Rubber Elasticity

High molecular weight polymers that exhibit rubber-like behavior are known as elastomers. Above the glass transition temperature, the rubber-like polymers are in a liquid-like state and the mer or repeat units change their position readily and continuously due to Brownian motion. Thus, each polymer chain takes up random conformations in a stress-free state.

In an entangled or cross-linked elastomer, the chains form a loose three-dimensional molecular network. For polymers of high-molecular-weight, the network could solely consist of entanglements by molecular intertwining (knots) with a spacing between knots characteristic for the particular polymer; or the chains could be chemically crosslinked, that is, the chains form a permanent network, as it is the case for vulcanized rubber. Thus for vulcanized rubber, the effective number n of network chains per unit volume is the sum of two terms, n_e and n_c, arising from physical (trapped) entanglements and chemical crosslinks, respectively:

n = n_c + n_e

n_c = N_A ρ / M_c;    n_e = N_A ρ / M_e

 

where ρ is the density of the elastomer, N_A is the Avogadro’s number, and M_e and M_c denote the average molecular weight between entanglements and between crosslinks, respectively.

To simplify the calculation of the retractive force, we assume that the junction points between the sub-chains move on deformation as they were embedded in an elastic continuum, that is, we assume the individual chains are strained in an identical manner to the entire network. If  α_i = L_i / L_i,0 are the extension ratios of the macroscopic rubber cube of volume V = L_x L_y L_z and if R_x,0² = R_y,0² = R_z,0² = Nl²/3 are the mean square end-to-end distances of a strand in the three principal directions, e_x, e_y, e_z, then the average end-to-end distance between two junction points, R, in the strained state in any direction may be written

R² = 1/3 R₀² [α_x² + α_y² + α_z²]

 where α_i ≈ R_i / R_i,0 are the extension ratios, and R is the average size of a rubber strand. Since α_x,0² = α_y,0² = α_z,0² = 1,

R² - R₀² = R₀²/3 · [(α_x² + α_y² + α_z²) - 3]

Then the change in free energy on deformation of N = nV rubber strands is given by:^1,2

ΔF_el = (Vn 3kT/2) · (R² - R₀² ) / R₀² = (VnkT/2) · [α_x² + α_y² + α_z² - 3]

where V = L_xL_yL_z is the volume of the rubber cube and n is the number of network subchains per unit volume. For an incompressible system, α_x α_y α_z = 1, and uniaxial deformation, 1/α^1/2 = α_y = α_z, this expression may be written¹

ΔF_el = (VnkT / 2) · [α² + 2/α - 3]

and for the tensile force:

f_el = ∂F_el / ∂(L_xα) = (VnkT / L_x) · (α - α^-2)

Conversion to stress yields

σ_xx = f_el / (L_yL_zα^-1) = nkT · (α² - α^-1)

The Youngs modulus is determined by the slope at the origin, α = 1. One obtaines

E = ∂σ_xx / ∂α (α = 1) = 3nkT

As these relations demonstrate, the stress-strain relationship of a rubbery material is non-Hookean.

Example:
The Young's modulus of a rubber is E = 4.0 MPa and its density 1000 kg/m³ at a temperature of T = 300 K. The molar crosslink density is then

n = E / 3RT = 4 x 10⁶ N/m² / (3 x 8.314 Nm/mol-K x 300 K)
= 475 mol/m³,                                 

and the average molecular weight between two crosslinks is

M_c = ρ / n = 1000 kg/m³ / 475 mol/m³ = 2105 g/mol

References

1. P.J. Flory, J. Chem. Phys., 18, 108 (1950)
2. P.J. Flory, Principles of Polymer Chemistry, New York 1953