Wormlike chain parameters

Figure 4 compares osmotic compressibility data for isotropic schizophyllan-water solutions [63] with the scaled particle theory. The ratios of the z-average to the weight-average molecular weights of these schizophyllan samples are ca. 1.2. The solid curves, calculated with d taken to be 1.52 nm and other molecular parameters (Lc, v, and c ) estimated from Mw and the wormlike chain parameters in Table 1, are seen to come close to the data points for all samples. 

The zero-shear viscosity r 0 has been measured for isotropic solutions of various liquid-crystalline polymers over wide ranges of polymer concentration and molecular weight [70,128,132-139]. This quantity is convenient for studying the stiff-chain dynamics in concentrated solution, because its measurement is relatively easy and it is less sensitive to the molecular weight distribution (see below). Here we deal with four stiff-chain polymers well characterized molecu-larly schizophyllan (a triple-helical polysaccharide), xanthan (double-helical ionic polysaccharide), PBLG, and poly (p-phenylene terephthalamide) (PPTA Kevlar). The wormlike chain parameters of these polymers are listed in Tables... 

In this article, we have surveyed typical properties of isotropic and liquid crystal solutions of liquid-crystalline stiff-chain polymers. It had already been shown that dilute solution properties of these polymers can be successfully described by the wormlike chain (or wormlike cylinder) model. We have here concerned ourselves with the properties of their concentrated solutions, with the main interest in the applicability of two molecular theories to them. They are the scaled particle theory for static properties and the fuzzy cylinder model theory for dynamical properties, both formulated on the wormlike cylinder model. In most cases, the calculated results were shown to describe representative experimental data successfully in terms of the parameters equal or close to those derived from dilute solution data. 

Moha et al. (54) considered that polypeptides in helicogenic solvents can assume intact helices but that the resulting rods can be weakly bent, and proposed to represent such a flexible rod by a wormlike chain. It was shown that the nonlinear dependence of 1/2 on N as depicted in Fig. 21 can be fitted by a suitable choice of the two parameters characterizing the wormlike-chain model. However, the necessary value for the length per monomeric unit forces us to accept the conclusion that the helical conformation is not of the a-type but of the 310-type. This is at variance with the ample experimental evidence now available for many synthetic polypeptides. 

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. 

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... 

Figure 5-4 compares particle scattering function data for PHIC in hexane [27] with the theory corresponding to the indicated values of q and Ml, though the comparison is limited to the samples exhibiting unperturbed wormlike chain behavior. It is seen that a close agreement between theory and experiment can be obtained with the parameter values consistent with those estimated above from (5 ) data. 

Figure 5-5 shows that radius of gyration data [31] for double-stranded DNA, another typical stiff polymer, can be described accurately by eq 2.4 with q = Q8 nm and Ml = 1970 nm . These parameter values have been estimated by the method of Murakami et al., and the Ml value is in close agreement with 1950 nm that can be derived from the well-established geometry of the DNA double helix. Kirste and Oberthiir [32] showed that the k dependence of k P(k) for a DNA sample measured by light and small-angle X-ray scattering can be represented by the theory of unperturbed wormlike chains. 

For wormlike chains at least two of the three parameters q, Ml, and So need be known to determine (5 from S )a.pp and 6. Sakuiai et al. [40] proposed an iteration method which permits all the unknowns S ),q, Ml, and to be estimated at one time if data for (5 )app and 6 are available as fiinctions of M and the sign of is known in advance. A similar iteration method can be developed also for the evaluation of these unknowns from the measured M dependence of (5 )uvapp and S. The solid curve in Figure 5-14 is actually the final result obtained by its application the convergent values of q, Ml, and So are shown in the caption. Sakurai et al. [40] found that when substituted into the theory of Yamakawa et al. for [ ] [3, 4], these q and Ml values along with a diameter value of 0.48 nm lead to a close fit to the data plotted in Figure 5-8. The Ml value of 185 nm is favorably compared with 190 nm from crystallographic data [56] and 180-190 nm from SAXS measurements [57]. 

The ideally flexible continuous chain generated from the spring-bead chain retains no microscopic feature of actual polymer molecules and hence it is the most abstract of polymer models. The wormlike chain, though a continuous chain, is more realistic since through the parameter q it allows for the stiffness possessed by actual molecules. Up to this point we have seen a number of examples which substantiate the usefulness of these chain models for the quantitative description of global behavior of polymers in dilute solution. But this never means that no other chain model need be considered. 

