Enhancing High-Pressure Capillary Rheometer Viscosity Data Calculation with the Propagation of Uncertainties for Subsequent Cross-Williams, Landel, and Ferry (WLF) Parameter Fitting

Abstract

Measuring the shear viscosity of polymeric melts is an extensive effort frequently performed in high-pressure capillary rheometers, where the pressures required to push the melt through a capillary at various temperatures and volumetric flow rates are recorded. Then, the viscosity values are obtained through Bagley and Weissenberg–Rabinowitsch corrections involving parameter fitting. However, uncertainties in those conversions due to pressure variations and measurement inaccuracies (random errors) affect the accuracy of the consequently calculated viscosities. This paper proposes quantifying them through a propagation of uncertainties calculation. This has been experimentally demonstrated for a polycarbonate melt. In addition, the derived viscosity uncertainties were used for the weighted residual sum of squares parameter estimation of the Cross-WLF viscosity model and compared with the coefficients obtained using the standard residual sum of squares minimization approach. The motivation was that, by comparison, individual poorly measured viscosity values should have a less negative impact on the overall fit quality of the former. For validation, the rheometer measurements were numerically simulated with both fits. The simulations based on the Cross-WLF fit, including the derived viscosity uncertainties, matched the measured pressures ~16% more closely for shear rates below 1500 1/s. Considering the uncertainties led to more precise coefficients. However, both fits showed substantial deviations at higher shear rates, probably due to substantial non-isothermal flow conditions that prevailed during these measurements. A capillary rheometer experiment was also simulated using arbitrarily chosen Cross-WLF parameters to exclude such systematic errors. A normally distributed error was then applied to the simulated pressures before re-fitting the parameters. Again, taking advantage of the derived viscosity uncertainties, the fit could recover the initial parameters better.