Freely rotating tetrahedral chain

Before concluding the discussion on the equivalent random link it may be mentioned that other model systems, such as the freely rotating tetrahedral chain model which was used to give eqn (3.6) may be corresponded to an rjc model. It can be shown that ... 

The unperturbed dimensions of various condensation polymers obtained by the present method are listed in Table 10. A polyelectrolyte chain, sodium polyphosphate, has been included because theta-solvent results are available. The freely-rotating chain dimension (Lzyof of poly(dimethylsiloxane) in the table is due to Flory and his coworkers (705), that for the polyphosphate chains is taken directly from the paper of Strauss and Wineman 241 ), while most of the others have been calculated in the standard manner with the convenient and only negligibly incorrect assumption that all the aliphatic bond angles are tetrahedral. The free-rotation values for the maleate and fumarate polyesters are based on parameters consistent with those of Table 6 for diene polymers. 

As discussed further in the following section, it can be shown that the statistical distribution of end-to-end distances for any real chain reduces to the Gaussian form if the number of rotatable links is sufficiently large. By suitably choosing n and / for the freely jointed random-link model, both rms and the fully extended length can be made equal to the corresponding values for the real chain. These values define the equivalent freely jointed random chain. For example, if it is assumed that in a real polyethylene chain (i) the bonds are fixed at the tetrahedral angle and (ii) there is free... 

Qualitatively similar behavior was seen by Darinskii et al. [59]. They performed BD simulations of a freely rotating chain with tetrahedral bond angles. This basic model resulted in mode relaxation rates similar to those shown in Fig. 6 in the absence of a barrier. Darinskii et al also performed simulations for the same model with side groups added on alternating carbons. For this case the results were more similar to those in Fig. 6 with a barrier of 4 kgT. Thus alterations of the potential or the structure which tend to slow local motions have similar effects on mode relaxation rates. 

A limitation of the freely jointed chain model is that the bond angle 9 linking polymer repeat units is constant (e.g., 9 = 109.5 for tetrahedrally connected carbons), which results in some degree of correlation between repeat unit orientations. A model termed the freely rotating chain model fixes the bond angle between repeat units, but allows free rotation about bonds connecting them (Figure 7.5). In this case, it can be shown that... 

While the freely jointed chain is a simple model from which to begin predictions of chain dimensions, it is physically unrealistic. Since each carbon atom in a real polymer chain is tetrahedral with fixed valence bond angles of 109.5°, the links are subject to bond angle restrictions. Moreover, the links do not rotate freely because, as we have seen earlier, there are energy differences between diflferent conformations (cf. Fig. 2.3). Both of these effects cause to be larger than that predicted by the freely jointed... 

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