Stiff chains wormlike

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. 

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. 

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). 

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. 

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 ... 

The Yamakawa-Fujii wormlike cylinder model [25,26] has been widely used for estimation of I (or q) of stiff chain materials from intrinsic viscosities. This approach is based on the equation... 

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. 

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. 

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 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 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. 

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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