Freely jointed segments model

Let us now at first treat in more detail the ideal chains, on the basis of the introduced freely jointed segments model. We choose numbers, from 0 to iVg,... 

The coarse-grained representation of a given polymer by the freely jointed segments model is not unambiguous. There is only the requirement to select the subchain so that its contour length is large compared to the persistence length /pg. Different choices of subchains imply diflFerent values of Ns and Ug. All pairs of values, however, have to lead to the same value of Rq. For two different choices, with parameters iVg,ag and iVg, a, we have... 

It is possible to remove the arbitrariness in the choice of the freely jointed segments-model by imposing a second condition. For this purpose, the length of the real chain in the fully extended straight form, denoted jRmaxj may be employed, and a second condition formulated as... 

Considering the large variation of / for the poly[2]catenand 51b, it is expected that little correlation will exist between the spatial orientation of neighboring monomer segments and that it will represent the closest synthetic equivalent of the freely jointed chain model [63]. In this model, a real polymer chain is replaced by an equivalent chain consisting of N rectilinear segments of length Z, the spatial orientations of which are mutually independent (Scheme 24) [63]. 

Figure 6.14 Picture of a linear polymer in the ideal freely jointed segments chain model.

One such model is the ideal freely jointed segments chain (Fig. 6.14). In this model the polymer is considered to consist of a chain of n links. We call each chain link a subunit . Each subunit has a length l. This parameter / can correspond to the length of a monomer but it can also be shorter or longer. The angle between adjacent chain links is taken to be arbitrary. The chain forms a random coil. To characterize the size and volume of such a coil we use the mean square of the end-to-end distance R2. The square-root of this value — we call it the size of a polymer chain — is given by... 

For ease of calculation, we make a number of simplifying assumptions. These are relaxed in advanced treatments of the subject. First, rather than requiring tetrahedral bonds at each vertex of the chain, we allow all bond angles and assume that these are randomly distributed. Second, we ignore any excluded volumes or interactions between the segments of the chain. In this sense, our calculation is similar to the Bernoulli model of the ideal gas, which neglects intermolecular interactions. Our approximation is called the freely jointed chain model. 

Formulae (1.2) and (1.3) determine the model of a freely-jointed segment chain, which is frequently used in polymer physics as a microscopic heuristic model (Mazars 1996, 1998, 1999). A Kuhn segment in the flexible polymers (polyethylene, polystyrene, for example) usually includes a few monomer units, so that a typical length of the Kuhn segment is about 10 A or 10-7 cm and, at the number of segments 2 = 104, the end-to-end distance (R2)1 2 of a macromolecule is about 10-5 cm. 

In such a way, there are two universal, (that is, irrespective of the chemical nature) methods of description of a macromolecule either as a flexible thread or as freely-jointed segments. Either model reflects the properties of each macromolecule long enough to be flexible. A relation... 

It is noteworthy that Eq. (3.1), as well as the other results of this section, can be applied not only to the model of freely jointed segments but also to any other model of semiflexible macromolecules it is necessary only to replace p in all equations by the ratio of the effective segment length to its width. In fact, the translational entropy... 

In the simplest picture, the polymer molecule can be considered as a chain consisting of N segments with length b, each of which is free of any constraint to orient in an arbitrary direction. In other words, the orientation of a segment is totally independent of other segments in the chain and is random. Such a chain is referred to as the freely jointed chain model. 

We have proved Equation (6.11) only for a particular model of polymer, with independent, freely-jointed segments. Is the formula valid for other models (including random walks) We would need a special investigation to find out. The investigation, however, can be reduced to a very simple argument. 

The right-hand side of (132) indicates that the segment volume fractions are uniquely computed from the segment potentials. As mentioned above, we implement a freely jointed chain model, which ensures the chain connectivity, but which does not prevent backfolding of the chain to previously occupied lattice sites. For this chain model, the volume fractions can be computed efficiently using a propagator formalism, which is intimately related to the Edwards equation [91]. 

The first SCF result that we discuss here is shown in Fig. 7. In this case, bd has been varied from 0 to 1.5, with steps of 0.5, for the case that the lengths of the two chains is varied under the constraint that the sum of the two is fixed to 200. The most simple system is found for Nb=Nd = 100 and Xwi = 0. In this case, we have a spherical homodisperse, athermal brush. For this situation, the (dimensionless) segment potential is simply given by u r) = ln[l - (ps(r)] [73]. In short, within a freely jointed chain model, we generate all possible conformations of the polymer chains with the constraint that the first segment is positioned at r = 6 (next to the surface). Depending on the positions visited by, e.g. a conformation c, we can exactly enumerate the potential felt by this conformation Mc. The statistical weight of... 

The model used to describe these results was the freely jointed chain model of the polymer molecule, originally due to Kuhn and Gruen, which treats the molecule as a chain of independent segments of length Z. The force of retraction is then entropic and given by the Langevin function... 

Models for Transverse Relaxation. The simplest model to describe chain statistics from the NMR point of view is a chain of freely jointed segments of fixed length (12). Such a chain maybe rescaled in several hierarchical steps according to the time scale of the motions which takes place at different space scales, compared to the time scale defined by the NMR spin interactions. All intrachain motions are assumed to be fast enough to average elementary interactions, whereas junction average positions are static. 

The freely-jointed chain model (Fig. 2a) is the simplest it has been described as the ideal gas of polymer physics, as all interactions between the chain segments, except the chemical bonds connecting them, are neglected. The model represents a chain as a sequence of steps, or rectilinear statistical (Kuhn) segments, connected together at their ends. Each statistical segment represents a sequence of several chemical bonds. 

Equations (2) and (3) determine the model of a freely-jointed segment chain which is frequently used in polymer physics as a microscopic heuristic model [16]. 

To consider the relative permittivity tensor of polymeric system, one makes use of the heuristic model of a macromolecule as freely-jointed segments each macromolecule consists of z segments and is surrounded by solvent molecules (Sect. 2.1). Then, the simple old-fashion [111, 112] speculations allow us to determine the relative permittivity tensor of polymeric system in terms of the mean orientation of anisotropic segments of the macromolecules (6,6 ). The relative permittivity tensor is formulated below to within first-order terms in the orientation tensor... 

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