Infinite chain models

This representation fits into the more general treatment of prochirality and prostereoisomerism (42, 50-52). Using the infinite chain model the + and — bonds must be considered enantiotopic (53) because they are related by a mirror plane perpendicular to the chain axis and passing through every tertiary carbon atom. 

It is evident that consideration of terminal groups brings the model nearer to the state of real polymers but reduces its predictive ability in regard to chirality. The set of chiral stractures in accord with the infinite chain model is much more restricted and more clearly defined than that predicted using other models. Such a model represents the most severe test to be passed in the design of a chiral polymeric structure. The following analysis will, therefore, be carried out using the infinite chain model. 

The treatment of irregular and atactic polymers, and, in general, real polymeia (with statistical distribution of molecular weight and defects) is much more complex. With regard to atactic vinyl polymers it is not possible to recognize the presence of mirror symmetry, not even in the infinite chain model. 

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. 

Once the Lanczos coupling constants an,pn have been computed, we could construct the Green function TZ(u) = c0(< 0 ul — UJ cho), which is defined in Eq. (49). If in the formula (49) for 7Z(u), we use the Lanczos representation for the matrix Ml - U based on Eq. (60) within the infinite chain model, we shall have ... 

By combining the information drawn from ab initio (mode symmetry and frequency) and the infinite chain model (ky and frequency), the LAM modes above 200 cm can be unambiguously assigned [14]. Table 10.1 lists the assignments from pentane (n = 5) to pentacosane (n = 25). Fig. 10.8 shows the observed and calculated LAM modes, the dashed line is a polynomial fit and is the best available approximation to the V5 dispersion curve of polyethylene. The INS data for k > 0.7 is absent because it is not possible to assign the modes owing to the spectral... 

The Newtonian equations of motion for individual segments are integrated numerically using the Verlet algorithm [11] with a time step of 10 fs (or 6 fs in the case of infinite chain model) at constant energy and volume. The temperature T is evaluated from the average kinetic energy per atom, and the pressure by... 

If we restrict ourselves to the infinite chain model, the analysis of the derived Fisher projection of an infinite chain indicates that this is chiral when the symmetry plane containing the chain, as well as those perpendicular to the chain and that with translation containing the chain are all lacking. For finite length chains, chirality is granted by the lack of a symmetry plane containing the chain and of the plane perpendicular to the chain in its central point. 13... 

All these observations suggest that tactic macromolecules of the above type are substantially, even sometime not formally, achiral and their symmetry properties are better represented by the infinite length chain. Thus in order to prepare macromolecules capable of displaying stable and appreciable chiro-optical properties in all conditions, the symmetry elements rendering the infinite chain model achiral must be eliminated. 

In Fig. 4.11 we illustrate the alternating and in-phase stretching modes of an infinite chain model in a planar zig-zag form. Using the same procedure... 

Recently, des Cloizeaux has conceived a rubber -like model for the transition regime to local reptation [56]. He considered infinite chains with spatially fixed entanglement points at intermediate times. In between these... 

In contrast to the concise account of Bose and Foo (1974), here, a detailed derivation is provided of the matrix elements of the Greenian for an infinite semiconductor, modelled as a 1-dimensional chain, with s-orbitals on the (even) A sites and p-orbitals on the (odd) B sites (see Fig. K.l). The site energies on the even (odd) sites are taken to be (Xa((Xb) and the bond energies to be / 2, resulting in a Hamiltonian of the form... 

Rubber-like models take entanglements as local stress points acting as temporary cross finks. De Cloizeaux [66] has proposed such a model, where he considers infinite chains with spatially fixed entanglement points at intermediate times. Under the condition of fixed entanglements, which are distributed according to a Poisson distribution, the chains perform Rouse motion. This rubber-like model is closest to the idea of a temporary network. The resulting dynamic structure factor has the form ... 

In this article the use of formula 3 will always refer to rotated Fischer projections. In any case, the formulas must represent a section of chain long enough to illustrate the structural features excluding, unless explicitly required, the terminal groups. This representation corresponds to the use of a macromolecular model with an infinite chain length. 

Recently, Bockris and co-workers (6, 7, 9) have modified the above model to satisfy the experimental findings outlined in (1) to (5). In place of infinite chains and sheets, smaller anions the sizes of which are directly proportional to the concentration of Si02 in the melt, were postulated. Thus at 33 mole-% metal oxide, the anions present may be (SieOnn)6- or (SisCho)8-- Although this later ionic model is entirely compatible with available physicochemical data, it must be emphasized that, at present, direct proofs of the existence of the anions postulated are lacking. 

The generator matrix treatment of simple chains with excluded volume described earlier S 010) properly reproduces the known chain length dependence of the mean-square dimensions in the limit of infinite chains. The purpose of this paper is to compare the behaviour of finite generator matrix chains with that of Monte-Carlo chains in which atoms participating in long-range interactions behave as hard spheres. The model for the unperturbed chain is that developed by Flory et at. for PE (S 027). 

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