The Oldroyd 8-constant model is a continuum framework containing, as special cases, many important constitutive equations for elastic liquids. When polymeric liquids undergo large-amplitude oscillatory shear flow, the shear stress responds as a Fourier series, the higher harmonics of which are caused by the fluid nonlinearity. We choose this continuum framework for its rich diversity of special cases (we tabulate 14 of these). Deepening our understanding of this Oldroyd 8-constant framework thus at once deepens our understanding of every one of these special cases. Previously [C. Saengow et al., Macromol. Theory Simul. 24, 352 (2015)], we arrived at an exact analytical solution for the corotational Maxwell model. Here, we derive the exact analytical expression for the Oldroyd 8-constant framework for the shear stress response in large-amplitude oscillatory shear flow. Our exact solution reduces to our previous solution for the special case of the corotational Maxwell model, as it should. Our worked exampl...

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Exact analytical solution for large-amplitude oscillatory shear flow from Oldroyd 8-constant framework: Shear stress

When polymeric liquids undergo large-amplitude shearing oscillations, the shear stress responds as a Fourier series, the higher harmonics of which are caused by fluid nonlinearity. Previous work on large-amplitude oscillatory shear flow has produced analytical solutions for the first few harmonics of a Fourier series for the shear stress response (none beyond the fifth) or for the normal stress difference responses (none beyond the fourth) [JNNFM, 166, 1081 (2011)], but this growing subdiscipline of macromolecular physics has yet to produce an exact solution. Here, we derive what we believe to be the first exact analytical solution for the response of the extra stress tensor in large-amplitude oscillatory shear flow. Our solution, unique and in closed form, includes both the normal stress differences and the shear stress for both startup and alternance. We solve the corotational Maxwell model as a pair of nonlinear-coupled ordinary differential equations, simultaneously. We choose the corotational Maxwell model because this two-parameter model (η0 and λ) is the simplest constitutive model relevant to large-amplitude oscillatory shear flow, and because it has previously been found to be accurate for molten plastics (when multiple relaxation times are used). By relevant we mean that the model predicts higher harmonics. We find good agreement between the first few harmonics of our exact solution, and of our previous approximate expressions (obtained using the Goddard integral transform). Our exact solution agrees closely with the measured behavior for molten plastics, not only at alternance, but also in startup.

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Exact Analytical Solution for Large-Amplitude Oscillatory Shear Flow

Rheologists have expectations for signatures of linear viscoelastic properties, such as shapes of G′(ω) and G″(ω). Medium amplitude (or asymptotically-nonlinear) oscillatory shear (MAOS) provides additional nonlinear rheological information with low dimensional, well-defined material functions [Ewoldt and Bharadwaj, Rheol. Acta 52, 201–209 (2013)]. Here, we develop expectations of signatures (or fingerprints) for the four asymptotically-nonlinear material functions associated with MAOS, [e1](ω), [e3](ω), [v1](ω), [v3](ω). Although the linear fingerprints may be identical for different models, the asymptotically-nonlinear fingerprints may be different in magnitude, frequency-scaling, curve shapes, and sign changes. To perform the analysis, we collect/translate a library of available analytical strain-controlled MAOS fingerprints for seven different constitutive models. Using this library, we identify general trends and highlight key differences of asymptotic-nonlinear viscoelasticity. Asymptotic nonlineari...

Constitutive model fingerprints in medium-amplitude oscillatory shear

To solve problems in polymer fluid dynamics, one needs the equations of continuity, motion, and energy. The last two equations contain the stress tensor and the heat-flux vector for the material. There are two ways to formulate the stress tensor: (a) One can write a continuum expression for the stress tensor in terms of kinematic tensors, or (b) one can select a molecular model that represents the polymer molecule and then develop an expression for the stress tensor from kinetic theory. The advantage of the kinetic theory approach is that one gets information about the relation between the molecular structure of the polymers and the rheological properties. We restrict the discussion primarily to the simplest stress tensor expressions or constitutive equations containing from two to four adjustable parameters, although we do indicate how these formulations may be extended to give more complicated expressions. We also explore how these simplest expressions are recovered as special cases of a more general framework, the Oldroyd 8-constant model. Studying the simplest models allows us to discover which types of empiricisms or molecular models seem to be worth investigating further. We also explore equivalences between continuum and molecular approaches. We restrict the discussion to several types of simple flows, such as shearing flows and extensional flows, which are of greatest importance in industrial operations. Furthermore, if these simple flows cannot be well described by continuum or molecular models, then it is not necessary to lavish time and energy to apply them to more complex flow problems.

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Polymer Fluid Dynamics: Continuum and Molecular Approaches.

General rigid bead-rod theory [O. Hassager, “Kinetic theory and rheology of bead-rod models for macromolecular solutions. II. Linear unsteady flow properties,” J. Chem. Phys. 60(10), 4001–4008 (1974)] explains polymer viscoelasticity from macromolecular orientation. By means of general rigid bead-rod theory, we relate the complex viscosity of polymeric liquids to the architecture of axisymmetric macromolecules. In this work, we explore the zero-shear and complex viscosities of 24 different axisymmetric polymer configurations. When nondimensionalized with the zero-shear viscosity, the complex viscosity depends on the dimensionless frequency and the sole dimensionless architectural parameter, the macromolecular lopsidedness. In this work, in this way, we compare and contrast the elastic and viscous components of the complex viscosities of macromolecular chains that are straight, branched, ringed, or star-branched. We explore the effects of branch position along a straight chain, branched-chain backbone length, branched-chain branch-functionality, branch spacing along a straight chain (including pom-poms), the number of branches along a straight chain, ringed polymer perimeter, branch-functionality in planar stars, and branch dimensionality.

Macromolecular architecture and complex viscosity

We examine the simplest relevant molecular model for large-amplitude shear (LAOS) flow of a polymeric liquid: the suspension of rigid dumbbells in a Newtonian solvent. We find explicit analytical expressions for the shear rate amplitude and frequency dependences of the first and third harmonics of the alternating shear stress response. We include a detailed comparison of these predictions with the corresponding results for the simplest relevant continuum model: the corotational Maxwell model. We find that the responses of both models are qualitatively similar. The rigid dumbbell model relies entirely on the dumbbell orientation to explain the viscoelastic response of the polymeric liquid, including the higher harmonics in large-amplitude oscillatory shear flow. Our analysis employs the general method of Bird and Armstrong ["Time-dependent flows of dilute solutions of rodlike macromolecules," J. Chem. Phys. 56, 3680 (1972)] for analyzing the behavior of the rigid dumbbell model in any unsteady shear flow. We derive the first three terms of the deviation of the orientational distribution function from the equilibrium state. Then, after getting the "paren functions," we use these for evaluating the shear stress for LAOS flow. We find the shapes of the shear stress versus shear rate loops predicted to be reasonable.

R. B. Bird

Papers

Dilute rigid dumbbell suspensions in large-amplitude oscillatory shear flow: shear stress response.