Semiflexible chains

Let us consider a wormlike chain discussed in Section 2.4 to be trapped inside a cylindrical pore of diameter Dp. The bending energy of a wormlike chain is 

Similarly, the mean square projection of the end-to-end distance vector on the direction u(0) of the first bond is obtained from Equation 2.55 (Yamakawa 1997)as 

With z-coordinate along the pore axis, and x- and y-axes being perpendicular to the pore axis, the mean square end-to-end distance is given by 

We define the arc length s, at which a semiflexible chain-contour collides with the pore wall on an average, as the deflection length (Odijk 1983), satisfying the relation of Equation 5.55 

Let us now consider a semiflexible chain of length L larger than its persistence length tp, which in turn is larger than the deflection length Xd 

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A.G. Cherstvy and R.G. Winkler, Complexation of semiflexible chains with oppositely charged cylinder. J. Chem. Phys. 120, 9394-9400 (2004). 

The focus of this chapter is on an intermediate class of models, a picture of which is shown in Fig. 1. The polymer molecule is a string of beads that interact via simple site-site interaction potentials. The simplest model is the freely jointed hard-sphere chain model where each molecule consists of a pearl necklace of tangent hard spheres of diameter a. There are no additional bending or torsional potentials. The next level of complexity is when a stiffness is introduced that is a function of the bond angle. In the semiflexible chain model, each molecule consists of a string of hard spheres with an additional bending potential, EB = kBTe( 1 + cos 0), where kB is Boltzmann s constant, T is... 

Figure 3. Normalized density profiles of semiflexible chains at a hard wall for N = 20 and T] = 0.3.

Figure 5. End-site density profiles for semiflexible chains normalized to the value in the middle region.

Flexible and semiflexible chains for the connection of carbazole units have been used by Braun et al. [43, 44]. However, it can clearly be seen that introducing longer alkyl spacers lowers the Tg. In addition to carbazoles, the... 

An increase of g in the theta state with respect to the ideal values is similarly obtained by Ganazzoli et al. [52,53] through the use of a theoretical approach based on the self-consistent minimization of the intramolecular free energy. Their results indicate a significant expansion of the star arms due to the core effects. The same type of calculations have later been used to describe the star contraction in the sub-theta regime [54]. Guenza et al. [55] described a star chain at the 0 point as a semiflexible chain with partially stretched arms that take into account the star core effect. Their results are also consistent with experimental data. 

In the second half of this article, we discuss dynamic properties of stiff-chain liquid-crystalline polymers in solution. If the position and orientation of a stiff or semiflexible chain in a solution is specified by its center of mass and end-to-end vector, respectively, the translational and rotational motions of the whole chain can be described in terms of the time-dependent single-particle distribution function f(r, a t), where r and a are the position vector of the center of mass and the unit vector parallel to the end-to-end vector of the chain, respectively, and t is time, (a should be distinguished from the unit tangent vector to the chain contour appearing in the previous sections, except for rodlike polymers.) Since this distribution function cannot describe internal motions of the chain, our discussion below is restricted to such global chain dynamics as translational and rotational diffusion and zero-shear viscosity. 

The conformational properties of an uncharged molecular chain are well described by a (discrete) semiflexible chain model [33]. The chain is comprised of mass points, each one may represent several monomers, at positions r, (z =0,..., N). The (average) length of a bond is l. The partition function of such a chain is given by... 

Ozbas, B., Rajagopal, K., Schneider, J.P., and Pochan, D.J. "Semiflexible chain networks formed via self-assembly of fl-hairpin molecules". Phys. Rev. Lett. 93(26), 268106 (2004). 

Various models are available [20] to describe semiflexible chains. Some are based on expansions either close to the gaussian coil or to the rigid rod limits, while others interpolate between these two chain stiffness limits. One of these, the sliding rod model [19], is described here because of its inherent simplicity. 

Fig. 3.6. Dependences of the chain size (gyration radius) on the inverse temperature calculated through Monte Carlo simulations. (Top) A semiflexible chain with contour length L/a = 512 and Kuhn length l/a 20, and (bottom,) a flexible chain (l/a 2) with the same contour length. The error bars represent the standard deviations. The insets show snapshots of (a) coil states and (b) folded states...

Fig. 3.7. (Left) Schematic image of a torus. (Right) Double-logarithmic plot of the torus size, average radius R, and thickness r vs. chain length L (3.4) with parameters 7a2/T = 4 and l/a = 15. Also shown are the results for a charged semiflexible chain (cf. Sect. 3.3.4 and [32] for more details)...

Using the results of Sects. 2 and 3, we will consider in Sects. 4 and 5 the intramolecular liquid-crystalline ordering of the segments of one semiflexible chain. 

Thus, in the athermal limit the only difference between the equilibrium free energies of the solutions of separate rods and long chains of rods is due to the translational entropy term. Consequently, we can immediately conclude (analogously to Sect. 2) that the liquid-crystalline transition for the athermal solution of semiflexible chains takes place at 1/p. 

This system of rigid blocks with flexible spacers may serve as a model of polymers with a limited flexibih ty. In his early work Flory considered the behavior of semiflexible chain polymers by introducing the flexibility parameter / which represents the fraction of bonds which are not in a colhnear position in the... 

V2 = 1). The transition (partial or complete) into the liquid crystalline state occurs only after the system is heated above the glass-transition point. For real polymeric systems with semiflexible chains, the liquid crystalline state in the initial solution often is not realized, so the formation of nonequilibrium amorphous polymer upon the introduction of a nonsolvent is quite probable. 

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