The freely rotating chain

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. 

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... 

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 can now be written in terms of cosines ... 

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 ... 

Table 2.2 summarizes the assumptions of the ideal chain models. The worm-like chain model is a special case of the freely rotating chain with a small value of the bond angle 6. Moving from left to right in Table 2.2, the models become progressively more specific (and more realistic). As more constraints are adopted, the chain becomes stiffer, reflected in larger Coo-... 

Osmotic pressure measurements for the determination of MW were used in 1900 to characterize starch. Twenty years later, the solution viscosity measurements were introduced by Staudinger for this purpose. However, it was Mark and his collaborators who developed the concept of the intrinsic viscosity ([r ]) and demonstrated that it provides information on the volume of individual colloidal particles, thus on MW. For the freely rotating chains the dependence (today known as Mark-Houwink-Sakurada equation) was obtained [Guth and Mark, 1934] ... 

The most unrealishc feature of the freely jointed chain model is the assumption that the bond angles can vary continuously. In the freely rotating chain model the bond angles are held fixed but free rotation is possible about the bonds, such that any torsion angle value between 0°... 

The characteristic ratio approximately indicates how extended the chain is. For the freely rotating chain the characteristic ratio 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). 

The freely rotating chain is formed by randomly distributed segments around some reference origin F = 0. The segment distribution density is given by the... 

The freely-rotating chain of a polymer in dilute solution is represented by a statistical distribution of conformational structures of the chain. The resultant excimer fluorescence emission is a broad Gaussian band. The excimer fluorescence spectrum from a solid polymer results in a very broad spectrum which is characteristic of the distribution of conformations adopted by the polymer in the preparation process. This distribution depends upon the thermal history of the sample and the conditions used for the casting these dependencies were presented by Frank and co-workers [26]. The complexity of photophysical processes accompanying the excimer formation and the inherent complexity... 

In most molecular theories of rubberUke elasticity, the individual chains are approximated by the freely jointed or the freely rotating chain model. In reality, however, rotations about each bond are subject to potentials that arise... 

Though the freely jointed chain is a very simple model, the result R ) general models. Consider for example the model shown in Fig. 2.2, called the freely rotating chain, in wMch the n-th bond is connected to the (n - l)-th bond with a fixed angle 6 and can rotate freely around the (n - l)-th bond. 

The results of an illustrative calculation are depicted in Fig. 3.4. The chain has 0 = 112°, i = 3, and < > = 180° and 60°. The statistical weight matrix for all internal bonds is given by Eq. (3.9) with t = if/ = 1. When cr and freely rotating chain with the same bond angle. Imposition of a symmetric torsional potential that penalizes the g states, with tr = 0.4, increases the C . Introduction of a pair-wise interdependence, via cr = 0.4 and w = 0.1, produces a further increase in C . Obviously, the interdependence of the bonds can have a strong effect on the unperturbed dimensions of the chain. 

The projection of the end-to-end vector of a chain r on the direction of the first bond /l for the freely rotating chain is... 

The quantity a is called the persistence length and is a measure of chain stiffiiess. The wormlike chain model (sometimes called the Porod-Kratky chain) is a special continuous curvature limit of the freely rotating chain, such that the bond length I goes to zero and the number of bonds n goes to infinity, but the contour length of the chain L = nl and the persistance length a are kept constant. In this limit... 

For the freely rotating chains, the dependence (today known as Mark-Houwink-Sakurada equation) was obtained (Guth and Mark 1934) ... 

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