Scaling Representation of a Macromolecule

A number of methods exist for the modelling of polymers at a molecular level. One of the most widely used methods is scaling presentation [54], according to which the diameter 0 of the tube in which the macromolecule is enclosed (it is equal to the distance between the entanglement points) can be estimated from the ratio [75]  

The main difference between physical entanglements (loops) and chemical crosslinks is that the former allow creeping of chains. The density of both types of link are determined using the same rubber elasticity equation, a factor of 0.8 being frequently added to take creeping into account [75]. 

As noted above, a change in the density of chemical crosslinking vf) in epoxy polymers results in an extremal variation of the characteristic ratio (C ) with the minimum K 1.0 [118]. If the section of a macromolecule between chemical crosslinks is modelled by a freely jointed chain, the use of known and values results in a typical fractal dependence [122]  

With the assumption that = const and K = 1.0, = 0.85 was obtained with L = const and K 1.0, the calculations gave % = (0.85) = 0.926. The calculation of the dimension D from Equation (11.55) gave 1.17. Thus, when the condition mentioned above is fulfilled, the section of a macromolecule between chemical crosslinking points can be represented as a fractal. Variation of K changes the crosslinking density and, hence, L. As a consequence, the chain ceases to be a self-similar fractal. Nevertheless, it can still be modelled by a non-homogeneous fractal with the same fragmentation step but with a variable D value, determined from relationship (11.39). The D values calculated in this way are listed in Table 11.6. 

The results of calculation of the characteristic ratio from Equation (11.54) are inconsistent with the data obtained by Jean and co-workers [131]. Indeed, the maximum rather than the minimum is attained at K = 1.0. Since  

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