Segment chain model

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

Doi and Edwards (1978, 1979, 1986). They started with the Rouse-segmented chain model for a polymer molecule. Because of the presence of neighboring molecules, there are many places along the chain where lateral motion is restricted, as shown in Fig. 21. To simplify the representation of these restrictions, Doi and Edwards assume that they are equivalent to placing the molecule of interest in the tube as shown in Fig. 22. This tube has a diameter d and length L. The mean field is represented by a three-dimensional cage. The primitive chain can move randomly forward or backward only along itself. For a monodisperse polymer, the linear viscoelasticity is characterized by... 

A definite relationship exists between the end-to-end chain distance and the radius of gyration for linear (nonbranched) chains (with or without fixed valence angles or free rotation). The relationship is derived in the appendix of this chapter for the segment chain. It can be seen from Equations (4-24), (4-25), and (4-26) that on transferring from the segment-chain model to either... 

This model was introduced by Frisch, Schuerch, and Szwarc (38) who discussed the chirality of vinyl polymers assuming that effects due to the chain length and the nature of the terminal groups were negligible. Arcus (58) spoke of nonterminal chain segments. The infinite chain model was explicitely used by Natta, Danusso, Corradini, Farina, and others (30-32). Natta, Pino, and Mazzanti cited this model in 1957 (35). It represents a simplification of the one with different terminal groups used by Natta in his first paper on isotactic polymers. 

Clark, M.B., Zimm,B.H. A linearized chain model for dielectric loss in polymers. ACS Polymer Preprints 12, 116-120 (1971). See also Tobolsky,A.V., DuPre,D.B. Macromolecular relaxation in the damped torsional oscillator and statistical segment models. Advan. Polymer Sci. 6,103-127 (1969). 

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]. 

Thus all segment correlations of our noninteracting model take the form of Gaussian functions. We stress that for our Gaussian chain model all these results are valid rigorously for all n > 1 or k2 — k] > 1, respectively. 

Even for d < 4 the question of existence of the continuous chain limit is not completely trivial. The problem is most easily analyzed by taking a Laplace transform with respect to the chain length, which results in the held theoretic representation of polymer theory. In field theory it is not hard to show that the limit — 0 can be taken only after a so-called additive renormalization we first have to extract some contributions which for — 0 would diverge. The extracted terms can be absorbed into a 1 renormalization he. a redefinition of the parameters of the model. Transfer riling back to polymer theory we find that this renormalization just shifts the chemical potential per segment. We thus can prove the following statement after an appropriate shift of the chemical potential the continuous chain limit for d < 4 can be taken order by order in perturbation theory. In this sense the continuous chain model or two parameter theory are a well defined limit of our model of discrete Gaussian chains. 

The Edwards Hamiltonian is an appealing but most formal object. To mention a simple fact, shrinking to zero the segment size of the discrete chain model as done in the continuous chain limit, we in general get a continuous but not differentiable space curve. Strictly speaking the first part, of Vj, does not exist. Further serious mathematical problems are connected to the (5-function interaction. Hie question in which sense Ve is a mathematically well defined object beyond its formal perturbation expansion is ari interesting problem of mathematical physics. 

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... 

The visual user interface is clustered into four main segments (cf. Fig. 35) general setup, value chain model, external parameters and evaluation. The general setup items and a subset of the external parameters (e.g., transportation costs, exchange rates) can be used across multiple value... 

In real systems, the number of spin centers N is, of course, too large to deterministically solve the corresponding eigenvalue problem of a corresponding spin Hamiltonian. Thus, no exact solution exists. Several approximate expressions were developed, such as the Bonner Fisher finite-chain model for equidistant antiferro-magnetically coupled S = 1/2-based chains.20 Here, the susceptibility is calculated for finite chain segments (ca. N 10 spin centers) and extrapolated to an infinite chain (N > oo). The extrapolated expression for the susceptibility is as follows ... 

Another method leading to nonaggregating copolymers may be connected with the molecular design of their monomeric units. We have discussed an extended variant of the HP model, the HA side-chain model [97], that explicitly takes into account the amphiphilic nature of hydrophilic segments. 

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. 

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