The Rheology Handbook. Thomas Mezger
showing the 2nd zero-normal stress coefficient as a plateau value in the low-shear range
Equation 5.8
ψ2,0 [Pa ⋅ s2] = lim γ ̇ → 0 ψ2( γ ̇ ) = const
In order to determine the first normal stress difference, the raw data measured by a rheometer are the values of the normal force FN in [N] in axial direction (y-direction). Normal forces of samples are forces acting into the direction of the shaft of the measuring bob, trying to push the upper plate or the cone upwards or the lower plate downwards, respectively (when using a parallel-plate or cone-and-plate measuring geometry). For Information on tests using the normal force control (NFC) option, see Chapters 10.4.6 and 10.7b.
Effects of normal forces occur in different forms. They may cause large problems in several application fields in industry. Examples (see Figures 5.3, 5.4, 5.10 and 5.11):
1 When performing rotational tests with viscoelastic samples, the corresponding effects may occur in the form of streaks and similar defects on the surface of cylinder measuring geometry or on the edges of cone-and-plate and parallel-plate systems. For stiff samples, these effects are also influenced by the stiffness of as well the measuring geometry as well as of the measuring instrument used.
2 Post-extrusion swell or die swell effects and as melt fracture when extruding polymer melts
3 The Weissenberg effect or rod-climbing effect when stirring
Figure 5.10: Cone-and-plate (CP) measuring geometry in side view during a rotational test, with a certain constriction of the sample at the edge of the cone. In the direction along the cone axis (axial direction), the normal force tries to push apart cone and plate, and with a stiff bottom plate system available, it may be measured on top as FN
Figure 5.11: CP or PP geometry in top view during a rotational test on a polymer sample: (1) with coiled macromolecules when at rest or under low shear conditions, and (2) with stretched molecules under the shear force FS working in x-direction causing a high shear deformation or shear rate, respectively. Due to the resetting elasticity of the molecules, as a consequence, normal forces are resulting, as well in y-direction (axial, along the axis of the measuring geometry, see Figure 5.10) as well as in z-direction (towards the axis)
Further information on normal stresses can be found e. g. in DIN 13316 [5.4] [5.13].
Note: Lodge/Meissner relation
In 1976, Arthur S. Lodge (1942 to 2007) and Joachim Meissner (1929 to 2011) presented the following relation [5.20] [5.21]:
Equation 5.9
N1,LM = γ ⋅ τ
Using this LM-relation, with the values of shear strain γ [1] and shear stress τ [Pa] which were preset or determined via relaxation tests (see Chapter 7), the 1st normal stress difference N1 [Pa] can be calculated. Numerous tests with many standard polymers have confirmed this empirical rule. Comparisons have shown good correlation between
1 N1 values which are measured directly by the normal stress sensor of a rheometer, and
2 N1,LM values, which are calculated by use of the LM relation from data which are measured by stress relaxation tests [5.22]
In the following Chapters 6 to 8, the most important types of tests are presented which can be performed to measure viscoelastic behavior: creep tests, relaxation tests, and oscillatory tests.
5.4References
[5.1]Maxwell, J. C., Constitution of solids, Encyclopaedia Britannica, 1878
[5.2]Giesekus, H., Die Elastizität von Flüssigkeiten, Rheol. Acta, 1966; Phänomenologische Rheologie, Springer, Berlin, 1994
[5.3]Boger, D. V., Walters, K., Rheological phenomena in focus, Elsevier, Amsterdam, 1993
[5.4]Böhme, G., Strömungsmechanik nichtnewtonscher Fluide, Teubner, Stuttgart, 2000 (2nd ed.)
[5.5]Joseph, D. D., Fluid dynamics of viscoelastic liquids, Springer, New York, 1990/2007
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[5.7]Weissenberg, K., A continuum theory of rheological phenomena, Nature, 1947; Rheology of hydrocarbon gels, Proc. Roy. Soc., 1950; The testing of materials by means of the Rheogoniometer, Sangamo Controls Ltd., 1964
[5.8]Ottersbach, J., Bedruckstoff und Farbe, Verl. Beruf u. Schule, Itzehoe, 1995
[5.9]Weipert, D., Tscheuschner, H.-D., Windhab, E., Rheologie der Lebensmittel, Behr’s, Hamburg, 1993
[5.10]Meyer, O. E., Theorie der elastischen Nachwirkung, Pogg. Ann. Physik, 1874
[5.11]Thomson, W. (Kelvin), On the elasticity and viscosity of metals, Proc. Roy. Soc., 1865; Papers, London, 1890
[5.12]Voigt, W., Über innere Reibung fester Körper insbesondere der Metalle, Ann. Phys., 1892
[5.13]Barnes, H. A., Hutton, J. F., Walters, K., An introduction to rheology, Elsevier, Amsterdam, 1989; Barnes, H. A., A handbook of elementary rheology, Univ. of Wales Inst. Non- Newtonian Fluid Mechanics, Aberystwyth, 2000; Barnes, H. A., Viscosity, dto., 2002
[5.14]Pahl, M., Gleissle, W., Laun, H.-M., Praktische Rheologie der Kunststoffe und Elastomere, VDI, Düsseldorf, 1995
[5.15]Kohl, P., Ruhe im Betrieb – Maßnahmen des Schallschutzes, J. Maschinen-Markt, 3/2006
[5.16]J. ADAC Motorwelt, 3/2016
[5.17]Ilschner, B., Singer, R. F., Werkstoffwissenschaften und Fertigungstechnik, Springer, Berlin, 2010 (5th ed.)
[5.18]Khademhosseini, A., Vacanti, J. P., Langer, R., Progress in tissue engineering, J. Sci. Am., 05/2009
[5.19]Cauchy, A. L., De la pression ou tension dans un corps solide, 1827; Sur les équations differentielles ou de mouvement pour les points matériels, 1829
[5.20]Lodge, A. S. Meissner, J., J. Rheol. Acta, 1976; L., On-line measurement of elasticity and viscosity in flowing polymeric liquids, J. Rheol. Acta, 1996; L., Normal stress differences from hole pressure measurements, in: A. A. Collyer, D. W. Clegg, Rheological Measurement, Chapman, London, 1998 (2nd ed.)
[5.21]Wissbrun, K. F., Transient rheometry, in: A. A. Collyer, D. W. Clegg, Rheological Measurement, Chapman, London, 1998 (2nd ed.)
[5.22]Läuger, J., Heyer, P., Validation of empirical rules for a standard polymer solution by different rheological tests, Ann. Trans. Nordic Rheol. Soc. 15, 2007
[5.23]Freundlich, H., Über Thixotropie, Kolloid-Zeitschrift, 1928; Thixotropie, Hermann, Paris, 1935
[5.24]Messow, U., Wolfgang Ostwald und der kolloid-disperse Zustand, in: Zur Entwicklung erster Messgeräte der physikalischen Chemie an der Universität Leipzig, Leipziger Univ. Verl., 2013