Scaling analysis of the viscoelastic response of linear polymers

Viscoelastic response in terms of the complex shear modulus G*(omega) 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'(omega tau(alpha)) and G ''(omega tau(alpha)) 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 tau(alpha) 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 '' (omega tau(alpha)) /proportional to omega(epsilon ''). The exponent epsilon '' = epsilon '' (omega tau(alpha)) equivalent to d lnG '' (omega tau(alpha))/d ln(omega tau(alpha)) 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 epsilon '' (omega tau(alpha)). The high-frequency flank of the terminal relaxation closely follows the prediction for CLF (epsilon '' = 0.25), i. e., G '' (omega) proportional to omega(-0.21 +/- 0.02). At lower frequencies, a second regime with lower exponent epsilon '' is observed signaling the crossover to coherent reptation. Application of the full Likhtman-McLeish calculation provides a quantitative interpolation of epsilon '' (omega tau(alpha)) at frequencies below those of the Rouse regime. The derivative method also allows identifying the entanglement time tau(e). However, as the exponent in the Rouse regime (omega tau(e) > 1) varies along epsilon(eRouse) = 0.66 +/- 0.04 (off the Rouse prediction epsilon(Rouse) = 0.5) and that at omega tau(e) < 1 is similar, only a weak manifestation of the crossover at tau(e) is found at highest M. Yet, calculating tau(e)/tau(alpha) = (M/M-o)(2), we find good agreement among the polymers when discussing epsilon '' (omega tau(alpha)). The terminal relaxation time tau(t) is directly read off from epsilon '' (omega tau(alpha)). Plotting tau(t)/tau(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 '' (omega tau(alpha)) scales as tau(min) proportional to 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 '' (omega), 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. Published by AIP Publishing.