Polymeric fractal

Muthukumar and Winter [42] investigated the behavior of monodisperse polymeric fractals following Rouse chain dynamics, i.e. Gaussian chains (excluded volume fully screened) with fully screened hydrodynamic interactions. They predicted that n and d (the fractal dimension of the polymer if the excluded volume effect is fully screened) are related by... 

The authors [4, 5] considered random walks without self-intersections or self-avoiding walks (SAW) as the critical phenomenon. Monte-Carlo s method within the framework of computer simulation confirmed SAW or polymer chain self-similarity, which is an obligatory condition of its fac-tuality. Besides, in Ref [4], the main relationship for polymeric fractals at their treatment within the framework of Flory conception was obtained ... 

The first stage of the considered treatment is the formulation of polymeric fractal dimension ZT and its spectral (fracton) dimension d, which characterizes fractal object coimectivity degree [41], intercommunication. In this case Ihe value linear polymer chain and <7 =1.33 for very branched (cross-linked) chain. Using Flory-de Gennes mean field approximation, Vilgis obtained the following equation for/recalculation [39] ... 

Hence, it can be concluded, that the polymeric fractals at the condition... 

Up to now we considered pol5meric fiiactals behavior in Euclidean spaces only (for the most often realized in practice case fractals structure formation can occur in fractal spaces as well (fractal lattices in case of computer simulation), that influences essentially on polymeric fractals dimension value. This problem represents not only purely theoretical interest, but gives important practical applications. So, in case of polymer composites it has been shown [45] that particles (aggregates of particles) of filler form bulk network, having fractal dimension, changing within the wide enough limits. In its turn, this network defines composite polymer matrix structure, characterized by its fractal dimension polymer material properties. And on the contrary, the absence in particulate-filled polymer nanocomposites of such network results in polymer matrix structure invariability at nanofiller contents variation and its fractal dimension remains constant and equal to this parameter for matrix polymer [46]. 

The fraction filling of slit between two plates by polymeric fractal 3 is given as follows [57] ... 

Mlgis, T. A. Flory theory of polymeric fractals—intersection, saturation and condensation. PhysicaA 1988,153(2), 341-354. 

LhuUher, D. A simple model for polymeric fractals in a good solvent and an improved version of Floiy approximation. J. Phys. France, 1988, 49(5), 705-710. 

These results have been generalized further in [31], for charged polymeric fractals with arbitrary connectivity characterized by spectral dimension, d, (the latter relates the longest path in the fractal, / max - aN /, to its mass -- N) and arbitrary fractal dimension df (in the absence of ionic charges) in d-dimensional space. For ideal (Gaussian) fractals df = 2ds/ (2 - ds). For charged fractals ... 

Kozlov, G. V Ozden, S. Malkanduev, Yu. A. Zaikov, G. E. Autoacceleration in the process of radical polymerization fractal analysis. In book Fractals and Local Order in Polymeric Materials. Ed. Kozlov, G. Zaikov, G. New York, Nova Science Publishers, Inc., 2001,11-19. 

The development of new molecular closure schemes was guided by analysis of the nature of the failure of the MSA closure. In particular, the analytic predictions derived by Schweizer and Curro for the renormalized chi parameter and critical temperature of a binary symmetric blend of linear polymeric fractals of mass fractal dimension embedded in a spatial dimension D are especially revealing. The key aspect of the mass fractal model is the scaling relation or growth law between polymer size and degree of polymerization Ny cr. The non-mean-field scaling, or chi-parameter renormalization, was shown to be directly correlated with the average number of close contacts between a pair of polymer fractals in D space dimensions N /R if the polymer and/or... 

Muthukumar, M. (1985) Dynamics of polymeric fractals, J. Chem. Phys. 83, 3161-3168... 

In this section, we review some of the recent progress in simulating polymeric fractals, with particular emphasis on tethered membranes made of linear polymer segments connected together to form a two-dimensional surface. After a brief review of the theory, we present results from a munber of groups which show that two-dimensional tethered membranes remain flat and do not crumple. We then consider the effect of changing the solvent quality by adding attractive interactions between nonbonded monomers. While there is clear evidence for a collapsed phase at low T, the nature of the crossover from flat to compact state remains unclear. 

Two-body terms are irrelevant for d > d a as mentioned above. Three-body terms are irrelevant for d > Ad/ 6 + three-body terms are irrelevant for d < 4/3. This explains why there is no induced bending rigidity for linear chains and gelation/percolation clus-... 

In the case of realisation of the second of the variants of crosslinked polymer morphology indicated above, its structure can be presented as a mixture of arbitrary polymeric fractals in the gelation point, having spectral dimensions d (globules) and 8 (interglobular zones). Then the following relationship can be used for 8 estimation [63] ... 

---