Rubber elasticity

In a flexible crosslinked polymer network, deformation restricts the configurations of chain segments between cross-links. The resulting decrease of entropy produces a restoring entropic force. For an ideal affine Gaussian network, the shear modulus is $G={\nu }_s{k}_{B}T$, where ${\nu }_s$ is the number density of elastically active strands. A nearly incompressible isotropic rubber has Young's modulus $E\simeq 3G$. At fixed network density within the rubbery regime, both increase approximately linearly with absolute temperature. This statement does not describe crossing the glass transition or chemical degradation.

A network of flexible thermally fluctuating strands stores an order-$k_{B}T$ elastic free-energy change per active strand under an order-one deformation. For active-strand number density $n$, its shear modulus is therefore of order $n{k}_{B}T$. If one active element occupies a mesh volume $m^3$, this becomes $k_{B}T/m^3$. The numerical coefficient depends on network connectivity, defects and the deformation model. A mesh size alone does not fix the modulus of a general athermal or bending-dominated network; strand stiffness is then additional input.

A chain of $N$ up/down segments of length $a$ has extension $L=Nax$ and binary configurational entropy. An external force $f(L)=f_{0}L/a$ has potential $-f_0{L}^2/(2a)$, obtained by integrating the force. Its mean-field free-energy minima obey the displayed equation. Expansion gives $F-F(0)=N{k}_B(T-T_c)x^2/2+N{k}_{B}T{x}^4/12+\cdots$, so the ordered extension emerges continuously, $|x|\sim \sqrt{3(T_c-T)/T}$, rather than jumping. The equation matches zero-field mean-field theory of the Ising model; finite systems round the ideal singular behavior.

The ideal rubber elasticity relation infers active strand number density from the shear modulus. If a quoted modulus is instead Young's modulus, nearly incompressible behavior gives ${\nu }_s\simeq E/(3k_{B}T)$. This is not automatically the junction density: for junction functionality $f$ and no dangling ends or loops, counting strand ends gives ${\nu }_s=f{\nu }_j/2$. Network fluctuations and inactive strands modify the modulus, so converting it into a chemical cross-link count requires a network model.