Nonlinear Rheology Research Articles

The nonlinear rheology of complex yield stress foods

Many foods have a yield stress that allows them to retain a desired shape at rest, but transition into a viscous fluid when being served or consumed. The determination of the yield stress of the food dictates how the foods are formed and packaged, how they are served, and how they are perceived when being eaten. Oscillatory shearing provides an ideal test protocol to map the rheology across a range of time and flow strength scales. We couple oscillatory shearing and an iterative recovery procedure to show that the yielding process is a continuous transition for two common yield stress foods. We show that unrecoverable processes from oscillatory tests are equivalent to the steady shear flow behavior. We show that this yielding behavior can be well approximated by a recently published model that treats yield stress materials as continuous viscoelastic fluids with a rate-dependent relaxation time and has parameters that can be obtained from the linear viscoelastic oscillatory frequency sweep and the steady shear flow curve.

Non-linear rheology reveals the importance of elasticity in meat and meat analogues

The interest in plant-based meat analogues as an alternative to meat is currently growing. Rheological benchmarking is used to reveal how closely meat analogues resemble the original meat products. Texture maps and dissipation colour schemes were used to reveal similarities in and differences between rheological responses of meat and meat analogues (especially chicken analogues). Under heating, meat analogues differ in terms of their lower elasticity compared with heated meat. The changes caused by heating meat and meat analogues were different as well. Heating of meat resulted in a tougher and more elastic material, while heating has a minor effect on meat analogues. Future developments should therefore focus on routes to create more elasticity and possibly allow heating effects on texture to mimic meat characteristics even better.

Centrifugal spinning of viscoelastic nanofibres

The centrifugal spinning method is a recently invented technique to extrude polymer melts/solutions into ultra-fine nanofibres. Here, we present a superior integrated string-based mathematical model, to quantify the nanofibre fabrication performance in the centrifugal spinning process. Our model enables us to analyse the critical flow parameters covering an extensive range, by incorporating the angular momentum equations, the Giesekus viscoelastic constitutive model, the air-to-fibre drag effects and the energy equation into the string model equations. Using the model, we can analyse the dynamic behaviour of polymer melt/solution jets through the dimensionless flow parameters, namely, the Rossby ($Rb$), Reynolds ($Re$), Weissenberg ($Wi$), Weber ($We$), Froude ($Fr$), air Péclet ($Pe^*$) and air Reynolds ($Re^*$) numbers as well as the viscosity ratio ($\delta _s$), corresponding to rotational, inertial, viscous, viscoelastic, surface tension, gravitational, air thermal diffusivity, aerodynamic and viscosity ratio effects. We find that the nonlinear rheology remarkably affects the fibre trajectory, radius and normal stresses. Increasing $Wi$ leads to a thicker fibre, whereas increasing $\delta _s$ shows an opposite trend. In addition, by increasing $Wi$, the fibre curvature is enhanced, causing the fibre to spiral closer to the rotation centre.

Large-amplitude oscillatory shear to investigate the nonlinear rheology of polymer glasses – PMMA

Large-amplitude oscillatory shear to investigate the nonlinear rheology of polymer glasses – PMMA

Microstructural changes of concentrated Newtonian suspensions in the first oscillation cycles probed with linear and non-linear rheology.

Carotenuto et al. (Rheol Acta, 2021, 60, 309) recently showed that the complex viscosity of a Newtonian non-Brownian suspension is smaller than the steady shear one, whatever the imposed strain amplitude. Oscillatory shear can alter the microstructure through a shear induced particle diffusion mechanism. This mechanism needs time to show its effect and cannot be invoked to explain the observed mismatch between the steady shear and the complex viscosity. Moreover, in the limit of vanishing strain amplitudes and of very large ones, where the oscillatory shear is equivalent to consecutive steady flow reversals, the oscillatory shear should not alter the microstructure and the Cox-Merz rule should hold. With a combination of approaches exploiting the Lissajous-Bowditch plots, the Fourier transform rheology and the Sequence of Physical Processes, we investigate the microstructure changes induced in the first oscillatory cycles. The results from the different analyses agree with the microstructure rearranging mechanisms proposed by Carotenuto et al.: at small amplitudes, the oscillatory shear rotates couples of touching particles towards the flow direction, at medium amplitudes it breaks particle clusters and at very large amplitudes it reshuffles and reorients all the particles. We show that the vast majority of the microstructure rearrangement occurs soon after the flow inversion of the first cycle, while before it the microstructure is not altered. This allows us to suggest a procedure to "recover" the Cox-Merz rule: a single cycle of oscillation must be imposed and the stress response of the sole first quarter of oscillation must be analysed.

