Analytical and Empirical Consequences of the Postulated Agreement Between the Cox/Merz-Relation and the Carreau Function

摘要

We postulate that the Cox/Merz relation and the Carreau function both describe identically the shear-rate-dependent viscosity of a polymer melt. We find that an exponentially decreasing relaxation time distribution and an exponentially increasing relaxation strength distribution can describe the experimen- tal rheological data with respect to storage G' and loss modulus G'' in the literature. We have therefore applied such exponential distributions as well as trial functions to calculate G' ,G'' and the corresponding shear-rate-dependent viscosity utilizing the Cox/Merz relation. The expansion of the viscosity in a Taylor series, utilizing the Cox/Merz relation and the Carreau function, makes it possible to compare the frequency and shear-rate coefficients, and this results in a system of coupled transcendental equations.We justify the empirical value of m=1/3 mostly utilized for the Carreau function and prove the square (n=2) in the denominator of the Carreau function to be a fundamental fact, independent of the above-mentioned formulation of the relaxation time and relaxation strength distributions. Moreover, the empirical relation γ_c · η_o ≈ 10~4 Pa is confirmed by our pure mathematical approach.