Polymers reaction-diffusion system models

Poehlein and Degraff [336] extended the derivation of Gershberg and Long-field [330] to the calculation of both molecular weight and particle size distribution in the continuous emulsion polymerization of St in a CSTR. On the other hand, Nomura et al. [163] carried out the continuous emulsion polymerization of St in a cascade of two CSTRs and developed a novel model for the system by incorporating their batch model [ 14], which introduced the concept that the radical capture efficiency of a micelle relative to a polymer particle was much lower than that predicted by the diffusion entry model (pocd -°). The assumptions employed were almost the same as those of Smith and Ewart (Sect. 3.3), except that the model did not assume a constant value of p. The elementary reactions and their rate expressions employed in the first stage are as follows ... 

Peskin et al [1993] have proposed the Brownian ratchet theory to describe the active force production. The main component of that theory was the interaction between a rigid protein and a diffusing object in front of it. If the object undergoes a Brownian motion, and the fiber undergoes polymerization, there are rates at which the polymer can push the object and overcome the external resistance. The problem was formulated in terms of a system of reaction-diffusion equations for the probabilities of the polymer to have certain number of monomers. Two limiting cases, fast diffusion and fast polymerization, were treated analytically that resulted in explicit force/velocity relationships. This theory was subsequently extended to elastic objects and to the transient attachment of the filament to the object. The correspondence of these models to recent experimental data is discussed in the article by Mogilner and Oster [2003]. 

A detailed description of AA, BB, CC step-growth copolymerization with phase separation is an involved task. Generally, the system we are attempting to model is a polymerization which proceeds homogeneously until some critical point when phase separation occurs into what we will call hard and soft domains. Each chemical species present is assumed to distribute itself between the two phases at the instant of phase separation as dictated by equilibrium thermodynamics. The polymerization proceeds now in the separate domains, perhaps at differen-rates. The monomers continue to distribute themselves between the phases, according to thermodynamic dictates, insofar as the time scales of diffusion and reaction will allow. Newly-formed polymer goes to one or the other phase, also dictated by the thermodynamic preference of its built-in chain micro — architecture. 

The beginning of modeling of polymer-electrolyte fuel cells can actually be traced back to phosphoric-acid fuel cells. These systems are very similar in terms of their porous-electrode nature, with only the electrolyte being different, namely, a liquid. Giner and Hunter and Cutlip and co-workers proposed the first such models. These models account for diffusion and reaction in the gas-diffusion electrodes. These processes were also examined later with porous-electrode theory. While the phosphoric-acid fuel-cell models became more refined, polymer-electrolyte-membrane fuel cells began getting much more attention, especially experimentally. 

The effect of the polymer backbone on the intrinsic chemical reactivity of metal complexes has been studied in aqueous solution and in Nafion (perfluorocarbon sulfonic acid) film 44). Using a model catalyst-substrate system, the independent kinetic effects of reaction site homogeneity, substrate diffusion into the polymer film, and changes on activation parameters have been addressed. The ligand substitution reaction (6), was chosen for this purpose (Py = pyridine and its derivatives). 

Summing up, the two-phase model is physically consistent and may be applied for designing industrial systems, as demonstrated in recent studies [10, 11], Modeling the diffusion-controlled reactions in the polymer-rich phase becomes the most critical issue. The use of free-volume theory proposed by Xie et al. [6] has found a large consensus. We recall that the free volume designates the fraction of the free space between the molecules available for diffusion. Expressions of the rate constants for the initiation efficiency, dissociation and propagation are presented in Table 13.3, together with the equations of the free-volume model. 

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