By G. V. Kozlov
Fractal research supplies simply common mathematical description of polymers, i.e. it doesn't establish structural devices (elements), from which any actual polymer is shaped. actual description of thermodynamically non-equilibrium polymer constitution within the framework of the neighborhood order principles supplies the cluster version of the polymer amorphous nation constitution that quantitatively establish its elements.Since those versions examine the polymer constitution a bit from aspects, they're very good finishing each other. it's normal wisdom that buildings showing fractal behaviour on small size scale and being homogeneous on huge size scale are named homogeneous fractals. those fractals are percolation clusters on the percolation threshold. As proven lower than, the cluster constitution represents the percolation process and, because of the above-said, the homogeneous fractal. to place it another way, the presence of the neighborhood order within the condensed section of polymers determines fractality in their buildings.
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Extra info for Cluster Model Interrelation With Modern Physical Concepts
Clusters as Dissipative Structures 23 the structure of the polymeric sample approaches thermodynamically equilibrium one. To put it differently, for the current polymers, equations (30) and (42) give minimal df values . Further on, variation limits for С∞, determined from the equation (42), will be appraised. As is known [1, 18], df values for bulky samples at d = 3 vary within the limits as follows: 2 ≤ df < 3. At df = 2, parameter С∞ possesses the minimal value equal 2. As observed from the literature [45, 53, 59], many polymers do not display С∞ < 2.
Actually, βТ value is the function of molecular 56 G. V. Kozlov and G. E. Zaikov mobility, characterized by the fractal dimensionality (D) of the chain segment between clusters . Some other important points should be noted, associated with the possibility of describing the polymer structure in the framework of equation (72). For the "rigidity segment", the statistical segment with the length lst, is accepted in the cluster model. The value of lst is several times lower than the Kuhn segment length, A, which characterizes rigidity of freely joint chain .
The same result was obtained for glassy polymer networks  and amorphous-crystalline polymers . For example, in the framework of these general laws, the number of cross-link points, S, in a finite cluster, which can be the local order zone consisting of F/2 closely packed segments (where F is the cluster functionality ), depends upon dimensionless deviation of concentration, τ, from its critical value (τ = (х – хс)/хс ) at τ → 0 as follows: S~τ −γ n . (73) As mentioned above, the number of cross-link points, S, in the local order zone (classified as the cluster [34, 90]) equals F/2, and for τ parameter (Тg – Т)/Тg should be accepted.