Pair correlation function fractal

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... 

Tiencc, Eq. (2.121) is a special case of this result, with P — 2 for an ideal chain. The fractal dimension of a rod polymer is — 1 and the pair correlation function is g(r) ... 

The intramolecular pair correlation function grows with increasing r because the fractal dimension is larger than the space dimension ... 

The fractal dimension may be observed experimentally by light or neutron scattering [8]. The scattered intensity S( is the Fourier transform of the pair correlation function... 

Equation (6.10) was checked directly, using polymers with different masses. It was also tested using scattering experiments, by measuring the Fourier transform S(q) of the pair correlation function. As above Eq. (6.7), one can show that the scattered intensity is related to the wave vector q by the fractal dimension. n d = 3 one finds, using Eq. (6.10), that... 

The pair correlation function of fractal aggregates reflects their self-similarity for medium interpaiticle distances (a r Rg) hy a power-law relationship (Eq. (4.8)). At large arguments (r > Rg), the pair correlation function decays steeply, which can be modelled by a cut-off function. For medium and large arguments, one obtains ... 

Fig. 4.18 Fractal dimension with regard to the hydrodynamic diameters of translation and rotation and with regard to the diameter of gyration, the maximum Feret-diameter, and the pair correlation function for DLCA (left) and RLCA (right) aggregates...

A swollen coil (expanded by excluded-volume interactions) is a fractal. The size of any internal chain segment (blob) of g rmits (N < g < N) is g . The mean number of rmits in a sphere of radius r around any unit is Nr for b < r < R, that is, the chain fractal dimension is df= 1/v. The pair correlation function (cf. eqn [18]) of a swollen chain is... 

For a wave transfer momentum q, the intensity I(m,q) scattered by a cluster of mass m and characteristic size R(m) is proportional to the Fourier transform of its pair correlation function - (r /r )g(r/R(m)) -where D is the fractal dimension, R(m)... 

Schematic representation of two-polymer molecule correlation pathways. The wiggly bonds denote long range chemical bonding pair correlations due to chain connectivity, while the short solid line denotes the spatially local dii correlation function. There are of order a + bN pair contacts between two interpenetrating polymer chains of degree rf pcdymerization N which are embedded in a correlation volume of linear dimension R, oc N . Here dp is the mass fractal dimension of the polymer, 0 = 2 — (D/dp), D is the spatial dimension, and a and b are numerical constants of order unity... 

This strategy is common in atomic fluid theory at low to moderate densities, and for Coulombic systems, and corresponds to the reference idea ubiquitous in liquid state theory [5], The purely hard core problem is treated using the accurate [27] PY closure. (2) The construction of a closure approximation for the tail part of the potential is subject to the constraint of exactly describing the weak coupling limit. In physical terms, for fractal-like interpenetrating molecules these indirect processes may strongly couple the direct correlation functions associated with those pairs of sites which are in simultaneous contact The number of such two-molecule pair contacts. Me. scales with N as [23] ... 

Fractal aggregate, fractal agglomerate aggregates or agglomerates with a non-uniform distribution of the constiment particles, which typically coincides with a very porous, branch-like morphology fractal aggregates are characterised by a power-law decrease of the pair-correlation density function g(v) (Eq. (4.8)) and a power-law relationship between mass and size (Eq. (4.9)), in which the exponent is less than the Euclidean dimension fractal aggregates are not ideal fractal objects, but rather obey the fractal relationships only in a statistical sense (cf. Sect. 4.2.1). 

---