Ideal chains self-similarity

An increase of g in the theta state with respect to the ideal values is similarly obtained by Ganazzoli et al. [52,53] through the use of a theoretical approach based on the self-consistent minimization of the intramolecular free energy. Their results indicate a significant expansion of the star arms due to the core effects. The same type of calculations have later been used to describe the star contraction in the sub-theta regime [54]. Guenza et al. [55] described a star chain at the 0 point as a semiflexible chain with partially stretched arms that take into account the star core effect. Their results are also consistent with experimental data. 

This description is a manifestation of the self-similarity (fractal nature) of polymers, discussed in Section 1.4. The fractal nature of ideal chains leads to the power law dependence of the pair correlation function g(r) on distance r. This treatment for the ideal chain can be easily generalized to a linear chain with any fractal dimension V. The number of monomers... 

Since both ideal and real chaitis are self-similar fractals, the same scaling applies to subsections of the chains of size r containing n monomers ... 

The examples of self-similar functions considered above fall into an especially simple category all are functions of the generalized time or group parameter and functionals (either linear or nonlinear) of the initial condition. However, there are many self-similar physical properties and mathematical objects that depend on additional, unsealed variables. For example, the probability density for the distribution of end-to-end distances of a linear, ideal (phantom) polymer chain is given by the expression... 

According to (5.29) the (FT of the) probability densities for the end-to-end vectors of all K-bond macrosegments are functionally similar to the corresponding probability density for the entire chain. These macrosegments differ from the complete chain only by virtue of the number of monomers from which they are constituted. The fact that the segment is part of the larger chain does not affect its statistical properties. This is a feature common to all strictly self-similar, ideal systems. 

In the theory of many-body systems, the Haitiee approximations can generally be augmented by an applicatirm of self-consistent methods, not to one body concentration c(r) but to two-body properties, such as the pair c( relatiphase approximation (RPA) introduced by Bohm, Pines, and Noziires. In election systems, RPA is useful iruunly for nearly free electrons. Similarly, for our chain systems, RPA will work for nearly ideal chains— i.e., in melts. The corresponding experiments are essentially based on neutron scattering with labeled molecules and are summarized briefly in Chapter II. 

The possibility of a rescaling of the representative freely jointed chain, as formulated by these equations, expresses an important basic property of ideal polymer chains, namely their self-similarity . Self-similarity here means that independent of the chosen length scale, i.e. the resolution, an ideal chain always exhibits the same internal structure, one for which all internal distance vectors are distributed like Gaussian variables. A change of the length scale leaves this structure s characteristics invariant. 

Self-similarity is the basic property of fractal objects and ideal chains do indeed represent a nice example. The fractal dimension can be directly derived. If one proceeds rig segmental steps, starting from a point in the interior of the chain, there results on average a displacement in the order of... 

Equation (2.54) expresses a power law behavior. In the last section we addressed the self-similar nature of ideal polymer chains. The power law reflects exactly this property, since the function g(r) 1/r maintains its shape if we alter the unit length employed in the description of r. 

The latter result, as well as eqn [2], can be rationalized on the basis of profound analogy between conformation of ideal (self-intersecting) polymer chain and a random walk in 3D space. A similar Gaussian probability distribution applies to the vector connecting any pair of segments inside the chain, provided that they are separated by sufficiently large number of segments. 

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