Abstract
In a recent paper, we derived expressions for determining the rate-dependent response spectra directly from parallel superposition rheometry data for the case of a certain Lodge-type integral constitutive model. It was shown that, within the confines of linear Yamamoto perturbation theory, the corresponding parallel superposition moduli satisfy the classical Kramers-Kronig relations. Special bases were presented to convert parallel superposition moduli to orthogonal superposition moduli. In the current paper, we obtain similar results for the integral models of Wagner I and, more generally, K-BKZ. These results facilitate the physical interpretation of parallel superposition moduli and direct model-based comparison of parallel and orthogonal superposition moduli in the study of weak nonlinear response.
Highlights
Superposition rheometry is a technique for exploring the nonlinear rheological properties of complex fluids,1,2 which involves superposition of a small amplitude oscillatory perturbation, of amplitude γ0 and angular frequency ω, upon a unidirectional flow with a constant strain-rate γ
The oscillatory parts of the stress and strain waveforms generated by the kinematics may be used to define a superposition complex modulus G∗∥ (ω, γ) or G∗(ω, γ)
We show that the same is true for incompressible K-BKZ integral constitutive models
Summary
Superposition rheometry is a technique for exploring the nonlinear rheological properties of complex fluids, which involves superposition of a small amplitude oscillatory perturbation, of amplitude γ0 and angular frequency ω, upon a unidirectional flow with a constant strain-rate γ. Despite the ease of implementing PSR experiments on commercial rheometers, OSR has (for the past 20 years) been the preferred methodology Such experiments require specific hardware, e.g., the TA Instruments Orthogonal Superposition accessory, which employs the rheometer’s normal force transducer to generate the oscillatory component, and the availability of relatively large quantities of materials (approximately 50 ml). In Ref. 18, it was shown that, for certain Lodge-type constitutive models, under the constraint of oscillatory perturbations of small amplitude, (i) the real and imaginary parts of G∗ and G∗∥ satisfy the Kramers-Kronig relations and (ii) the relationships between the superposition moduli can be derived that may be used as a basis for a quantitative comparison of PSR and OSR data. The Wagner I model, studied by Vermant et al., merits a separate study as a special case of K-BKZ, and we begin with this model
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