The Rheology Handbook. Thomas Mezger

lg η1) – (lg lg η2) / (lg T1 – lg T2)], with η in mPas and T in K,

      until the maximum of the GI function is exceeded finally. See also ASTM D341:

      Empirical η(T) equation by MacCoull, Walther and Wright [3.72].

      Presentation in a diagram as [lg lg y(lg x)] curve, the so-called gelation curve.

      Analysis: The GI value is reached at the maximum of the GI curve, and the corresponding GI temperature (GIT) is determined. GI is therefore the value at the maximum viscosity increase.

      Comment: Corresponding modern testing methods are oscillatory tests to determine the sol/gel transition temperature (see Chapter 8.6.3b).

      Note 2: Cloud point , pour point , freezing point , dropping point of petrochemicals

      In order to analyze cooling behavior of petrochemicals there are many, and often very simple, measuring and analysis methods. Samples include fuels such as kerosene, gasoline, diesel oils and heating oils, lubricants such as mineral oils and lubricating greases, paraffins and waxes.

      3.1.2.1.1Examples (here in alphabetic order):

      1 Borderline pumping temperature (BPT)

      2 Cloud point (CP; ISO 3015; ASTM D2500, D7467, D5771 – stepped cooling method, D5772 – linear cooling rate method, D5773 – constant cooling rate method, D7397; DIN EN 23015)

      3 Cold filter plugging point (CFP; ASTM D4539, D6371, DIN EN 116, DIN EN 16329, IP 309)

      4 Congealing point (ISO 2207; ASTM D938; DIN ISO 2207)

      5 Crystallization point

      6 Drop melting point (ASTM D127)

      7 Dropping point (ISO 2176; ASTM D566, D2265, D3954; DIN ISO 2176)

      8 Flocculation point

      9 Freezing point (ISO 3013; ASTM D2386)

      10 Gelation point

      11 Melting point (ASTM D87)

      12 No-flow point (ASTM D7346)

      13 Pour point (PP; ISO 3016; ASTM D97, D5950; DIN ISO 3016; IP 15). When cooling under the specified conditions, PP is the lowest temperature at which a petrochemical sample is still able to flow.

      14 Solidification point

      15 Wax appearance point (ASTM D3117)

      Here, some frequently used methods of determination and criteria are explained briefly.

      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,