The Rheology Handbook. Thomas Mezger

only, and to avoid in this context the terms dilatant and dilatancy [3.81] [3.82].

       Figure 3.20: Toothpaste – our daily struggle with the yield point

3.3.4Yield point

      Experiment 3.2: Squeezing toothpaste out of the tube (see Figure 3.20)

      A certain amount of force must be applied before the toothpaste starts to flow. A sample with a yield point begins to flow not before the external forces Fext acting on the material are larger than the internal structural forces Fint. Below the yield point, the material shows elastic behavior, i. e. it behaves like a rigid solid, exhibiting under load only a very small degree of deformation, which however recovers completely after removing the load. The following applies: If Fext < Fint the material is deformed to such a small degree only that it is hardly perceptible to the human eye. The sample does not begin to flow before Fext > Fint. The yield point is also referred to as yield stress or yield value . Also, other terms were proposed formerly, e. g. such as stiffness-at-rest (see Chapter 14.2: T. Schwedoff in 1880).

      Experiment 3.3: Sticking rods into hand cream and silicone

      Small rods are put into the following two materials, in order to observe their motion.

      1 Hand cream: The rod remains standing straight in the cream, thus here, a yield point exists.

      2 Silicone polymer (uncrosslinked PDMS): The rod moves very slowly to the side, i. e., although the highly viscous silicone displays a clear flow resistance, there is no yield point. It exhibits behavior of a viscoelastic liquid indicating a high value of the zero-shear viscosity in the low-shear range (see also Chapter 3.3.2.1a).

      3.1.2.1.3Examples of materials which may show a yield point

      Gels, dispersions with a high concentration of solid particles such as plastisol pastes, conductor pastes (electrotechnics), toothpaste, sealants, putties, emulsion paints, printing pastes, ceramic masses, lipstick, creams, ketchup, mayonnaise, chocolate melts, margarine, yogurts; semi-solid materials, concentrated surfactant systems

      Yield points have great importance for practical users, and therefore, various methods for the acquisition of appropriate measurement values have been developed over the years – with quite a lot of creativity, obtaining more or less useful results. See also Chapters 3.3.6.4 (model functions, e. g. according to Bingham), 11.2.3d (slump test), 11.2.4a/c (inclined plate), 11.2.6d/e (inclined channel, Casagrande Apparatus), 11.2.7c (Kasumeter), 11.2.8b2 (falling rod), 11.2.9 (penetrometers), 11.2.11a/e/i (gelation test, Mini-Rotary), as well as Chapter 12.4.1a (guideline) and the index. Sometimes, there is distinguished between tests to determine the “apparent” or the “really existing” yield point (for more information on this discussion, see also [3.9] [3.24] [3.82]). Materials having a yield point often show “plastic behavior”, as they tend to flow inhomogeneously, and then, wall-slip effects should be taken into account (see also Chapter 3.3.4.3c).

       3.3.4.1Yield point determination using the flow curve diagram

      a) With controlled shear rate (CSR): Yield point calculation via curve fitting models

      Here, rotational speeds (or shear rates, resp.) are preset in the form of steps or as a ramp (see Figures 3.1 and 3.2). However, using this kind of testing, a yield point cannot be determined directly. Therefore, it is calculated by use of a fitting function which is adapted to the available measuring points of the flow curve. Curve fitting is carried out using one of the various model functions, e. g. according to the models of Bingham, Casson or Herschel/Bulkley (see Chapter 3.3.6.4). For all these approximation models, the yield point is determined by extrapolation of the flow curve towards the shear rate value γ ̇ = 0, or at the intersection point of the fitting function and the τ-axis, respectively (as described in the meanwhile withdrawn DIN 53214). The different model functions usually produce different yield point values because each model uses a different basis of calculation. Today, this method should only be used for simple QC tests but no longer for modern research and development work since a yield point value obtained in this way is not measured but merely calculated by a more or less exactly fitting approximation.

      b) With controlled shear stress (CSS): Yield point as the stress value at the onset of flow

      This is the “classic” method for the determination of a yield point: When increasing the shear stress with time in the form of steps or as a ramp (similar to Figures 3.1 and 3.2), the shear stress value is taken as the yield point, at which the measuring device is still detecting no sign of motion. This is the last measuring point at which the rotational speed n (or shear rate γ ̇ , resp.) is still displayed as n = 0 (or as γ ̇ = 0, respectively). The yield point τ0 occurs as an intersection on the τ-axis when plotted on a linear scale (see Figure 3.21). If presented on a logarithmic scale, the yield point τ0 is the τ-value at the lowest measured shear rate, e. g. at γ ̇ = 1 or 0.1 or 0.01 s-1 (see Figure 3.22).

       Figure 3.21: Flow curve showing a yield point

      (on a linear scale)

       Figure 3.22: Flow curve showing a yield point

      (on a logarithmic scale)

      Summary: Using the flow curve analysis methods mentioned above, the resulting yield point is dependent on the speed resolution of the viscometer or rheometer used. An instrument which can detect lower rotational speeds (e. g. nmin = 10-4 min-1) will display a lower yield point value compared to a device which cannot detect such low minimum speeds (e. g. displaying nmin = 0.5 min-1 only). Of course, the latter device cannot detect any motion below its measuring limits, therefore still evaluating any speed in this range as n = 0. As a result, a lower value of the yield point will be obtained by the more sensitive instrument. This can be illustrated clearly when presenting flow curves on a logarithmic scale (see Figure 3.22): The lower the smallest shear rate which can be detected, the lower is the corresponding shear stress. Therefore counts the following: A yield point is not a material constant since this value is always dependent on the options of the measuring instrument used.

      For this reason, the two methods (a and b) mentioned before should only be taken for simple quality assurance tests, thus, just for a rough estimation of a yield point.

      Note: Yield point and flow point

      For users in R & D, however, more modern methods are recommended compared to the simple methods as explained above, using flow curves. See Chapter 4.4 for analysis of yield points by a logarithmic shear stress/deformation diagram; or even better, see Chapter 8.3.4 to determine both yield point and flow point (oscillatory tests, amplitude sweeps). An overview on even further methods which might be used for yield point determination is given in [3.25].

       3.3.4.2Further information on yield points

      3.3.4.2.1a) Time-dependence of the yield point

      Yield point values depend on the duration of the test. With each new measuring point at the beginning of each new step on the stress ramp, the structure of the sample is stretched at first under the applied constant shear load. As a consequence, a constant, steady-state measuring