade theory of network formation for various crosslink functionalities and maximum number of potoitial junction zones curve C, which gives the best fit, f = 6 and maximum number potential juiK tion zones = 8 [17,489j. Broken lines (a, b, c) results of calculaticms with the network model [39-44] for crosslink functionalities f = 4,6 and 8, plotted vs y/f i...

Recently, Clark [570] calculated the molecular weight between crosslinks for the results shown in Fig. 223 by making use of the cascade theory. He concluded that molecular weights between crosslinks are 2.35-3.55 kg/mol, which is much... 

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