Large Amplitude Oscillatory Shear Flow Research Articles

Harmonic balance for differential constitutive models under oscillatory shear

Harmonic balance (HB) is a popular Fourier–Galerkin method used in the analysis of nonlinear vibration problems where dynamical systems are subjected to periodic forcing. We adapt HB to find the periodic steady-state response of nonlinear differential constitutive models subjected to large-amplitude oscillatory shear flow. By incorporating the alternating-frequency-time scheme into HB, we develop a computer program called FLASH (acronym for Fast Large Amplitude Simulation using Harmonic balance), which makes it convenient to apply HB to any differential constitutive model. We validate FLASH by considering two representative constitutive models, viz., the exponential Phan-Thien–Tanner model and a nonlinear temporary network model. In terms of accuracy and speed, FLASH typically outperforms the conventional approach of solving initial value problems by numerical integration via time-stepping methods often by several orders of magnitude. Exceptions to this rule are sometimes encountered for materials that are strongly shear thinning or described by constitutive models with discontinuous derivatives. We discuss how FLASH can be conveniently extended for other nonlinear constitutive models, which opens up potential applications in model calibration and selection, and stability analysis.

Can numerical methods compete with analytical solutions of linear constitutive models for large amplitude oscillatory shear flow?

Can numerical methods compete with analytical solutions of linear constitutive models for large amplitude oscillatory shear flow?

Characterizing blood hysteresis via tensorial thixo-elasto-viscoplastic modeling

A characteristic feature of human blood rheology is a distinctive stress hysteresis during shear ramp up in the shear rate from zero, followed by a ramp back to zero. This is a result of the fact that human blood has a longer characteristic time of shear-induced rouleaux breakdown compared to the shear aggregation of the rouleaux. We demonstrate this telltale phenomenon of human blood rheology during the triangle ramp, time-dependent change in the shear rate. The unique hysteresis data are then used along with steady state data to fit parameters of a recently published thixo-elasto-viscoplastic rheological model, the tensorial enhanced structural stress thixotropic-viscoelastic (t-ESSTV) model. These best-fit parameter values from the hysteresis ramps are then used to predict step-up/down in shear, small amplitude oscillatory shear, uni-directional large amplitude oscillatory shear, and large amplitude oscillatory shear flow. Additionally, correlations between the calculated fitting parameters and physiological data are analyzed to inform the interpretation of model behavior in physical terms. The fit adherence to the triangle ramp and rheological hysteresis data are then evaluated alongside recently developed techniques to assess thixotropy via hysteresis loop area, indicating the efficacy of the t-ESSTV model in potentially predicting blood's complex characteristics in useful ways for future use in modeling blood flows under a variety of mechanical and biological conditions and predicting pathologies.

Steady elongational flow from rotarance theory

The physics of macromolecular orientation has been used to explain the elasticity of polymeric liquids. Specifically, by first sculpting a rigid bead–rod likeness of the macromolecule, we can then derive its hydrodynamic resistance to orientation. The solution for the orientation distribution function has then been used, by integration in phase space, to get rheological material functions in both (i) small- and (ii) large-amplitude oscillatory shear flow, including its limiting case, and (iii) steady shear flow. However, rheological material functions in steady homogeneous extension from rigid bead–rod theory remain elusive. In this paper, we derive the orientation distribution function, and the rheological material functions, for suspensions of general rigid bead–rod structures. We focus on the time-steady viscosities in extension, and we first do so for general extensional kinematics. We then obtain the viscosities in steady extension for (iv) uniaxial extension, (v) planar extension, and (vi) biaxial extension. We close with a worked example, in which we use our new result for the steady uniaxial extensional viscosity to build a bridge between the macromolecular theory and the Oldroyd framework for rheological constitutive models. We, thus, arrive at a constitutive equation whose parameters are deducible from the moments of inertia of the macromolecule, and thus, deducible from macromolecular architecture alone. Our model is accurate up to third order for time-independent flows and is accurate to second order for time-dependent ones.

