Fractal Characteristics of Macromolecules

Condensed systems start to behave as polymers after a certain molecular mass has been attained and a network of macromolecular entanglements or chemical nodes has been formed [57]. In consideration of condensed polymers, it is especially important to describe the sections of the macromolecule between the points of chemical crosslinking or entanglements [56]. The deformability and mobility of these sections largely determine the macroscopic properties of polymers [57, 104]. The general physical grounds that account for the fractality of these chain sections are considered next in relation to epoxy polymers. 

For most polymers, /q = 0.125 nm [93], the minimum = 2 [94] therefore, the minimum possible is 0.25 nm. This value is in good agreement with the experimental estimate of the lower limit of fractality in polymers ( 0.3 nm [22]) and should be accepted as the for the chain section between chemical crosslinks. The length of this section can be taken as the upper limit. For most of the epoxy polymers considered, this value is in the range from 1.6 to 7.4 nm, which is also consistent with the data of [22, 36]. 

Finally, the third aspect concerns determination of the fractal dimension D of an object [12]. For fractals, the condition  

By dividing both parts of Equation (11.30) by A, we find that  

Let us consider conditions under which d = 2. For a real polymer, with allowance for the effect of excluded volume, one can write that [57]  

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