# Thermodynamics - the entropy spring

When a stress is applied to a sample of cross-linked rubber, equilibrium is established fairly rapidly. Once at equilibrium, the properties of the rubber can be described by thermodynamics.

Consider an element of rubber placed under uniaxial tension. The first law of thermodynamics states that

dU = dQ - dW

where dU is the change in the system's internal energy, and dQ and dW are the heat and work exchanged between system and surroundings as the system undergoes differential change.

We are going to look at the specific case of uniaxial tension. Work done is given by force multiplied by distance, so the work done by a uniaxial force f is given by

dW_{f} = –f dL

where dL is the differential change in the system's length due to the force f. (The negative sign implies that the work is done on the system).

If the deformation process is assumed to occur reversibly (in a thermodynamic sense), then

dQ = TdS

where S is the system's entropy. Combining the above equations gives (for uniaxial tension with V and T constant)

dU = TdS + f dL.

From this, the tensile (retractive) force

F = (dU/dL)_{T,V} - T(dS/dL)_{T,V}

The first term on the RHS is the energy contribution to the tensile (retractive) force, or energy elasticity. In rubbers, this represents the storage of energy resulting from rotation about bonds and the straining of bond angles and lengths from equilibrium values. The second term on the RHS is the entropy contribution to the tensile (retractive) force, or entropy elasticity. It is caused by the decrease in entropy upon uncoiling of the chain segments.

When rubber is extended, the change in length (and energy) comes almost entirely from a change in conformation, i.e. the rotation of bonds, and there is negligible stretching of the bonds. Therefore, at constant temperature, it can be approximated that the internal energy of the bonds does not change.

dU = 0

F = -T(dS/dL)

As the rubber is stretched, the chain is moving from a more probable (higher entropy) to a less probable (lower entropy) state. It is this lowering of entropy of the conformation that causes the retractive force, so rubber is described as an entropy spring.