Elasticity of a Single Chain

By Boltzmann s distribution law the probability that a segment makes an angle 6 with the x-axis is proportional to  

In the rubbery state the segment will continually be undergoing thermal motion and the average value of the x-component will be, as shown in Appendix 3B  

The function in the brackets on the right-hand side of eqn (3.16) was encountered by Langevin in his theory of the ordering of magnetic dipoles in a magnetic field and is today known as the Langevin function and using the above symbols is usually represented by the notation (aFIkT). 

If there are n segments with an average -component of (a ) the average x-component of chain end separation will be  

Since there is no tension in the y- and z-directions this result is equal to the actual average chain end separation, r, where  

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Our present theories of rubber elasticity are mostly of the kinetic theory type. In some of these theories the elasticity of a single chain is multiplied by the number of effective chains in the network to provide the total elasticity. In these theories no information concerning the statistical properties of the network structure can be inferred since either no structural aspects are evaluated or a simple structural nature is assumed. These structuril properties should be of great interest in furthering our understanding of polymers in bulk. Thus the strain dependence of X-ray scattering measurements from networks, which have heavy atoms at the cross-links and/or the chain ends, should provide some correlation between the structural and elastic properties of polymer networks. 

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