The Rheology Handbook. Thomas Mezger

γ ̇ *)] + η∞ ⋅ γ ̇

      This analysis model was developed to evaluate chocolate melts at T = +40 °C in the range of

       γ ̇ = 2 to 50 s-1 (or in an extended range of γ ̇ = 1 to 100 s-1, resp.) [3.15].

      Flow curve model function with yield point τ0 [Pa]; shear stress τ1 [Pa] which leads to the “maximum shear-induced structural change” (at the intersection of τ-axis and the straight line fitted to the flow curve in the high shear rate range); shear rate γ ̇ * [s-1] at the point

      τ* = τ0 + (τ1 - τ0) ⋅ (1 – 1/e), and slope value of the flow curve at high shear rates termed η∞ [Pas], (“final viscosity“, “steady-state viscosity”; this approaches usually to a constant value for most chocolate melts in the range of γ ̇ = 60 to 100 s-1); see Figure 3.31.

       Figure 3.31: Flow curve fitting according to Windhab

      The following applies for γ ̇ = 0:

      τ = τ0 + (τ1 - τ0) ⋅ (1 – e0) + 0 = τ0 + (τ1 - τ0) ⋅ (1 – 1) = τ0 + 0 = τ0

      Result: “Structural strength at rest” is represented by the “yield point” in the range of τ ≤ τ0.

      The following applies for γ ̇ = ∞:

      τ = τ0 + (τ1 - τ0) ⋅ (1 – e-∞) + η∞ ⋅ γ ̇ = τ0 + (τ1 - τ0) ⋅ (1 – 0) + η∞ ⋅ γ ̇ = τ1 + η∞ ⋅ γ ̇

      Result: In the “high-shear” range, the slope of the flow curve is constant, which corresponds to “Bingham behavior”.

      The following applies for γ ̇ = γ ̇ *:

      τ = τ0 + (τ1 - τ0) ⋅ (1 – 1/e) + 0 = τ0 + 0.632 ⋅ (τ1 - τ0) = τ*

      Result: Between τ0 and τ1 is the “range of shear-induced structural change” (with structural decomposition at increasing shear load, i. e. in the “upwards ramp”; or with structural regeneration at decreasing load, i. e. in the “downwards ramp”). Above the yield point τ0 the point τ* (or γ ̇ *, resp.) is reached at the shear stress value τ = 63.2 % (τ1 - τ0).

      3.3.6.4.8e) Tscheuschner:τ = τ0 + c1 ⋅ γ ̇ p + c2 ⋅ γ ̇

      Flow curve model function with yield point τ0 [Pa], coefficient c1 [Pas] for the low and medium shear range to describe the shear-induced structure change, coefficient c2 [Pas] as slope value of the flow curve at high shear rates, and exponent p. This model function was designed for chocolate melts at T = +40 °C [3.53].

      3.3.6.4.9f) Polynomials

      These model functions are purely mathematical descriptions of flow curve functions (e. g. according to Williamson , Rabinowitsch, Weissenberg). Polynomial models represent the most general approach to curve analysis and can be fitted to each type of curves. However, when using this method, a relatively large number of coefficients have to be determined, although this is no longer a problem with current software analysis programs.

      3.3.6.4.10Example: third order polynomial τ = c1 + c2 ⋅ γ ̇ + c3 ⋅ γ ̇ 2 + c4 ⋅ γ ̇ 3

      with the coefficients c1 [Pa] as 0th order coefficient, representing the yield point; c2 [Pa ⋅ s], as 1st order coefficient; c3 [Pa ⋅ s2], as 2nd order coefficient; c4 [Pa ⋅ s3], as 3rd order coefficient. If a second order polynomial is used for analysis, then the last term of the function is ignored. Polynomials of higher orders (e. g. 5th) show correspondingly more terms.

      There are also polynomials which are solved to the shear rate:

       γ ̇ = c1 ⋅ τ + c2 ⋅ τ2

      with the coefficients c1 [1/Pas] and c2 [1/Pa2 ⋅ s]

      A comparable model is the Steiger/Ory model (see Chapter 3.3.6.2b).

3.3.7The effects of rheology additives

      in water-based dispersions

      Legislative restrictions concerning industrial products increasingly are forcing the reduction of volatile organic compounds (VOC). As a result, solutions and dispersions containing organic solvents are more and more replaced by water-based dispersions; examples are coatings of all kinds such as paints and adhesives. Therefore, adapted rheology additives have been developed which of course also influence flow behavior, sometimes even considerably [3.59] [3.60] [3.61] [3.62]. Typical examples of nanostructures and microstructures of rheology additives are shown in Figure 3.32 [3.63]; the black bar in the Figure indicates the dimension of 100 nm to illustrate the order of magnitude when comparing the material systems.

      3.1.2.1.1a) Water-based dispersions containing clay as the thickening agent

      Inorganic primary particles of clay exhibit the shape of thin platelets. A “house of cards structure” with a network of secondary forces is built up between the surfaces and edges of the platelets when at rest; see Figure 3.32. This is due to the fact that the large flat surfaces of the platelets show negative electrical charge, whereas the narrow sides and edges are positively charged. Therefore, at rest a superstructure in the form of a gel structure is occurring if the additive is incorporated in an appropriate way. In this case, in the low-shear range there is a yield point or an “infinitely high” viscosity, respectively.

      The network of forces breaks if the yield point is exceeded under sufficiently high shear force. With increasing shear rates the viscosity values are decreasing continuously since finally, the “house of cards structure” will be completely destroyed. Increasingly, the individual sheet-like and very flat particles of the additive are oriented into shear direction. At high shear rates, and at a typical concentration of around 0.1 to 0.3 % as it is used in practice, the clay particles in the flowing state display no longer a significant thickening effect.

       Figure 3.32: Nano- and microstructures in water-based coating systems (the black bar indicates the

      dimension of 100 nm.). Left side: at rest, and right side: in a shearing state. 1) clay (as an inorganic

      gellant), 2) polymer molecules in solution, 3a) dispersed polymer particles, without an additive,

      3b) polymer dispersion with a polymeric associative thickener, here, also surfactant molecules are integrated in the bridge-like clusters

      Dimensions: Primary particles of bentonite are around 800 ⋅ 800 ⋅ 1 (in nm) and those of hectorite are around 800 ⋅ 80 ⋅ 1; with an organic surface modification the distance between individual platelets is around 4 nm.

      Figure 3.33 presents a typical viscosity function of a pigmented water-based coating with clay as the thickener. On the one hand there is a very pronounced thickening effect in the low-shear range at γ ̇ < 1 s-1 which for example may cause problems when starting to pump the dispersion from a state of rest. On the other hand, there is no more thickening effect at high shear rates, e. g. at γ ̇ > 1000 s-1, and this may cause uncontrolled splashing and spattering of