Chain freely-rotating

If we consider the size of a polymer molecule, assuming that it consists of a freely rotating chain, with no constraints on either angle or rotation or of which regions of space may be occupied, we arrive at the so-called unperturbed dimension, written (r)o Such an approach fails to take account of... 

Ratio of the root-mean-square end-to-end distance of a macromolecular chain with unperturbed dimensions,, to that of a freely rotating chain with the same structure,... 

The structure of tha polycarbonate chain Is analyzed from tha point of view of the spatial configurations. It follows that the molecule can be treated as a freely rotating chain consisting of a succession of virtual bonds (lvp) 7.0 A In length, jointed at angles (SI of ca. 112°. 

A comparison of the conformational freedom of rotation of the contiguous phenyl groups in polycarbonates, with various substituents at the Ca atom, is presented. Conformational maps are calculated for the polymer shown above. Synchronous rotation of the phenyls with a low-energy barrier is possible for 1, 4, 5, and 8. Although the extent of freedom of rotation depends on the nature of the substituent, there is very little difference in the characteristic ratio of the unperturbed end-to-end distance for these polycarbonates, and the temperature coefficient of the characteristic ratio is extremely small. In spite of the limited conformational freedom, it is shown that the steric symmetry and the geometric asymmetry of the chain segments enable the treatment of these chains in the framework of the freely rotating chain. 

The characteristic ratio changes from 1.3 to 2.8 with o changing from 0 to 1, when the virtual bond is used. On the other hand, when each bond of the phenylene group is taken into account individually, the two extreme values are 3.41 and 7.40. By assuming all the statistical weight factors to be unity, which corresponds to the freely-rotating chain, the characteristic ratio is 1.60 when the virtual bond is used, and 4.22 if it is not. 

Hoiyst R, Schick M (1992) J Chem Phys 96 731. In this paper, freely jointed chains are referred to as freely rotating chains... 

Table 6. Unperturbed dimension of freely-rotating chains...

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. 

The value of a for poly(propylene oxide) was obtained from the data of Moacanin 189) for polyurethanes prepared by condensation of toluene-2,4-diisocyanate with atactic polypropylene glycols of molecular weights about 1000 and 2000. The small quantity of diisocyanate present in these compositions can make only a negligible effect on the chain dimensions (less than one per cent on the freely rotating chain, as is... 

Calculation of the dimension FR/N of a chain with N segments with free rotation about valence bonds has not yet been carried out because of uncertainties in the dimensions of the six isomers of the repeat unit. The measured values of correspond to /IV on the order of 100 A.2, where a repeat unit is equated to a segment. It is expected that the freely rotating chain with fixed valence bonds will be a good model for BBB owing to the inappreciable steric interference to rotation about the single bonds connecting the repeat units. Such behavior has... 

Three-membered ring formation is fast—the product is strained so AH is large but this is offset by the reacting atoms being as close as they can get in a freely rotating chain... 

Cellulose acetate (CA) molecules become most rigid at <(F)> 2.5, if they are dissolved in a common solvent (DMAc or DMSO). By doubly extrapolating A values of CA to = 0 and s = 1, the cellulose molecule, hypothetically dissolved in a non-polar solvent, should be expected to be a freely rotating chain. 

To compensate for one of the most unrealistic aspects of the freely jointed chain model, the freely rotating chain model was developed. In this model one removes the assumption of continuously variable bond angles. However, the energy of the chain is independent of rotational angles, and therefore they may still assume any value from 0° to 360°. The characteristic ratio of the freely rotating chain is given by... 

The amount of meso and racemic dyads in PS was 0.43 and 0.57, respectively. The conformational parameters for PDMS were taken from ref. 112.) The strong difference in the half-peak widths q /2 is a clear indication of a higher conformational rigidity for PS, which is also suggested by the characteristic ratios relative to the freely rotating chain = o/ [see Eqn. (3.3.4)], the value of which is 4.4 and 1.9 for PS and PDMS, respectively. 

As the name suggests, this model ignores differences between the probabilities of different torsion angles and assumes all torsion angles - 7T< < 7t to be equally probable. Thus, the freely rotating chain model... 

The mean-square end-to-end distance of the freely rotating chain is a simple function of the number of bonds in the chain backbone n, the length of each backbone bond I and the bond angle 9 ----------------------------... 

Polymer chains are never as flexible as the freely rotating chain model predicts, since the most flexible polymers with 9 = 68° have Coo > 4... 

The worm-like chain model (sometimes called the Kratky-Porod model) is a special case of the freely rotating chain model for very small values of the bond angle. This is a good model for very stiff polymers, such as double-stranded DNA for which the flexibility is due to fluctuations of the contour of the chain from a straight line rather than to trans-gauche bond rotations. For small values of the bond angle ( < 1), the cos 9 in Eq. (2.23) can be expanded about its value of unity at = 0 ... 

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