Chain, wormlike

The rigidity depends on the solvent, temperature, and additives. When the chain is straight, the polymer is called a rodlike molecule. 

Two limiting cases are interesting. When Lp L i.e., the chain is either sufficiently rigid or short, 

3 Radius of Gyration To calculate the radius of gyration of the wormlike chain, we use a slightly different version of Eq. 1.25  

In the short-chain or the rigid-chain limit (LfLp 1), 

In the long-chain limit or the flexible-chain limit (LjLp 1), 

Analogous to Equation 2.1, the mean square end-to-end distance of the Kuhn chain is 

Since the adjacent Kuhn steps are freely rotating, the projection of a Kuhn step onto the direction of the preceding step is cos 0, so that the correlation function for the segmental orientations becomes 

By using this result and taking the definition of the persistence length to be the projection of the end-to-end distance vector on the first step (Cantor and Schimmel 1980, Yamakawa 1997), 

Clearly, as the parameter 0 approaches zero, the persistence length becomes very large. For fixed values of ip and the contour length L = Ni for a specific 

Therefore, the orientational correlation function follows from Equations 2.51 and 2.54 as 

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Hongmei Jian. A Combined Wormlike-Chain and Bead Model for Dynamic Simulations of Long DNA. PhD thesis. New York University, Department of Physics, New York, New York, October 1997. 

The sedimentation coefficient for wormlike chains was first worked out by Hearst and Stockmayer [123], later improved by Yamakawa and Fujii [124] to give this expression for s ... 

For poly electrolyte solutions with added salt, prior experimental studies found that the intrinsic viscosity decreases with increasing salt concentration. This can be explained by the tertiary electroviscous effect. As more salts are added, the intrachain electrostatic repulsion is weakened by the stronger screening effect of small ions. As a result, the polyelectrolytes are more compact and flexible, leading to a smaller resistance to fluid flow and thus a lower viscosity. For a wormlike-chain model by incorporating the tertiary effect on the 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. 

The most austere representation of a polymer backbone considers continuous space curves with a persistence in their tangent direction. The Porod-Kratky model [99,100] for a chain molecule incorporates the concept of constant curvature c0 everywhere on the chain skeleton c0 being dependent on the chemical structure of the polymer. It is frequently referred to as the wormlike chain, and detailed studies of this model have already appeared in the literature [101-103], In his model, Santos accounts for the polymer-like behavior of stream lines by enforcing this property of constant curvature. 

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... 

H. Yamakawa, and W. H. Stockmayer, Statistical mechanics of wormlike chains. II. Excluded volume effects. J. Chem. Phys. 57, 2843-2854 (1972). 

Klenin, K., Merlitz, H., and Langowski, J. (1998) A Brownian dynamics program for the simulation of linear and circular DNA and other wormlike chain polyelectrolytes. Biophys. J. 74, 780-788. 

Allison, S.A. (1986) Brownian dynamics simulation of wormlike chains. Fluorescence depolarization and depolarized light scattering. Macromolecules 19, 118-124. 

Allison, S.A., Sorlie, S.S., and Pecora, R. (1990) Brownian dynamics simulations of wormlike chains Dynamic light scattering from a 2311 base pair DNA fragment. Macromolecules 23, 1110-1118. 

Thermotropic liquid crystalline polymers, like polyesters containing mesogenic units on the main chain, may not be described by the wormlike chain model (cf. Sect. 1.2). The present article does not consider this type of polymers. 

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. 

Liquid-Crystalline Polymers Viewed as the Wormlike Chain... 

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 . 

Hentschke [45] and DuPre and Yang [46] compared Kubo and Ogino s data with Lee s theory [53] extended to a (monodisperse) wormlike chain system by using ct(N) (cf. Sect. 2.3). Hentschke took d to be 1.6 nm, which is consistent with the value estimated from the partial specific volume (cf. Table 2). Though good for the middle and highest molecular weight samples in the... 

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. 

Miller and Flory (43) calculated of a-helices by taking bond-angle fluctuations into account. This approach gives the molecular rod a flexibility similar to that of wormlike chains but retains the a-helical conformation of the chain. Thus, in a sense, it may be regarded as a refinement of the treatment of Moha et al. The calculated results gave a nonlinear chain-length dependence of experimental results shown in Fig. 21. 

Investigations are presented concerning the relationship between the RIS and wormlike chain models. 

A comparison is presented between the behavior of unperturbed stars of finite size whose configurational statistics are evaluated by R1S theory and the Kratky-Porod wormlike chain model. Emphasis Is placed on the initial slopes of the characteristic ratio, C, or g when plotted as a function of the reciprocal of the number of bonds, n. 

By now it may have dawned on the reader that the long-time Rouse spectrum (i.e., proportionality of xp to p 2) is to be expected for any chain model in which the correlation lengths for both equilibrium conformations and frictional processes are small compared to the chain dimensions (and thus to the wavelength of the slow normal modes). A possible exception is that of the continuous wormlike chain of invariant contour length, which has been studied by Saito, Takahashi, and Yunoki.33 In this latter case, the low-frequency spectrum makes xp proportional to p A, which resembles our special one-dimensional model in the limit 1 — p 1. 

The first important question to be answered concernes the lower limit of molecular weight down to which the concepts obtained for long Gaussian or latticelike chains were applicable. It is clear that a Gaussian chain does not adequately describe the conformational properties of short oligomer chains. Other models e.g. a model of the wormlike chain may be more suitable. The introduction of this model may lead to considerable mathematical complications and the determination of Kd may become difficult. 

Garcia Molina, J.J., Lopez Martinez, M.C. and Garcia de la Torre, J. (1990) Computer simulation of hydrodynamic properties of semiflexible macromolecules Randomly broken chains, wormlike chains, and analysis of properties... 

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 ( ) ... 

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