Large-amplitude oscillatory shear flow from general rigid bead-rod theory

Oscillatory shear flow, performed at small-amplitude, interrogates polymeric liquids in their equilibrium states. The fluid responds in sinusoidal shear stress waves whose amplitude and phase lead depend on the dimensionless frequency (called the Deborah number). By contrast, this same flow field, performed at large-amplitude, probes departures from the equilibrium state, and the fluid responds with shear stress in the form of a Fourier series, whose component amplitudes and phase leads depend on both the dimensionless frequency (called the Deborah number) and the dimensionless shear rate amplitude (called the Weissenberg number). The physics of these departures from equilibrium in an oscillatory shear flow may be explained by (i) chain disentanglement or (ii) motion along the polymer chain axes (called reptation) or (iii) macromolecular orientation. Of these radically different and yet otherwise equally effective approaches, only (iii) allows the macromolecular structure to be varied arbitrary so that the effect of molecular architecture on the rheology can be explored. Though much has been written about a large-amplitude oscillatory shear flow, we understand little about the role of molecular structure on the measured behaviors, and this has limited its usefulness. In this work, we explain the higher harmonics of both the shear stress (first and third), the first normal stress differences (zeroth, second, and fourth), and the second normal stress differences (zeroth and second) arriving at analytical expressions for all three. These expressions, written in dimensionless form, express the dimensionless rheological responses in large-amplitude oscillatory shear flow in terms of the ratio of the two principal macromolecular moments of inertia. To get these expressions, we derive the first five terms of the orientation distribution function, by solving the general diffusion equation in Euler coordinates. We then integrate in phase space with this orientation result to arrive at our expression for the first seven terms of the polymer contribution to the extra stress tensor. From this tensor, we next write down the Fourier coefficients for the shear stress responses, and the normal stress difference responses, in large-amplitude oscillatory shear flow for a suspension of macromolecules sculpted from a rigid bead-rod structure of any arbitrary axisymmetric shape.

Multiscale modeling of complex fluids under SAOS and LAOS using a combined FENE transient network model

The multiscale modeling of complex fluids under small and large amplitude oscillatory shear flow using non-linear kinetic and transient network models is presented. The kinetics of microstates is analogous to chemical kinetics, which defines the physical macromolecule interaction in a Newtonian fluid, and the concentration of microstates defines a variable maximum length of extension for each microstate. The effect of important parameters like viscosity ratio, chain length, viscoelasticity, kinetic rate constants, for different initial entanglement scenarios (entangled, disentangled and aleatory) are analyzed. The Lissajous curves for the shear stress and the first normal stress difference versus the instantaneous strain or strain-rate are shown. The self-intersection of the Lissajous curves or secondary loops is shown to depend on the kinetic rate constants, the maximum extension length, and the elasticity.

Oscillatory behavior with fiber orientation in the flow–vorticity plane for fiber suspension under large amplitude oscillatory shear flow

In fiber suspensions, fibers with three-dimensional orientation states are dramatically flow-oriented in the flow–flow gradient plane when the flow starts. In contrast, large strains are required for flow orientation in the flow–vorticity plane. Under oscillatory shear flow, when the strain amplitude is small, the flow orientation in the flow–vorticity plane is weakly induced, unlike that in the flow–flow gradient plane. The orientation in the flow–vorticity plane increases with the strain amplitude. At large strain amplitudes, fibers are oriented in the flow–flow gradient plane; thus, the rotational motion of fibers in the flow–flow gradient plane is dominant, i.e., fibers are almost flow-oriented in the flow–vorticity plane. However, contributions of the oscillatory behavior of fiber orientation in the flow–flow gradient and flow–vorticity planes to complex viscosity are unclear. In this study, the objective is to determine the contributions of fiber orientation in each plane to complex viscosity with the two initial orientation states (random and flow-oriented states) for the strain sweep test. The two initial orientation states were controlled by manipulating the upper disk of the parallel disk flow path. The initial random state was induced by sliding the upper disk up and down. The initial flow-oriented state was induced by rotating the upper plate. Furthermore, phase transition behaviors from the random to flow-oriented state in the flow–vorticity plane with increasing strain amplitude were qualitatively estimated as the orientation angle via visualization. Consequently, when the initial orientation was random, the fibers gradually vibrated in a medium strain amplitude region, and complex viscosity was higher than that of the initial flow-oriented state. In the conclusion, we divided the complex viscosity behavior of the strain sweep test into five strain amplitude regions and clarified the dominant orientation state in each region. It is conceivable that this result will help in understanding the relationship between fiber orientation and dynamic viscoelastic properties of fiber suspension like fiber-reinforced thermoplastics.

