Nonlinear Relaxation of Unentangled Associative Polymers:\nStrain-Induced Hardening and Softening

Nonlinear\nstress relaxation under step strain γ was examined\nfor two types of ionomers, sulfonated polystyrene (SPS) having either\nzinc (Zn) or sodium (Na) as the counter-cation. These SPS samples\nwere of low molecular weight (<i>M</i><sub>w</sub> = 11\nkg mol<sup>–1</sup>) and in the unentangled state by themselves\nbut had, on average, two associating salt groups (stickers) <i>per</i> chain thereby forming a transient network sustained\nby aggregates of those stickers. The thermal dissociation of the stickers\nfrom those aggregates was much slower than the intrinsic Rouse motion\nof the chain backbone between the stickers because of a large energy\nrequired for the stickers to escape from their aggregates. Thus, for\nboth SPS-Zn and SPS-Na ionomers, the terminal relaxation was much\nslower than the intrinsic Rouse relaxation. For small γ (=0.5)\nin the linear viscoelastic regime, this terminal relaxation was almost\nsingle-Maxwellian, possibly because the ionomer backbone had only\ntwo stickers (on average) and the dissociation of one sticker allowed\nthe whole backbone to relax. For large γ (still below 2), both\nSPS-Zn and SPS-Na ionomers exhibited strain-hardening at short times <i>t</i> followed by a fast modulus decay and the strain-softening\nin the terminal relaxation zone at long <i>t</i>. In this\nterminal relaxation zone, the nonlinear relaxation modulus <i>G</i>(<i>t</i>,γ) was found to obey the time-strain\nseparability, <i>G</i>(<i>t</i>,γ) = <i>G</i>(<i>t</i>)<i>h</i>(γ) with <i>G</i>(<i>t</i>) and <i>h</i>(γ) being\nthe linear relaxation modulus and the damping function, respectively.\nThese nonlinear features were discussed in relation to strain-induced\nchanges in the associative network structure. The hardening at short <i>t</i> was <i>not</i> attributable to the finite extensible\nnonlinear elasticity (FENE), because the chain backbone between the\nstickers was too long to exhibit a significant FENE effect at γ\n< 2. Instead, the hardening was related to strain-induced enhancement\nof aggregation of the stickers not involved in large and stable aggregates\nbefore imposition of the strain. Correspondingly, the fast stress\ndecay after the hardening was attributed to mechanical rupture of\nthe aggregates formed by those stickers. After this rupture, the transient\nnetwork sustained by the surviving aggregates would have relaxed through\nthe thermal dissociation of those aggregates. This thermal process\nshould have been very similar to that at equilibrium, which resulted\nin the time-strain separability of <i>G</i>(<i>t</i>,γ) at long <i>t</i>. These structural arguments\nof the nonlinearities were in harmony with a double-step reverse strain\ntest as well as a model analysis assuming the mechanical rupture and\nsuccessive equilibration of the aggregates.