Muthukumar-Edwards theory

Even at very low concentrations, the viscoelastic properties are affected by in-termolecular interactions as evidenced by the extrapolation plots of Fig. 9-10. A simple analysis of the initial concentration dependence has suggested that the first effect is an increase in the longest relaxation time. Recent theoretical calculations of Muthukumar and Freed, based on the Freed-Edwards theory of concentration dependence of viscosity in polymer solutions, show that the pth.relaxation time has a concentration dependence of the form... 

Here, omitting the mathematical details, we sketch a relatively new dieoty due to Muthukumar and Edwards [43]. The M-E theory considers a uniform solution consisting of a pure solvent and the Edwards continuous chains with a Kuhn segment length b. It is formulated on the assumption that the Hamiltonian H of the solution is given by... 

Further, the theory envisages renormalization of the segment length and interaction with due account of the segment density fluctuations. Thus, the quantitic s A and Zi are the key functions of theory. Their definition provides a possibility to calculate the free energy of the system. Solving the problem has led to the following results (Muthukumar and Edwards, 1982)... 

The starting point is the probability of finding a conformation i (s) for the linear chain. This problem is discussed in detail in the chapter by Edwards and Muthukumar, and we give a brief description in Section 8.3.2 within this chapter, because of use in rubber theory, and we refer the mathematically interested reader to these parts of this series (Volume 2, Chapter 9). Assuming Gaussian conformations, it is given by the Wiener measure " ... 

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