M4 Entropic Elasticity (Rubber Elasticity)

Topic

When this elastomer—a cross-linked network of the polymer's molecular chains—is stretched, each molecular chain within that network is extended beyond the coiled, disordered configuration it would spontaneously assume at rest; consequently, the conformational entropy of each chain is reduced. The fewer distinct coiled configurations a chain can adopt, the more stretched it is relative to its resting state, and the lower its conformational entropy becomes. Stretching the elastomer further causes a further reduction in the conformational entropy of its molecular chains.

It is this loss of conformational entropy—rather than chemical bond energy between atoms, as seen in crystalline solids—that gives rise to the elastomer's elasticity. When the stretching force is removed, each molecular chain spontaneously reverts to its coiled configurations, thereby regaining the conformational entropy it had lost; this spontaneous tendency to recover conformational entropy generates the elastomer's elastic restoring force. A greater loss of conformational entropy—resulting from stretching the elastomer with greater force—leads to a correspondingly greater elastic restoring force. Consequently, the stretching of this network—and the associated entropic elasticity—means that the network's cross-link density (the number of chemical cross-links connecting molecular chains per unit volume) determines the elastomer's network modulus; this is the same modulus described by the Neo-Hookean stress-elongation relationship found in another model within this set. The more cross-links connecting the molecular chains within the network, the shorter the average length of the free molecular chain segment between consecutive cross-links; this limits the amount of conformational entropy that segment can lose before becoming fully extended, thereby increasing the network modulus. The fact that this entropic elasticity—which has no analogue in crystalline solids like metals or ceramics—is precisely what distinguishes this elastomer makes it the defining characteristic that sets the material apart from any other state of the polymer.

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