Bond rotational potential

Since the bonds of this chain are free to rotate about one another (bond rotational potentials are zero), g = 1. 

Up to this point it has been assumed that there are no restrictions to free rotations ateut bonds. This is physically unrealistic. Both macromolecules and minimolecules display bond rotational potentials that depend strongly upon the respective torsional angle. These restrictions can be incorporated into the expression for the unperturbed dimensions viz... 

It can be shown that the rms size of a polymer molecule of high molecular weight subject to independent bond rotational potentials is given by... 

The characteristic ratio of the poly(methylene) chain, which might be considered to be the simplest polymer molecule, as a function of chain length is shown in Fig. 4.4 for the simple models considered thus far. All save the simple freely jointed chain display end effects, manifest by an increase in Ci with chain length at small n, which will not be considered further. In what follows, only the asymptotic limit of the characteristic ratio for 00 (Coo) will be discussed. The values of for the freely jointed chain, the freely rotating chain and the chain with independent bond rotational potentials increase in value from 1 through 2 to ca 3-5. The latter value is, however, only ca one-half of the experimentally determined value of Coo=6-9 for poly(methylene). This serious discrepancy points to the fact that the bond rotational potentials are definitely not independent, i.e. the conformation of bond i depends upon the conformations of bonds (/—I) and (/-i-1). 

Fig. 4.4. Plots of the characteristic ratio of poly(methylene) at 140 °C as a function of the number of bonds using different theoretical models (1) the freely jointed chain (2) the rotating chain (3) independent bond rotational (4) interdependmt bond rotational potentials (after Flory, 1969).

Interdependent bond rotational potentials The origins of the interdependence of the bond rotational potentials in poly(methylene) can be understood in terms of the conformations of n-pentane (hence the phenomenon is termed the pentane effect). 

Flory (1969 1971 1974) has developed the rotational isomeric state theory for predicting the conformation of polymer molecules. This incorporates the interdependence of the bond rotational potentials. This theory is fully explained in Flory s comprehensive monograph (Flory, 1969) and only the barest details will be mentioned here. 

Fig. 7a, k Second oidm- transition jn-obabilities a C,( Pa) for internal bo mis in polyethylene, obtmned from BD simulations, using independent bond rotational potentials, b C,( a T]P) for internal braids in polyethylene, obtained from BD simulations, using independent bond rotational potentiids... 

Clearly, in dealing with the conformational properties of a given polymer, only the latter group of degrees of freedom has to be considered. A discussion of the conformational states of a given macromolecule therefore first of all requires an analysis of the bond rotation potentials. 

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