Self-similar chain

After the necessary excursion into astronomy, enough evidence now exists to repeat that chemistry occupies a central position in a self-similar chain between sub-atomic matter and super-galactic structures. By virtue of this unique position in the natural sciences chemical theory should represent the link between physics, biology and the earth sciences. However, compared to the fundamental theories of physics, chemistry has no credible independent theory of its own. 

Li KR, Stockman MI, Bergman DJ (2003) Self-similar chain of metal nanospheres as an efficient nanolens. Phys Rev Lett 91(22) 227402... 

The close link of elemental periodicity with the golden ratio, which can scarcely be accidental, suggests an intimate relationship between atoms, botanical structures and spiral galaxies. The intermediate position of the solar system in this self-similar chain prompted a re-investigation of the commonality between atoms and celestial structures, previously invoked by Nagaoka and Bohr to formulate their atomic models. 

Fig. 1.3-10 1 -D Fibonacci sequence. Moving downwards corresponds to an inflation of the self-similar chains, and moving upwards corresponds to a deflation...

Size of the chains between cross-links The randomly coiled chains exhibit self-similar behavior and the transition from the self-similar critical gel to the self-similar chain (between network junctions) is difficult to detect experimentally. 

Coarse-grained models have a longstanding history in polymer science. Long-chain molecules share many common mesoscopic characteristics which are independent of the atomistic stmcture of the chemical repeat units [4, 5 and 6]. The self-similar stmcture [7, 8, 9 and 10] on large length scales is only characterized by a single length scale, the chain extension R. 

A wide variety of polymeric materials exhibit self-similar relaxation behavior with positive or negative relaxation exponents. Positive exponents are only found with highly entangled chains if the chains are linear, flexible, and of uniform length [61] the power law spectrum here describes the relaxation behavior in the entanglement and flow region. 

Power law relaxation behavior is also expected (or has already been found) for other critical systems. Even molten polymers with linear chains of high molecular weight relax in a self-similar pattern if all chains are of uniform length [61]. 

The question of whether proteins originate from random sequences of amino acids was addressed in many works. It was demonstrated that protein sequences are not completely random sequences [48]. In particular, the statistical distribution of hydrophobic residues along chains of functional proteins is nonrandom [49]. Furthermore, protein sequences derived from corresponding complete genomes display a distinct multifractal behavior characterized by the so-called generalized Renyi dimensions (instead of a single fractal dimension as in the case of self-similar processes) [50]. It should be kept in mind that sequence correlations in real proteins is a delicate issue which requires a careful analysis. 

The self-similarity of the viscoelastic behaviour offlexible chains... 

This self-similarity of the viscoelastic behavioiir of monodisperse linear chains, whatever their chemical structure, may be extended to polydisperse species having the same shape of the molecular weight distribution (i.e., the same... 

Fractal Model. We consider in more detail the issue of relaxation times relevant to our fractal model of anomalous relaxation. As appears from the potential energy landscape for a system under the action of an external electrical field (Fig. 67), the energy differences between minima separated by energy maxima of different levels of self-similarity diminish, the larger the number of a selfsimilarity level. In view of the standard definition of a relaxation time, x eu "r. one may write the following chain of inequalities,... 

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 concentration in the adsorbed layer decreases away from the surface as (f) (z/b) for exponent 0.588. This power law concentration profile in an adsorbed polymer layer was proposed by de Gennes and is called the de Gennes self-similar carpet. This profile of adsorbed polymer can be described by a set of layers ot correlation blobs with their size of order of their distance to the surface z (see Fig. 5.11). The self-similar concentration profile starts at the adsorption blob ads in the first layer. The self-similar profile ends either at the correlation length of the surrounding solution if it is semidilute or at the chain size Rf bN if the surrounding solution is dilute. 

Consider multichain adsorption in a 0-solvent, of dilute chains with AT monomers of size b, and with monomer surface interaction of 6kT. Calculate the density profile of the de Gennes self-similar carpet. Calculate the thickness ads of the adsorbed layer and the coverage P. 

Above the gel point there is a macroscopic molecule (the gel). The structure of the gel is also self-similar (fractal) in the same range of length scales between the average distance between crosslinks (the linear chain size) and the correlation length ... 

Note that power-law behaviour is prevalent at gelation. This has been proposed to be due to a fractal or self-similar character of the gel. Note that the exponent )f is termed the fractal dimension. For any three-dimensional structure D = 3) the exponent Df<3 (where Df < 3 indicates an open structure and Df= 3 indicates a dense strucmre). Also Muthu-kumar (Muthukumar and Winter, 1986, Muthukumar, 1989) and Takahashi et al. (1994) show explicitly the relationship between fractal dimension (Df) and power-law index of viscoelastic behaviour (n). Interestingly, more recent work (Altmann, 2002) has also shown a direct relationship between the power-law behaviour and the mobility of chain relaxations, which will be discussed further in Chapter 6. 

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