Medium Amplitude Oscillatory Shear Research Articles

Nonlinear oscillatory rheology of aqueous suspensions of cellulose nanocrystals and nanofibrils

In this work, the nonlinear rheological behavior of aqueous suspensions composed of two typical nanocellulose [rod-like cellulose nanocrystals (CNCs) and filamentous cellulose nanofibrils (CNFs)] was examined and compared by using various large-amplitude oscillatory shear (LAOS) analysis methods, such as Fourier-transform rheology, stress decomposition, Chebyshev polynomials, and the sequence of physical processes. From our analysis, the nonlinear rheological parameters of higher harmonics, dissipation ratio, strain hardening ratio, shear thickening ratio, transient modulus, and cage modulus were obtained and quantitatively analyzed. CNCs tend to assemble to form anisotropic structures in an aqueous medium while the CNFs are entangled to form gels. The CNF suspensions demonstrated a significant viscous modulus overshoot and had stronger yield stresses, but the yield of CNC suspensions was more ductile. In the case of low concentrations, the CNF suspensions demonstrated stronger intracycle shear thickening behavior in medium-amplitude oscillatory shear region and lower dissipation ratios at small strain amplitudes. Although both nanocellulose suspensions revealed the existence of four intracycle rheological transition processes (viscoplastic deformation, structural recovery, early-stage yielding, and late-stage yielding), the CNF suspensions exhibited a stronger structural recovery ability. Larger strain amplitudes did not invariably result in a broader range of intracycle rheological transitions, which are also affected by the excitation frequency. The application of the various LAOS analysis methods provided valuable intracycle nonlinear rheological insights into nanocellulose suspensions, which are of great importance for enhancing their industrial perspectives.

Investigating the phase behaviour of binary suspensions of cellulose nanocrystals and montmorillonite with nonlinear rheology, SAXS and polarized optical microscopy

Using microscopy imaging techniques, X-ray scattering, and nonlinear rheological measurements, we have examined microstructural changes, and phase behavior of cellulose nanocrystals (CNC)/montmorillonite (MMT) composite colloidal systems. Fourier-transform (FT) rheology-stress decomposition methods, Lissajous-Bowditch plots, and other techniques are used to both qualitatively and quantitatively determine the various phases that the suspensions exhibit. In the MAOS (medium amplitude oscillatory shear) region, multiple scaling regions, as well as quadratic scaling regions are obtained. The dependence of the normalized third relative intensity (I3/1) values on angular frequency (ω) also provided new insights into how nonlinear parameters depend on the various CNC/MMT phases. These findings allow us to differentiate various phases exhibited by the composite system, including the isotropic, biphasic, and chiral nematic liquid crystalline (LC) phases. The CNC/MMT nanocomposite system exhibits a variety of liquid crystalline phases, which may contribute to improved mechanical properties for the system, including increased mixture strength, stiffness, and toughness because of the particle alignment in the LC phase. Additionally, the field of optics and photonics may find use for this nanocomposite system in the fabrication of sensors, polarizers, and optical films; energy storage applications where the aligned structures facilitate ion transport; and the formulation of paints and coatings where the strain stiffening and shear thinning behavior of the CNC/MMT composite system are critical.

Unified interpretation of MAOS responses via experimentally decomposed material functions

Oscillatory shear testing, used to characterize the viscoelastic response of soft materials, is often divided into small, medium, and large amplitude oscillatory shear (SAOS, MAOS, and LAOS) regions. SAOS is a common test that gives us a unified analysis and interpretation of linear viscoelastic behavior, whereas understanding MAOS and LAOS is still an active area of research. While numerous mathematical techniques have been proposed, a consensus interpretation is still missing. Recently, our understanding of nonlinear behavior in the LAOS regime has been developed using iterative recovery tests. Recovery rheology decomposes the strain into two components, allowing an unambiguous interpretation of the nonlinear behavior in terms of sequences of recoverable and unrecoverable processes. In this work, we revisit the MAOS material functions for polyvinyl alcohol-borax hydrogel and worm-like micelles using recovery rheology. We show that two mathematical formalisms, the Chebyshev and sequence of physical processes analyses, provide competing physical interpretations when they are derived from the total strain, but provide unified interpretations when describing the decomposed strains. We, therefore, show that what has often been treated as a mathematical problem can instead be solved experimentally by acknowledging the extra information provided by recovery rheology.

