Measurement of the first and second normal stress differences: Correlation of four experiments on a polyisobutylene/decalin solution “D1”

Abstract

Pressure distribution measurements for a polyisobutylene/decalin solution “D1” in the Truncated Cone-and-Plate (TCP) apparatus are combined with elastic hole pressures obtained for the same solution on the Lodge Stressmeter^® in order to provide two independent estimates of the second normal-stress difference (N ₂). The values ofN ₂ from the TCP apparatus, obtained by numerical differentiation of a function of the center-hole pressure and the pressure gradient, are in good agreement with measurements made on the same sample by Tanner et al. with a direct method, namely the Tilted Trough Experiment, and by Christiansen et al. with a method that requires an extrapolation to the pressure at the free surface of coneand-plate rheogoniometer data obtained with flush-mounted pressure transducers. The viscosities from the modified Stressmeter for low shear rates extend over five decades of shear rate, including a zero-shear-rate region, and agree with the data of Christiansen on a torque-driven flow. The Higashitani-Pritchard-Baird-Lodge (HPBL) equation relatingN ₁–N ₂ to the hole pressure gives good agreement with the data over a certain range of shear stress. The Newtonian hole pressures for several liquids at 20 and 46 °C compare well with a finite-element calculation for a two-dimensional Poiseuille flow. When the elastic hole pressures from the Stressmeter are combined with the extrapolated rim pressures from the TCP Apparatus in order to extract the value ofN ₂, an agreement betweenN ₂ from the center-hole pressure andN ₂ from the rim pressure can only be obtained up to a shear rate of about 40 s⁻¹, beyond which the value of −N ₂ from the rim pressure diverges abruptly to negative values. It is possible that this constitutes the first quantitative estimate of an edge effect in cone-and-plate rheometry. Alternatively, the elastic hole pressure in cone-and-plate flow is not equivalent to the elastic hole pressure in Poiseuille flow, at least at high shear rates. The data of Christiansen et al. with flush-mounted pressure transducers appear to confirm this second possibility. Finally, a single set of shift factors obeying the Williams, Landel and Ferry equation superposes the viscosity, the first and the second normal-stress difference within experimental scatter, which can be less than 1% for a certain combination of normal-stress differences. The data were recorded at 3, 20, 30, and 46 °C in the shear rate range 1–260 s⁻¹.