Scaling analysis of the viscoelastic response of linear polymers

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 T_g (174 K-373 K). Master curves G'(ωτ_α) and G″(ωτ_α) are constructed for the polymer-specific relaxation. Segmental relaxation occurring close to T_g is independently addressed by single spectra. Altogether, viscoelastic response is effectively studied over 14 decades in frequency. The structural relaxation time τ_α used for scaling is taken from dielectric spectra. We suggest a derivative method for identifying the different power-law regimes and their exponents along G″(ωτ_α) ∝ ω^ε″. The exponent ε″ = ε″(ωτ_α) ≡ d ln G″(ωτ_α)/d ln(ωτ_α) 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″(ωτ_α). The high-frequency flank of the terminal relaxation closely follows the prediction for CLF (ε″ = -0.25), i.e., G″(ω) ∝ 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″(ωτ_α) at frequencies below those of the Rouse regime. The derivative method also allows identifying the entanglement time τ_e. However, as the exponent in the Rouse regime (ωτ_e > 1) varies along ε_eRouse = 0.66 ± 0.04 (off the Rouse prediction ε_Rouse = 0.5) and that at ωτ_e < 1 is similar, only a weak manifestation of the crossover at τ_e is found at highest M. Yet, calculating τ_e/τ_α= (M/M_o)², we find good agreement among the polymers when discussing ε″(ωτ_e). The terminal relaxation time τ_t is directly read off from ε″(ωτ_α). Plotting τ_t/τ_e as a function of Z = M/M_e, 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″(ωτ_α) scales as τ_min ∝ M^1.0±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.