Worm-like polymer chains

FIGURE 21.4 Nanofishing of a single polystyrene (PS) chain in cyclohexane. The solvent temperature was about 35°C (0 temperature). A cantilever with a 110 pN nm spring constant was used. The worm-like chain (WLC), solid line, and the freely jointed chain (FJC), dashed line models were used to obtain fitting curves. (From Nakajima, K., Watabe, H., and Nishi, T., Polymer, 47, 2505, 2006.)... 

As for further analysis, curve fitting against the worm-like chain (WLC) model was conducted and indicated as a solid line in Figure 21.4. The model describes single polymer chain mechanics ranging from random-coil to fully extended forms, as follows ... 

As discussed in the last 3 years, polysaccharides behave in solution under a worm like chain [26] the local stiffness of the chain is characterized by a persistance length (Ip) the larger Ip is, the larger the chain deviates from the gaussian behaviour in the usual molecular weight range of these natural polymers [27], This makes difficult to use the relations given in litterature for synthetic... 

The molar mass dependence of the intrinsic viscosity of rigid chain polymers cannot be described by a simple scaling relation in the form of Equation (36) with molar mass independent of K and a. over a broad molar mass range. Starting from the worm-like chain model, Bohdanecky proposed [29] the linearizing equation... 

The analysis described above is useful for modelling colligative properties but does not address polyelectrolyte conformations. Polyelectrolyte conformations in dilute solution have been calculated using the worm-like chain model [103,104], Here, the polymer conformation is characterized by a persistence length (a measure of the local chain stiffness) [96]. One consequence of the... 

Equations (4) and (5) show that when the parameter x = 2 L/A changes from 0 to the hydrodynamic properties of a worm-like chain change from those of a thin straight tod to those of an undrained Gaussian coil. In accordance with this the dependence of intrinsic viscosity (nl and diffusion coefficient D on molecular weight M of a rigid-chain polymer cannot be described by the usual Mark-Kuhn dependence... 

Vm 0.57 and poo =, ym 1, respectively. Curve C corresponds to the Noda-Hearst theory. Experimental data are in good agreement with Curve A showing that for these polymers a kinetic-ally rigid worm-like chain is an adequate model for the dynamooptical properties of their molecules. 

Table 8). This permits the interpretation of experimental data by using the electro-optical properties of flexible-chain polymers in terms of a worm-like chain model However, EB in solutions of polyelectrolytes is of a complex nature. The high value of the observed effect is caused by the polarization of the ionic atmosphere surrounding the ionized macromolecule rather than by the dipolar and dielectric structure of the polymer chain. This polarization induced by the electric field depends on the ionic state of the solution and the ionogenic properties of the polymer chain whereas its dependence on the chain structure and conformation is slight. Hence, the information on the optical, dipolar and conformational properties of macromoiecules obtained by using EB data in solutions of flexible-chain polyelectrolytes is usually only qualitative. Studies of the kinetics of the Kerr effect in polyelectrolytes (arried out by pulsed technique) are more useful since in these... 

The orientational mechanism of EB in solutions of r id-chain polymers and the possibility of determining rotatory diffusion constants of their molecules from dispersion curves may be utilized for the characterization of equilibrium conformational properties of their drains. The theory of rotational friction of kinetically rigid molecules developed by Hearst makii% use of the statistics of worm-like chains can be employed for this purposes. The results of this theory for the two limiting cases of molecular conformation refering to the slightly bent rod and the worm-like coil are expressed by Eqs. (27) and (28) (Sect. 2.3). 

Equation (85) represents a general relationship between the Kerr constant K and the dipolar and optical properties of a kinetically rigid particle. To establish the quantitative dependence of K on the conformation and structure of a rigid-chain polymer molecule, the molecular model describing its electro-optical properties should be specified. For this purpose, we use a kinetically rigid worm-like chain, just as for the study of the FB problem. 

The values of S obtained in this manner (Table 14) and those obtained by other methods (Tables 1 and 9) are close to each other within experimental error whereas the values of mo (Table 14) and the values that could be expected taking into account the structure of the main chain of tl se polymers are in reasonable agreement This means that equilibrium dielectric properties of rigid-chain polymer solutions can be adequately described in terms of the model of a worm-like chain according to Eqs. (86) and (87). 

Problem 2.4 Suppose a worm-like chain with diameter = 1.5 nm, persistence length kp = 100 nm, and length L = 20 nm is dissolved in a solvent. Estimate 02, the minimum volume fraction"of polymer needed to form a wholly nematic phase, it L is increased to 500 nm, and d and kp are kept the same, what is 02 ... 

The freely jointed chain model is most appropriate for synthetic polymers, such as polyethylene and polystyrene. For other molecules, such as DNA and polypeptides, the molecular flexibility is better described by the worm-like chain model (described in Section 2.2.4), whose force law can be approximated by a simple expression due to Marko and Siggia (1995), namely. 

Figure 11.3 Schematic drawings of a worm-like polymer chain (with continuous flexibility) and a Kuhn chain (with rigid links joined at hinges that allow free rotation about the angle between the links). The length of the links has been chosen so that the contour length and mean-square end-to-end length of the Kuhn chain are the same as those of the worm-like chain. (From Donald and Windle 1992, with permission from Cambridge University Press.)...

A qualitatively different mechanism of flexibility of many polymers, such as double-helix DNA is uniform flexibility over the whole polymer length. These chains are well described by the worm-like chain model (see Section 2.3.2). 

The worm-like chain model (sometimes called the Kratky-Porod model) is a special case of the freely rotating chain model for very small values of the bond angle. This is a good model for very stiff polymers, such as double-stranded DNA for which the flexibility is due to fluctuations of the contour of the chain from a straight line rather than to trans-gauche bond rotations. For small values of the bond angle ( < 1), the cos 9 in Eq. (2.23) can be expanded about its value of unity at = 0 ... 

A comparison of the theoretical expectations of worm-like liquid crystalline polymers and experimental data is made in Figure 2.14. The abscissa is the ratio of total length to persistence length L/l the ordinate is the critical volume fraction in the unit of the ratio of molecular diameter to persistence length. The theoretical expectation is taken from Khokhlov et aVs theory (Khokhlov Semenov, 1982 Odijk, 1986) on the worm-like chains. 

Unfortunately, there are very few discussions about the viscosity behavior for semi-flexible liquid crystalline polymers. Semenov (1988) calculated the viscosities of worm-like chains and found that 0 3/02 of this liquid crystalline polymers is always positive and decreases as the order parameter increases. The conclusion is different from that of rod systems. 

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