The Rheology Handbook. Thomas Mezger

c3 [s2].

      Assumptions: For very low shear rates the viscosity plateau value c1 is reached, corresponding to η0; and for very high shear rates the viscosity plateau value is (c1 ⋅ c2/c3) [Pas], corresponding to η∞.

      3.3.6.3.8h) Reiner/Philippoff:

      Flow curve model function with the coefficients c1 [Pa] and c2 [Pa], (of 1936 [3.30] [3.52]).

      The following applies for γ ̇ → ∞:

      ( γ ̇ /τ) = (1/η∞) = (τ ⋅ c1)2 / [(c2 ⋅ τ)2 ⋅ η0] = (c1/c2)2/η0

      or

      (η0/η∞) = (c1/c2)2 or η∞ = (c2/c1)2 ⋅ η0

      End of the Cleverly section

       3.3.6.4Model functions for flow curves with a yield point

      Detailed information on the yield point can be found in Chapter 3.3.4.1 Figure 3.21 presents a possible shape of flow curves explained below in sections b) to e).

      a) Bingham :τ = τB + ηB ⋅ γ ̇

      Flow curve model function with the “Bingham yield point” τB which occurs at the intersection of “Bingham straight line” and τ-axis in a diagram on a linear scale (see Figure 3.30), and “Bingham viscosity“ηB (according to E. C. Bingham, of 1916 [3.4]). Note: Despite of the denomination, this equation first was proposed by T. Schwedoff already in 1880 [3.12] [3.53].

      Note: ηB is not a viscosity value of an investigated sample, since it is not more than a calculated coefficient used for curve fitting (it would be better to speak of the “Bingham flow coefficient”).

      Before computers became widely used for analysis of flow curves, the Bingham model was often selected because analysis is very simple via the “Bingham straight line”, merely requiring a ruler. However, the “Bingham yield point” describes the transition from the state of rest to flow rather inaccurately. This model should therefore only be used for very simple QC tests (see also Chapter 3.3.4.2c: Plastic behavior, and Chapter 14.3: 1916, Bingham).

       Figure 3.30: Flow curve fitting according to Bingham

      Note: Simple evaluation methods (“according to Bingham”)

      Some users perform the following simple test and analysis method consisting of two intervals. In the first part, a constantly high rotational speed nH [min-1] is preset for a period of t10 = 10 min, and in the second part, a constantly low speed nL for another 10min = t20 (e. g. for ceramic suspensions: with nL = nH /10, for example, at nH = 100 min-1 and at nL = 10 min-1). Please be aware that the values of τ and η are relative stress and viscosity values if the test is performed with a spindle which is a relative measuring system (see also Chapter 10.6.2). Here, instead of the shear stress often is used dial reading DR (which is the relative torque value Mrel in %), and the viscosity values are calculated then simply as η = DR/n (with the rotational speed n in min-1). Usually here, all units are ignored [3.7].

      3.3.6.4.11) “Yield stress” (YS) using a straight line through two measuring points

      Preparation of a flow curve diagram with DR on the y-axis and the speed n on the x-axis (both on a linear scale). The two measuring points are plotted: (DRH; nH) and (DRL; nL) showing the values from the end of the two test intervals. A straight line is drawn through the two points, and then extrapolated towards n = 0. YS is the point at which this line crosses the y-axis (or DR-axis, respectively).

      3.3.6.4.2Example: with nH = 100, nL = 10, DRH = 50, DRL = 40, then: YS is read off as 38.

      3.3.6.4.32) “Plasticity index” (PI)

      Calculation: PI = YS/ηB

      with the “Bingham plastic viscosity” ηB = (DRH - DRL)/(nH - nL), or ΔDR/Δn

      3.3.6.4.4Example: with nH, nL, DRH, DRL and YS as above, then:

      ηB = (50 - 40) / (100 - 10) = (10/90) = 0.90, and PI = 38/0.9 = 42.2

      Another, but also very simple evaluation method called “Bingham build up” is mentioned in Chapter 3.4.2.2c: structure recovery and thixoptropic behavior.

      3.3.6.4.5b) Casson:

      or in the following form:τ1/2 = τC 1/2 + (ηC ⋅ γ ̇ )1/2

      Flow curve model function with “Casson yield point” τC, i. e. where the fitting curve meets the τ-axis in a diagram on a linear scale, and “Casson viscosity” ηC (of 1959 [3.54]). For ηC the same applies as above for ηB, thus, ηC is not more than a coefficient to be used for curve fitting but it is not a real viscosity value.

      This model function was originally designed for printing pastes. In 1973, OICC (Office International du Cacao et du Chocolat) or IOCCC (International Office of Cocoa, Chocolate and Confectionery), respectively, recommended for chocolate melts at T = +40 °C a testing and analysis method in the range of γ ̇ = 5 to 50 s-1 (60 s-1), by use of the Casson model. Evaluated are only the “Casson plastic viscosity“ and the “Casson yield value“ [3.55] [3.79].

      Modifications of the Casson model:

      1 “Generalized Casson model”: Here, instead of exponent 1/2 (which corresponds to the square root), the exponent 1/p is used.

      2 Casson/Steiner : with form-factor a = (Ri/Re) = (1/δcc) of a cylinder measuring geometry, bob radius Ri (inner radius) and cup radius Re (external radius), and ratio of radii δcc (of 1958 [3.56]). For ISO cylinder measuring systems applies:

      δcc = 1.0847, or a = (1/δcc) = 0.9219 (see Chapter 10.2.2.1a)

      Then:

      Summary: Using the Casson/Steiner model function, there is only a change in the “Casson yield point” when comparing to the original Casson model.

      3.3.6.4.6c) Herschel/Bulkley:τ = τHB + c ⋅ γ ̇ p

      Flow curve model function with “yield point according to Herschel/Bulkley” τHB, “flow coefficient” c [Pas], (also called “Herschel/Bulkley viscosity” ηHB), and exponent p (also called “Herschel/Bulkley index”; of 1925 [3.57]). It counts: p = 1 for “Bingham behavior”, p < 1 for shear-thinning, p > 1 for shear-thickening.