RESUMO
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 Gâ³(ωτα) are constructed for the polymer-specific relaxation. Segmental relaxation occurring close to Tg is independently addressed by single spectra. Altogether, viscoelastic response is effectively studied over 14 decades in frequency. The structural relaxation time τα used for scaling is taken from dielectric spectra. We suggest a derivative method for identifying the different power-law regimes and their exponents along Gâ³(ωτα) â ωεâ³. The exponent εⳠ= εâ³(ωτα) ≡ d ln Gâ³(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 εâ³(ωτα). The high-frequency flank of the terminal relaxation closely follows the prediction for CLF (εⳠ= -0.25), i.e., Gâ³(ω) â ω-0.21±0.02. At lower frequencies, a second regime with lower exponent εⳠis observed signaling the crossover to coherent reptation. Application of the full Likhtman-McLeish calculation provides a quantitative interpolation of εâ³(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 τe is found at highest M. Yet, calculating τe/τα= (M/Mo)2, we find good agreement among the polymers when discussing εâ³(ωτe). The terminal relaxation time τt is directly read off from εâ³(ωτα). Plotting τt/τe as 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 Gâ³(ωτα) scales as τmin â M1.0±0.1. An M-independent frequency marks the crossover to glassy relaxation at the highest frequencies. Independent of the amplitude of Gâ³(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.
RESUMO
An experimental study of the viscosity of a macroscopic suspension, i.e., a suspension for which Brownian motion can be neglected, under steady shear is presented. The suspension is prepared with a high packing fraction and is density matched in a Newtonian carrier fluid. The viscosity of the suspension depends on the shear rate and the time of shearing. It is shown that a macroscopic suspension shows thixotropic viscosity, i.e., shear thinning with a long relaxation time as a unique function of shear. The relaxation times show a systematic decrease with increasing shear rate. These relaxation times are larger when decreasing the shear rates, compared to those observed after increasing the shear. The time scales involved are about 10 000 times larger than the viscous time scale tau(visc)=a2/nu and about 1000 times smaller than the thermodynamic time scale tau(therm)=Pe/gamma. (a is the gap width of the viscometer, nu is the kinematic viscosity, Pe=6pi(eta)gamma;tau)3/(k(B)T) is the Péclet number and gamma; is the shear rate.) The structure of the suspension at the outer cylinder of a viscometer is monitored with a camera, showing the formation of a hexagonal structure. The temporal decrease of the viscosity under shear coincides with the formation of this hexagonal pattern.
