Gaussian functional link network

The properties of a polymer network depend not only on the molar masses, functionalities, chain structures, and proportions of reactants used to prepare the network but also on the conditions (concentration and temperature) of preparation. In the Gaussian sense, the perfect network can never be obtained in practice, but, through random or condensation polymerisations(T) of polyfunctional monomers and prepolymers, networks with imperfections which are to some extent quantifiable can be prepared, and the importance of such imperfections on network properties can be ascertained. In this context, the use of well-characterised random polymerisations for network preparation may be contrasted with the more traditional method of cross-linking polymer chains. With the latter, uncertainties can exist with regard to the... 

For a network of Gaussian chains having the same number n of links, uniaxially stretched by an amount L/Lo = A., the assumptions of affine displacement of jimction points and initial Gaussiein distribution of end-to-end vectors allows one to calculate the optical anisotropy of the network by integrating Eq.lO over the distribution of end-to-end vectors in the stretched state. By taking Treloar s expansion [11] for the inverse Langevin function, the orientation distribution function for the network can be put into the form of a power series of the number of Unks per chain ... 

As indicated previously, an elastomer can be identified with an assembly of chains connected through randomly distributed cross-links that are separated from one another by a quadratic average end-to-end distance ((> o) satisfying a Gaussian distribution function P(n, r) (see Chapter 5). In the following treatment, the network is considered ideal, without dangling chains and entanglements. 

The statistical theory is remarkable in that it enables the macroscopic deformation behaviour of a rubber to be predicted from considerations of how the molecular structure responds to an applied strain. However, it is important to realize that it is only an approximation to the actual behaviour and has significant limitations. Perhaps the most obvious problem is with the assumption that end-to-end distances of the chains can be described by the Gaussian distribution. This problem has been highlighted earlier in connection with solution properties (Section 3.1) where it was shown that the distribution cannot be applied when the chains become extended. It can be overcome to a certain extent with the use of more sophisticated distribution functions, but the use of such functions is beyond the scope of this present discussion. Another problem concerns the value of N. This will be governed by the number of junction points in the polymer network which can be either chemical (cross-links) or physical (entanglements) in nature. The structure of the chain network in an elastomer has been discussed earlier (Section 4.4.5). There will be chain ends and loops which do not contribute to the strength of the network, but if their presence is ignored it follows that if all network chains are anchored at two cross-links then the density, p, of the polymer can be expressed as... 

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