Semiflexible chain models

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... 

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... 

Table 2. Mean-square end-to-md distance, in units of the hard core diameter, as a function of packing fraction and N. pmc f the results of [99) using self-consistent PRESM/Monte Carlo based on Eq. (10.2) for a hard core tangent semiflexible chain model [24]. c are the results of many chain exact Monte Carlo simulations of Yethiraj and Hall [29]. The corresponding density-independent self-avoiding walk (ideal walk) values of are 50.78 (30.79), 152.03 (80.8), and 348.51 (164.12) for N = 20, 50, and MX), respwlively...

The thermodynamic behavior of binary blends of the semiflexible chain model (4 = dl2), discussed in Sections III.A.2 and IV.C, have been thoroughly investigated numerically using PRISM theory within the HTA framework." " Calculations have focused on the experimentally rele-... 

The basic theory of star polymer fluids developed by Grayce and Schweizer is general in its ability to treat polymer models of variable chemical detail. For simplicity, we discuss the theory in the context of the tangent, semiflexible chain model. As true for most of the results discussed in Section VIII, the bare bending energy is set equal to zero, and pure hard-core interactions (athermal or good solvent conditions) are employed in numerical studies carried out so far. 

The radial distribution functions from SC/PRISM calculations and MD simulations are shown in Fig. 4 for the semiflexible chain model with N=50 repeat units per chain for different bending potentials, Eq. (5). The fully flexible chain shows... 

The fact that the agreement is somewhat better in this FJC polyethylene model indicates that internal constraints cause some difficulty in SC/PRISM theory. As seen in the semiflexible chain model, the bending and torsion constraints probably cause some local nematic ordering in the melt which cannot be captured in the present theory [59]. 

The wall-PRISM equation has been implemented for a number of hard-chain models including freely jointed [94] and semiflexible [96] tangent hard-sphere chains, freely rotating fused-hard-sphere chains [97], and united atom models of alkanes, isotactic polypropylene, polyisobutylene, and polydimethyl siloxane [95]. In all implementations to date, to my knowledge, the theory has been used exclusively for the stmcture of hard-sphere chains at smooth structureless hard walls. 

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. 

Now let us discuss the applicability of the results obtained for other models of semiflexible macromolecules. It is clear that the qualitative form of the phase diagram does not depend on the model adopted. The low-temperature behavior of the phase diagram is independent of the flexibility distribution along the chain contour as well, since at low temperatures the two coexisting phases are very dilute, nearly ideal solution and the dense phase composed of practically completely stretched chains. The high temperature behavior is also universal (see Sect. 3.2). So, some unessential dependence of the parameters of the phase diagram on the chosen polymer chain model (with the same p) can be expected only in the intermediate temperature range, i.e. in the vicinity of the triple point. 

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... 

In the second type of semiflexible polymer molecule, rigid units are interspersed with flexible ones. Some examples of molecules of this kind are given in Chapter 11. The freely jointed chain model might be a suitable model for such semiflexible polymers. If a persistently flexible molecule and a freely jointed molecule are characterized by the same values of L and Xp (or, equivalently, of bx and Nk), then the gross statistical measures of the coil dimensions of the two such isolated chains will be the same, despite the differences in the type of flexibility. 

The real liquid crystalline polymers exhibit finite flexibility. This kind of polymer was studied extensively by Khokhlov and his co-workers. Assume that the semiflexible chain has the total contour length L, and Kuhn length l and diameter D, and L l D. Analogous to the Onsager model, the free energy of the polymer in solution is composed of the conformational... 

We have discussed the ideal-chain model in Sect. 2.2 by incorporating short-range restrictions into the freely-jointed-chain model first the fixed bond angles, then the hindered internal rotation. In this way, we reached the description of semiflexibility of the real polymer chains. The mean-square end-to-end distances of chains in different models are given below. 

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