The rotational isomeric state method

At present the only rigorous way of treating isolated polymer molecules at an interface is by a combination of rotational isomeric state theory and Monte Carlo procedures. Some results for tails have been published by Feigin and Napper (1979) and Mattice and Napper (1981). These strictly apply only in a 

0-solvent or a melt of the polymer, although it was noted in the previous section that the expansion factor for isolated chains of at least modest molecular weight is predicted theoretically to be only marginally smaller than that in free solution. 

One striking feature of the conformation of isolated tails is that the spatial extension of the chain is expanded in a direction normal to the interface whilst being unaffected parallel to the interface. This is illustrated by Figs 4.8 and 4.9. These show that for poly(methylene) the mean square displacement normal to the interface for high molecular weight polymers asymptotes in value to twice that for chains in free solution. This result was first derived by DiMarzio and McCrackin (1965) for a 1-D random walk normal to an impenetrable interface. The doubling of the mean square displacement can be understood in terms of the halving of the available space normal to the interface by the presence of the impenetrable barrier. The components of the mean square displacement parallel to the interface are unaffected by its presence. 

A third striking feature of the results for tails is the divergence of the persistence vector a= r . This arises because the x-component of the persistence vector is divergent. A comparison is presented in Table 4.2 of the effects of intramolecular excluded volume with that due to an interface. It is apparent that the two different types of excluded volume exert different effects on the conformation of the polymer chain. 

Comparison of the effects of intramolecular excluded volume and that due to an interface 

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Experimental molar cyclization equilibrium constants for cyclics in the PDA n%lt and the P1 melt at 423 K are own dotted as log gainst Ic x in Fig. 12 and 13. They are compared with theoretical values calculated by the Jacobson and Stockmayer ex esrion Eq. (6) with - 2jc- Values of < / >q required by this expression were comfHited by the exact mathematical methods of Flory and Jemi-gan 37,30) using the rotational isomeric state models for the polyesters set up by Flory and Williams 27,128). Agreement between experiment and theory is excel-... 

In a subsequent paper Brant and Flory (55) have successfully correlated their experimental data with polypeptide chain structure using the rotational-isomeric state model and statistical mechanical methods applicable to linear systems of interacting subunits. 

Between the appearance of the MD studies by Vishnyakov and Neimark [56], and that by Urata et al. [60], Khalatur, Khokhlov and co-workers published two coarse-grained molecular modelling studies of Nafion [64, 65] that utilised rather different approaches to studying the morphology of PFSIs. The first of these [64] was based on a hybrid Monte Carlo/reference interaction site model (MC/RISM). The principle behind this method was to use MC simulations, based on the rotational isomeric state (RIS)... 

P. J. Flory, Macromolecules, 7, 381 (1974). Foundations of the Rotational Isomeric State Theory and General Methods for Generating Conformational Averages. 

The above studies indicate that many conformational transitions occur as isolated transitions. For these motions, schematic representations like those shown in Fig. 7 are completely inadequate. Such representations assume that the rotational isomeric states (RIS) provide a reasonable basis set for understanding conformational dynamics. The observation that many transitions occur as isolated transitions cannot be explained within the RIS framework. In other words, many conformational transitions cannot be explained in terms of "cartoon pictures like Fig. 7. More general methods for discussing cooperativ-ity are examined in the next section. 

A totally different approach to rubber elasticity has been developed by Stepto and co-workers [15, 16], which also accounts for the Mooney-Rivlin softening. Their approach is not phenomenological, but is based on structural considerations that give an accurate description of the moleeular eonformational states of the units in the polymer chains as the network is stretched. They have proposed a method for calculating the free energy of a stretched molecular network based on the rotational isomeric state of the network chains, with conformational energies determined from observations of conformational properties. 

Whilst some of these cases have been treated previously, Eichinger s method allows the calculation not only of statistical averages, but also of the distribution function itself. For polymer molecules with real bond angles and restricted bond rotation, the rotational isomeric state (RIS) model has proved powerful. Flory has summarized some of the most important results of the treatment, and Mark has considered the applications, particularly to bulk polymers and networks, pioneered by his group. A recent paper uses the RIS model to calculate the distribution function of the end to-end vector distribution for short polymer chains. 

Flory reviewed in 1969 the development and applications of the rotational isomeric state scheme calculations, which allow, by matrix algebra, the statistical mechanical averaging over the rotational states of chain properties which may be expressed as a vector or tensor quantity associated with the chain bonds, and estimations of the probabilities of chosen conformational sequences. The methods were generalized and schemes for reducing the dimensions of certain generator matrices were presented in 1974, when comparisons were also made with an alternative Fourier expansion method, currently in use for atactic polypropylene. These techniques have greatly contributed to an understanding... 

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