Gaborheometry: Applications of the discrete Gabor transform for time resolved oscillatory rheometry

Oscillatory rheometric techniques such as small amplitude oscillatory shear (SAOS) and, more recently, medium amplitude oscillatory shear and large amplitude oscillatory shear (LAOS) are widely used for rheological characterization of the viscoelastic properties of complex fluids. However, in a time-evolving or mutating material, the build-up or breakdown of microstructure is commonly both time- and shear-rate (or shear-stress) dependent, and thixotropic phenomena are observed in many complex fluids including drilling fluids, biopolymer gels, and many food products. Conventional applications of Fourier transforms for analyzing oscillatory data assume the signals are time-translation invariant, which constrains the mutation number of the material to be extremely small. This constraint makes it difficult to accurately study shear-induced microstructural changes in thixotropic and gelling materials, and it is becoming increasingly important to develop more advanced signal processing techniques capable of robustly extracting time-resolved frequency information from oscillatory data. In this work, we explore applications of the Gabor transform (a short-time Fourier transform combined with a Gaussian window), for providing optimal joint time-frequency resolution of a mutating material’s viscoelastic properties. First, we show using simple analytic models and measurements on a bentonite clay that the Gabor transform enables us to accurately measure rapid changes in both the storage and/or loss modulus with time as well as extract a characteristic thixotropic/aging time scale for the material. Second, using the Gabor transform we demonstrate the extraction of useful viscoelastic data from the initial transient response following the inception of oscillatory flow. Finally, we consider extension of the Gabor transform to nonlinear oscillatory deformations using an amplitude-modulated input strain signal, in order to track the evolution of the Fourier–Tschebyshev coefficients of thixotropic fluids at a specified deformation frequency. We refer to the resulting test protocol as Gaborheometry (Gabor-transformed oscillatory shear rheometry). This unconventional, but easily implemented, rheometric approach facilitates both SAOS and LAOS studies of time-evolving materials, reducing the number of required experiments and the data postprocessing time significantly.

Kramers–Kronig relations for nonlinear rheology. Part I: General expression and implications

The principle of causality leads to linear Kramers–Kronig relations (KKR) that relate the real and imaginary parts of the complex modulus G∗ through integral transforms. Using the multiple integral generalization of the Boltzmann superposition principle for nonlinear rheology, and the principle of causality, we derived nonlinear KKR, which relate the real and imaginary parts of the nth order complex modulus Gn∗. For n=3, we obtained nonlinear KKR for medium amplitude parallel superposition (MAPS) rheology. A special case of MAPS is medium amplitude oscillatory shear (MAOS); we obtained MAOS KKR for the third-harmonic MAOS modulus G33∗; however, no such KKR exists for the first harmonic MAOS modulus G31∗. We verified MAPS and MAOS KKR for the single mode Giesekus model. We also probed the sensitivity of MAOS KKR when the domain of integration is truncated to a finite frequency window. We found that (i) inferring G33′′ from G33′ is more reliable than vice versa, (ii) predictions over a particular frequency range require approximately an excess of one decade of data beyond the frequency range of prediction, and (iii) G33′ is particularly susceptible to errors at large frequencies.

Kramers–Kronig relations for nonlinear rheology. Part II: Validation of medium amplitude oscillatory shear (MAOS) measurements

The frequency dependence of third-harmonic medium amplitude oscillatory shear (MAOS) modulus G33∗(ω) provides insight into material behavior and microstructure in the asymptotically nonlinear regime. Motivated by the difficulty in the measurement of MAOS moduli, we propose a test for data validation based on nonlinear Kramers–Kronig relations. We extend the approach used to assess the consistency of linear viscoelastic data by expressing the real and imaginary parts of G33∗(ω) as a linear combination of Maxwell elements: the functional form for the MAOS kernels is inspired by time-strain separability (TSS). We propose a statistical test based on fitting a sum of Maxwell elements using LASSO (least absolute shrinkage and selection operator) regression, and call it the SMEL test. It works well on a broad range of materials and models including those that do not obey TSS. It successfully copes with experimental data that are noisy or confined to a limited frequency range. When Maxwell modes obtained from the SMEL test are used to predict the first-harmonic MAOS modulus G31∗, it is possible to identify the range of time scales over which a material exhibits TSS.

