Rod-climbing rheometry revisited

TL;DR

This paper revisits the rod-climbing experiment to better measure normal stress differences in polymeric fluids, combining experiments with rheometry and accounting for inertial effects to improve accuracy at low shear rates.

Contribution

It introduces a comprehensive approach combining rod-climbing experiments with rheometry, including inertial effects, to accurately measure normal stress differences in polymeric fluids.

Findings

The climbing constant can be measured even when fluids descend due to inertial effects.

A new predictive model for rod-climbing versus rod-descending behavior based on elasticity and inertia.

Rotating rod rheometry can measure normal stress differences at low shear rates.

Abstract

The rod-climbing or Weissenberg effect in which the free surface of a complex fluid climbs a thin rotating rod is a popular and convincing experiment demonstrating the existence of elasticity in polymeric fluids. The interface shape depends on the rotation rate, fluid elasticity, surface tension, and inertia. By solving the equations of motion in the low rotation rate limit for a second-order fluid, a mathematical relationship between the interface deflection and the fluid material functions, specifically the first and second normal stress differences, emerges. This relationship has been used in the past to measure the climbing constant, a combination of the first ($\Psi_{1,0}$) and second ($\Psi_{2,0}$) normal stress difference coefficients from experimental observations of rod-climbing in the low inertia limit. However, a quantitative reconciliation of such observations with the capabilities of modern-day torsional rheometers is lacking. To this end, we combine rod-climbing experiments with both small amplitude oscillatory shear flow measurements and steady shear measurements of the first normal stress difference from commercial rheometers to quantify the values of both normal stress differences for a series of polymer solutions. Furthermore, by retaining the oft-neglected inertial terms, we show that the climbing constant $\hat{\beta}=0.5\Psi_{1,0}+2\Psi_{2,0}$ can be measured even when the fluids, in fact, experience rod descending. A climbing condition derived by considering the competition between elasticity and inertial effects accurately predicts whether a fluid will undergo rod-climbing or rod-descending. The analysis and observations presented in this study establish rotating rod rheometry as a prime candidate for measuring normal stress differences in polymeric fluids at low shear rates that are often below commercial rheometers' sensitivity limits.