Cox–Merz rules from phenomenological Kelvin–Voigt and Maxwell models

Highlights

• Modified Cox–Merz rules for dynamic viscosity were derived.

• Nonlinear versions of Kelvin–Voigt and Maxwell models were considered.

• Linear component via first-harmonic balance for power-law fluids was used.

• The modified Cox–Merz rule depends only of the viscous modulus.

• Poor viscosity predictions were obtained for liquid-like behavior.

Abstract

The Cox–Merz rule (CMR) is commonly used for identifying similarities between the rheological results obtained from incremental and oscillatory fluid deformation tests. The CMR states that the steady state viscosity ${\eta }_{ss}\left(\dot{\gamma }\right)$ and the complex viscosity η^∗(ω) are similar when the equivalence $\omega \leftrightarrow \dot{\gamma }$ is considered. The empirical applicability of the CMR for diverse food material has been tested in recent decades, showing non-conclusive results for a diversity of foods. This work used the Kelvin–Voigt and Maxwell phenomenological models of power-law fluids to obtain equivalences between the steady state and the complex viscosities. Modified CMR rules (MCMR) were derived using first-harmonic balances for the underlying differential equations governing the strain dynamics. It is shown that the structure and applicability of the MCMR depended on the underlying model. Only the loss modulus was involved in the viscosity estimation for Kelvin–Voigt models while the traditional CMR was consistent with Maxwell model in the low-frequency regimen.

Introduction

Incremental and oscillatory deformations are two commonly used rheological tests for estimating the viscoelastic properties of complex fluids. In the first case, the fluid is allowed to reach a steady state incremental deformation under the application of a constant shear rate, $\stackrel{·}{\gamma }$, and the developed shear stress, τ, is measured, leading to estimates of the steady state viscosity ${\eta }_{ss}\left(\stackrel{·}{\gamma }\right)=\frac{\tau }{\stackrel{·}{\gamma }}$. In the second case, the fluid is deformed by the application of either shear stress, τ(t), or shear strain, γ(t), induced by the harmonic periodic oscillation of frequency ω to obtain the transfer function $\gamma \left(\omega j\right)\stackrel{G\left(\omega \right)}{\to }\tau \left(\omega j\right)$, where G^∗(ω)=G′(ω) + G″(ω)j, where G′(ω) and G″(ω) are the elastic (i.e., storage) and viscous (i.e., loss) modulus, respectively.

Cox and Merz (1958) reported that the steady state viscosity ${\eta }_{ss}\left(\stackrel{·}{\gamma }\right)$ and the complex viscosity defined as ${\eta }^{\ast }\left(\omega \right)=\frac{\left|G^{\ast }\left(\omega \right)\right|}{\omega }$ , where $\left|G^{\ast }\left(\omega \right)\right|=\sqrt{G^{\prime }{\left(\omega \right)}^2+G^{″}{\left(\omega \right)}^2}$ is the magnitude of G^∗(ω), are similar in the sense that

$${{\eta }_{ss}\left(\stackrel{·}{\gamma }\right)\approx {\eta }^{\ast }\left(\omega \right)|}_{\omega =\stackrel{·}{\gamma }}$$

Bistany and Kokini, 1983a, Bistany and Kokini, 1983b pioneered the application of the CMR to foodstuff material, finding that in many instances the dynamic viscosity η^∗(ω) is much larger than the steady viscosity ${\eta }_{ss}\left(\stackrel{·}{\gamma }\right)$, suggesting a nonlinear nature of the biomaterial response. Following Bistany and Kokini's work, much work has been devoted to tests the application of the CMR in biomaterials intended. Biomaterials that follow the Cox–Merz rule (CMR) are salep glucomannan milk beverages (Yaşar et al., 2009), tomato juice (Augusto et al., 2013) and sage seed gum (Razavi et al., 2014). Instances of materials that do not obey the CMR are worm-like micellar systems (Manero et al., 2002), inulin-waxy maize starch mixtures (Zimeri and Kokini, 2003), lyotropic polymeric liquid crystals (Davis et al., 2004), salad-dressing-type emulsions (Riscardo et al., 2005), ice cream mixes (Dogan et al., 2013), salep (Karaman et al., 2013), rocket (Eruca sativa) puree (Ahmed et al., 2013), among many other systems. Model-based and physical arguments have been considered for explaining the differences between the steady state and complex viscosities. For materials exhibiting yield stress, Doraiswamy et al. (1991) proposed a modification to the CMR in terms of an “effective shear rate”. Soltero et al. (1995) argued that in some systems, such as surfactant-based lamellar liquid crystals concentrated dispersions, the CMR rule was not satisfied due to orientation, breaking and aggregation of the liquid-crystalline microdomains at high shear rate conditions. Marrucci (1996) reported that the Cox–Merz viscosity equivalence for polymer melts depended strongly of the dynamics of entanglement. Extended versions of the CMR have been also proposed for closing the gap between the steady state and complex viscosity predictions. Doraiswamy et al. (1991) proposed the use of a horizontal shift factor γ_m to overlap steady state and complex viscosities as ${\eta }^{\ast }\left({\gamma }_m\omega \right)\approx {\eta }_{ss}\left(\stackrel{·}{\gamma }\right)$. Tattiyakul and Rao (2000) found that a modified CMR given by ${\eta }^{\ast }\left({\gamma }_m\omega \right)\approx C{\eta }_{ss}{\left(\stackrel{·}{\gamma }\right)}^{\alpha }$ described well the behavior of cross-linked starch dispersions. Although the CMR is widely used in a diversity of fields, a limited number of analytical studies have focused on the derivation of the viscosity relationships. Marrucci (1996) found that a model of dynamics of entanglements was consistent with the CMR. Manero et al. (2002) used a model that coupled the Oldroyd-B constitutive equation with a kinetic equation that accounted for the structural changes induced by the flow to derive the CMR. Mead (2011) considered a model for polydisperse linear polymers to find conditions of applicability of the CMR. The works commented above used basic models (i.e., momentum balance) with involved stress relationships to assess the applicability and limitations of the CMR. The analysis to derive the CMR requires advanced mathematical tools to approximate the solution of the resulting motion equations. In this regard, there is a tradition of using phenomenological models based on simple mechanical analogues to describe the rheological response of colloidal systems in a large diversity of science and engineering fields. However, the link between simple phenomenological models and the CMR or modified CMR has not been explored yet. Studies regarding this issue should provide valuable insights about the scope and range of applicability of the CMR. In this regard, the aim of this work is to consider Kelvin–Voigt and Maxwell phenomenological models of power-law fluids to obtain equivalences between the steady state and the complex viscosities. The derivation of the modified CMR rules (MCMR) uses a first-harmonic balance approach for the underlying differential equations governing the strain dynamics.