Chain freely rotating model

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  

Note that (cos ) decays rapidly as the number of bonds between bond 

The final relation defines 5 p as the number of main-chain bonds in a persistence segment, which is the scale at which local correlations between bond vectors decay  

Since the decay is so rapid, the summation in Eq. (2.21) can be replaced by an infinite series over k  

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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... 

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 ... 

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°... 

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... 

Values of the parameter used are listed in Table 1. For comparison purposes we have also investigated, to a limited extent, a so-called freely-rotating chain model, which is identical to the above polyethylene model except that k,j, is set equal to zero, i.e., the torsional barrier to conformational transitions is eliminated. 

Fig. 22a,b. The points are the time-correlation functions Mj (t) and M2(t) chain axis reorientation angle xW obtained with polyethylene and freely-rotating chain models at 300 K. The solid curves are the fit by the theoretical model of restricted rotational diffusion due to Wang and Pecora [44]... 

Wang and Pecora [51] earlier presented an analytical solution applicable to a restricted rotational diffusion model, in which a rigid rod undergoing a rotational diffusion is allowed to change its direction only up to an angle o and not beyond. The solution contains two parameters, o and the diffusion coefficient D,. In Fig. 22 the solid lines are those fitted by the analytic equations using nearly identical numeric valu of o and D, for both Mi(t) and Mzft) in either the polyethylene or the freely-rotating chain model. Similarly in Fig. 23 the observed time correlation functions are fitted by analytical solutions for... 

The Kratky and Porod model is an extension of the freely rotating chain model in which the valence angles between two units are close to 180°. This confers to such chains a certain persistence which prevents them from behaving like a random coil such chains are semi-rigid and are also called worm-like. The persistence length (fl) is defined as... 

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... 

From the restricted rotation model, C , = (1 + cos 0)/(l - cos 0) (1 + Z>)/(1 - f>), whereas from the freely rotating chain model, C , = (1 + cos 0)/(l - cos0). Thus, the restricted motion at the junctions connecting polymer repeat units has the physical effect of stiffening the links, which increases the number of links along a chain one must travel before orientation vectors become uncorrelated. values for some common polymers are provided in Table 7.4. A more extensive list is available in the Polymer Handbook [37], It is apparent from the table that polymers such as polystyrene, which possess naturally bulky repeat units, have large values, whereas polymers connected by flexible links (e.g., polyethylene oxide and polyisoprene) have the lowest values. 

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