Читать книгу The Rheology Handbook - Thomas Mezger - Страница 72

For M < Mcrit applies:

Equation 3.2

η/M = c2 = const

with the material-specific factor c2, the molar mass M [g/mol], and the critical molar mass Mcrit for the formation of effective entanglements between the macromolecules.

Polymers with smaller molecules, therefore showing no effective entanglements between the individual molecule chains, display ideal-viscous flow behavior within the entire shear rate range. Therefore, this counts also for the low-shear range, as illustrated by the bottom curve of Figure 3.12. In this case, viscosity is directly proportional to the molar mass.

For M > Mcrit

Equation 3.3

η0 = c2 ⋅ M 3.4

This relation is often associated with the name of T. G. Fox [3.11]. Polymers with larger molecules, therefore showing effective entanglements between the individual molecule chains, display a zero-shear viscosity in the low-shear range. The higher the molar mass, the higher is the plateau value of η0. Proportionality of η0 and M usually shows the exponent 3.4 (to 3.5). This value is approximately the same for all uncrosslinked polymers (although it is possible to find also values between 3.2 and 3.9). Using the above relation, it is possible to estimate the average molar mass from the η0-value, if the factor c2 is known. Information about this factor is documented for almost all polymers.

Summary: The higher the average molar mass M, the higher is the η0-value (see Figure 3.12).

Figure 3.11: Polymer solutions: dependence of the low-shear viscosity values on the polymer concentration (with the critical concentration ccrit)

Figure 3.12: Polymer solutions and melts: Dependence of the zero-shear viscosity values on the average molar mass

Note: Different M crit values for diverse polymers

Mcrit is considered in physics a limiting value between materials showing a low molar mass and polymers. In order to make a rough estimate, often is taken here Mcrit = 10,000 g/mol. However, this value depends on the kind of polymer; see the following Mcrit-values (in g/mol) [3.12]: poly­ethylene PE (4000); butyl rubber BR (polybutadiene, 5600); polyisobutylene PIB (17,000); poly methylmethacrylate PMMA (27,500); polystyrene PS (35,000)

End of the Cleverly section

Figure 3.13: Structural changes of dispersions showing shear-thinning behavior