Polymer ideal chain model

This is the most successful ideal chain model used to calculate the details of conformations of different polymers. In this model, bond lengths I and bond angles d are fixed (constant). 

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. 

Due to the screening effect in the volume exclusion of polymer chains, singlechain conformation in the concentrated solutions will exhibit the size scaling similar to the ideal-chain model, as... 

The simplest model to describe polymers is the ideal-chain model. For books on polymer physics where all the relevant background material can be found see [11-19]. In this model the polymer consists of M subunits, each with a fixed bond length b, and their orientation is completely independent of the orientation and positions of previous monomers, even to the extent that two different monomers can occupy the same position in space there is no excluded volume. This model plays the same role in polymer physics as an ideal gas in molecular physics. It allows to describe the polymer chain as a (Gaussian) random walk of M steps, as depicted in Fig. 2.8. 

An exponent of more than 1/3 makes sense here for a three-dimensional (3D) polymer in solution because the polymer fills a volume space incompletely (a solid material fills space with v = 1/3). This ideal chain model for our polymer is modified, however, when we start to think of behavior of a real polymer chain, for which there can be monomer-monomer and monomer-solvent interactions. 

Ideal chain model In polymer science, a model for the polymer chain in which the monomers are freely joined with no interactions between monomers. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

The models discussed so far describe ideal chains and do not account for interactions between monomers which typically consist of some short-ranged repulsion and long-ranged attraction. Including these interactions will give a different scaling behavior for long polymer chains. The end-to-end radius,... 

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

The simplest model of this type is called the freely jointed chain, and is illustrated in Figure 2.21. In it, the skeletal bonds are joined end to end, but are completely unrestricted in direction. This is clearly a situation not found in a real polymer (bond angles in real polymers are relatively fixed). It is also assumed that the chains have zero cross-sectional area, that is that the chains are unperturbed by excluded-volume effects. These effects arise because atoms of a chain exclude from the space they take up all other atoms from all other chains. They are related to excluded-volume effects occurring even in systems as simple as real gases. The expression for the mean-square end-to-end distance of such an idealized chain is particularly simple ... 

The active centers in this process are free radicals, whose reaction with double bonds of monomers leads to the growth of a polymer chain. In the framework of the ideal kinetic model, the reactivity of a macroradical is exclusively governed by the type of its terminal unit. According to this model, the sequence distribution in macromolecules formed at any moment is described by the Markov chain with elements controlled by the instantaneous composition of the monomer mixture in the reactor as... 

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. 

Since these first calculations, a variety of theoretical methods have been studied and applied to ideal chains of polyethylene and other polymers. The books by Ladik (1988) and Andre et al. (1991) describe theoretical methods employed, and the literature published prior to 1990. While these calculations provided some insights into the microscopic origins of the electronic properties of the polymers modelled, the correlation of predicted properties, e.g. the band gap and bandwidths, with experimental data was poor. The quality of the experimental data was also variable. 

The conformation of an ideal chain, with no interactions between monomers, is the essential starting point of most models in polymer physics. In this sense, the role of the ideal chain is similar to the role of the harmonic oscillator or the hydrogen atom in other branches of physics. 

One of the simplest models of an ideal polymer is the freely jointed chain model with a constant bond length / = ri and no correlations between the directions of different bond vectors, (cos 0y) = 0 for i 7 j. There are only n non-zero terms in the double sum (cos 6 = 1 for i — j). The mean-square end-to-end distance of a freely jointed chain is then quite simple ... 

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