Energy Elasticity

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Entropy elasticity

Physical fundamentals

Entropy elasticity expresses the tendency of macromolecules to return to the entropically most favourable state, the tangled state, after deformation [1]. When a flexible-chain polymer material is subjected to mechanical stress, the macromolecules align themselves in the field of stress. The molecular state of order is accompanied by a reduction in the entropy of the system. If the irreversible slippage of the chain segments can be prevented, for example by cross-linking, the macromolecules strive to maximise entropy when the stress is relieved (2nd law of thermodynamics). They transition to the disordered state of equilibrium in a timeless manner.

Entropy-elastic behaviour is observed up to large strains of several hundred percent. The relationship between stress and deformation is non-linear. Simple continuum mechanical considerations as well as molecular statistical models [2] lead to a relationship of the form in the case of uniaxial stress:

$\sigma \,=\,{\frac {E}{3}}\cdot \left(\lambda -\lambda ^{-2}\right).$

(1)

Derivation of material parameters

The modulus of elasticity E as a material parameter is determined by the cross-linking density N or the average molar mass between the cross-linking nodes of the polymer M_C. It also depends on the temperature T, the Boltzmann constant k, the universal gas constant R and the density:

$E\,=\,3NkT\,=\,{\frac {3\rho }{{\overline {M}}_{C}}}RT.$

(2)

With the help of Eq. (1), essential phenomena of the mechanical behaviour of rubber vulcanisates can be represented. However, their quantitative validity is often limited to strains below 100 %. For this reason, the simple theory of rubber elasticity has undergone a series of further developments, which are reported on in [3], for example.

Entropy elasticity is not limited to covalently cross-linked polymers. It also plays an important role above the glass transition temperature in amorphous and semi-crystalline thermoplastics with sufficiently high molecular weight. Here, physical entanglements (see: polymer & structure) and interlockings take on the role of temporary cross-linking points [4–6].

See also

• Elasticity
• Rubber elasticity
• Cross-linking elastomers
• Glass transition temperarure
• Deformation
• Thermoelastic effect

References

[1]	Lüpke, Th.: Material Behavior and Constitutive Equations. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich, (2022) 3rd Edition, pp. 75–77 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)
[2]	Treloar, L. R. G.: The Physics of Rubber Elasticity. Clarendon Press, Oxford (1975)
[3]	Stavemann, A. J.: Properties of phantom networks and real networks. Adv. Polym. Sci. 44 (1982) 73–101 DOI: https://doi.org/10.1007/3-540-11471-8_3
[4]	Erman, B., Mark, J. E.: Structure and Properties of Rubberlike Networks. Oxford University Press, New York (1997) (ISBN 978-0-1950-8237-1)
[5]	Termonia, Y., Smith, P.: Kinetic model for tensile deformation of polymers. Macromolecules 20 (1987) 835–838 DOI: https://doi.org/10.1021/MA00170A023
[6]	Bensason, S., Stepanov, E. V., Chum, S., Hiltner, A., Baer, E.: Deformation of elastomeric ethylene-octene copolymers. Macromolecules 30 (1997) 2436–2444 DOI: https://doi.org/10.1021/ma961685j