Small Amplitude Oscillatory Shear Flow Research Articles

Smoothed Particle Hydrodynamics simulations of integral multi-mode and fractional viscoelastic models

To capture specific characteristics of non-Newtonian fluids, during the past years fractional constitutive models have become increasingly popular. These models are able to capture, in a simple and compact way, the complex behaviour of viscoelastic materials, such as the change in power-law relaxation pattern during the relaxation process of some materials. Using the Lagrangian Smoothed-Particle Hydrodynamics (SPH) method we can easily track particle history; this allows us to solve integral constitutive models in a novel way, without relying on complex tasks.Hence, we develop here a SPH integral viscoelastic method which is first validated for simple Maxwell or Oldroyd-B models under Small Amplitude Oscillatory Shear (SAOS) and start-up channel flows. By exploiting the structure of the integral method, a multi-mode Maxwell model is then implemented. Finally, the method is extended to include fractional constitutive models, validating the approach by comparing results with theory.

Viscoelasticity of suspension of red blood cells under oscillatory shear flow

We present a numerical analysis of the rheology of a suspension of red blood cells (RBCs) for different volume fractions in a wall-bounded, effectively inertialess, small amplitude oscillatory shear (SAOS) flow for a wide range of applied frequencies. The RBCs are modeled as biconcave capsules, whose membrane is an isotropic and hyperelastic material following the Skalak constitutive law. The frequency-dependent viscoelasticity in the bulk suspension is quantified by the complex viscosity, defined by the amplitude of the particle shear stress and the phase difference between the stress and shear. SAOS flow basically impedes the deformation of individual RBCs as well as the magnitude of fluid-membrane interactions, resulting in a lower specific viscosity and first and second normal stress differences than in steady shear flow. Although it is known that the RBC deformation alone is sufficient to give rise to shear-thinning, our results show that the complex viscosity weakly depends on the frequency-modulated deformations or orientations of individual RBCs but rather depends on combinations of the frequency-dependent amplitude and phase difference. The effect of the viscosity ratio between the cytoplasm and plasma and of the capillary number is also assessed.

Applicability of the Cox–Merz Relationship for Mayonnaise Enriched with Natural Extracts

The Cox and Merz rules are empirical correlations between the apparent viscosity of polymers with the effect of shear rate and the complex dynamic viscosity with the effect of frequency. In this study, the rheological properties of mayonnaise-type emulsions enriched with Averrhoa carambola extracts were investigated using small-amplitude oscillatory shear (SAOS) and steady shear flow. The results showed that the shear-thinning behavior of the samples was non-Newtonian with yield stress and had time-dependent characteristics, as evidenced by curves from non-oscillatory measurements. It was observed that the experimental data on the complex and apparent viscosity of the samples obeyed the Cox–Merz rule.

General rigid bead-rod theory for steady-shear flow

General rigid bead-rod theory yields uniquely the relation between macromolecular architecture and complex viscosity. For this, it relies on the analytical solution of the general diffusion equation for small-amplitude oscillatory shear flow of Bird et al. [Dynamics of Polymeric Liquids, 2nd ed. (Wiley, New York, 1987), Vols. 1–2]. Unfortunately, this general diffusion equation has yet to be solved for any other flow field. In this paper, we do so for steady-shear material functions, namely, viscosity and first normal stress coefficient. We, thus, explain the non-Newtonian behaviors of macromolecular suspensions of any axisymmetric design in steady-shear flow.

Confined steady simple shear flow of polymeric liquids

In a confined simple shear flow, the macromolecules of a polymeric liquid reorient near the walls so that the measured viscosity decreases. For instance, in a small-amplitude oscillatory shear flow, the real part of the complex viscosity decreases with confinement, and macromolecular orientation explains this. These effects in oscillation have been explained analytically for a rigid dumbbell suspension and, for a confined small-amplitude oscillatory shear flow, the summation coefficients have been determined. By contrast, for the confined steady shear flow, the summation coefficients are undetermined. In this paper, we determine these coefficients and use them to evaluate the steady shear (i) viscosity and (ii) normal stress coefficients for a rigid dumbbell suspension. We find that the zero-shear viscosity and the zero-shear first normal stress coefficients decrease with confinement. We further find that the dimensionless (i) steady shear viscosity curve increases with confinement and (ii) first normal stress coefficient first decreases with light confinement and then increases with greater confinement. We confirm our theory, at low confinement, by comparing with our new measurements of the confined zero-shear viscosity of a polystyrene solution.

