The Gordon–Schowalter/Johnson–Segalman model in parallel and orthogonal superposition rheometry and its application in the study of worm-like micellular systems

Parallel and Orthogonal Superposition experiments may be employed to probe a material’s non-linear rheological properties through the rate-dependent parallel and orthogonal superposition moduli, $G_{\parallel }^{\ast }(\omega ,\dot{\gamma })$ and $G_{\perp }^{\ast }(\omega ,\dot{\gamma })$, respectively. In a recent series of publications, we have considered the problem of interconversion between parallel and orthogonal superposition moduli as a means of probing flow induced anisotropy. However, as noted by Yamomoto (1971) superposition flows may be used to assess the ability of a particular constitutive model to describe the flow of complex fluids. Herein, we derive expressions for the superposition moduli of the Gordon–Schowalter (or Johnson–Segalman) fluid. This model contains, as special cases, the corotational Maxwell model, the upper (and lower) convected Maxwell models, the corotational Jeffreys model, and the Oldroyd-B model. We also consider the conditions under which the superposition moduli may take negative values before studying a specific, non shear banding, worm like micellular system of cetylpyridinium chloride and sodium salicylate. We find that, using a weakly non-linear analysis (in which the model parameters are rate independent) the Gordon–Schowalter/Johnson–Segalman (GS/JS) model is unable to describe the superposition moduli. However, by permitting strong non-linearity (allowing the GS/JS parameters to become shear rate dependent), the superposition moduli, at all rates studied, are described well by the model. Based on this strongly non-linear analysis, the shear rate dependency of the GS/JS ‘slip parameter’, $a$, suggests that the onset of shear thinning in the specific worm-like micellular system studied herein is driven by a combination of microstructural modification and a transition from rotation dominated (as in the corotational Jeffreys model) to shear dominated (as in the Oldroyd-B model) deformation of the microstructural elements.