The Rheology Handbook. Thomas Mezger

sum of the shear rates results in the differential equation according to Maxwell .:

      Equation 5.1

       γ ̇ = γ ̇ v + γ ̇ e = τv /η + τ ̇ e /G = τ/η + τ ̇ /G

      The solution and use of this differential equation are described in Chapters 7.3.2c and 7.3.3.2 (relaxation tests) as well as in Chapter 8.4.2.1 (oscillatory tests/frequency sweeps).

      End of the Cleverly section

      5.1.2.1.85.2.1.2Examples of the behavior of VE liquids in practice

      5.1.2.1.9a) Die swell, or post-extrusion swelling effect (see Figure 5.3)

      5.1.2.1.10Experiment 5.2: Extrusion, using a small toy extruder, producing spaghetti-like strands

      1 Die swell does not occur when extruding plasticine, (or wax). Immediately after the extrusion, the plasticine strands exhibit a stable form. Therefore, this material shows behavior of a (viscoelastic) solid.

       Figure 5.3: Die swell of a viscoelastic material after an extrusion (with pressure p, diameter d of the die, and diameter increase Δd of the extrudate)

      1 Silicone polymer (uncrosslinked PDMS) clearly displays post-extrusion die swelling. After a certain time when at rest, the extruded strands are slowly flowing and spreading, penetrating one another finally. Therefore, the silicone is showing behavior of a (highly viscous, viscoelastic) liquid.

      Due to the high pressure in the extrusion die, the polymer molecules are deformed and oriented into shear direction. As soon as leaving the die, the molecules of a visco-elastic material are recoiling since they are in a stress-less state then. As a consequence, re-formation in shear direction occurs, and in order to compensate this, they are expanding into other directions, for example, right-angled to the direction of the extrusion. Therefore, the extrudate shows an increased dia­meter finally (see also Chapter 11.4.2.2b). Further information on die swell effects, including images, can be found in [5.3] [5.4] [ 5.5] [ 5.6].

      5.1.2.1.11Examples: Materials showing die swell

      Polymers, when extruding rods, pipes or profiles (e. g. window frames), and when spinning synthetic fibers.

      Sometimes, the following parameters are determined:

      Swell ratio (or swell factor):

      SR = d1/d = (d + Δd)/d

      Die swell (or relative swell):

      Δd/d = (d1 – d)/d = (d1/d) – 1

      with the diameters d of the die and d1 of the extrudate, and their change Δd (= d1 – d)

      5.1.2.1.12b) The Weissenberg effect when stirring

      5.1.2.1.13Experiment 5.3: The two stirrer vessels, containing water and a polymer solution (Figure 5.4)

      Water as an ideal-viscous liquid exhibits the usual flow behavior during stirring, forming a vortex around the stirrer axis showing the lowest water level in the center of the vessel. The highest level occurs at the wall of the vessel. In contrast to that, the viscoelastic polymer solution creeps up the stirrer shaft when stirring; this is called the Weissenberg effect.

      Karl Weissenberg (1893 to 1976) studied VE effects in detail [5.7]. In 1951, he was the first one who presented an instrument which really deserves to be called a “rheometer” (see also Chapter 14.4). When shearing VE liquids at certain conditions, such as sufficiently high rotational speeds and at low temperatures, they are displaying the Weissenberg effect, which is also called the “rod climbing effect ”. For these kinds of fluids, the pressure and flow conditions in the vessel are changing compared to the behavior of an ideal-viscous fluid, and as a consequence, reversed flow direction of the VE liquid can be observed in a bypass pipeline as illustrated in Figure 5.4. Of course, these effects might have an enormous impact on the effectiveness of a stirring process, and the success in mixing corresponding liquids might be greatly limited. By the way, this effect also occurs if a magnetic stirrer is used, i. e. without using a stirrer shaft. In this case, a rising “mountain of liquid” can be seen in the center of the container above the rotating stirring rod. Further information on the Weissenberg effect, including images, can be found in [5.3] [5.4] [5.5] [5.6].

       Figure 5.4: Liquids in two stirrer vessels, left: displaying ideal-viscous flow behavior, and right: visco­elastic behavior showing the Weissenberg effect (with the pressures pA at the inner wall and pB in the center of the vessel)

      5.1.2.1.14Examples: Materials showing the Weissenberg effect

      Dough, emulsion paints, polymer melts, highly concentrated polymer solutions and dispersions when stirring at a sufficiently high rotational speed

      5.1.2.1.15c) Tack and stringiness when performing coating processes

      Offset printing inks, adhesives, polymer-modified bitumen (PmB), grease and coatings often exhibit tacky behavior and stringiness with long filaments during application when printing, coating and painting with a brush or a roller, respectively (see Figure 5.5). See also Chapter 11.2.1 g: finger test (e. g. for luricating grease: evaluation as brittle, buttery, long fiber, short fiber, stringy).

       Figure 5.5: Comparison of two more or less tacky materials, one of them shows stringiness when strained at a high deformation [5.8]

      5.1.2.1.16d) Mouth sensation

      When sticky, tacky, stringy and ropy foodstuffs or pharmaceuticals containing thickening agents show a too high elastic portion, this may lead to unpleasant mouth sensation and might even cause problems when swallowing [5.9].

      For “Mr. and Ms. Cleverly“

      Note 1: Explanation of the Weissenberg effect

      For rotational systems, the circumferential velocity v increases with increasing distance from the axis of rotation: v [m/s] = ω ⋅ r, with the angular velocity ω [s-1] = (2π ⋅ n)/60, the radius r [m], and the rotational speed n [min-1]. Since many polymer molecules cannot follow the rapid motion, e. g. at the edge of the stirrer vessel, they gather in the middle. This causes the “mountain of liquid” in the center of the container. As a simple illustrative example you can imagine: On a roundabout (merry-go-round) rotating at a constant speed, it is better to sit closer to the center than more outside. On the latter position, the “stress by motion” is increased due to the higher circumferential velocity compared to a place which is closer to the