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.

Rod-climbing rheometry revisited.

The rod-climbing or "Weissenberg" effect in which the free surface of a complex fluid climbs a thin rotating rod is a popular and convincing experiment demonstrating the existence of elasticity in polymeric fluids. The interface shape and steady-state climbing height depend on the rotation rate, fluid elasticity (through the presence of normal stresses), surface tension, and inertia. By solving the equations of motion in the low rotation rate limit for a second-order fluid, a mathematical relationship between the interface deflection and the fluid material functions, specifically the first and second normal stress differences, emerges. This relationship has been used in the past to measure the climbing constant, a combination of the first (Ψ1,0) and second (Ψ2,0) normal stress difference coefficients from experimental observations of rod-climbing in the low shear rate limit. However, a quantitative reconciliation of such observations with the capabilities of modern-day torsional rheometers is lacking. To this end, we combine rod-climbing experiments with both small amplitude oscillatory shear (SAOS) flow measurements and steady shear measurements of the first normal stress difference from commercial rheometers to quantify the values of both Ψ1,0 and Ψ2,0 for a series of polymer solutions. Furthermore, by retaining the oft-neglected inertial terms, we show that the "climbing constant"  = 0.5Ψ1,0 + 2Ψ2,0 can be measured even when the fluids, in fact, experience rod descending. A climbing condition derived by considering the competition between elasticity and inertial effects accurately predicts whether a fluid will undergo rod-climbing or rod-descending. Our results suggest a more general description, "rotating rod rheometry" instead of "rod-climbing rheometry", to be more apt and less restrictive. The analysis and observations presented in this study establish rotating rod rheometry combined with SAOS measurements as a prime candidate for measuring normal stress differences in complex fluids at low shear rates that are often below commercial rheometers' sensitivity limits.

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.

Numerical simulations of small amplitude oscillatory shear flow of suspensions of rigid particles in non-Newtonian liquids at finite inertia

We perform immersed-boundary-method numerical simulations of small amplitude oscillatory shear flow of suspensions of monodisperse noncolloidal rigid spherical particles in non-Newtonian liquids from the dilute to the concentrated regime. We study the influence of suspending liquid inertia and rheology and particle concentration on the computationally measured storage and loss moduli of the suspensions. In particular, the rheology of the suspending liquid is modeled through the inelastic shear-thinning Carreau–Yasuda constitutive equation and the viscoelastic Giesekus and Oldroyd-B constitutive equations. The role of inertia is quantified by the Stokes number, St, whereas the relevance of the non-Newtonian effects of the suspension matrix is measured through the Carreau number, Cu, for the Carreau–Yasuda liquid and the Deborah number, De, for the viscoelastic liquids. In suspensions with a Carreau–Yasuda matrix, both the storage and the loss modulus increase with St and decrease with Cu, yet the order of magnitude of Cu has to be greater than unity for these effects to be visible. In suspensions with a viscoelastic matrix, both the moduli increase with St and have a nonmonotonic trend with De, showing a maximum with no quantitative differences between the results pertaining suspensions with Giesekus and Oldroyd-B constitutive equations.

Electrohydrodynamic Processing of PVP-Doped Kraft Lignin Micro- and Nano-Structures and Application of Electrospun Nanofiber Templates to Produce Oleogels

The present work focuses on the development of lignin micro- and nano-structures obtained by means of electrohydrodynamic techniques aimed to be potentially applicable as thickening or structuring agents in vegetable oils. The micro- and nano-structures used were mainly composed of eucalyptus kraft lignin (EKL), which were doped to some extent with polyvinylpyrrolidone (PVP). EKL/PVP solutions were prepared at different concentrations (10–40 wt.%) and EKL:PVP ratios (95:5–100:0) in N, N-dimethylformamide (DMF) and further physico-chemically and rheologically characterized. Electrosprayed micro-sized particles were obtained from solutions with low EKL/PVP concentrations (10 and 20 wt.%) and/or high EKL:PVP ratios, whereas beaded nanofiber mats were produced by increasing the solution concentration and/or decreasing EKL:PVP ratio, as a consequence of improved extensional viscoelastic properties. EKL/PVP electrospun nanofibers were able to form oleogels by simply dispersing them into castor oil at nanofiber concentrations higher than 15 wt.%. The rheological properties of these oleogels were assessed by means of small-amplitude oscillatory shear (SAOS) and viscous flow tests. The values of SAOS functions and viscosity depended on both the nanofiber concentration and the morphology of nanofiber templates and resemble those exhibited by commercial lubricating greases made from traditional metallic soaps and mineral oils.