Nonlinear dilatational rheology of different protein aggregates at the oil-water interface.

Proteins tend to self-assemble into different morphological aggregates such as nanoparticles or fibrils during heat treatment depending on the processing conditions. The protein aggregates exhibit excellent interfacial activity and even better ability to stabilize emulsions than native proteins. The interfacial rheological properties at the oil-water interface play a very important role in emulsion stability, among which the interfacial nonlinear rheology is closely related to their ability to resist large perturbation. However, there are very few studies reporting the nonlinear interfacial rheological behavior of protein aggregates at the oil-water interface. In this study, β-lactoglobulin fibrous aggregates (F) and nanoparticle aggregates (NP) were prepared, and the adsorption kinetics and dilatational nonlinear rheological behavior of β-lactoglobulin aggregates at the oil-water interface under large amplitude deformation were studied using a pendant drop tensiometer, and compared with those of native proteins. From the adsorption experiments, the adsorption of protein aggregates, especially fibrils, was faster than that of native proteins in the early stage, while in the late stage, the native proteins displayed a significantly higher degree of rearrangement than the fibrils. The surface hydrophobicity and the short fibrils present mainly determine the properties of the fibril interface, while the behavior of the nanoparticle interface was significantly influenced by the size and charge properties of the nanoparticles. From the dilatational experiment, the Lissajous plots revealed that the F interface at all pHs evaluated and the βlg interface at pH 5.8 displayed strain softening in both expansion and compression processes, while the NP interface at all pHs and βlg interface at pH 2 and pH 7 displayed strain softening in expansion and strain hardening in compression processes. The nonlinear response of the protein aggregates at the oil-water interface was more obvious at pH 5.8. The modulus change from frequency sweeps revealed that the fibril interface was strong but not very structured in contrast to that formed by the native proteins which displays high structuration although weak in strength, whereas the strength of the interface formed by protein nanoparticles is in between, but more sensitive to the surface charge.

Subduction earthquake sequences in a non-linear visco-elasto-plastic megathrust

SUMMARYWe present a 2-D thermomechanical computational framework for simulating earthquake sequences in a non-linear visco-elasto-plastic compressible medium. The method is developed for a plane-strain problem and incorporates an invariant formulation of the classical rate- and state-dependent friction law and an adaptive time-stepping, which allows the time step to vary by many orders of magnitude during a simulation. Long-term tectonic convergence is imposed by displacing a boundary at a constant rate, whereas temperature-dependent viscosity is solved using a rapidly converging Newton–Raphson scheme. The 2-D volume is discretized using finite differences on a fully staggered grid and marker-in-cell techniques. An adaptive free-surface approximation is used to modulate the air viscosity with the time step, which allows stresses to vanish on the free surface during the propagation of fast slipping events. We present a set of increasingly complex models in which we investigate how inertia, radiation damping, thermally activated non-linear rheology and off-megathrust splay-fault events affect sequences of seismic and aseismic slip on a simplified subduction megathrust. The new method provides a unique computational framework to analyse earthquake sequences and to connect forearc deformation with the dynamic properties of the megathrust, thus providing a physical link between observations spanning from slow interseismic strain accumulation to localized coseismic slip of individual earthquakes and post-seismic viscoelastic relaxation.