A comparative study of the nonlinear rheological properties of three different cellulose nanofibril suspensions

This study focused on the nonlinear rheological characterization of three types of cellulose nanofibril (CNF) suspensions under large amplitude oscillatory shear (LAOS) flow. Three different CNFs were produced, two by mechanical fibrillation alone under different conditions [here named microfibrillated cellulose (MFC) and U-CNF] and the other by mechanical fibrillation after carboxymethylation (CM-CNF). MFC and U-CNF had broad width distributions, whereas CM-CNF had narrower fibril width and width distribution due to the presence of charged carboxymethyl groups. Nonlinear stress responses of the prepared suspensions were analyzed using the sequence of physical processes method. All CNF suspensions exhibited intracycle rheological transitions composed of three physical processes: (1) structure recovery, (2) elastic deformation to early stage yielding, and (3) late-stage yielding. MFC and U-CNF suspensions exhibited similar rheological transitions overall. However, CM-CNF suspension had a higher network recovery rate within a shorter time and showed an additional yielding step due to the complex interplay between recovery and yielding dynamics. This result originated from complete nanofibrillation and charged functional groups on fibril surfaces. Rapid reformation of effective fibril–fibril contacts in CM-CNF suspension was attributed to electrostatic repulsions and complete nanosized lateral dimensions. In addition, excitation frequency was found to influence intracycle rheological transitions. A range of intracycle rheological transitions became narrower on increasing frequency because the time period for each transition was not enough under faster flow conditions. In particular, the characteristic yielding step of CM-CNF suspension disappeared on increasing frequency, which suggested that high-frequency excitation might be unfavorable for the nonlinear viscoelastic characterization of soft materials under LAOS flow.

Correlating Rheological Experiments, TEVP Models, and Microstructure with Small Angle Light Scattering

The existence of an evolving microstructure in a 2.9 vol% fumed silica in paraffin oil and polyisobutylene is demonstrated experimentally and via rheological modeling during steady state and large amplitude oscillatory shear flow. The continuously evolving, rebuilding, and breaking down of the microstructure is shown, and correlated through the rheology experiments, thixo-elastovisco-plastic modeling, and small angle light scattering (SALS). All elements are then connected via a global, stochastic optimization algorithm that will provide parameter estimation with a “best-fit” of the steady state and transient data using the well-known Modified Delaware Thixotropic Model, allowing for the comparison of SALS results with experimental values.

Novel tensorial Thixo-Visco-Plastic framework for rheological characterization of human blood

Characterizing human blood, a complex material with a spectrum of thixo-elasto-visco-plastic properties, through the development of more effective and efficient models has achieved special interest of late. This effort details the development a new approach, the tensorial-enhanced-Thixo-Visco-Plastic model (t-e-TVP), which integrates elements from the proven Bingham and generalized Maxwell systems to create a more robust framework and subsequently cast into a tensorial format. Here, the elastic and viscoelastic stress contributions from the microstructure are superimposed upon the viscoelastic backbone solution for stress offered by the modified TVP frame. The utility of this novel model is tested against the contemporary tensorial-ethixo-mHAWB (t-ethixo-mHAWB) framework, a similar model with a greater number of parameters, using rheological data of human blood collected on an ARESG2 strain-controlled rheometer. The blood samples are parametrically and statistically analyzed, entailing the comparison of the t-e-TVP and t-ethixo-mHAWB models with their capacity to accurately predict small and large amplitude oscillatory shear as well as unidirectional large amplitude oscillatory shear flow in blood.

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.

Linear and nonlinear oscillatory rheology of chemically pretreated and non-pretreated cellulose nanofiber suspensions

Linear and nonlinear rheological properties of cellulose nanofiber (CNF) suspensions were measured under small and large amplitude oscillatory shear (SAOS and LAOS) flow. Four different CNFs were produced, two by only mechanical disintegration and two with chemical pretreatments. Linear viscoelastic properties distinguished chemically treated CNFs from two untreated fibers via a different scaling exponent of the elastic modulus. However, different mechanical fibrillation degree was not characterized via linear viscoelastic properties. In contrast, nonlinear viscoelastic properties reflected both effects of chemical pretreatments and mechanical fibrillation. More fibrillated CNFs exhibited nonlinear rheological phenomena at larger deformations. In addition, chemically treated CNFs exhibited greater network stiffness and higher network recovery rates due to the presence of charged functional groups on the fiber surfaces. A material-property co-plot showed that network stiffness and recovery rate were in a trade-off relationship.

