Stiffness Wormlike, chain

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Before proceeding to a review of both scaled particle theory and fuzzy cylinder model theory, it would be useful to mention briefly the unperturbed wormlike (sphero)cylinder model which is the basis of these theories. Usually the intramolecular excluded volume effect can be ignored in stiff-chain polymers even in good solvents, because the distant segments of such polymers have little chance of collision. Therefore, in the subsequent reference to wormlike chains, we always mean that they are unperturbed . 

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. 

This result may be compared with Eq. (9.7), from which it follows that the persistence length is equal to half the length of a statistical chain element or Kuhn length ap = i A. This representation of the wormlike chain is of particular importance for the description of stiff polymers. 

The qnD values of cellulose and its derivatives lie between 3 and 25 nm and are larger than those of typical vinyltype polymer ( 1 nm), but markedly smaller than those of typical stiff chain polymers, such as DNA (Table 14)67). Thus, the chains of cellulose and its derivatives can be considered as semi-flexible. It may be concluded that both the pearl-necklace chain and the wormlike chain models are adequately applicable to these polymers. 

Note that. In a sense, the polyelectrolyte behaves now as If it were an "ideal" (Gaussian) chain with a relatively small number of Kuhn lengths. This coil size Is determined by the local stiffness, but not by long-range excluded volume. Should q Increase even furher (approaching L), then the wormlike chain would be better described by a slightly curved rod (as expressed by (5.2.21)) than by a random-flight chain. 

At a given force, the elasticity of covalent bonds of the amino acid backbone gives rise to a length increase. But thermal fluctuations act on the backbone, which on an average pulls the cantilever closer to the membrane, a phenomenon referred to as entropic elasticity of linear polymers. The wormlike chain model [50] describes the polymer as an elastic rod with bending stiffness submitted to thermal fluctuations that decrease the end-to-end distance of the rod. Alternatively, the freely jointed chain model calculates the... 

The relatively small region of allowable values of tp and j/ in polysaccharides linked between rings makes the wormlike chain model realistic for them. The polymer behaves as a random flight one over contour lengths S Lp, but as a stiff rod if S Lp. Persistence lengths can be fairly large 350 50 A for xanthan gum, 80-100 A for alginate or hyaluronate. ... 

Shear thinning behavior of dilute polymer solutions can be qualitatively explained by two distinct models (1) the molecule may be a non-deformable, highly elongated prolate ellipsoid which becomes oriented at high shear stresses (2) the molecule may be a stiff but nevertheless wormlike chain wTiich becomes oriented and deformed under high shear stresses. 

In contrast, a recent study from this laboratory ( 3) concludes that native xanthan molecules are better viewed as stiff but wormlike chains. This conclusion follows from measurements of zero-shear intrinsic viscosity for a homologous series of xanthans of different molecular weight for native xanthan the exponent z in the relation [n ] = KM is only 0.96 rather than 1.8 as expected for rigid rods. It is the goal of this paper to explore whether a wormlike model is consistent with other experimental data, especially the dependence of intrinsic viscosity on shear stress (non-Newtonian behavior). 

Up to this point we have confined ourselves to ideally flexible chains. Thus, the theories developed on the models of such chains (for example, the spring-bead chain) should no longer be adequate for polymers whose chemical stmcture suggests considerable stiffness of the chain backbone. Many cheiin models may be used to formulate a theory of stiff or semi-flexible polymers in solution, but the most frequently adopted is the wormlike chain mentioned in Section 1.3 of Chapter 1 it is sometimes called the KP chain. This physical model was introduced long ago by Kratky and Porod [1] to represent cellulosic polymers. However, significant progress in the study of its dilute solution properties, static and dynamic, has occurred in the last two decades. 

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. 

It can be shown that q is equal to the average projection of R, the end distance vector, of an infinitely long wormlike chain onto the tangent vector at the chain end. Hence, on average, an infinite wormlike chain with a larger q is more extended in the direction of its end tangent vector. This suggests that q can be taken as a measure of chain stiffness. 

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. 

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. 

Many other semi-flexible polymers are known which can be reasonably well modeled on wormlike chains. The Ml values obtained for most of such polymers are close to those calculated from crystallographic data, suggesting that stiff polymers in dilute solution maintain their crystalline conformations at least locally. 

Unquestionably, Yamakawa and collaborators have made a substantial contribution to the understanding of transport behavior of semi-flexible polymers in dilute solution. However, their theories still leave something to be desired, as revealed by the recent careful experiments mentioned above. Their formulation is essentially the combination of the the Kirkwood-Riseman hydrodynamics and the statistics of wormlike chains. As mentioned in Chapter 2, this hydrodynamics fails to be good for flexible chains, but we have seen that it seems to work well for stiff chains. The reason is that the Kirkwood-Riseman formalism gives the exact solution in the limit of rigid rods. 

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. 

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... 

When the chain length L is much larger than the persistance length a, the effect of chain stiffness becomes negligible. In the limit L oo we have P-)q/L 2a and the wormlike chain reduces to a freely-jointed chain. 

Another caveat was raised by theory, which revealed the main chain stiffiiess to decrease from the middle of the chain towards the ends. This relates to the fact that the steric repulsion between the side chains becomes less at both ends due to the hemispherical volume accessible. In addition, the motion of segments near the chain ends is less restricted [66, 70, 75-77]. Accordingly, application of the wormlike chain model with uniform chain stiffness is somewhat questionable, but for long main chains this effect is most probably not pronounced. 

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