Confinement and complex viscosity

Whereas much is known about the complex viscosity of polymeric liquids, far less is understood about the behavior of this material function when macromolecules are confined. By confined, we mean that the gap along the velocity gradient is small enough to reorient the polymers. We examine classical analytical solutions [O. O. Park and G. G. Fuller, “Dynamics of rigid and flexible polymer chains in confined geometries. II. Time-dependent shear flow,” J. Non-Newtonian Fluid Mech. 18, 111–122 (1985)] for a confined rigid dumbbell suspension in small-amplitude oscillatory shear flow. We test these analytical solutions against the measured effects of confinement on both parts of the complex viscosity of a carbopol suspension and three polystyrene solutions. From these comparisons, we find that both parts of the complex viscosity decrease with confinement and that macromolecular orientation explains this. We find the persistence length of macromolecular confinement, Lp, to be independent of both λω and λγ̇0.

Dilute dispersion of compound particles: deformation dynamics and rheology

Compound particles are a class of composite systems in which solid particles encapsulated in a fluid droplet are suspended in another fluid. They are encountered in various natural and biological processes, for e.g. nucleated cells, hydrogels, microcapsules etc. Generation and transportation of such multiphase structures in microfluidic devices is associated with several challenges because of the poor understanding of their structural stability in a background flow and the rheological characteristics of their dispersions. Hence, in this work, we analyse the flow in and around a concentric compound particle and investigate the deformation dynamics of the confining drop and its stability against breakup in imposed linear flows. In the inertia-less limit (Reynolds number, $Re \ll 1$ ) and assuming that the surface tension force dominates the viscous forces (low capillary number, $Ca$ , limit), we obtain analytical expressions for the velocity and pressure fields up to ${O}(Ca)$ for a compound particle subjected to a linear flow using a domain perturbation technique. Simultaneously, we determine the deformed shape of the confining drop correct up to ${O}(Ca^2)$ , facilitating the following. (i) Since ${O}(Ca^2)$ calculations account for the rotation of the anisotropically deformed interface, the reorientation dynamics of the deformed compound particles is determined. (ii) Calculations involving the ${O}(Ca^2)$ shape of the confining interface are found to be important for compound particles as ${O}(Ca)$ calculations make qualitatively different predictions in generalised extensional flows. (iii) An ${O}(Ca)$ constitutive equation for the volume-averaged stress for a dilute dispersion of compound particles was developed to study both shear and extensional rheology in a unified framework. Our analysis shows that the presence of an encapsulated particle always enhances all the measured rheological quantities such as the effective shear viscosity, extensional viscosity and normal stress differences. (iv) Moreover, linear viscoelastic behaviour of a dilute dispersion of compound particles is characterised in terms of complex modulus by subjecting the dilute dispersion to a small-amplitude oscillatory shear (SAOS) flow. (v) Various expressions pertaining to a suspension of particles, drops, and particles coated with a fluid film are also derived as limiting cases of compound particles.

Cole–Cole relation for long-chain branching from general rigid bead–rod theory

Empirically, we find that the parametric plots of the imaginary vs real parts of the complex viscosity may depend neither on temperature nor on average molecular weight. Moreover, for a fixed polydispersity, these viscosity Cole–Cole curves amplify both rightward and upward with long-chain branching content. In this paper, we find that general rigid bead–rod theory [O. Hassager, “Kinetic theory and rheology of bead–rod models for macromolecular solutions. II. Linear unsteady flow properties,” J. Chem. Phys. 60(10), 4001–4008 (1974)] can explain these rightward and upward amplifications. We explore the effects of branching along a straight chain in small-amplitude oscillatory shear flow. Specifically, we explore the number of branches, branch length, branch position, and branch distribution.

