Helical wormlike chain

Yamakawa and co-workers have formulated a discrete helical wormlike chain model that is mechanically equivalent to that described above for twisting and bending/79111 117) However, their approach to determining the dynamics is very different. They do not utilize the mean local cylindrical symmetry to factorize the terms in r(t) into products of correlation functions for twisting, bending, and internal motions, as in Eq. (4.24). Instead, they... 

Helical Wormlike Chains in Polymer Solutions, Yamakawa, H., Springer, Berlin, 1997 Yamakawa, H. Abe, F. Einaga, Y. Macromolecules 1994, 27, 5704. 

Yamakawa, H., Helical Wormlike Chains in Polymer Solutions, Springer-Verlag, Berlin, 1997. 

Yamakawa, H., and Fujii, M., Statistical mechanics of helical wormlike chains I. Differential equations and moments, J. Chem. Phys., 64, 5222-5228 (1976). 

We may constrain a freely rotating chain in such a way that the internal rotation of each segment is allowed to occur only over a limited range of rotational angle. If the limiting process leading to the wormlike chain is carried out with this additional constraint taken into account, we obtain a continuous chain model first considered by Miyake and Hoshino [2] and also by Burchard [3]. Yet another continuous chain model called the helical wormlike chain was proposed by Yamakawa and Fuji [4]. This chain is derived from a discrete chain somewhat different from the random flight chain [5]. 

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. 

Yamakawa H (1997) Helical wormlike chains in polymer solution. Springer Verlag, Berlin... 

Finally, some rather recent devdopments must be noted. Several years ago, Yamakawa and co-workers [25-27] developed the wormlike continuous cylinder model. This approach models the polymer as a continuous cylinder of hydrodynamic diameter d, contour length L, and persistence length q (or Kuhn length / ). The axis of the cylinder conforms to wormlike chain statistics. More recently, Yamakawa and co-workers [28] have developed the helical wormlike chain model. This is a more complicated and detailed model, which requires a total of five chain parameters to be evaluated as compared to only two, q and L, for the wormlike chain model and three for a wormlike cylinder. Conversely, the helical wormlike chain model allows a more rigorous description of properties, and especially of local dynamics of semi-flexible chains. In large part due to the complexity of this model, it has not yet gained widespread use among experimentalists. Yamakawa and co-workers [29-31] have interpreted experimental data for several polymers in terms of this model. 

Yoshizaki T, Yamakawa H. Scattering functions of wormlike and helical wormlike chains. Macromolecules 1980 13 1518-1525. 

M. Fujii, K. Nagasaka, J. Shimada, and H. Yamakawa. More on the model parameters of helical wormlike chains. Macromolecules, 16 (1983), 1613-1623. 

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