Worm like chain

The Gaussian chain is obviously an oversimplification of the real polymer chain. The description of the shape of a polymer chain can in fact vary greatly according to the size scale with which we examine it. 

On a slightly larger scale, we would recognize the presence of monomeric units. 

As the size scale is increased further, a short sequence of monomers strung together would look like a more or less rigid rod. 

we would recognize the polymer molecule to be a flexible random coil. 

Finally, we would see the whole molecule as a particle characterized by its radius of gyration. 

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FIGURE 21.4 Nanofishing of a single polystyrene (PS) chain in cyclohexane. The solvent temperature was about 35°C (0 temperature). A cantilever with a 110 pN nm spring constant was used. The worm-like chain (WLC), solid line, and the freely jointed chain (FJC), dashed line models were used to obtain fitting curves. (From Nakajima, K., Watabe, H., and Nishi, T., Polymer, 47, 2505, 2006.)... 

As for further analysis, curve fitting against the worm-like chain (WLC) model was conducted and indicated as a solid line in Figure 21.4. The model describes single polymer chain mechanics ranging from random-coil to fully extended forms, as follows ... 

The worm like chain behaviour of the pectins is developped allowing to predict the dimensions of the chain as soon as the persistence length is known. The experimental data are compared with the theoretical prediction obtained from conformational analysis. 

As discussed in the last 3 years, polysaccharides behave in solution under a worm like chain [26] the local stiffness of the chain is characterized by a persistance length (Ip) the larger Ip is, the larger the chain deviates from the gaussian behaviour in the usual molecular weight range of these natural polymers [27], This makes difficult to use the relations given in litterature for synthetic... 

The other limiting case concerns locally smooth chains. For the case of a Gaussian chain in a network of rods, Edwards found d (L/V) 1/2 [66] which agrees with the above discussed binary contact model. Finally, considering worm-like chain bridges the differences in the power laws between the two limiting cases and exponents between — 1 and — 1/2 may be obtained. 

The molar mass dependence of the intrinsic viscosity of rigid chain polymers cannot be described by a simple scaling relation in the form of Equation (36) with molar mass independent of K and a. over a broad molar mass range. Starting from the worm-like chain model, Bohdanecky proposed [29] the linearizing equation... 

This conjecture was found to hold for flexible chains, stiff worm-like chains, star macromolecules, and hard spheres [156]. A few results are shown in Fig. 32. 

C. Bouchiat, M.D. Wang, J.-F. Allemand, T. Strick, S.M. Block, and V. Croquette Estimating the Persistence Length of a Worm-Like Chain Molecule from Force-Extension Measurements. Biophys. J. 76, 409 (1999). 

Note The persistence length is the basic characteristic of the worm-like chain. 

The analysis described above is useful for modelling colligative properties but does not address polyelectrolyte conformations. Polyelectrolyte conformations in dilute solution have been calculated using the worm-like chain model [103,104], Here, the polymer conformation is characterized by a persistence length (a measure of the local chain stiffness) [96]. One consequence of the... 

Finally, it should be mentioned that an interesting interpretation of the third eq. (5.4) is obtained by introducing the model of the worm-like chain, as developed by Kratky and Porod (153, 154). As is well-known the characteristic quantity of this model is the persistence length, i.e. the projection of the end-to-end distance of an infinitely long coiled worm... 

Up to this point, rather flexible molecules are considered. However, it is necessary to note that the theory of Gotlib and Svetlov should in first instance be applicable to rather stiff molecules, as very flexible molecules cannot be approximated by a worm-like chain. The applicability to stiff molecules is demonstrated in Fig. 5.6, with the aid of results obtained... 

From their light-scattering measurements Holtzer, Benoit, and Doty (126) concluded that the short-range interactions control the dimensions of cellulose nitrate chains, and they discussed their results in terms of the worm-like chain model of Kratky and Porod (142), obtaining a persistence length of about 34.7 A. In Fig. 21 these data are shown as a plot of (S yjMw against Mw. The open circles are the experimental points and the broken curve is that calculated from the equations for the worm-like chain model. The theoretical curve is claimed to reproduce the data to within the probable experimental error in all but two cases. 

Fig. 9.17 Models for a disordered polyacetylene chain (a) the Kuhn model and (b) the worm-like chain.

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