The Model of a Network Polymer

The model of network polymers considered next describes quantitatively the formation of structure (and, hence, properties) and is based on fundamental principles of physics. The initial parameters used for crosslinked systems are the v value (topological characteristics) and some generally accepted molecular parameters. Since the chemical crosslinking processes are complex, the model makes use of the simplest scheme of structure formation, which does not take into account, for example, the decrease in the rate of structure formation with time when one structure element starts to influence the conditions of formation of an adjacent element. An example of taking account of effects of this type has been reported [117]. Although this scheme is simple, it provides a quantitative description of the processes considered and the structures formed. 

One point of the model under discussion is estimation of the relationship between v and the level of local order in the structure of epoxy polymers, which is characterised by the 

As a first approximation, it can be assumed that = df, where df is the fractal dimension of the topological structure of the polymer, which can be found from Equation (11.27). 

It is clear that the chain fragments with the molecular mass are considered in a plane (d = 2) and the formation of clusters is discussed in a volume (d = 3). The D value calculated from Equation (11.39) for a two-dimensional fractal can be used to find the value for a three-dimensional fractal by applying Equation (11.27) to the dimension df. This method is not highly accurate but simple. Equation (11.51) makes it possible to determine the fraction of volume P occupied by vortices of the scale. Assuming that the P and (p,i values are equal, the value can be calculated from the relationship  

Since the temperature-time superposition is correct for polymers, we assume that and write relationship (11.51a) in the following form  

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