Rubber Elasticity

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Rubber elasticity; entropy elasticity

Definition

The term rubber elasticity, or the thermodynamically more accurate term entropy elasticity used in physics, refers to the resistance of rubber-like materials (elastomers) to deformation [1, 2].

Description of deformation

When rubber-elastic materials are deformed and oriented, chain segments are moved from their equilibrium position into a less favourable entropic state, i.e. under load, they transition into a more ordered structure. Due to the weak cross-linking, chain segments cannot slide completely against each other, which means that the entropically elastic bodies are reversibly deformable by very large amounts (several hundred percent).

When the load is removed, the chain segments return from an ordered to a disordered state, i.e. the entropy increases again. Thermodynamically, rubber elasticity is associated with a reduction in entropy in the deformed state.

From a molecular point of view, the deformation forces an orientation in parts of the molecule, i.e. a change in conformation. Even non-cross-linked macromolecules exhibit entropy-elastic behaviour under certain loads. In this case, the interlocking and entanglement during rapid deformation act like cross-links, but the entropy-elastic behaviour of these materials is overlaid by a more or less pronounced flow.

The thermodynamics of rubber elasticity

The temperature dependence of the elastomer material group is of crucial importance. The glass transition temperature T_G of elastomers is well below normal operating temperatures, i.e. within the operating temperature range, the chain mobility is so high that the stress during deformation occurs practically without delay.

From a thermodynamic point of view, rubber elasticity is essentially based on a decrease in entropy S in the general equation for the change in free energy $F=U-TS$ at a given strain.

The free energy F in the tensile test can be calculated using Eq. (1)

$dF=-S\cdot dT-p\cdot dV+f\cdot dl$

(1)

where

F – free energy	T – temperature
S – entropy	V – volume
p – pressure	dl – change in length
f – tensile force

can be calculated.

The measured force is

$f=\left({\frac {dF}{dl}}\right)_{T,V}$

(2)

Since, in the case of a reversible isothermal process, the change in energy is equal to the mechanical work dW/dl applied by external forces, Eq. (3) applies.

${\frac {dW}{dl}}={\frac {dF}{dl}}=-T\cdot {\frac {dS}{dl}}$

(3)

A statistical analysis of the relationship ultimately leads to the conclusion that, in the case of a network with N chains per unit volume, Eq. (4) applies to the mechanical work W [3‒6]:

$W=-T\cdot dS={\frac {1}{2}}N\cdot k\cdot T\cdot \left(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}-3\right)$

(4)

with

W – mechanical work

k – Boltzmann constant k = 1.38∙1023 JK-1

λ – deformation (strain ratio)

The parameter NkT is defined as the shear modulus G. Taking into account the molar mass M_c between the cross-linking points, the calculation Eq. (5) for the shear modulus G is obtained with R = k ∙ N_L and N = ρ / M_c ∙ N_L:

$G=N\cdot k\cdot T={\frac {\rho \cdot R\cdot T}{M_{c}}}$

(5)

with

R – gas constant

NL – Loschmidt number

ρ – density of the material

The uniaxial tensile test

In the case of uniaxial tensile loading, a deformation $\lambda _{1}=\lambda \!$ is applied, and the lateral deformations are then $\lambda _{2}=\lambda _{3}=\lambda ^{-{\frac {1}{2}}}\!$ (at constant volume, i. e. $\lambda _{1}\cdot \lambda _{2}\cdot \lambda _{3}=1$). The following Eq. (6) then applies to the stress applied to a test specimen (see also: test piece)

$\sigma ={\frac {dW}{d\lambda }}=-T\cdot {\frac {dS}{d\lambda }}=G\ \left(\lambda -{\frac {1}{\lambda ^{2}}}\right)$

(6)

,

which in many cases corresponds quite well with experiments. At higher degrees of deformation λ > 1.5, the empirical MOONEY-RIVLIN Eq. (7) describes the behaviour of elastomers more precisely. The following then applies

$\sigma =C_{1}\cdot \left(\lambda ^{2}-{\frac {1}{\lambda }}\right)+C_{2}\cdot \left(\lambda -{\frac {1}{\lambda ^{2}}}\right)$

(7)

with C₁ and C₂ being elasticity constants.

From Eq. (5), it can be concluded that the shear modulus increases with the number of cross-links or the cross-link density.

See also

• Elasticity
• Entropy elasticity
• Cross-linking elastomers
• Glass transition temperature
• Elastomers

References

[1]	Guth, E., Mark, H.: Zur statistischen Theorie der Kautschukelastizität. Zeitschrift Elektrochemie 43 (1937) 683−686
[2]	Schmiedel, H. (Ed.): Handbuch der Kunststoffprüfung. Carl Hanser, Munich Vienna (1992) pp. 58‒59 (ISBN 3-446-16336-0; see AMK-Library under A 3)
[3]	Kausch, H. H., Stalder, B., Barbezat, M.: Introduction aux matieres plastiques. Lausanne: Ecole Polytechnique Federale de Lausanne (1986)
[4]	Batzer, H. (Ed.): Polymere Werkstoffe, Vol. I, Chemie und Physik. Stuttgart, G. Thieme Publishing (1985) (ISBN 9783136481011; AMK-Library under G 17)
[5]	Aklonis, J. J., MacKnight, W. J., Shen, M. (Eds.): Introduction to Polymer Viscoelasticity. New York: J Willey & Son. Inc. (1972), DOI: http://onlinelibrary.wiley.com/doi/10.1002/pol.1973.130110114/abstract
[6]	Paufler, P., Schulze, G. E. R.: Physikalische Grundlagen mechanischer Festkörpereigenschaften, Vol. 1. Berlin: Akademie Publishing (1978)

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