Wormlike coil model

Yamakawa and co-workers developed the quasi-two-parameter (QTP) theory from the earlier two-parameter (TP) theory, to incorporate chain stiffness into the model. Specifically, the QTP theory computes C o via the helical wormlike coil model (HW),... 

To describe the intrinsic viscosity of wormlike coils in the absence of excluded volume, Yamakawa and co-workers developed theoretical descriptions based, first, on the KP model [Yamakawa and Fujii, 1974] and subsequently on its later adaptation, the HW model [Yoshizaki et al., 1988]. Using the cylindrical wormlike coil model, Yamakawa and Fujii [1974] obtained the following expressions, with L expressed in... 

A consideration of the molecular conformation using the wormlike chain model suggests that the curdlan molecule may contain helical portions but, as a whole, takes a random-coil conformation ( ) ... 

In the range of y where the helical and wormlike models display their absolute maxima in Figure 1, the presumably more realistic random coil model R has a much lower and broader... 

Figure 1. F(ii) = NiJP(ii) vi. ii = (4t/ K) sin (e/2) for helical amylosic chain nwdels A, B, and C, wormlike amylosic chain model W, jointed helical model J, and realistic random coil model R. Details of the models are described in the text.

Yoshizaki, T., Nitta, L, and Yamakawa, H., Transport coefficients of helical wormlike coils 4. Intrinsic viscosity of the touched-bead model. Macromolecules, 21, 165-171 (1988). 

A perfect helical main chain conformation always leads to a rodlike or cylindrical external shape. But each monomeric unit in such a rod contributes a certain flexibility. So, the flexibility of the rod, as a whole, must increase with increasing degree of polymerization, even when the flexibility per monomeric unit remains constant. A macroscopic example of this would be the flexibility of steel wires of equal diameter but different lengths. Thus, even a perfect helix will adopt coil shape if the molecular mass is very high. Because of this, helically occurring macromolecules, and other stiff macromolecules, can often be well represented by what is known as the wormlike screw model for macromolecular chains at low molecular masses, the chains behave like a stiff rod, but for high molecular masses, the behavior is more coil-like. Examples are nucleic acids, many poly(a-amino acids), and highly tactic poly(a-olefins). 

Explains how and when to apply rotational isomeric state, Gaussian chain, wormlike chain, and random coil models... 

As may be seen in Figure 8.7, approaches the rodlike limit of unity for L /a<< 1 and the coil limit 2d 3L for LJd >> 1, with transition developing for LJd l. A numerically similar result is obtained for the model with free rotation about N bonds with valance angles n - a ). As seen in Figure 8.7, the results for the model, with a=cos(ct ), are essentially equivalent to the results for the wormlike chain model [88]. 

The former is valid for Gaussian chains (Chapter 2) and the latter for straight rods. Hence the wormlike chain takes on a variety of conformations intermediate between Gaussian coils and rods depending on the value of a dimensionless parameter L/q. It is due to this property that the wormlike chain is used to model polymer molecules with stiffness. However, what can be obtained with wormlike chains is only a fraction of the infinitely numerous conformations realized by actual polymer molecules. 

Comparison of eqns [10] and [ 11 ] to eqns [2] and [3] points out that long wormlike chain acquires on the large scale the random coil conformation and its large-scale properties coincide with those of an equivalent freely jointed chain comprising Ni = L/2lp statistical segments each of length t = 2lp. Hence, both the freely jointed and the wormlike persistence chain models can be applied for desaiption of large-scale conformational properties of flexible and semiflexible chain polymers. 

The wormlike chain of Kratky and Porod [49K1] is characterized by a contour length L and a persistence length a. The latter increases with increasing stiffness, but is (on the basis of the model) independent of L. The relation between the radius of gyration and L for worm-like linear coils without excluded volume is[53Bl] ... 

Many rigid-chain polymers have been investigated in the molecular-weight range in which their molecules exhibit a conformation intermediate between the rodlike conformation and the Gaussian coil. The best model for these polymers is the wormlike chain. Under these conditions the shear optical coefficient An/A-r varies with the molecular weight M of the polymer and is given by the equation (100)... 

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