Temperature effects on real chains

As a final remark it must be mentioned that theoretical and experimental works have been dedicated to investigating the effect of the finite size of the chains [65]. In fact, as grows exponentially, at low temperatures it can become comparable with the distance between two consecutive defects (e.g. impurities and vacancies) which are always present in real systems and hardly separated by more than 103 -104 elementary units. In case of Z < , the nucleation of the DW is energetically favoured if occurring at the boundaries, because the energy cost is halved. However the probability to have a boundary spin is inversely proportional to L thus the pre-exponential factor becomes linearly dependent on L, as experimentally found in doped SCMs. As doping occurs at random positions on the chain, a distribution of lengths is observed in a real system. However, as the relaxation time is only linearly dependent on L, a relatively narrow distribution is expected. 

Note that the chain dimension thus obtained is independent of the temperature. For this reason, the real chains are often called athermal chains. The independence results from the fact that both the elasticity and the excluded volume are entropic in origin. The two terms on the right-hand side of Eq. 1.63 are independent of T. In some polymer solutions, however, the interaction is enthalpic. Dividing the interaction by k T makes the interaction term in Eq. 1.63 reciprocally proportional to T. Consequently, the polymer chain dimension depends on the temperature. In Section 2.3, we will see this effect. 

The feature which is unique to the chain-branching system is the paradoxical, upper, or second explosion limit. Plere one observes that a reaction proceeding with explosive speed at pressures below the limit is effectively (picnched on raising the pressure. In addition, the pressure limit increases if the temperature increases, just opposite to the behavior at the first and third limits. It is the existence of this limit that is the real evidence of the branching chain. It is observed that the limit is much less sensitive to surface-volume effects than is the first limit, while added inert gases always tend here to lower the limit (i.e., quench the explosion). 

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