Dynamic mechanical properties of cross‐linked rubbers. V. An approximate analysis of the modulus contributions of trapped and untrapped entanglements

Using the criterion that a coupling entanglement in a polymer network is trapped if all four strands radiating from it terminate in chemical cross‐links (an approximation to a more rigorous treatment of Langley), an equation is derived relating the equilibrium modulus to the magnitude of the compliance in the frequency region where all coupling entanglements, whether trapped or not, contribute to the elasticity. The latter value is estimated from the storage compliance at the frequency where the storage compliance of the uncross‐linked polymer corresponds to the entanglement compliance derived from integration over the loss compliance. The theory agrees rather well with data on vulcanizates of natural rubber. For 1,4‐polybutadiene and styrene–butadiene rubbers, the agreement is somewhat less satisfactory, but the results support the hypothesis that the low‐frequency losses observed in lightly cross‐linked rubbers are due to relaxation of untrapped entanglements.