The Fixed Junction Model of Ideal Rubbers

Let us first consider a single polymer chain with a mean squared end-to-end distance Rq and inquire about the force that arises if the two ends become separated. As sketched in Fig. 9.5, we assume that one end-group of the chain is located at the origin of a cartesian coordinate system and the second end-group can be moved along the y-ax s. In order to keep this second end fixed at a distance y, a non-vanishing force has to be applied. This can be calculated using the same equation as for the macroscopic piece of rubber and follows as 

In our case, the partition function is determined by the number of conformations available for the chain if the second end is at a distance y. Zp y) can be 

This in an interesting result. It states that this entropic force increases linearly with the distance between the two end-groups, just as if they were connected by a mechanical spring. The stiffness constant, denoted 6, increases with temperature and decreases with increasing size of the chain. 

The force disappears only for y = 0. This, however, does not imply that the end-groups are coupled together in thermal equilibrium. A harmonic oscillator with a stiffness constant b in contact with a heat bath at a temperature T shows a non-vanishing mean squared displacement (y ). Straightforward application of Boltzmann statistics yields 

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