Particle contact dynamics as the origin for noninteger power expansion rheology in attractive suspension networks

We show that Hertzian particle contacts are the underlying cause of the as-yet-unexplained noninteger power laws in weakly nonlinear rheology. In the medium amplitude oscillatory shear (MAOS) region, the cubic scaling of the leading order nonlinear shear stress (σ3∼γ0m3, m3=3) is the standard expectation. Expanding on the work by Natalia et al. [J. Rheol. 64, 625–635 (2020)], we report an extensive data set of noncubical, noninteger power law scalings m3 for particle suspensions in two immiscible fluids with a capillary attractive interaction, known as capillary suspensions. Here, we show that distinct power law exponents are found for the storage and loss moduli and these noninteger scalings occur at every secondary fluid concentration for two different contact angles. These compelling results indicate that the noninteger scalings are related to the underlying microstructure of capillary suspensions. We show that the magnitude of the third harmonic elastic stress scaling m3,elastic originates from Hertzian-like contacts in combination with the attractive capillary force. The related third harmonic viscous stress scaling m3,viscous is found to be associated with adhesive-controlled friction. These observations, conducted for a wide range of compositions, can help explain previous reports of noninteger scaling for materials involving particle contacts and offers a new opportunity using the variable power law exponent of MAOS rheology to reveal the physics of particle bonds and friction in the rheological response under low deformation instead of at very high shear rates.

Dielectrorheology of Aspect-Ratio-Tailored Carbon Nanotube/Polyethylene Composites under Large Deformations: Implications for High-Temperature Dielectrics

In this work, advanced custom-synthesized carbon nanotube (CNT)-based polymer nanocomposites were designed and prepared for electrical applications, with an innovative strategy in terms of catalyst preparation. The physical structures of the CNTs were fine-tuned by varying synthesis time to investigate the effect of aspect ratio on the electrical and rheological properties of the nanocomposites with state-of-the-art dielectro-nonlinear-rheological spectroscopy. The network structures of the nanotubes within the polymer matrix were studied utilizing a wide range of characterization techniques (e.g., linear rheology, electrical conductivity, dielectric properties, and electromagnetic interference shielding effectiveness). The results showed distinct differences in the microstructural features of the samples upon changing the aspect ratio of the CNTs. Then, for the first time, a correlation was developed between rheological and electrical properties with respect to deformation, under small, medium, and large amplitude oscillatory shear (SAOS, MAOS, and LAOS, respectively) flows. Dielectrorheological results depicted that the electrical properties of the nanocomposites containing longer CNTs are strain invariant while their rheological behaviors are highly sensitive to deformation. However, nanocomposites with shorter CNTs followed an opposite trend, which means there is a high dependency of electrical properties and a low sensitivity of rheological behavior to deformation. The output stress waveforms and Lissajous–Bowditch plots confirmed the aforementioned results. Our results disclose the importance of the physical properties of the nanomaterials on the performance of the high-temperature dielectrics and reveal that nanocomposites containing higher aspect ratio nanomaterials provide a more stable dielectric response when subjected to large deformations.

First- and third-order shear nonlinearities across the structural relaxation peak of the deeply supercooled pharmaceutical liquid indomethacin

Nonlinear rheological properties of viscous indomethacin are studied in the frequency range of its structural relaxation, that is, in a range so far inaccessible to standard techniques involving medium-amplitude oscillatory shear amplitudes. The first- and third-order nonlinearity parameters thus recorded using a sequence of small and large shear excitations in a time efficient manner are compared with predictions from rheological models. By properly phase cycling the shear amplitudes, build-up and decay transients are recorded. Analogous to electrical-field experiments, these transients yield direct access to the structural relaxation times under linear and nonlinear shearing conditions. To demonstrate the broader applicability of the present approach, transient analyses are also carried out for the glass formers glycerol, ortho-terphenyl, and acetaminophen.

