Branching process, theory

The basis of model calculations for copolymerization, branching and cross-linking processes is the stochastic theory of Flory and Stockmayer (1-3). This classical method was generalized by Gordon and coworkers with the more powerful method of probability generating functions with cascade substitution for describing branching processes (4-6). With this method it is possible to treat much more complicated reactions and systems (7-9). 

In this study computational results are presented for a six-component, three-stage process of copolymerization and network formation, based on the stochastic theory of branching processes using probability generating functions and cascade substitutions (11,12). 

In the stochastic theory of branching processes the reactivity of the functional groups is assumed to be independent of the size of the copolymer. In addition, cyclization is postulated not to occur in the sol fraction, so that all reactions in the sol fraction are intermolecular. Bonds once formed are assumed to remain stable, so that no randomization reactions such as trans-esterification are incorporated. In our opinion this model is only approximate because of the necessary simplifying assumptions. The numbers obtained will be of limited value in an absolute sense, but very useful to show patterns, sensitivities and trends. 

POLYMQ is similar to POLYM, but with the additional tetrafunc-tional 0 monomers in stage 1. These two programs contain the formulae derived with the stochastic theory of branching processes which are also specified elsewhere (12). 

A general theory of the equilibrium polycondensation of an arbitrary mixture of monomers, described by the FSSE model, has been developed [75]. Proceeding from rigorous thermodynamic considerations a branching process has been indicated which describes the chemical structure of condensation polymers and expressions have been derived which relate the probability parameters of this stochastic process to the thermodynamic parameters of the FSSE model. 

Harris TE (1963) The theory of branching processes. Springer, Berlin Heidelberg New York... 

According to the common (mean field) theory of branching processes, one has... 

The reacting system can be represented by graphs (trees) In which the nodes represent monomer units. In the theory of branching processes this collection of graphs (Figure 2) - a molecular forest -Is transformed Into another forest - the forest of rooted trees. 

T.E. Harris, The Theory of Branching Processes (Springer, Berlin 1963) P. Jagers, Branching Processes with Biological Applications (Wiley, London 1975). 

Dobson, G. R-, andM. Gordon Theory of branching processes and statistics of rubber elasticity. J. Chem. Phys. 43, 705 (1965) Rubber Chem. Technol. 39, 1472 (1966). 

The theory of branching processes and kineticaHy controlled ring-chain equilibria. IUPAC Symposium on Macromolecular Chemistry, Prague 1965. Preprint P 513. 

Harris, T. E. Theory of branching processes. Berlin-Gdttingen-Heidelberg Springer, 1963. 

Parker, T. G. and Dalgleish, D. G. 1977B. The potential application of the theory of branching processes to the association of milk proteins. J. Dairy Res. 44, 79-84. 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

Harris, T. E., "The Theory of Branching Processes". Springer, Berlin (1963). 

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