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Register a new userMedium Amplitude Parallel Superposition (MAPS) Rheology, Part 2: Experimental Protocols and Data Analysis
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An experimental protocol is developed to directly measure the new material functions revealed by medium amplitude parallel superposition (MAPS) rheology. This experimental protocol measures the medium amplitude response of a material to a simple shear deformation composed of three sine waves at different frequencies. Imposing this deformation and measuring the mechanical response reveals a rich data set consisting of up to 19 measurements of the third order complex modulus at distinct three-frequency coordinates. We discuss how the choice of the input frequencies influences the features of the MAPS domain studied by the experiment. A polynomial interpolation method for reducing the bias of measured values from spectral leakage and variance due to noise is discussed, including a derivation of the optimal range of amplitudes for the input signal. This leads to the conclusion that conducting the experiment in a stress-controlled fashion possesses a distinct advantage to the strain-controlled mode. The experimental protocol is demonstrated through measurements of the MAPS response of a model complex fluid: a surfactant solution of wormlike micelles. The resulting data set is indeed large and feature-rich, while still being acquired in a time comparable to similar medium amplitude oscillatory shear (MAOS) experiments. We demonstrate that the data represents measurements of an intrinsic material function by studying its internal consistency, its compatibility with low-frequency predictions for Coleman-Noll simple fluids, and its agreement with data obtained via MAOS amplitude sweeps. Finally, the data is compared to predictions from the corotational Maxwell model to demonstrate the power of MAPS rheology in determining whether a constitutive model is consistent with a materials time-dependent response.
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A new mathematical representation for nonlinear viscoelasticity is presented based on application of the Volterra series expansion to the general nonlinear relationship between shear stress and shear strain history. This theoretical and experimental framework, which we call Medium Amplitude Parallel Superposition (MAPS) Rheology, reveals a new material property, the third order complex modulus, which describes completely the weakly nonlinear response of a viscoelastic material in an arbitrary simple shear flow. In this first part, we discuss several theoretical aspects of this mathematical formulation and new material property. For example, we show how MAPS measurements can be performed in strain- or stress-controlled contexts and provide relationships between the weakly nonlinear response functions measured in each case. We show that the MAPS response function is a super-set of the response functions that have been previously reported in medium amplitude oscillatory shear and parallel superposition rheology experiments. We also show how to exploit inherent symmetries of the MAPS response function to reduce it to a minimal domain for straightforward measurement and visualization. We compute this material property for a few constitutive models to illustrate the potential richness of the data sets generated by MAPS experiments. Finally, we discuss the MAPS framework in the context of some other nonlinear, time-dependent rheological probes and explain how the MAPS methodology has a distinct advantage over these others because it generates data embedded in a very high dimensional space without driving fluid mechanical instabilities, and is agnostic to the flow protocol.
The weakly nonlinear rheology of a surfactant solution of wormlike micelles is investigated from both a modeling and experimental perspective using the framework of medium amplitude parallel superposition (MAPS) rheology. MAPS rheology defines material functions, such as the third order complex compliance, which span the entire weakly nonlinear response space of viscoelastic materials to simple shear deformations. Three-tone oscillatory shear deformations are applied to obtain feature-rich data characterizing the third order complex compliance with high data throughput. Here, data for a CPyCl solution are compared to the analytical solution for the MAPS response of a reptation-reaction constitutive model, which treats micelles as linear polymers that can break apart and recombine in solution. Regression of the data to the model predictions provides new insight into how these breakage and recombination processes are affected by shear, and demonstrates the importance of using information-rich data to infer precise estimates of model parameters.
Mixing a small amount of liquid into a powder can give rise to dry-looking granules; increasing the amount of liquid eventually produces a flowing suspension. We perform experiments on these phenomena using Spheriglass, an industrially-realistic model powder. Drawing on recent advances in understanding friction-induced shear thickening and jamming in suspensions, we offer a unified description of granulation and suspension rheology. A liquid incorporation phase diagram explains the existence of permanent and transient granules and the increase of granule size with liquid content. Our results point to rheology-based design principles for industrial granulation.
In a previous work, we proposed an integrability setup for computing non-planar corrections to correlation functions in $mathcal{N}=4$ super Yang-Mills theory at any value of the coupling constant. The procedure consists of drawing all possible tree-level graphs on a Riemann surface of given genus, completing each graph to a triangulation, inserting a hexagon form factor into each face, and summing over a complete set of states on each edge of the triangulation. The summation over graphs can be interpreted as a quantization of the string moduli space integration. The quantization requires a careful treatment of the moduli space boundaries, which is realized by subtracting degenerate Riemann surfaces; this procedure is called stratification. In this work, we precisely formulate our proposal and perform several perturbative checks. These checks require hitherto unknown multi-particle mirror contributions at one loop, which we also compute.
A general framework for Maxwell-Oldroyd type differential constitutive models is examined, in which an unspecified nonlinear function of the stress and rate-of-deformation tensors is incorporated into the well-known corotational version of the Jeffreys model discussed by Oldroyd. For medium amplitude simple shear deformations, the recently developed mathematical framework of medium amplitude parallel superposition (MAPS) rheology reveals that this generalized nonlinear Maxwell model can produce only a limited number of distinct signatures, which combine linearly in a well-posed basis expansion for the third order complex viscosity. This basis expansion represents a library of MAPS signatures for distinct constitutive models that are contained within the generalized nonlinear Maxwell model. We describe a framework for quantitative model identification using this basis expansion, and discuss its limitations in distinguishing distinct nonlinear features of the underlying constitutive models from medium amplitude shear stress data. The leading order contributions to the normal stress differences are also considered, revealing that only the second normal stress difference provides distinct information about the weakly nonlinear response space of the model. After briefly considering the conditions for time-strain separability within the generalized nonlinear Maxwell model, we apply the basis expansion of the third order complex viscosity to derive the medium amplitude signatures of the model in specific shear deformation protocols. Finally, we use these signatures for estimation of model parameters from rheological data obtained by these different deformation protocols, revealing that three-tone oscillatory shear deformations produce data that is readily able to distinguish all features of the medium amplitude, simple shear response space of this generalized class of constitutive models.
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