Time-dependence and thermal activation

Relaxation processes, explained in section 8.1.1 above, play an important role in the deformation of polymers. As they are thermally activated at sufficiently high temperatures, the probability of a relaxation process increases exponentially with the temperature and also with the time available for the process. 

To discuss this in detail, we consider a segment of the polymer chain that has to overcome an energy barrier Q of a relaxation process (see section 8.1.1) to slide past a neighbouring segment and enable the deformation. The probability P to overcome this barrier by thermal activation is, according to appendix C.l, P oc exp —Q/kT), with the temperature T and Boltzmann s constant k. 

Analogous to section 6.3.2, an external stress a eases overcoming the barrier. If this stress acts on a molecular segment of cross section A and if the width of the barrier is d, the external stress does the work W = a Ad when the segment overcomes the barrier. The probability to overcome the barrier is thus 

If the barrier is overcome, the back-reaction can also occur. However, it is less probable than the forward-reaction because work against the external stress has to be done. Its probability is 

The total probability to overcome the barrier is the difference of both contributions  

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