Statistical Mechanics of Fluids and Chain Systems

As in the case of the statistical mechanics of a fluid, these Boltzmann factors contain more information than is necessary in order to characterize the experimental properties of polymer systems. We therefore focus attention upon reduced distribution functions in order to make contact with the macroscopic observable properties of polymers. In the usual many-body problems encountered in statistical mechanics, the reduced distribution functions are the solutions to coupled sets of integro-differen-tial equations. - On the other hand, because a polymer is composed of several atoms (or groups of atoms) that are sequentially joined together by chemical bonds, these reduced distributions for polymers will obey difference equations. Therefore, by employing the limit in which a polymer molecule is characterized by a continuous chain, these reduced probability distributions can be made to obey differential, instead of difference, equations. This limit of a continuous chain then enables the use of mathematical analogies between polymers and other many-body systems. The use of this limit naturally leads to the use of the technique of functional integration. 

When we incorporate termination reactions into the analysis of monomer sequence formation in copolymerization kinetics, we note that in general the development of the copolymer sequence can be prematurely stopped. If one monomer sequence is being formed during propagation and the chain is terminated by disproportionation, then the result is more of a homopolymer than a copolymer. If the chain is terminated by recombination of a similar molecule, then the same type of homopolymer is formed. In fact, termination reactions usually prevent the formation of block copolymers when both monomers are present in the reactor fluid. With the radical trapping mechanism of the FRRPP process that will be discussed in the next chapter, formation of certain block copolymers becomes feasible in statistical-based radical copolymerizations. This is an apparent contradiction in terms, but the FRRPP process has been shown to break new ground in polymerization systems. 

Experimental data including the acidic species in the vapor phase within the above concentration range are scarce. Only very few publications of VLE data in that range are available [168, 173]. In contrast, numerous vapor pressure curves are accessible in literature. Chemical equilibrium data for the polycondensation and dissociation reaction in that range (>100 wt%) are so far not published [148]. However, a starting point to describe the vapor-Uquid equilibrium at those high concentratirMis is given by an EOS which is based on the fundamentals of the perturbation theory of Barker [212, 213]. Built on this theory, Sadowski et al. [214] have developed the PC-SAFT (Perturbed Chain Statistical Associated Fluid Theory) equation of state. The PC-SAFT EOS and its derivatives offer the ability to be fuUy predictive in combination with quantum mechanically based estimated parameters [215] and can therefore be used for systems without or with very little experimental data. Nevertheless, a model validation should be undertaken. Cameretti et al. [216] adopted the PC-SAFT EOS for electrolyte systems (ePC-SAFT), but the quality for weak electrolytes as phosphoric... 

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