Zero-Shear Viscosity of Ideal Stars and Combs

1 Zero-Shear Viscosity of Ideal Stars and Combs 

We see clearly that as/increases, g, and thus T7q, decrease. (Zimm and Stockmeyer [86] arrived at the same result by a different method see Eq. 2.17.) This relationship has been found to be reasonably correct as long as the molecular weight of the arm is less than about M /l. Bonchev et al. [87] used graph theory to derive formulas for g for branching structures of high complexity. 

In the above expression for the viscosity of entangled stars, we note that M/f is the molecular weight per arm of the star polymer, and this implies that the viscosity of symmetric stars with entangled branches depends only on the length of the branch and not the number of branches. It has been found that this holds for functionalities up to at least 33 [90], except for a modest deviation from the rule for /= 3. Thus, data for stars with various numbers of arms can be plotted versus M3 to form a master curve. The dependence of rjg on M3 is found to be approximately exponential, and theoretical arguments [91, 92] suggest a relationship of the following form  

Fetters et al. [93] found that their data for polyisoprene stars followed this relationship with a = 3/2 and V = 0.47. Ngai and Roland [94] found that this form also fitted their data for polyisoprene stars but that when MJM reached 44, the viscosity fell well below that indicated by Eq. 5.44. 

The exponential increase of viscosity with M is consistent with the picture in which relaxation occurs primarily by means of primitive path fluctuations (sometimes called arm retraction). In Chapter 9 we will see that this effect can be explained quantitatively by a tube model. The exp onential increase of t]q with M results from the fact that the branch point prevents reptation, so that the principal mechanism of relaxation is primitive path fluctuation, which becomes exponentially slower with increasing arm length. The energy of activation for the zero-shear viscosity is little affected by star branching, except in the case of polyethylene and its close relative, hydrogenated polyisobutylene. 

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