Comparison with scaling and exponential models

We began in Section 1.2 by observing that the large number of theoretical models could with a modest number of exceptions be partitioned into two major phenomenological classes, based on whether the models predicted scaling (power-law) or exponential dependences of transport coefficients on polymer concentration, molecular weight, or other properties. What do the data say about the relative merit of these classes of theoretical model  

An obvious first question is whether the precision of experimental measurement, as viewed through the lens of our data analysis methods, is adequate to say which models are acceptable. Can we distinguish between power laws and stretched exponentials The answer is unambiguously in the affirmative. The most transparent demonstration of this claim is given by single studies in which some results show power-law behaviors, some results show exponential behaviors, and measurements can be unambiguously partitioned into power-law and stretched-exponential 

What then is the relative importance in the real world of stretched-exponential or power-law concentration and molecular weight dependences For polymer solutions, the overwhelming majority of measurements of each transport coefficient follow stretched exponentials in c and M. Scaling behavior is found only as a rare exception. Theoretical models that lead to exponential behavior are therefore desired. Theoretical models that predict scaling behavior at some crude level of approximation appear to be less than useful. Theoretical models of polymer solutions that simply assume scaling as the normal observable behavior over extended ranges of c or M are not consistent with experiment. 

It is sometimes suggested that the success of the stretched exponential form in describing D (c) arises from a peculiar flexibility of the stretched exponential, so that the systematic successes shown in previous chapters are accidental. Claims that Doexp(—ac ) is unusually flexible, relative to other functional forms, are not consistent with basic mathematics the stretched exponential describes the concentration dependence with three free parameters. The function is not singular for real c and positive v. Therefore, the region of function space spanned by the set of aU stretched exponentials can be no larger than the region of function space spanned by any other function with three free parameters. 

Where does one not find stretched-exponential behavior In dilute solution, very modest deviations - concentration dependences weaker than expected - are sometimes seen. A few cases of re-entrant behavior, in which Ds c)t] c) 7 D (0) y(0) over some limited concentration range, have been noted. For melts, extensive reviews of the literature(2,3) generally find scaling behavior for tj and D, at least for adequately large polymers. It is then reasonable to expect that as the melt is approached there should be a transition to power-law behavior. Experiments of Tao, et al. are consistent with this expectation(4). 

---