Fourier decomposition of polymer orientation in large-amplitude oscillatory shear flow.

A J Giacomin,A M Schmalzer,P H Gilbert

doi:10.1063/1.4914411

A J Giacomin, A M Schmalzer + Show 1 more

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https://doi.org/10.1063/1.4914411

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Journal: Structural dynamics (Melville, N.Y.)	Publication Date: Mar 1, 2015
Citations: 17	License type: cc-by

Affiliation: Queen's University

Abstract

In our previous work, we explored the dynamics of a dilute suspension of rigid dumbbells as a model for polymeric liquids in large-amplitude oscillatory shear flow, a flow experiment that has gained a significant following in recent years. We chose rigid dumbbells since these are the simplest molecular model to give higher harmonics in the components of the stress response. We derived the expression for the dumbbell orientation distribution, and then we used this function to calculate the shear stress response, and normal stress difference responses in large-amplitude oscillatory shear flow. In this paper, we deepen our understanding of the polymer motion underlying large-amplitude oscillatory shear flow by decomposing the orientation distribution function into its first five Fourier components (the zeroth, first, second, third, and fourth harmonics). We use three-dimensional images to explore each harmonic of the polymer motion. Our analysis includes the three most important cases: (i) nonlinear steady shear flow (where the Deborah number is zero and the Weissenberg number is above unity), (ii) nonlinear viscoelasticity (where both and exceed unity), and (iii) linear viscoelasticity (where exceeds unity and where approaches zero). We learn that the polymer orientation distribution is spherical in the linear viscoelastic regime, and otherwise tilted and peanut-shaped. We find that the peanut-shaping is mainly caused by the zeroth harmonic, and the tilting, by the second. The first, third, and fourth harmonics of the orientation distribution make only slight contributions to the overall polymer motion.

Highlights

A rigid dumbbell has both length and width, the two key geometric properties of a polymer molecule, and these properties confer upon the polymer orientation
We learned that the zeroth order term in the expansion contributes only to the zeroth harmonic, the first order term to the first, the second order term to the zeroth and second, third order term to the first and third, and the fourth order term to the zeroth, second and fourth harmonics
We identify the even harmonics of the polymer motion with the differences in the normal components of the extra stress tensor, whose Fourier components are all even

Summary

Introduction

A rigid dumbbell has both length and width, the two key geometric properties of a polymer molecule, and these properties confer upon the polymer orientation (see Figure 1 and Figure 2). We have used the diffusion equation for molecular orientation to derive the expression for the orientation distribution function by expanding it in powers of the shear rate amplitude [37,38,39,40,41,42]. This expression contains zeroth, first, second, third and fourth harmonics. We have used the contributing parts to derive expressions for the shear stress and for both normal stress difference responses (see Eq (82) of [37] and Eqs. In our most recent work, we derived the full orientation distribution function, including the terms that make no contribution to the rheological responses [40,41,42]

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Results

Conclusion