The Rheology Handbook. Thomas Mezger

does not apply for polymer solutions with a concentration which is too low to form entanglements, and not for gels and paste-like dispersions exhibiting a network of chemical bonds or physical-chemical interactive forces between the molecules or particles. The three intervals are in detail:

      1 The first Newtonian range with the plateau value of the zero-shear viscosity η0

      2 The shear-thinning range with the shear rate-dependent viscosity function η = f( γ ̇ )

      3 The second Newtonian range with the plateau value of the infinite-shear viscosity η∞ (sometimes it is called plateau value of the limiting high-shear viscosity)

       Figure 3.10: Viscosity function of an uncrosslinked and unfilled polymer solution or melt

      In order to explain this, you can imagine a volume element containing many entangled polymer molecules. Of course, in each sample there are millions and billions of them.

      a) Shear range 1 at low-shear conditions: the “low-shear range”

      For many polymers, the upper limit of this range occurs around the shear rate γ ̇ = 1 s-1. For some polymers, however, this limit can be found already at γ ̇ = 0.01 s-1 or even lower.

      Superposition of two processes

      When shearing, a certain number of macromolecules are oriented into shear direction. For some of them, this results in partial disentanglements. As a consequence, viscosity decreases in these parts of the volume element. Simultaneously however, some other macromolecules which were already oriented and disentangled in the previous time interval are recoiling and re-entangling now again. This is a consequence of their visco-elastic behavior which the polymer molecules still are able to show under the occurring low-shear conditions. As a result, viscosity increases again in these parts of the volume element.

      Summary of this superposition

      In the observed period of time, the sum of the partial orientations and re-coilings of the macromolecules, and as a consequence, the sum total of all disentanglements and re-entanglements, results in no significant change of the flow resistance related to the behavior of the whole volume element. Therefore here, the sum of the viscosity decreases and increases results in a constant value. Thus finally, the total η-value in this shear rate range is still measured as a constant value, which is referred to as the zero-shear viscosity η0. For unfilled and uncrosslinked molecules, zero-shear viscosity η0 is occurring as a constant limiting value of the viscosity function towards sufficiently low shear rates which are “close to zero-shear rate”.

      For “Mr. and Ms. Cleverly”

      1) Zero-shear viscosity in mathematical notation

      Equation 3.1

      η0 = lim γ ̇ → 0 η( γ ̇ )

      Zero-shear viscosity is the limiting value of the shear rate-dependent viscosity function at an “infinitely low” shear rate (see also Chapter 6.3.4.1: η0 via creep tests and Note 3 in Chapter 8.4.2.1a: η0 via oscillation, frequency sweeps).

      2) Dependence of η0 on the polymer concentration c

      For a polymer solution in the low-shear range, the η0 value is attained only under the condition that the polymer concentration c [g/l] is high enough. Only in this case the molecule chains are able to get in contact with one another to form entanglements. For polymers having a constant molar mass and using the value of the critical concentration ccrit, the following applies (see Figure 3.11):

      For c < ccrit: η/c = c1 = const

      Low-concentrated, diluted polymer solutions having no effective entanglements between the individual molecule chains display ideal-viscous flow behavior, and this counts also for the low-shear range. In this case, the viscosity value is directly proportional to the concentration (with the

      material-specific proportionality factor c1).

      For c > ccrit applies:

      For concentrated solutions and melts of uncrosslinked polymers showing effective entanglements between the individual molecule chains each of them indicates a zero-shear viscosity value in the low-shear range. With increasing concentration, there is a stronger increase of the η0-value as can be seen by the higher slope of the η(c)-function in this concentration range (see Figure 3.11).

      3) Dependence of η0 on the average molar mass M [3.9] [3.10]

      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