An orientation-stretch coupled model for entangled comb polymers

A semi-empirical constitutive equation is suggested for the model comb polymer with the purpose to simultaneously describe the intrinsic medium amplitude oscillatory shear (MAOS) behaviors and transient behaviors in start-up elongational experiments. The molecular stress and hierarchical relaxation of comb polymers are discussed on the basis of the orientation-stretch coupled conformation tensor of both backbone and arm, which are governed by the same evolution equation with different relaxation times. The discrete Baumgaertel–Schausberger–Winter spectrum is adopted for the backbone relaxation. The model has four parameters that are fully determined from the linear spectrum. The equation set is solved analytically by the perturbation method under MAOS and solved numerically under start-up extensional flows. The predictions on the nonlinear rheological behaviors are well consistent with the experiments.

The weakly nonlinear response and nonaffine interpretation of the Johnson–Segalman/Gordon–Schowalter model

We derive new analytical solutions for the nonaffine Johnson–Segalman/Gordon–Schowalter (JS/GS) constitutive equation with a general relaxation kernel in medium-amplitude oscillatory shear (MAOS) deformation. The results show time-strain separable (TSS) nonlinearity, therefore providing new physically meaningful interpretation to the heuristic TSS nonlinear parameter in MAOS [Martinetti and Ewoldt, Phys. Fluids 31, 021213 (2019)]. The upper-convected, lower-convected, and corotational Maxwell models are all subsets of the results presented here. The model assumes that the microscale elements causing stress in the material slip compared to the continuum deformation. We introduce a visualization of the nonaffine deformation field that acts on stress-generating elements to reinforce the physical interpretation of the JS/GS class of models. Finally, a case study is presented where previously published results, from fitting TSS models to MAOS data, can be reinterpreted based on the concept of nonaffine motion of the JS/GS framework.

Effect of nitrogen doping on medium-amplitude oscillatory shear (MAOS) response of nanotube/polyvinylidene fluoride nanocomposites: Molecular simulations, rheology, and broadband electrical conductivity

This study sheds light on the effect of nitrogen (N) doping of carbon nanotubes (CNTs) on medium-amplitude oscillatory shear (MAOS) response of CNT/polyvinylidene fluoride (PVDF) nanocomposites within a rheologically percolated concentration regime. Custom-synthesized CNTs without and with nitrogen heteroatom (at a nitrogen atomic percent of 3.85 at. %) were incorporated into a PVDF matrix using a miniature melt-mixer at different concentrations. In both cases, as confirmed by TEM investigations, a nanoscopic state of dispersion in the PVDF matrix was achievable using the applied melt mixing procedure. Our results indicated that N-doped nanocomposites, well below their electrical percolation, form a hybrid, load-bearing network structure where network interconnectivity is driven by N-doped CNT domains and near-surface regions of the PVDF phase. This hybrid network formation behavior combined with N-doped CNTs inferior aspect ratio and their higher susceptibility to breakage and length loss during the melt mixing process have led to delayed electrical percolation. In contrast, localized clustering and contact aggregation in a submicrometer scale was the dominant mode of network formation in the undoped CNT/PVDF nanocomposites. These microstructural inferences were further validated in the frame of molecular simulations and optical microscopy investigations.

Chain-stretch relaxation from low-frequency Fourier transform rheology

Medium or large amplitude oscillatory shear (M/LAOS) is sensitive to polymer chain structure, yet poses unsolved challenges for 'a priori' structural characterisation. In this letter, we present a MAOS protocol applied to near-monodisperse linear polymer melts, from which chain-stretch relaxation, a key structural feature, is clearly discernible. The third harmonics of MAOS frequency sweeps are decomposed into real and imaginary components and found to obey time-temperature superposition. The GLaMM molecular tube-based model of linear entangled melt rheology and structure, which has no free parameters, closely follows the form of our experimental results for the third harmonics and contains discriminatory features which depend only on the polymer's chain stretch relaxation time. Significantly, these third harmonic features occur at low frequency and are readily accessible with standard rheometers. For materials where phase transitions restrict the use of time temperature superposition, this method greatly increases the scope of rotational rheometry for structural analysis of polymers. Although the theory shows good qualitative agreement with experimental data, we find fundamental differences in magnitude and the frequency dependence of the third harmonics which must be resolved in order to fully understand the molecular basis of the stress response.

