The Rheology Handbook. Thomas Mezger

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СКАЧАТЬ rest. After removing the load, any re- formation of the layers depends on the elastic portion. Related to the entire gap this re-formation is also inhomogeneously. In comparison to this, Figures 2.1, 2.9 (no.2) and 4.1 exhibit the behavior desired by rheologists at homogeneous deformation and flow conditions.

      3.3.4.2.9Examples of plastic materials

      Solids or dispersions with a high concentration of solids showing high interactive and cohesive forces; i. e. plasticine (see Experiment 1.1b of Chapter 1.2), wax, a piece of soap, sealants, sludge, loam, mocha coffee grounds, plasters, surfactant, systems showing shear-banding (see Chapter 9.2.2).

      When performing rheological experiments in a scientific sense, however, it is assumed that there are homogeneous shear conditions across the entire shear gap. Only then, a constant shear rate or shear deformation can be expected to occur during the whole shearing process and after removing the load, and viscous, viscoelastic or elastic behavior. And only under these preconditions, the behavior can be described formally and mathematically by the relations according to Newton, Hooke, Maxwell, Kelvin/Voigt and Burgers.

      3.3.4.2.10Summary

      Plastic behavior cannot be described unequivocally using the usual scientific fundamentals of mathematics and physics since it is an inhomogeneous behavior. Therefore, it can only be presented in terms of relative values obtained from empirical tests.

      Unfortunately, in many industrial laboratories the terms plastic, ideal-plastic, viscoplastic or elastoplastic still are used to mean a lot of different things. It is useful to understand what these terms might mean, but their use should be avoided when performing and analyzing scientific rheological tests. Within a limited deformation range, in most cases, samples can be characterized as viscoelastic (DIN 13343). However, if a material cannot be sheared homogeneously it is often necessary to use special relative measuring systems (see Chapter 10.6). In this case, it is better to work only with the measured raw data such as torque, rotational speed and deflection angle, instead of any rheological parameter such as shear stress, shear rate, shear deformation, viscosity, and shear modulus.

      3.3.4.2.11Note by the way

      Both, the plastic surgeon and the sculptor (working with wood, stone or plaster to create plastic sculptures), are performing usually inhomogeneous and irreversible deformation processes, related to the structure of the material as a whole. Hopefully, the patients, operated beauties and lovers of fine arts are pleased with the end-products of these kinds of plastic deformations.

      3.3.4.2.12d) Practical example: Yield point and wet layer thickness of a coating

      Using the yield point value, it is possible to make a simple, rough estimation of the wet layer thickness on a vertical wall (see Figure 3.23).

      The following holds: τ = F/A, with the area to be coated: A = b ⋅ c,

      and the weight force of the layer due to gravity: F = FG = m ⋅ g = V ⋅ ρ ⋅ g

      with the mass m [kg] and the volume

      V [m3] = a ⋅ b ⋅ c of the volume element,

      the gravitation constant g = 9.81 m/s2, and the density ρ [kg/m3] of the coating,

      it follows that: FG = a ⋅ b ⋅ c ⋅ ρ ⋅ g

      Result: τ = τ0 = FG/A = (a ⋅ b ⋅ c ⋅ ρ ⋅ g)/(b ⋅ c) = a ⋅ ρ ⋅ g

      3.3.4.2.13Summary

      A coating layer with the layer thickness “a” remains on the wall only then, if the limiting value of the shear stress between the states of rest and flow is not exceeded. This τ-value is the yield point τ0 [Pa]. As the calculation shows, the yield point is independent of the layer width and length. Thus, the wet layer thickness of a coating which remains on a vertical wall can be calculated as:

      Equation 3.5

      a = τ0 /(ρ ⋅ g)

      3.3.4.2.14Examples

      1 for 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

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