The Rheology Handbook. Thomas Mezger

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       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 and shear-thickening flow behavior

      Here are explained three model functions for flow curves without a yield point.

      3.3.6.2.1a) Ostwald/de Waele, or power-law: τ = c ⋅ γ ̇ p

      Flow curve model function according to W. Ostwald jun. (of 1925 [3.2]) and A. de Waele (of 1923 [3.39]) with “flow coefficient” c [Pas] and exponent p. Sometimes, c is referred to as “consistency”, and p to as “flow index” or “power-law index”. It counts: p = 1 for ideal-viscous flow behavior, p < 1 for shear-thinning, p > 1 for shear-thickening.

      A disadvantage of this model function is that for flow curves of most polymer solutions and melts it cannot be fitted as well in the low-shear range (since it was developed for linear scaled diagrams) as well as in the high-shear range. These are the ranges of zero-shear viscosity and infinite-shear viscosity. Despite this, the model function is often used in the polymer industry to be fitted in the medium shear rate range (see Figures 3.6 and 3.7, 3.15 and 3.16, and the curve overview of Chapter 3.3.5).

      For “Mr. and Ms. Cleverly”

      3.3.6.2.2b) Steiger/Ory: γ ̇ = c1 ⋅ τ + c2 ⋅ τ3

      Flow curve model function with the (Steiger/Ory) coefficients c1 [1/Pas] and c2 [1/Pa3 ⋅ s],

      (of 1961 [3.40])

      3.3.6.2.3c) Eyring/Prandtl/Ree or Ree-Eyring: γ ̇ = c1 ⋅ sinh(τ/c2)

      Flow curve model function with (EPR) factor c1 [1/s] and “scaling factor” c2 [Pa],

      (of 1936/1955 [3.41] [3.42])

       3.3.6.3Model functions for flow behavior with zero-shear viscosity and infinite-shear viscosity

      Listed below are ten model functions showing the following parameters:

      Zero-shear viscosity η0 = lim γ ̇ → 0 η( γ ̇ ) and infinite-shear СКАЧАТЬ