Theory of cross-linked polymerized material

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Abstract
A study is made of the statistical thermodynamics of cross-linked polymers. It is argued that several physical and mathematical hypotheses or approximations are involved in making progress and these are enumerated. The analysis concerns that part of the behaviour of the system solely due to cross links and entanglements, and it is shown that under certain conditions a cross link will have a distribution (2/πLl)-3/2 exp(-2r2/Lal) about its mean position, where La is the average length between links and l the monomer length. This value can be obtained in a calculation making the total entropy a bound. The elastic free energy has the form where N is the number of polymers, L the length of a polymer, i the strains, and small omega, macron satisfies where M is the number of cross links, V the volume and C a constant. This expression offers a form where the effect of entanglement is added to the effect of the cross links.

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