Scaling analysis of the viscoelastic response of linear polymers

© 2018 Author(s). Viscoelastic response in terms of the complex shear modulus G∗(ω) of the linear polymers poly(ethylene-alt-propylene), poly(isoprene), and poly(butadiene) is studied for molar masses (M) from 3k up to 1000k and over a wide temperature range starting from the glass transition temperature Tg(174 K-373 K). Master curves G′(ωτα) and G″(ωτα) are constructed for the polymer-specific relaxation. Segmental relaxation occurring close to Tgis independently addressed by single spectra. Altogether, viscoelastic response is effectively studied over 14 decades in frequency. The structural relaxation time ταused for scaling is taken from dielectric spectra. We suggest a derivative method for identifying the different power-law regimes and their exponents along G″(ωτα) â ωϵ″. The exponent ϵ″ = ϵ″(ωτα) ≡ d ln G″(ωτα)/d ln(ωτα) reveals more details compared to conventional analyses and displays high similarity among the polymers. Within a simple scaling model, the original tube-reptation model is extended to include contour length fluctuations (CLFs). The model reproduces all signatures of the quantitative theory by Likhtman and McLeish. The characteristic times and power-law exponents are rediscovered in ϵ″(ωτα). The high-frequency flank of the terminal relaxation closely follows the prediction for CLF (ϵ″ = -0.25), i.e., G″(ω) â ω-0.21±0.02. At lower frequencies, a second regime with lower exponent ϵ″ is observed signaling the crossover to coherent reptation. Application of the full Likhtman-McLeish calculation provides a quantitative interpolation of ϵ″(ωτα) at frequencies below those of the Rouse regime. The derivative method also allows identifying the entanglement time τe. However, as the exponent in the Rouse regime (ωτe> 1) varies along ϵeRouse= 0.66 ± 0.04 (off the Rouse prediction ϵRouse= 0.5) and that at ωτe< 1 is similar, only a weak manifestation of the crossover at τeis found at highest M. Yet, calculating τe/τα= (M/Mo)2, we find good agreement among the polymers when discussing ϵ″(ωτe). The terminal relaxation time τtis directly read off from ϵ″(ωτα). Plotting τt/τeas a function of Z = M/Me, we find universal behavior as predicted by the TR model. The M dependence crosses over from an exponent significantly larger than 3.0 at intermediate M to an exponent approaching 3.0 at highest M in agreement with previous reports. The frequency of the minimum in G″(ωτα) scales as τmin∞ M1.0±0.1. An M-independent frequency marks the crossover to glassy relaxation at the highest frequencies. Independent of the amplitude of G″(ω), which may be related to sample-to-sample differences, the derivative method is a versatile tool to provide a detailed phenomenological analysis of the viscoelastic response of complex liquids