Constitutive Modeling of Nonlinear Rheological Behavior of Carbon Nanotube-Filled Polypropylene Nanocomposites

Abstract The rheological behavior of multi-walled carbon nanotube (MWCNT)-filled polypropylene (PP) nanocomposites with different filler loadings was experimentally studied and simulated using constitutive modeling. Rheological behavior was characterized in small amplitude oscillatory shear (SAOS) flow, large amplitude oscillatory shear (LAOS) flow, startup of shear flow, steady shear flow, and stress relaxation after the imposition of a step shear strain. Virgin PP and PP with CNT loadings of 1, 3, and 5 wt% were used. The formation of a rheological percolation network was observed at these loadings. The Leonov and Simhambhatla-Leonov models were used to simulate the rheological behavior. In the linear region, the simulations provided good predictions of the experimental data for both the unfilled and filled PP. In the nonlinear region, the simulations also provided good results for the virgin PP and satisfactory results for the PP/1 wt%CNT nanocomposite under most flow conditions. However, for the other two nanocomposites the model showed mixed results.

Wall-anchored semiflexible polymer under large amplitude oscillatory shear flow.

The properties of semiflexible polymers tethered by one end to an impenetrable wall and exposed to oscillatory shear flow are investigated by mesoscale simulations. A polymer, confined in two dimensions, is described by a linear bead-spring chain, and fluid interactions are incorporated by the Brownian multiparticle collision dynamics approach. At small strain, the polymers follow the applied flow field. However, at high strain, we find a strongly nonlinear response with major conformational changes. Polymers are stretched along the flow direction and exhibit U-shaped conformations while following the flow. As a consequence of confinement in the half-space, frequency doubling in the time-dependent polymer properties appears along the direction normal to the wall.

Shear localization in large amplitude oscillatory shear (LAOS) flows of particulate suspensions

Significant shear localization, observed as substantial concentration gradients, is discovered in large amplitude oscillatory shear (LAOS) simulations of a complex fluid in a planar Couette setup. The localization is demonstrated to occur due to the inertial effects imposed by the oscillatory shear in LAOS, which serve as perturbations in the nonlinear equations describing the structural response of the fluid to shear. Additionally, a criterion for the onset of this shear localization is presented.

Correlations in rheological behavior between large amplitude oscillatory shear and steady shear flow of silica‐filled star‐shaped styrene‐butadiene rubber compounds: Experiment and simulation

AbstractThe large amplitude oscillatory shear (LAOS) and steady shear behavior of star‐shaped SSBR/silica 60 phr (21 vol%) compounds with various filler surface areas was measured and simulated. An SBR gum and SBR compounds containing four different silicas with surface areas of 55, 135, 160, and 195 m2/g were utilized. Rheological behavior indicated clear correlation with surface area. LAOS tests showed an increase in dynamic moduli, shear stress, and higher order harmonic contributions with surface area. Elastic and viscous Lissajous figures showed significant distortion at intermediate and higher strain amplitudes. Additionally, ratios of third and fifth order stress harmonics to the first stress harmonic (I3/1 and I5/1, respectively) showed a ''bump'' at intermediate strain amplitudes for the three highest surface area compounds. With regards to steady shear, all materials showed strong shear thinning behavior, and an increase in shear viscosity with surface area. The Cox‐Merz rule was shown to be valid for the SBR gum but not for the filled compounds. However, the complex viscosity as a function of shear rate amplitude at various frequencies at high strain amplitudes and the steady shear viscosity as a function of shear rate coincided. This correlation, referred to as the Philippoff approach, has important ramifications for the rubber industry, providing quick data for predicting processing behavior. The Simhambhatla‐Leonov model was successfully employed to simulate rheological behavior for the SBR gum and the lowest surface area silica compound, but the model yielded mixed results for the higher surface area silica compounds.

Integration of thixotropy into Giesekus model for characterization of human blood

Recent work modeling the rheological behavior of human blood indicates that blood has all the hallmark features of a complex material, including shear-thinning, viscoelastic behavior, yield stress, and thixotropy. Using a recently developed linear superposition technique to account for the effects of thixotropy with the Giesekus model and recently collected human blood rheological data from a strain-controlled rheometer, we perform parametric and statistical analysis of the parameter values of 5 donors. The work is validated with the incorporation of a recent thixotropic framework to model elastic and viscoelastic contributions from the microstructure. The elastic and viscoelastic stress contributions from the microstructure are then linearly superimposed with the viscoelastic backbone solution for stress given by the classic Giesekus rheological model. Demonstrated here are a parametric and statistical analysis and a comparison of the ability of the new enhanced thixotropic Giesekus model to predict large amplitude oscillatory shear and uni-directional large amplitude oscillatory shear flow. In addition, there is a new methodology to model the normal forces of blood. We compare this approach to other recently developed enhanced thixotropic Oldroyd-8 inspired models.

