Flory configurational partition function

We also have an interest in solution processing, that is in ternary phase diagrams of the type PLC + engineering polymer + solvent. A statistical mechanical theory of rigid-rod systems has been developed by Flory he started the work in 1956 (44.) when most of the present applications of PLCs were unknown and continued it more than two decades later (45). The Flory configurational partition function is... 

Both Flory and Huggins derived statistical mechanical expressions for aS . Their expressions are still among the best available. For this reason, Prigogine and his co-workers concentrated their efforts on revising the statistical mechanical configurational partition function which leads, among other things, toAH. ... 

Flory-Orwoll-Vrij [1964] The FOV equation of state was developed to correlate properties of macromolecular liquids, sacrificing rigor of mathematical derivation for simplicity. The authors followed Prigogine s derivation up to the configurational partition function and then modified the expression for the intermolecular energy, Eq. The final product is indeed algebraically simple ... 

The configuration partition function is denoted by Z. The matrix Qj is the generator matrix, for the quantity (r a jr)o is a rather large expression. The interested reader is encouraged to consult Flory s work on the subject (7). 

Models for Stiff-Chain Polymers.— Flory briefly reviewed some of the consequences of separating the configurational partition function for long chain molecules and their solutions into inter- and intra-molecular parts. In particular, he pointed out that this separation, and hence the partition function derived therefrom, are valid only for sufficiently flexible chains, or when the polymer concentration is sufiiciently low. He indicates how this can be rectified in statistical mechanical models for semi-rigid and rod-like polymer molecules. For the latter case this is pursued in considerable detail in a very recent series of papers by Flory and co-workers. ... 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

Probably the most powerful technique for sampling the conformation space of real polymers is the transfer matrix approach in combination with the RIS approximation discussed in the previous chapter. The approach was pioneered by Flory (e.g., [13]), but its potential for generating configurational averages was recognized very early (cf. the reference to E. Montrol in the seminal paper by Kramers and Wannier [14]). In the transfer matrix approach the partition function of a polymer is expressed in terms of a product of matrices, whose elements in the simplest case, e.g., polyethylene, have the form... 

The calculation of the partition function can be done by the standard Flory-Huggins lattice method. The lattice model predicts the existence of a true second-order transition at a temperature T2. This is shown schematically in Figure 13 for the entropy-pressure-temperature equation of state. As can be seen, the transition occurs at a critical value of the entropy (zero configurational entropy) and the Kauzmann paradox is resolved for thermodynamic reasons rather than kinetic ones, i.e. one is simply not permitted to extrapolate high temperature behavior through the glass transition. Rather, as the material is cooled, a break in the S-T (or V-T) curves occurs because of a second-order transition. 

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