Polymer Physics

dU¼dQþdW (3.11)

When the total volume is kept constant,dV¼0, we have

dW¼PdVþfdl¼fdl (3.12)

According to the second law of thermodynamics, the reversible heat exchange is
proportional to the entropic change under constant temperatures and pressures,

dQ¼TdS (3.13)

while

dU¼TdSþfdl (3.14)

Therefore, we have

f¼ð

@U

@l

ÞT:V:Tð

@S

@l

ÞT:V:¼fEþfS (3.15)

where

fE¼ð

@U

@l

ÞT:V:

fS¼Tð

@S

@l

ÞT:V:

Here,fEis calledthe energetic elasticity, andfSis calledthe entropic elasticity.
For instance, the spring exhibits a high elasticity mainly contributed by the ener-
getic elasticity due to the metallic bonds for the strong interactions of iron atoms,
while the rubber exhibits a high elasticity mainly contributed by the entropic
elasticity due to the chain conformations for the large deformations of polymers.

Fig. 3.4 Illustration of the stretching forcefimposed onto both ends of an elastic body with a
lengthland a cross-sectional areaA

36 3 Conformation Statistics and Entropic Elasticity