Shear Rate Dependent Relaxation Spectrum: Conversion from Dynamic Viscosity to Steady Flow Viscosity for Polyethylene Melts

Abstract

The steady flowviscosity η and the dynamic viscosity η′ of several linear polyethylene melts (190°C) were measured using the Weissenberg rheogoniometer and the Instron rheometer. In correlating the two viscosities, the use of Graessley's theory of viscosity in steady shearing flow in combination with the box distribution of relaxation times as suggested by Maruyama et al. has been critically examined. The new approach discussed in this paper uses Graessley's functions h(θ) and g(θ) in conjunction with the relaxation spectrum H(τ) derived from linear viscoelastic data using an iterative method. The required relation is η(γ̇)= ∫ H(τ)h(θ)g(θ) 3/2 τd ln τ, θ=γ̇τ/2; where γ̇ is the shear rate and τ is the relaxation time. Here g(θ)=(2/π) [ cot −1 θ+θ/(1+θ 2 )] and k(θ)=(2/π) [ cot −1 θ+θ (1−θ 2 )/(1+θ 2 ) 2 ]. A good agreement was obtained in all cases without involving any coordinate shift. The approach illustrated in this article enabled us to estimate quantitatively the effect of shear rate on the relaxation spectrum and also to estimate fractional contribution to the viscosity at a given relaxation time and shear rate.