Wormlike chains chain dimensions

By now it may have dawned on the reader that the long-time Rouse spectrum (i.e., proportionality of xp to p 2) is to be expected for any chain model in which the correlation lengths for both equilibrium conformations and frictional processes are small compared to the chain dimensions (and thus to the wavelength of the slow normal modes). A possible exception is that of the continuous wormlike chain of invariant contour length, which has been studied by Saito, Takahashi, and Yunoki.33 In this latter case, the low-frequency spectrum makes xp proportional to p A, which resembles our special one-dimensional model in the limit 1 — p 1. 

In the diaryl-substituted polysilane, the global dimensions are approximated by a wormlike chain model with a persistence length of 100 A. Thus, for the highest molecular weight studied, LIq — 100, and the chain dimensions are similar to those of a flexible-model Kuhn chain with bond lengths IkOf —200 A, that is,... 

The subject of polymer size or chain dimensions is concerned with relating the sizes and shapes of individual polymer molecules to their chemical structure, chain length, and molecular environment. The shape of the polymer molecule is to a large extent determined by the effects of its chemical structure upon chain stiffness. Polymers with relatively flexible backbones tend to be highly coiled and can be represented as random coils. But as the backbone becomes stiffer, e.g., in polymers with more aromatic backbone chain, the molecules begin to adopt a more elongated wormlike shape and ultimately become rodlike. However, the theories which are presented below are concerned only with the chain dimensions of linear flexible polymer molecules. More advanced texts should be consulted for treatments of wormlike and rodlike chains. 

The present chapter aims to describe some typical contributions from recent studies on stiff polymers in dilute solution. We will be mainly interested in (1) applicability of the wormlike chain model to actual polymers, (ii) validity of the hydrodynamic theories [2-4] recently developed for this model, and (iii) the onset of the excluded-volume effect on the dimensions of semi-flexible polymers. Yamakawa [5, 6] has generalized the wormlike chain model to one that he named the helical wormlike chain. In a series of papers he and his collaborators have made a great many efforts to formulate its static and dynamic properties in dilute solution. In fact, the theoretical information obtained is now comparable in both breadth and depth to that of the wotmlike chain (see Ref. [6] for an overview). Unfortunately, however, most of the derived expressions are too complex to be of use for quantitative anal) sis and interpretation of experimental data. Thus, we only have a few to be considered with reference to the practical aspects of the helical wormlike chain, and have to be content with mentioning the definition and some basic features of this novel model. 

As can be seen from the above discussions, the chain dimensions and the particle scattering function of ein unperturbed wormlike chain at infinite dilution are expressed as functions of two independent parameters L and q. Since L is proportional to the molecular weight M, it turns out that these properties of a series of homologous polymers in the unperturbed state are determined by M, q, and Ml. Here, Ml is defined by... 

Chain stiffness prevents the monomer units of a polymer molecule from having contact with one another. Thus, we may expect that there exists for a semi-flexible polymer a certain contour length Lc below which the excluded-volume effect on the chain dimensions disappears (in a statisitical sense). This prediction is borne out by the data of Figure 5-3, which show that (5 )/M follows the curve for an unperturbed wormlike chain until Mw reaches 3 x 10 (L 4 X 10 ). Obviously, Lc ought to be larger for a stiffer polymer, i.e., one with larger persistence length q. 

Equation 10 shows that in the small coverage limit the dimension of macromolecules covered with surfactant increases slightly with a. In the high coverage limit, a complex is described as a wormlike chain with persistent length... 

Unlike other flexible chains, the dimensions of the wormlike chain are greatly influenced by the shape of the molecules. The parameter A, is a measure of the chain stiffness. 

In this article, we will review major methods for the evaluation of unperturbed dimensions of both highly flexible and less flexible (wormlike) chains. Methods for extracting the temperature dependence of unperturbed dimensions will also be described. Experimental results are tabulated and discussed. The emphasis here is on developing structure/chain-flexibility relationships for a few series of carefully chosen polymers rather than providing an exhaustive tabulation. Furthermore, new developments and unresolved issues are stressed in an attempt to identify areas needing further attention. Finally, interrelationships between structure, flexibility, and mechanical properties will be discussed. 

The wormlike chain model affords a continuous description of chain character ranging from flexible p/L small) to rodlike p/L large). The unpertubed mean-square dimensions RI q and... 

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