# How does temperature affect the elasticity and spring constant of a rubber band

Rubber bands expand when chilled and contract when heated (no stretching, just when heated/chilled and at rest) to my knowledge. Why are hot rubber bands able to be stretched longer than cold rubber bands when acted upon by the same force? For example, a hot rubber band will stretch 10 cm when a weight it pulling it down, but a cold rubber band will stretch 5 cm when the same weight is pulling it down. Is it because of entropy? Or potential energy? Is it Hooke's Law, where K-constant decreases if temperature increases?

• Rubber bands don't actually follow Hooke's Law. – David White Dec 18 '18 at 3:04
• @DavidWhite Almost anything follows Hooke's Law approximately for small displacements. – Buzz Dec 18 '18 at 3:16

It sounds like you're describing two separate effects:

1. Elastomers can have a negative thermal expansion coefficient, i.e., $$\alpha=\frac{1}{L}\left(\frac{\partial L}{\partial T}\right)_F<0,$$ because the higher-entropy state (corresponding to a higher temperature) is one in which the polymer chains move around more, which has the consequence of shortening their end-to-end length. When characterizing this material property, the force $$F$$ is kept constant.

2. Most materials (including elastomers) generally get more compliant when they are heated; thus, their stiffness $$k=\left(\frac{\partial F}{\partial L}\right)_T,$$ which is characterized at a constant temperature, decreases with increasing temperature:

In polymers, the origin is the melting of weak bonds between the chains. However, exceptions can exist; for example, the stiffness of a purely entropic system increases with temperature, so once melting is complete, we might see an increase in the elastic modulus with heating:

(Hosford, Mechanical Behavior of Materials)

Therefore, if you hang a weight on a rubber band (to achieve a change in length of $$\Delta L$$) and heat it up, the weight will probably rise because of (1). If you then remove the weight at the same higher temperature, the rubber band may contract by more or less than $$\Delta L$$ as mediated by the mechanisms discussed in (2). If you then cool the system down to the original temperature, the rubber band will return to its original length, $$\Delta L$$ above the original position of the weight, if it is not already there and if the entire loading time was short enough that irreversible flow was negligible.

Put another way, since you're asking about elasticity in the context of a hot and a cold rubber band loaded by the same weight, I should emphasize that one can't directly measure a system's stiffness by keeping the force constant and observing the displacement when changing other things. One measures the stiffness by changing the force while watching the displacement (or vice versa) while keeping other things constant.

The elastic modulus of materials decreases with temperature.

• Does the property of entropy influence the elasticity changes in the rubber bands? – Bryan C. Dec 18 '18 at 0:00
• If you increase the temperature, you increase entropy... – nodarkside Dec 18 '18 at 0:37
• It's not changing bond lengths that change the behavior of the rubber band with temperature. – Buzz Dec 18 '18 at 3:15