Statistical properties of an unperturbed HW chain in equilibrium are determined with five parameters chain length L, ao,l3o,Ko, and tq (the latter four characterize U s) of the chain). This should be contrasted to the fact that only two parameters L and q are needed for the description of these properties of an unperturbed wormlike chain. In adapting the HW chain to actual polymers, the shift factor Ml(= M/L), instead of L, may be chosen as a parameter since M can be determined experimentally. Thanks to eq 2.10 the stiffness parameter (2A) may be used for ao. For HW chains we have no equation corresponding to eq 2.2. Hence (2A) may not be equated to q according to eq 2.14. It should be noted that the persistence length q is the concept associated only with wormlike chains. The Poisson ratio ao of the HW chain is expressed in terms of o and j3o as... 

Thus we see that A,ao,KO) o. and Ml may be used as the basic parameters of the HW chain. These five parameters give the HW a greater flexibility flian that obtained with the wormlike chain and should allow it to describe a wider range of physical properties of actual polymer molecules. However, in actuality, there would be considerable difficulty in evaluating so many parameters from comparisons between theory and experiment. 

The effect of the flexibility has been studied for a fixed chain length. The order parameters and transition temperatures are found to decrease with increasing flexibility (fig.2 ) The temperature dependence of the peristence length can be of importance. From the wormliKe chain q 1/T, although excluded volume... 

We have discussed a model of stiff polymer chains and have shown that it gives results very similar to those descriptive of the wormlike chain in the limits of small and large stiffness. Presumably, for intermediate ranges of stiffness, the presence of three basic parameters in this model should enable it to reflect the character of real chains. One of the important virtues of the present model is the fact that the relevant distribution functions may be obtained in a simple closed-form analytic expression. It therefore represents a possible zeroth-order model for discussions of... 

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 the part devoted to neutral polymers, we mentioned that semiflexible and stiff chains do not obey the behavior predicted by the Kuhn model. Restricted flexibility of the chain can be caused by the presence of stiff units with multiple bonds or bulky pendant groups, but it can be a result of external conditions or stimuli. In the preceding part, it was explained in detail that repulsive interactions together with entropic forces increase the stiffness of PE chains. Hence, a sudden pH change can be used as a stimulus affecting the stiffness of annealed PE chains. The properties of semiflexible polymers are usually treated at the level of the wormlike chain (WLC) model developed by Kratky and Porod [31]. The persistence length, /p, is an important parameter strongly related to the WLC model and has been used as the most common characteristic of chain flexibility—in both theoretical and experimental studies. It is used to describe orientational correlations between successive bond vectors in a polymer chain in terms of the normalized orientation correlation function, C(s) = (r,.r,+j). For the WRC model, this function decays exponentially ... 

Although a variety of models can be used to interpret the dependence of the parameters A2, , 6, and [x]] on M, it will suffice for our purpose to consider the relations among these parameters according to the wormlike chain model.The persistence length p and the mass per unit length are important parameters in this model. For example. 

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. 

Many polymer chains are not completely flexible under the usual experimental conditions of interest. In order to incorporate the local chain stiffness, the Kuhn model is modified slightly by introducing a bond angle 180-0 between the consecutive Kuhn steps, as sketched in Figure 2.12a. Obviously, this angle is a parameter to capture the backbone stiffness of the chain. Further, let us assume that the Kuhn steps are freely rotating, and now the model is called the Kratky-Porod or wormlike chain model. 

An alternate interpretation of the parameter 9 that enters in the definition of ip can be reached by considering a continuous representation of the chain backbone. Let us consider a space curve (Figure 2.12b) of contour length L to represent a wormlike chain. Here, the arc length variable s represents a segment along the contour (0 < < L), whose position, local tangent, and local... 

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

Figure 4 Characteristic ratio Cp (equation 5-2) vs. degree of polymerization solid lines, calculated for a wormlike chain with the persistence lengdi (in number of mmiomer units) as the parameter points, results fiom ILS. (from rdierence 11)...

The simplest model for free (or linker) DNA is the wormlike-chain or Kratky-Porod model [48], It is based on the assumption that changing the contour of a linear chain by bending costs energy. If we describe the contour of length I by introducing the contour parameter s e [0, /], an infinitesimal segment of the contour (arc length) can be expressed in local coordinates by... 

How well do predicted and observed non-Newtonian intrinsic viscosity agree for a wormlike model of xanthan Fixman (Ref. J2 Fig. 4) gives the non-Newtonian intrinsic viscosity for a flexible chain model at various values of the excluded volume parameter a, as a function of the normalized shear rate parameter The parameter Kn, which incorporates the effects of molecular weight and chain stiffness, equals 1.71[n]oMnog/RT where [nJo is the polymer intrinsic viscosity at zero shear stress, o is the solvent viscosity, g is the shear rate in sec"i and the other symbols have their usual meaning. 

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