Shear‐Dependent Relaxation Spectrum: Conversion from Steady‐Flow Viscosities to Relaxation Spectrum for Polymer Melts

The steady‐flow viscosity η(γ), the loss modulus G″(ω), and the storage modulus G′(ω) for five linear polyethylene melts (190°C) were measured using the Weissenberg rheogoniometer and the Instron rheometer. An excellent agreement was observed between the experimental G′(ω) and G″(ω) values and those calculated from the relaxation spectrum obtained from η(γ) using the relation η(γ)=∫H(lnτ)h(θ)g(θ)3/2τd lnτ, θ=γτ/2. Here, γ is the shear rate, ω is the radian frequency, and τ is the relaxation time. The functions g(θ) and h(θ) are expressed as g(θ)=(2/π)[cot−1θ+θ/(1+θ2)] and h(θ)=(2/π)[cot−1θ+θ(1−θ2)/(1+θ2)2]. No coordinate shift was required. An interative procedure was used in such an interconversion. The first approximation to the spectrum H1(lnτ) was (2/π)[η(γ)×γ] and the corresponding viscosity was η1(γ). The successive approximations were expressed by the equation HK(lnτ)=HK−1(lnτ)+pγ[η(γ)−ηK−1(γ)], where p=2/π and K≥2. In all cases, only six iterations were required to achieve satisfactory convergence. The relaxation spectra so obtained from viscosity measurements at large shearing deformation were in excellent agreement with those calculated from G″(ω) measured in oscillatory experiments carried at small deformation.