A simple approach for adding thixotropy to an elasto-visco-plastic rheological model to facilitate structural interrogation of human blood

• Simple thixotropic term introduced that allows thixo-elasto-visco-plasticity modeling. • Use of step up/down in shear rate experiments with human blood to fit parameters . • Use of oscillatory rheological experiments with human for model comparison. We demonstrate refinements of a simple elastic-viscous-plastic rheological model with the addition of a thixotropic term to better model rheological flow of human blood. The alteration of this model will contribute to better representation of lightly structured complex materials. The simple elastic-viscous-plastic (EVP) framework requires thixotropy for more accuracy in fitting and predicting the thixo-elasto-visco-plasticity of rheological flows with a microstructure. We demonstrate the performance of the added thixotropic features by comparing the TEVP construct to similar peer models using a cost function generated from predictions of rheological flows from triangle ramps to large amplitude oscillatory shear flow. To verify the efficacy of these thixotropic enhancements, the performance of the model is then measured against the other models with a penalty assessed based on number of model parameters. The framework holds the name “Enhanced Thixotropic-Elasto-Visco-Plastic” to reflect this methodology of adding thixotropy to the elastic-viscous-plastic capabilities of the standard construct.

Normal Stress Differences of Human Blood in Unidirectional Large-Amplitude Oscillatory Shear Flow

Abstract This work analyzes normal stress difference responses in blood tested in unidirectional large-amplitude oscillatory shear flow (udLAOS), a novel rheological test, designed for human blood. udLAOS mimics the pulsatile flow in veins and arteries, in the sense that it never reverses, and yet also nearly stops once per heartbeat. As for our continuum fluid model, we choose the Oldroyd 8-constant framework for its rich diversity of popular constitutive equations, including the corotational Jeffreys fluid. This work arrives at exact solutions for normal stress differences from the corotational Jeffreys fluid in udLAOS. We discover fractional harmonics comprising the transient part of the normal stress difference responses, and both integer and fractional harmonics, the alternant part. By fractional, we mean that these occur at frequencies other than integer multiples of the superposed oscillation frequency. More generally, predictions from the Oldroyd 8-constant framework are explored by means of the finite difference method. Finally, the generalized versions of both the Oldroyd 8-constant framework and the corotational Jeffreys fluid are employed to predict the nonlinear normal stress responses for the model parameters fitted to udLAOS measurements from three very different donors, all healthy. From our predictions, we are led to expect less variation in normal stress differences in udLAOS from healthy donor to donor, than for the corresponding measured shear stress responses.

Delay of re-entanglement kinetics by shear-induced nucleation precursors in isotactic polypropylene melt

Rheological measurements were carried out to in-situ monitor the re-entanglement process of large amplitude oscillatory shear flow (LAOS) modified partially disentangled isotactic polypropylene (i-PP) melt. The characteristic entanglement recovery time (τent) was essentially longer than the average reptation time (τD) of fully entangled melt. At temperatures above 200 °C, the Arrhenius temperature dependence of τent and τD are the same, displaying a normal chain diffusion-controlled mechanism with flow activation energy of 41.4 kJ/mol. However, when the temperature is lower than 200 °C, τent is unexpectedly longer than expectation extrapolated from high temperatures, and a much higher activation energy of 98.6 kJ/mol is observed for the re-entanglement process. Based on our understanding, the re-entanglement process of shear-induced disentanglements in i-PP melt was postponed by the existence of quasi-ordered clusters created by shear at relatively low temperatures and their dissipation turns to be the rate-determining step for the whole recovery. Moreover, morphological study under polarized optical microscope showing much higher nucleation density which originates from the flow-induced precursors corroborates our hypothesis. Meanwhile, the surviving time of LAOS-generated nucleation precursors is comparable to the recovery time of disentangled chains, suggesting that the delay effect of nucleation precursors on chain diffusion may last over the entire re-entanglement process. Notwithstanding the obstruction of “extra” physical entanglements for chain diffusion, the molecular weight (Mw) dependence of entanglement recovery still follows the reptation model-based power law: τent=(a+b)Mw3.4, where a and b is the contribution of chain diffusion and dissipation of precursors, respectively.