Medium amplitude parallel superposition (MAPS) rheology. Part 2: Experimental protocols and data analysis

An experimental protocol is developed to directly measure the new material functions revealed by medium amplitude parallel superposition (MAPS) rheology. This protocol measures the medium amplitude response of a material to a simple shear deformation composed of three sine waves at different frequencies, revealing a rich dataset consisting of up to 19 measurements of the third-order complex modulus at distinct three-frequency coordinates. We discuss how the choice of 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 reducing 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 dataset is indeed large and feature-rich, while still acquired in a time comparable to similar medium amplitude oscillatory shear (MAOS) experiments. We demonstrate that the data represent measurements of an intrinsic material function by studying its internal consistency, compatibility with low-frequency predictions for Coleman–Noll simple fluids, and agreement with data obtained via MAOS amplitude sweeps. Finally, the data are 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 material’s time-dependent response.

Intra-Cycle Elastic Nonlinearity of Nitrogen-Doped Carbon Nanotube/Polymer Nanocomposites under Medium Amplitude Oscillatory Shear (MAOS) Flow.

This study seeks to unravel the effect of carbon nanotube’s physical and chemical features on the final electrical and rheological properties of polymer nanocomposites thereof. Nitrogen-doped carbon nanotubes (N-CNTs) were synthesized over two different types of catalysts, i.e., Fe and Ni, employing chemical vapor deposition. Utilizing this technique, we were able to synthesize N-CNTs with significantly different structures. As a result, remarkable differences in the network structure of the nanotubes were observed upon mixing the N-CNTs in a polyvinylidene fluoride (PVDF) matrix, which, in turn, led to drastically different electrical and rheological properties. For instance, no enhancement in the electrical conductivity of poorly-dispersed (N-CNT)Ni/PVDF samples was observed even at high nanotube concentrations, whereas (N-CNT)Fe/PVDF nanocomposites exhibited an insulative behavior at 1.0 wt%, a semi-conductive behavior at 2.0 wt%, and a conductive behavior at 2.7 wt%. In terms of rheology, the most substantial differences in the viscoelastic behavior of the systems were distinguishable in the medium amplitude oscillatory shear (MAOS) region. The stress decomposition method combined with the evaluation of the elastic and viscous third-order Chebyshev coefficients revealed a strong intra-cycle elastic nonlinearity in the MAOS region for the poorly-dispersed systems in small frequencies; however, the well-dispersed systems showed no intra-cycle nonlinearity in the MAOS region. It was shown that the MAOS elastic nonlinearity of poorly-dispersed systems stems from the confinement of N-CNT domains between the rheometer’s plates for small gap sizes comparable with the size of the agglomerates. Moreover, the intra-cycle elastic nonlinearity of poorly-dispersed systems is frequency-dependent and vanished at higher frequencies. The correlation between the microstructure and viscoelastic properties under large shear deformations provides further guidance for the fabrication of high-performance 3D-printed electrically conductive nanocomposites with precisely controllable final properties for engineering applications.

Evaluating predictability of various constitutive equations for MAOS behavior of entangled polymer solutions

To predict the nonlinear viscoelastic behavior of polymer melts and solutions under medium amplitude oscillatory shear (MAOS) flow, we summarized known analytical MAOS material functions and derived new analytical solutions from several continuum- or molecular-based constitutive models [Johnson–Segalman, Phan-Thien–Tanner, Leonov, Larson, Pom-Pom, coupled double-convection-reptation with chain stretch (cDCR-CS), and Rouse linear entangled polymers (Rolie-Poly)]. All constitutive models discussed in this study were classified as time-strain separable or inseparable models. Whereas linear viscoelastic behavior of all constitutive models under small amplitude oscillatory shear flow was identical, the frequency-dependent signatures of four MAOS moduli (first-harmonic elastic and viscous MAOS moduli and third-harmonic elastic and viscous MAOS moduli) were model-specific, though all analytical solutions exhibited negative first-harmonic elastic MAOS moduli. In particular, tube-based constitutive models displayed a sign change in first-harmonic viscous MAOS modulus when chain retraction of the models after deformation was not instantaneous. In addition, two models for entangled linear polymers, that is, the Rolie-Poly and cDCR-CS models, displayed a second sign change in third-harmonic elastic MAOS modulus when finite chain extensibility was introduced. These results suggest that chain relaxation dynamics can be reflected in signatures of MAOS moduli. We extended single-mode predictions into multimode versions to predict the behavior of weakly or moderately entangled polystyrene solutions (effective entanglement number < 5). All models described the two elastic MAOS moduli well, but different predictability was obtained for viscous MAOS moduli. Therefore, viscous MAOS moduli can be a measure of evaluating the predictability of constitutive models for MAOS.

