The Rheology Handbook. Thomas Mezger

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      M1) Optical method: When cooling, the cloud point is reached at the temperature at which the previously transparent sample becomes turbid and “cloudy” due to the precipitation of paraffin crystals.

      M2) Filter method: When cooling, the period of time increases which is needed to pump a certain amount of sample through a fine-meshed filter (e. g. CFP point).

      M3) Sagging method: When cooling, the pour point (PP) is reached at the temperature at which the sample still flows off a vertical surface. First the solidification point (SP) is determined as the point at which the sample is no longer able to flow at the transition from the liquid to the solid state. Afterwards is added ΔT = 3 K, thus: PP = SP + 3 K.

      M4) Yield point method: When cooling, the yield point occurs at the critical temperature when measuring with a simple rotational viscometer, for example, at a constant low shear rate or shear stress, respectively. This gives insight into problems with pumping processes, such as start-up of pumping and continued pumping (e. g. BPT point).

      M5) Flow cup method: When heating, the dropping point is reached at the transition temperature from the solid state to the liquid state, i. e., when the sample begins to flow through the orifice in the bottom of the flow cup (e. g. dropping point acc. to Ubbelohde [3.84]).

      Comment: All the above-mentioned simple methods are dependent on the test conditions and the skill of the tester. The methods therefore can only deliver relative values, as well as those obtained by the simple tests covering cooling behavior, described in Chapter 11.2.11 (d to f and h to k). Recommended methods for determining the crystallization or melting temperature are explained in Chapter 8.6.2.2a (oscillatory tests).

3.5.4Fitting functions for curves of the

      temperature-dependent viscosity

      The advantage of a fitting function is that it can be used to characterize the shape of a whole measuring curve using only a few model parameters, although the curve actually may consist of a large number of individual measuring points. A variety of viscosity/temperature fitting functions are mentioned in specialized literature, e. g. in ASTM D341, DIN 51563 and DIN 53017 [3.69]. As an example, explained is below the Arrhenius relation which usually is used for low-viscosity liquids.

      For “Mr. and Ms. Cleverly”

      The relations described here only apply to thermo-rheologically simple materials , i. e. materials, which do not change their structural character in the observed temperature range. Therefore, they do not change from the sol state to the gel state or vice versa (see also Chapter 8.2.4a, Note 1). In the temperature range which is related to practice, most polymer solutions and polymer melts are showing thermo-rheologically simple behavior. However, this applies usually not to dispersions and gels.

      3.1.2.1.1a) Arrhenius relation, flow activation energy EA, and Arrhenius curve

      An approximation model for kinetic activity in chemistry was developed in the general form by Svante A. Arrhenius (1859 to 1927) who introduced an activation constant (see also Chapter 14.2: 1884) [3.74]. The Arrhenius relation in the form of a η(T) fitting function describes the change in viscosity for both increasing and decreasing temperatures:

      Equation 3.6

      η(T) = c1 ⋅ exp (-c2 / T) = c1 ⋅ exp [(EA / RG) / T]

      with the temperature T in [K], (i. e. using the unit Kelvin), and the material constants c1 [Pas] and c2 [K] of the sample (where c2 = EA / RG), the flow activation energy EA [kJ/mol], and the gas constant RG = 8.314 ⋅ 10-3 kJ / (mol ⋅ K)

      Conversion between the temperature units:

      Equation 3.7

      T [K] = T [°C] + 273.15 K

      At a certain temperature, the flow activation energy E A characterizes the energy needed by the molecules to be set in motion against the frictional forces of the neighboring molecules. This requires exceeding the internal flow resistance, with other words, a material-specific energy barrier, the so-called potential barrier [3.27].

      The exponential curve function (Equation 3.6) occurs in a semi-logarithmic diagram as a straight line showing a constant curve slope if (1/T) is plotted on a linear scale on the x-axis (with the unit: 1/K), and η on a logarithmic scale on the y-axis. In this lg η / (1/T) diagram, the so-called Arrhenius curve, temperature-dependent behavior occurs as a downwardly or upwardly sloping straight line for a heating or a cooling process, respectively.

      Note : Recommended temperature range for fitting functions (Arrhenius and WLF)

      The Arrhenius relation is useful for low-viscosity liquids and polymer melts in the range of T > Tg + 100K (with the glass-transition temperature Tg, see Chapter 8.6.2.1a) [3.10] [3.34]. For analysis of polymer behavior at temperatures closer to Tg, it is better to use the WLF relation. For more information on this time/temperature shift method TTS, see Chapter 8.7.1.

      3.1.2.1.2b) Viscosity/temperature shift factor aT, and Arrhenius plot

      The following holds for the viscosity/temperature shift factor aT in general:

      Equation 3.8

      aT = η(T) / η(Tref)

      The dimensionless factor aT is the ratio of the two viscosity values at the temperature T and at the reference temperature Tref. For polymers, this relation is only valid for the values of the zero-shear viscosity η0. The following holds for the temperature shift factor according to Arrhenius (with T in [K]):

      Equation 3.9

      The semi-logarithmic, so-called Arrhenius plot presents the temperature shift factor aT on the y-axis on a logarithmic scale versus the reciprocal temperature 1/T on the x-axis (with the unit: 1/K).

      In order to estimate viscosity values at temperatures at which no measuring values are available, proceed as follows:

      1 Select Tref (e. g. a temperature, at which an η-value is available).

      2 Calculate the shift factor aT for another available η(T)-value (using Equation 3.8).

      3 Calculate the flow activation energy value EA (using Equation 3.9).

      4 Calculate the shift factor aT for the desired η(T)-value (using Equation 3.9).

      5 Result: Calculate the desired η(T)-value (using Equation 3.8).

Table 3.5: Temperature-dependent viscosity values, see the example of Chapter 3.5.4b
T [°C]	50	60	70	80
T [K]	323	333	343	353
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