Direct simulation of fiber suspensions under smalland large amplitude oscillatory shear flow

Experimentally, oscillatory shear flow is used to investigate polymeric viscoelastic properties. However, the knowledge of the microstructure’s orientation evolution (kinematics) with time is essential in order to predict those properties, and is difficult experimentally. Therefore it is often required to conduct numerical simulations to get insight on the kinematics evolution of fiber suspensions. The main aim of the current work is to use direct numerical simulations to investigate the time evolution of the orientation and kinematics of fiber suspensions under small and large amplitude oscillatory shear flow. Keywords: fiber suspensions; direct numerical simulation; oscillatory shear flow; fiber-fiber interactions;

A charged finitely extensible dumbbell model: Explaining rheology of dilute polyelectrolyte solutions

A robust non-Newtonian fluid model of dilute polyelectrolyte solutions is derived from kinetic theory arguments. Polyelectrolyte molecules are modeled as finitely elongated nonlinear elastic dumbbells, where effective charges (interacting through a simple Coulomb force) are added to the beads in order to model the repulsion between the charged sections of polyelectrolyte chains. It is shown that the relative strength of this repulsion is regulated by the electric-to-elastic energy ratio, E, which is one of the key parameters of the model. In particular, E accounts for the intrinsic rigidity of polyelectrolyte molecules and can be used to explain the impact of solvent salinity on polyelectrolyte rheology. With two preaveraging approximations, the constitutive equations of the resulting fluid model are formulated in closed form. Material functions predicted by the model for steady shear flow, steady extensional flow, small-amplitude oscillatory shear flow, and start-up and cessation of steady shear flow are obtained and investigated using a combination of analytical and numerical methods. In particular, it is shown how these material functions depend on E. The two limiting cases of the model—uncharged dumbbells (E = 0) and rigid dumbbells (E → ∞)—are included in the analysis. It is found that despite its simplicity, the model predicts most of the experimentally observed rheological features of polyelectrolyte solutions.

Polymer branching and first normal stress differences in small‐amplitude oscillatory shear flow

AbstractGeneral rigid bead‐rod theory explains polymer viscoelasticity from macromolecular orientation. By means of general rigid bead‐rod theory, we relate the normal stress differences of polymeric liquids to the branch position on a backbone branched macromolecule. In this work, we explore the first normal stress differences coefficients of different axisymmetric polymer configurations. When non‐dimensionalized with the zero‐shear first normal stress difference coefficient, the normal stress differences depend solely on the dimensionless frequency. In this work, in this way, we compare and contrast the normal stress differences of macromolecular chains that are branched. We explore the effects of branch position, length, functionality, spacing, and multiplicity, along a straight chain, in addition to rings and star‐shaped macromolecules in small‐amplitude oscillatory shear flow.

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.

Large-amplitude oscillatory shear flow loops for long-chain branching from general rigid bead-rod theory

General rigid bead-rod theory [O. Hassager, “Kinetic theory and rheology of bead-rod models for macromolecular solutions. II. Linear unsteady flow properties,” J. Chem. Phys. 60, 4001–4008 (1974)] explains polymer viscoelasticity from macromolecular orientation. By means of this theory, we relate the complex viscosity of polymeric liquids to the architecture of axisymmetric branched macromolecules. In this work, we explore how adding long-chain branching to polymers affects the shapes of large-amplitude oscillatory shear (LAOS) flow loops. By loops, we mean plots of the alternant part of the shear stress response vs the cosinusoidal shear rate. We choose LAOS for its ability to amplify subtle differences in small-amplitude oscillatory shear flow at a high Weissenberg number. When non-dimensionalized with the product of the zero-shear viscosity and the shear rate amplitude, the loop shapes depend on the sole dimensionless architectural parameter, the macromolecular lopsidedness of the long-chain branched macromolecule. In this work, in this way, we compare and contrast the loop shapes of macromolecular chains that are straight with those branched. Specifically, we explore symmetric branch multiplicity, branch functionality, branch length, branch position, branch distribution, and multiple branch asymmetry. We find that adding branching collapses and distorts the loops. We then find that so long as branch length, branch position, and branch distribution are held constant and so long as the branching is symmetric about the center of mass, the peak shear stress increases with branch multiplicity. We also find that branch functionality hardly affects the loops. The structural details explored in this paper have yet to be explored in the laboratory.