The Rheology Handbook. Thomas Mezger

τ0 = 200 Pa and ρ = 2.0 g/cm3 = 2000 kg/m3, results: a = 10 mmfor τ0 = 3 Pa and ρ = 1.0 g/cm3 = 1000 kg/m3, results: a = 0.3 mm

      Comment: The obtained values should only be seen as a rough estimation since other factors, such as roughness of the wall, more or less (in-)homogeneous sagging behavior, and surface tension of the coating additionally may have a crucial effect on the result. If the value of the yield point is determined according to Chapter 3.3.4.1, the restrictions should also be taken into consideration as discussed in that section. However, the thickness of the wet layer also depends strongly on its time-dependent behavior during structural regeneration directly after the shear-intensive application (thixotropic behavior). In order to obtain useful results for R & D it is recommended to evaluate leveling and sagging behavior as explained in Chapter 3.4.2.2 (and Table 3.3), or even better as shown in Chapter 8.5.2.2 (and Table 8.4).

      End of the Cleverly section

       Figure 3.23: Volume element of a coating layer on a wall, with weight FG , wet layer thickness a, layer width b and layer length c

3.3.5Overview: flow curves and viscosity functions

      In this section, an overview is presented by Figures 3.24 to 3.29, showing the above discussed flow and viscosity curves. The labels used are as follows: (1) ideal-viscous (Newtonian), (2) shear-thinning, (3) shear-thickening, (4) without a yield point, (5) showing a yield point

      3.1.2.1.1a) Diagrams on a linear scale

      See Figures 3.24 to 3.26.

      3.1.2.1.2b) Diagrams on a logarithmic scale

      Logarithmic scaling is recommended if it is desired to present the shape of the curves also at very low values of τ and γ ̇ , see Figures 3.27 to 3.29. In this case, the diagrams usually are displayed on a double-logarithmic scale.

      3.1.2.1.3c) Three-dimensional diagrams of flow curves and viscosity functions

      Results of several shear tests measured at different, but constant temperatures (i. e. isothermal) can be presented in the form of a three-dimensional (3D) diagram, for example, with the shear rate γ ̇ on the x-axis, shear stress τ or viscosity η on the y-axis, and the temperature T on the z-axis.

       Figure 3.24: Comparison of flow curves

       Figure 3.25: Comparison of viscosity functions

       Figure 3.26: Comparison of flow curves with and without a yield point

       Figure 3.27: Comparison of flow curves:

      (1) showing the slope s = 1:1 = 1, (2) with s < 1, (3) with s > 1

       Figure 3.28: Comparison of viscosity functions: (1) showing the slope s = 0, (2) with s < 0,

      (3) with s > 0

       Figure 3.29: Comparison of viscosity functions; (4) showing a zero-shear viscosity plateau (i. e. there is no yield point), (5) without a zero-shear viscosity plateau (i. e. there is a yield point)

3.3.6Fitting functions for flow and viscosity curves

      After a test, measuring data mostly are available in the form of diagrams and tables. For each single measuring point there are usually values available for temperature, measuring time, shear rate, shear stress, and, calculated from these, viscosity. If measuring data should be compared, for example, when performing quality assurance tests, it is not useful – and in most cases it is also not possible – to compare all values of one test with those of another due to the mostly large number of individual measuring points.

      Mathematical model functions for curve fitting are therefore used to characterize complete flow or viscosity curves resulting in a small number of curve parameters only. This simplifies to compare measuring curves, since there are only a few model parameters left for comparison then. Fitting is also called approximation and the corresponding functions are often referred to as regression models.

      In the past, these fitting functions were used more frequently than today because only few users had access to rheometers which enabled the user to control or to detect such low deflection angles or rotational speeds as required to determine technically important parameters like yield points and zero-shear viscosities with sufficient accuracy. At that time, these kinds of rheometers were usually too expensive for industrial users. They therefore resorted to these model functions to characterize samples in terms of the above parameters. In this way, analysis was at least possible using approximately calculated data. Since around 1985, the increased use of computers in industrial laboratories has facilitated analysis of flow and viscosity curves, and above all, made it less time-consuming. Before, analysis of curves had to be performed manually with the aid of rulers, multicurves or so-called nomograms.

      Fitting functions are still used today in many laboratories, especially for quality assurance, where financial support is at a minimum, and therefore small, inexpensive instruments are still in use. However, if data in the low-shear range below 1 s-1 are of real interest, it is better to measure in this range instead of calculating whatsoever values via fitting functions. Appropriate instruments are affordable meanwhile, even for small companies due to considerable improvements in the price/performance ratio.

      Not each model function can be used for each kind of flow behavior. If the correlation value (e. g. in %) indicates insufficient agreement between measuring data and model function, it is useful to try another model function. It is also important to keep in mind that both, model-specific coefficients and exponents are purely mathematical variables, and in principle, do not represent real measuring data completely.

      Since there are a lot of fitting functions – because it seems in the past almost each rheologist designed his own one – it is only possible to mention frequently used models here. In the following there are listed more than 20 models. Often are used those of Newton, Ostwald/de Waele, Carreau/Yasuda and Herschel/Bulkley. Further information on model functions can be found e. g. in DIN 1342-3 and [3.9] [3.10] [3.32] [3.33] [3.34] [3.38].

       3.3.6.1Model function for ideal-viscous flow behavior

      according to Newton

      Newton :τ = η ⋅ γ ̇

      (see also Chapter 2.3.1a, with Figures 2.5 and 2.6)

       3.3.6.2Model functions for shear-thinning