Theoretical study of micropolar fluid flow in porous media

In this paper, a mathematical model of Hele-Shaw flow was developed on the basis of the field equation of micropolar fluid. The motion equation of micropolar fluid was extended to flow in porous media, contributing to the mathematical model of the unsteady radial flow of micropolar fluid in an infinite reservoir. The relations between flow velocity, pressure gradient and flow rate were obtained by solving the mathematical model under fixed production internal boundary and fixed-pressure external boundary conditions. In addition, the effect of the physical properties of micropolar fluid on its flow was analyzed. The results indicated a significant dependence between the apparent permeability and physical properties of micropolar fluid, which clearly deviated from the Newtonian fluid. The results showed that, as the ratio of rotational viscous force to Newtonian viscous force in the micropolar parameter and the characteristic length of the micropolar metamaterial increase, the non-Newtonian characteristics of the fluid become gradually more significant. The non- Newtonian behavior of micropolar fluid consequently becomes unneglected, leading to a decreasing apparent permeability. Compared with a Newtonian fluid, a micropolar fluid causes obvious flow resistance, resulting in larger energy consumption and lower flow rate. This study can provide an academic reference for nonlinear rheology and the flow of heterogeneous fluids in the fields of petroleum engineering and biofluid flow. Cited as: Zhu, W. Theoretical study of micropolar fluid flow in porous media. Advances in Geo-Energy Research, 2021, 5(4): 465-472, doi: 10.46690/ager.2021.04.11

Nonlinear rheology of cellular networks.

Morphogenesis depends crucially on the complex rheological properties of cell tissues and on their ability to maintain mechanical integrity while rearranging at long times. In this paper, we study the rheology of polygonal cellular networks described by a vertex model in the presence of fluctuations. We use a triangulation method to decompose shear into cell shape changes and cell rearrangements. Considering the steady-state stress under constant shear, we observe nonlinear shear-thinning behavior at all magnitudes of the fluctuations, and an even stronger nonlinear regime at lower values of the fluctuations. We successfully capture this nonlinear rheology by a mean-field model that describes the tissue in terms of cell elongation and cell rearrangements. We furthermore introduce anisotropic active stresses in the vertex model and analyze their effect on rheology. We include this anisotropy in the mean-field model and show that it recapitulates the behavior observed in the simulations. Our work clarifies how tissue rheology is related to stochastic cell rearrangements and provides a simple biophysical model to describe biological tissues. Further, it highlights the importance of nonlinearities when discussing tissue mechanics.

Stability bounds of a delay visco-elastic rheological model with substrate friction.

Cells and tissues exhibit sustained oscillatory deformations during remodelling, migration or embryogenesis. Although it has been shown that these oscillations correlate with intracellular biochemical signalling, the role of these oscillations is as yet unclear, and whether they may trigger drastic cell reorganisation events or instabilities remains unknown. Here, we present a rheological model that incorporates elastic, viscous and frictional components, and that is able to generate oscillatory response through a delay adaptive process of the rest-length. We analyse its stability as a function of the model parameters and deduce analytical bounds of the stable domain. While increasing values of the delay and remodelling rate render the model unstable, we also show that increasing friction with the substrate destabilises the oscillatory response. This fact was unexpected and still needs to be verified experimentally. Furthermore, we numerically verify that the extension of the model with non-linear deformation measures is able to generate sustained oscillations converging towards a limit cycle. We interpret this sustained regime in terms of non-linear time varying stiffness parameters that alternate between stable and unstable regions of the linear model. We also note that this limit cycle is not present in the linear model. We study the phase diagram and the bifurcations of the non-linear model, based on our conclusions on the linear one. Such dynamic analysis of the delay visco-elastic model in the presence of friction is absent in the literature for both linear and non-linear rheologies. Our work also shows how increasing values of some parameters such as delay and friction decrease its stability, while other parameters such as stiffness stabilise the oscillatory response.