Medium amplitude parallel superposition (MAPS) rheology. Part 1: Mathematical framework and theoretical examples

A new mathematical representation for nonlinear viscoelasticity is presented based on the 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 superset 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 datasets 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.

Small and Medium Amplitude Oscillatory Shear Rheology of Model Branched Polystyrene (PS) Melts.

Linear and nonlinear rheological properties of model comb polystyrenes (PS) with loosely to densely grafted architectures were measured under small and medium amplitude oscillatory shear (SAOS and MAOS) flow. This comb PS set had the same length of backbone and branches but varied in the number of branches from 3 to 120 branches. Linear viscoelastic properties of the comb PS were compared with the hierarchical model predictions. The model underpredicted zero-shear viscosity and backbone plateau modulus of densely branched comb with 60 or 120 branches because the model does not include the effect of side chain crowding. First- and third-harmonic nonlinearities reflected the hierarchy in the relaxation motion of comb structures. Notably, the low-frequency plateau values of first-harmonic MAOS moduli scaled with (total molecular weight), reflecting dynamic tube dilution (DTD) by relaxed branches. Relative intrinsic nonlinearity Q0 exhibited the difference between comb and bottlebrush via no low-frequency Q0 peak of bottlebrush corresponding to backbone relaxation, which is probably related to the stretched backbone conformation in bottlebrush.

First-Order and Third-Order Nonlinearities from Medium-Amplitude Oscillatory Shearing of Hydrogen-Bonded Polymers and Other Viscoelastic Materials

Medium-amplitude oscillatory shear experiments are employed to study the supramolecular relaxation modes in hydroxyl-terminated polydimethylsiloxane. For these associating melts, this allowed us to identify the nonlinear rheological signature of hydrogen-bond-mediated structures occurring in addition to those of the covalent chains. First-order and third-order nonlinearity parameters are determined for the supramolecular and chainlike modes to compare them with the predictions of an array of rheological models. Fingerprints of the two supramolecular modes can be recognized in the nonlinear response. By collecting third-order nonlinearity parameters 3Q0 for about 35 materials including polymer melts, polymer solutions, and other viscoelastic fluids, it is demonstrated that the maximum value of 3Q0 correlates with the magnitude of the shear modulus at the frequency at which this maximum occurs. The implications of this experimental finding are discussed.

Nonlinear material functions under medium amplitude oscillatory shear (MAOS) flow

Dynamic oscillatory shear flow has been widely used to investigate viscoelastic material functions. In particular, small amplitude oscillatory shear (SAOS) tests have become the canonical method for characterizing the linear viscoelastic properties of complex fluids based on strong theoretical background and plenty of experimental results. Recently, there has been increasing interest in the use of large amplitude oscillatory shear (LAOS) tests for the characterization of complex fluids. However, it is difficult to define material functions in LAOS regime due to an infinite number of higher harmonic contributions. For this reason, many recent studies have focused on intrinsic nonlinearities obtained in medium amplitude oscillatory shear (MAOS) regime, which is a subdivision of the full LAOS regime. In this study, we reviewed recent experimental and theoretical results of nonlinear material functions in the MAOS regime, which contain four MAOS moduli (two first-harmonic moduli and two third-harmonic moduli) from Fourier and power series of shear stress, and a nonlinear material function Q0 and its elastic and viscous parts from Fourier-transform rheology (FT rheology). Furthermore, to identify linear-to-nonlinear transitions in stress response of model polystyrene (PS) solutions, we presented Pipkin diagrams in frequency ranges from the rubbery plateau region to the terminal region.