Measurement of the dynamic viscoelastic properties of polymer solutions using the Birnboim-Schrag multiple lump resonator. A theoretical and numerical study

Abstract

The Birnboim-Schrag multiple lumped resonator (MLR) is the most sensitive and precise instrument in existence for dynamic viscoelastic measurements on dilute low-viscosity polymer solutions in the frequency range 100–8000 Hz. Estimation of polymer viscoelastic spectra from MLR-data has so far been carried out using an approximate dumbbell theory for description of the torsional dynamics of the resonator itself. Here, we report on the development of a significantly improved mathematical description of the resonator dynamics in the absence and presence of surrounding polymer solution. For some of the lumps the relative torsional displacements predicted by dumbbell theory and the new analysis differ by as much as a factor of 2. In an accompanying paper, we show that the predictions of our new theoretical analysis is in excellent agreement with experimental measurements of resonator dynamics. The new analysis does not rest on any particular choice of number of lumps, relative lump size, which lumps are surrounded by polymer solution, or the length or the diameter of each torsional spring. The mathematical analysis takes fully into account the mass and the energy loss of each torsional spring as well as the mechanical coupling between each spring and the surrounding medium. This part of the analysis is carried out in terms of damped torsional waves. An analytical analysis in terms of normal modes as well as the results of numerical analysis of the full set of simultaneous differential equations describing the resonator torsional dynamics are presented. The relationship between idealized dumbbell theory and normal mode theory is discussed. We also demonstrate how more accurate estimates of the viscoelastic parameters for polymers from experimental data can be obtained using the improved theory.