Composite entanglement topology and extensional rheology of symmetric ring-linear polymer blends

Extensive molecular simulations are applied to characterize the equilibrium dynamics, entanglement topology, and nonlinear extensional rheology of symmetric ring-linear polymer blends with systematically varied ring fraction ϕR. Chains with degree of entanglement Z≈14 are mixed to produce 10 well-entangled systems with ϕR varying from neat linear to neat ring melts. Z is large enough that except for very large ϕR, the rings are threaded by multiple linear chains in equilibrium. Primitive path analysis is used to visualize and quantify the structure of the composite ring-linear entanglement network. We measure the quantity of ring-linear threading and linear-linear entanglement as a function of ϕR and identify with simple arguments a ring fraction ϕR≈0.4 where the topological constraints of the composite entanglement network are maximized. These topological analyses are used to rationalize the ϕR-dependence of ring and linear chain dynamics, conformations, and rheology. Simulations of startup uniaxial elongation flows demonstrate the extensional stress overshoot observed in recent filament stretching experiments and characterize how it depends on the blend composition and entanglement topology. The overshoot is driven by an overstretching and recoil of ring polymers due to the convective unthreading of rings from linear chains.

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.

Nonlinear rheological behavior of telechelic ionomers with the distribution of ionic stickers at the ends

This study examines the nonlinear rheology properties of telechelic ionomers with connecting ions at the chain ends. Our previous studies revealed that these telechelic ionomers relaxed in a hierarchical way depending on the average number of connecting ionic groups per chain end m. The terminal relaxation time increased with m because the dissociation was slower for the chain end with more connecting ions. In this study, the stress growth and damping functions are examined for these telechelic ionomer samples and a reference conventional telechelic sample having strictly one ion per chain end, both based on poly(isobutyl acrylate). During both the steady shear and elongation measurements, the stress overshoots are observed, and the corresponding strains at the stress overshoots, γpeak and εH,peak, are insensitive to the Weissenberg number Wi for Wi≫1. This feature suggests that γpeak or εH,peak are critical strains for the induction of dissociation, which is rationalized by the phenomenological analysis of the yielding stress based on the von Mises criterion. The approximate time-strain separability was surprisingly observed for the telechelic ionomer networks with a moderate m, which may be due to the fact that, although the high strain leads to a rupture of strand, the network structure has been maintained and its terminal relaxation is governed by the ionic dissociation (at least in the strain range of our measurements). These arguments are supported by the proper normalization of damping functions obtained for different samples using γ/γpeak as the horizontal axis.

Structure formation and non-linear rheology of blends of plant proteins with pectin and cellulose

Blends of proteins and carbohydrates are of interest because of their ability to form fibrous products that resemble meat products. Here the effect of pectin and cellulose addition is investigated on structuring potential of pea protein isolate (PPI), soy protein isolate (SPI). While SPI and PPI could only be transformed into a homogeneous of layered product, a pronounced fibrous structure could be formed with blends of SPI and pectin, and PPI and pectin (95:5 and 93:7). In the case of SPI blends, a fibrous product was also anisotropic in terms of mechanical properties. For PPI blends, a fibrous morphology was not always accompanied with mechanical anisotropy. Differences in structuring potential of the various blends were further analysed using oscillatory rheological experiments. PPI/carbohydrate and SPI/carbohydrates blends resulted in mushier products than products containing only PPI or SPI. This was most pronounced for blends with pectin, which was attributed to the combined effects of a decrease in pH and/or water redistribution between the two phases.

Enzymatic Hydrolysis of Triglycerides at the Water-Oil Interface Studied via Interfacial Rheology Analysis of Lipase Adsorption Layers.

The enzymatic hydrolysis of sunflower oil occurs at the water-oil interface. Therefore, the characterization of dynamic interfacial phenomena is essential for understanding the related mechanisms for process optimizations. Most of the available studies for this purpose deal with averaged interfacial properties determined via reaction kinetics and dynamic surface tension measurements. In addition to the classical approach for dynamic surface tension measurements, here, the evolution of the dilational viscoelasticity of the lipase adsorbed layer at the water-oil interface is characterized using profile analysis tensiometry. It is observed that lipase exhibits nonlinear dilational rheology depending on the concentration and age of the adsorbed layer. For reactive water-oil interfaces, the response of the interfacial tension to the sinusoidal area perturbations becomes more asymmetric with time. Surface-active products of the enzymatic hydrolysis of triglycerides render the interface less elastic during compression compared to the expansion path. The lipolysis products can facilitate desorption upon compression while inhibiting adsorption upon expansion of the interface. Lissajous plots provide an insight into how the hysteresis effect leads to different interfacial tensions along the expansion and compression routes. Also, the droplet shape increasingly deviates from a Laplacian shape, demonstrating an irreversible film formation during aging and ongoing hydrolysis reaction, which supports our findings via interfacial elasticity analysis.

