2 Properties of Polymers
2.4 Crystallisation and Solid Properties
2.4.4 Viscoelastic Properties in the Solid State and in the
Viscoelastic properties of polymers in the solid state are treated within the same framework as for the molten state (see Section 2.2.2). They can be measured in creep (constant stress or load) or in relaxation (constant strain) experiments. Very often, they are determined through dynamic mechanical analysis. The material is subjected to a small sinusoidal strain, e.g., a shear strain g applied in torsion:
)
Properties of Polymers
where G′ is the elastic (or storage) modulus and G′′ is the viscous (or loss) modulus.
The ratio G′′/ G′ is the tangent of the phase angle, tan d.
From experimental results, master curves can be established. In the case of semicrystalline polymers, the situation is more complicated compared to the molten state, since the material is not thermorheologically simple, because of its crystallinity.
A usual way to solve the problem is to introduce an additional vertical shift to the classical horizontal one used in the time-temperature superposition (see Section 2.2.3).
Figure 2.11 shows the tan d curves obtained for a LDPE and a HPDE used in pipe extrusion. The superposition can be considered as satisfactory for HDPE. This is not the case for LDPE in spite of the double shift.
0.3
Figure 2.11 tan d master curves (symbols) for a HDPE and a LDPE used in tube extrusion. Reference temperature: 35 °C. The solid lines have been calculated with
a multimode Maxwell model
The Maxwell model with one relaxation time presented in Section 2.2.2 is not able to describe the behaviour of solid polymers, but this can be done by using a multimode Maxwell model, consisting of N simple Maxwell models (see Figure 2.4) in parallel.
Each element is characterised by its modulus G(n) and its relaxation time l(n). The G′ and G′′ components of the complex modulus are given by:
∑
+where w is the angular frequency introduced in Equation 2.27. The values of G′, G′′ and tan d calculated with the model can be compared to the experimental ones.
This is done for tan d in Figure 2.11. Fifteen relaxation modes have been used for both polymers.
Multimode Maxwell models are able to describe, with appropriate coefficients, the polymer behaviour in both the liquid and the solid state. In spite of interesting works in the field, (e.g., [27, 28]), the treatment of the rheological behaviour in the liquid-solid transition zone remains an open question. An empirical solution for numerical simulation is to use a mixing rule combining quantities for the fluid (f) and the solid (s) states. For instance, in the case of PE [29], G′ has been written:
(2.32)
(2.33)
where xc denotes the crystallinity; the value xc = 0.25 corresponds to the end of primary crystallisation, which can be described by Ozawa’s model (Equation 2.25).
This means that the polymer is considered as solid at the end of primary crystallisation.
The same approach has been applied to polyamide 12 [30].
Properties of Polymers
2.5 Conclusion
Polymers are complex materials whose properties, both in the liquid and solid states, control the relationships between processing conditions, structure and final properties. We will see in the following chapters that the understanding of these specific properties is ofv vital importance for the control and the optimisation of the extrusion processes and the manufacturing of a variety of industrial products such as pipes, sheets, films and profiles.
References
1. W. Ostwald, Kolloid Zeitung, 1925, 36, 99.
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4. K.Y. Yasuda, R.C. Amstrong and R.E. Cohen, Rheologica Acta, 1981, 20, 163.
5. C.W. Macosko, Rheology, Principles, Measurements and Applications, Wiley-VCH, New York, NY, USA, 1994.
6. P.J. Carreau, D.C.R. De Kee and R.J. Chhabra, Rheology of Polymeric Systems. Principles and Applications, Hanser, Munich, Germany, 1997.
7. J.M. Dealy and K.F. Wissbrun, Melt Rheology and its Role in Plastics Processing, Kluwer Academic Publishers, Dordrecht, Netherlands, 1999.
8. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, MA, USA, 1988.
9. J.M. Dealy and R.G. Larson, Structure and Rheology of Molten Polymers.
From Structure to Flow Behavior and Back Again, Hanser, Munich, 2006.
10. M.L. Williams, R.F. Landel and J.D. Ferry, Journal of the American Chemical Society, 1955, 77, 3701.
11. A. Goubert, J. Vermant, P. Moldenaers, A. Göttfert and B. Ernst, Applied Rheology, 2001, 11, 26.
12. M. Mooney, Transactions of the Society of Rheology, 1931, 2, 210.
13. A. Allal and B. Vergnes, Journal of Non-Newtonian Fluid Mechanics, 2009, 164, 1.
14. S.G. Hatzikiriakos and K.B. Migler, Polymer Processing Instabilities:
Understanding and Control, Marcel Dekker, New York, NY, USA, 2005.
15. D. Kay, P.J. Carreau, P.G. Lafleur, L. Robert and B. Vergnes, Polymer Engineering and Science, 2003, 43, 78.
16. C. Venet and B. Vergnes, Journal of Rheology, 1997, 41, 873.
17. L. Robert, Y. Demay and B. Vergnes, Rheologica Acta, 2004, 43, 89.
18. C. Combeaud, Y. Demay and B. Vergnes, Journal of Non-Newtonian Fluid Mechanics, 2004, 121, 175.
19. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, NY, USA, 1960.
20. J.M. Haudin and B. Monasse in Structure Development During Polymer Processing, NATO Science Series, Series: Applied Sciences, Volume 370, Eds., A.M. Cunha and S. Fakirov, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000, p.93.
21. E. Piorkowska, A. Galeski and J.M. Haudin, Progress in Polymer Science, 2006, 31, 549.
22. M. Avrami, Journal of Chemical Physics, 1940, 8, 212.
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24. T. Ozawa, Polymer, 1971, 12, 150.
25. G. Strobl, The Physics of Polymers. Concepts for Understanding Their Structures and Behavior, 3rd Edition, Springer, Berlin, Heidelberg, Germany, 2007, p.544.
26. J.M. Haudin in Handbook of Polymer Crystallization, John Wiley, Hoboken, NJ, USA, 2012, Chapter 16.
27. K. Boutahar, C. Carrot and J. Guillet, Macromolecules, 1998, 31, 1921.
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Properties of Polymers
29. O. Parant, Etude Expérimentale et Calcul des Contraintes Résiduelles dans des Tubes Extrudés en Polyéthylène, Ecole Nationale Supérieure des Mines de Paris, 2002. [Ph.D Thesis]. [In French]
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Montezin, International Polymer Processing, 2008, 23, 55.
Jean-Marc Haudin, Michel Vincent, Bruno Vergnes
3.1 Introduction
The production of pipes is usually performed according to the scheme presented in Figure 3.1:
Circular saw
Haull-off
system Cooling bath
Pipe
Die Extruder
Calibrator
Figure 3.1 Extrusion line for pipe production
An extruder continuously feeds an axisymmetric die, by means of which the geometry of the pipe is obtained. At the die exit, the pipe passes through a calibrator, in order to cool down its surface and to freeze the outside diameter. Then, the pipe is pulled through different cooling baths, before being eventually cut and stored at the end of the extrusion line. In the present chapter, we will focus only on the die and the calibrator.
Different types of die may be used [1, 2], but the most common is the spider leg die, shown in Figure 3.2.