Incorporating Rheological Nonlinearity into Fractional Calculus Descriptions of Fractal Matter and Multi-Scale Complex Fluids

In this paper, we use ideas from fractional calculus to study the rheological response of soft materials under steady-shearing flow conditions. The linear viscoelastic properties of many multi-scale complex fluids exhibit a power-law behavior that spans over many orders of magnitude in time or frequency, and we can accurately describe this linear viscoelastic rheology using fractional constitutive models. By measuring the non-linear response during large step strain deformations, we also demonstrate that this class of soft materials often follows a time-strain separability principle, which enables us to characterize their nonlinear response through an experimentally determined damping function. To model the nonlinear response of these materials, we incorporate the damping function with the integral formulation of a fractional viscoelastic constitutive model and develop an analytical framework that enables the calculation of material properties such as the rate-dependent shear viscosity measured in steady-state shearing flows. We focus on a general subclass of fractional constitutive equations, known as the Fractional Maxwell Model, and consider several different analytical forms for the damping function. Through analytical and computational evaluations of the shear viscosity, we show that for sufficiently strong damping functions, for example, an exponential decay of fluid memory with strain, the observed shear-thinning behavior follows a power-law response with exponents that are set by the power-law indices of the linear fractional model. For weak damping functions, however, the power-law index of the high shear rate viscosity is set by the terminal behavior of the damping function itself at large strains. In the limit of a very weak damping function, the theoretical formulation predicts an unbounded growth of the shear stress with time and a continuously growing transient viscosity function that does not converge to a meaningful steady-state value. By determining the leading terms in our analytical solution for the viscosity at both low and high shear rates, we construct an approximate analytic expression for the rate-dependent viscosity. An error analysis shows that, for each of the damping functions considered, this closed-form expression is accurate over a wide range of shear rates.

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.

Nonlinear Rheology of Telechelic Ionomers Based on Sodium Sulfonate and Carboxylate

Nonlinear Rheology of Telechelic Ionomers Based on Sodium Sulfonate and Carboxylate

Non-Newtonian laminar flow in pipes using radius, stress, shear rate or velocity as the independent variable

The flow of a non-Newtonian fluid in a circular pipe is a classic introductory transport phenomena problem, familiar to readers of Robert Byron Bird textbooks. A characteristic of Bird's work was taking the time to explore alternative ways to describe a problem and refine the results into elegant and readable formulas. Inspired by that approach, we compare methods for pipe flow solutions that differ on the independent variable used (radius, stress, shear rate) to obtain flow rate and residence time distributions for generalized Newtonian fluids. We highlight cases where using the shear rate as the independent variable has advantages for analytical and numerical solutions. We describe a method to use velocimetry experimental data coupled with a pressure drop measurement to directly construct a curve of flow rate vs pressure drop without the need of fitting the data to any rheological models. We present a geometrical interpretation of velocity profiles as areas in the stress–shear rate plane and derive analytical solutions for a three-parameter model of soft glassy materials [Caggioni et al., “Variations of the Herschel–Bulkley exponent reflecting contributions of the viscous continuous phase to the shear rate-dependent stress of soft glassy materials,” J. Rheol. 64, 413 (2020)] and a four-parameter model for chocolate melts [H. D. Tscheuschner, “Rheologische eigenschaften von lebensmittelsystemen,” in Rheologie Der Lebensmittel, edited by D. Weipert, H. Tscheuschner, and E. Windhab (Behr's Verlag, Hamburg, 1993), pp. 101–172]. We also compare the speed of various numerical approaches for a fractional viscoelastic model [A. Jaishankar and G. H. McKinley, “A fractional K-BKZ constitutive formulation for describing the nonlinear rheology of multiscale complex fluids,” J. Rheol. 58, 1751 (2014)].

Medium amplitude parallel superposition (MAPS) rheology of a wormlike micellar solution

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.