An outline of the non-linear viscoelastic behaviour of wheat flour dough in shear

Creep and creep recovery, stress relaxation and small- and large-amplitude oscillatory shear experiments have been used to study the steady-state flow behaviour and the transient viscoelastic response of wheat flour dough in shear over large ranges of time, stress and strain. The results are discussed with reference to the limited body of reliable literature data. Dough does display a linear viscoelastic domain. The complex character of its non-linear viscoelastic properties is essentially due to the extremely low shear rate limit of the initial Newtonian plateau and to the onset of time-dependent flow behaviour above a certain strain threshold, which explain qualitatively the discrepancies observed in certain cases on a part of the range of the rheological variables explored, despite global self-consistency of the results. Comparison of gluten and dough linear viscoelastic properties shows that dough cannot be viewed simply as a concentrated suspension of starch granules in the hydrated viscoelastic gluten matrix.Paper presented at the second Annual European Rheology Conference (AERC 2005) held in Grenoble, France on April 21–23, 2005.     

Nonlinear viscoelastic properties of MR fluids under large-amplitude-oscillatory-shear

Nonlinear viscoelastic properties of the MR fluid, MRF-132LD, under large-amplitude oscillatory shear were investigated. This was accomplished by carrying out the experiments under the amplitude sweep mode and the frequency sweep mode, using a rheometer with parallel-plate geometry. Investigations under the influence of various magnetic field strength and temperatures were also conducted. MR fluids behave as nonlinear viscoelastic or viscoplastic materials when they are subjected to large-amplitude shear, where the storage modulus decreases rapidly with increasing strain amplitude. Hence, MR fluid behaviour ranges from predominantly elastic at small strain amplitudes to viscous at high strain amplitudes. Large-amplitude oscillatory shear measurements with frequency sweep mode reveal that the storage modulus is independent of oscillation frequency and approaches plateau values at low frequencies. With increasing frequency, the storage modulus shows a decreasing trend before increasing again. This trend may be explained by micro-structural variation. In addition, the storage modulus increases gradually with increasing field strength but it shows a slightly decreasing trend with temperature.     

Smectic rheology

We have studied the oscillatory shear response of three thermotropic smectic-A liquid crystalline materials with no external aligning field (other than the necessary presence of rheometer plates). Two are polymers (one main-chain and one side-chain) and the other is a small molecule smectic. All three exhibit the classical linear response to oscillatory shear characteristic of a viscoelastic solid at sufficiently small strain amplitudes and frequencies. However, for strain amplitudes exceeding a small critical value, these materials exhibit a strongly nonlinear response to strain, which is characterized in detail. While the low-strain moduli and the critical strain of the three smectics are considerably different, the nonlinear response has some universal character which is presumably related to the low energy for the formation of defects in smectic liquid crystals.     

Large amplitude oscillatory shear behavior of graphene derivative/polydimethylsiloxane nanocomposites

Rheological properties of three different nanocomposites, consisting of graphene oxide (GO), reduced graphene oxide (rGO), and polyhedral oligomeric silsesquioxane grafted reduced graphene oxide (rGO-POSS) as nanofillers and polydimethylsiloxane (PDMS), were investigated by large amplitude oscillatory shear (LAOS). The viscoelastic nonlinearity of the three nanofluids groups was studied by Lissajous curves, local nonlinear viscoelastic moduli of an oscillatory shear cycle, and Fourier transform rheology as a function of filler concentration and increasing and decreasing strain magnitude. The nonlinear behavior of the nanofluids was compared to understand the variation of internal microstructures. Firstly, GO/PDMS composites behave with higher moduli and smaller linear viscoelastic range comparing to that of other two composites. Secondly, the elastic stress Lissajous curves of these composites changed from elliptic to rectangular with round the corner with increasing the filler level and strain amplitude. Thirdly, all these three nanofluids exhibited intra-cycle strain stiffening with increasing strains and shear thickening at intermediate strain and then shearing thinning with increasing strain further. Fourthly, higher harmonic intensity of rGO/PDMS increased with increasing strain and came to a plateau, while that of other two nanofluids reached a maximum and then decreased. It suggested that different surface functionalization of nanoparticles will present different rheological behavior due to formed different network and LAOS could be used as a potential helpful method to characterize rheological properties of nanocomposites, especially at higher shear strain.     

Large amplitude oscillatory shear rheology of three different shear-thickening particle dispersions

We present a large amplitude oscillatory shear rheology (LAOS) investigation of three different shear-thickening particle dispersions - fumed silica in polyethylene oxide (FLOC), fumed silica in polypropylene glycol (HydroC), and cornstarch in water (JAM). These systems shear-thicken by three different mechanisms - shear-induced formation of particle clusters flocculated by polymer bridging, hydrocluster formation, and jamming. The viscoelastic non-linearities of the three fluids were studied as a function of strain and strain-rate space through the use of Lissajous-Bowditch curves and local nonlinear viscoelastic moduli of an oscillatory shear cycle. The nonlinear behaviors of the three fluids were compared and contrasted to understand the nonlinear shear-thickening mechanism of each. Both HydroC and JAM dispersions were found to exhibit strong strain stiffening of the elastic moduli and strain thickening of the loss moduli behavior associated with possible hydrocluster formation and particle jamming. However, the FLOC dispersion, in contrast, showed strong strain softening and strain thinning behavior at large strain amplitudes associated with yielding of the microstructure. The expected thickening of the loss modulus of FLOC in LAOS with increasing strain was not observed even though viscosity of FLOC was found to shear-thicken in steady-shear measurements. This disagreement is likely due to very large strain amplitudes required for shear-thickening to occur by shear-induced polymer bridging mechanism. The hypothesis was confirmed through stress growth experiments. Conversely, the HydroC and JAM dispersions required relatively small applied strains for shear-thickening to occur by hydrocluster and jamming mechanism. The comparison of local intra-cycle nonlinearity through Lissajous-Bowditch plots and nonlinear viscoelastic parameters indicated that the elastic nonlinearities of all three systems are primarily driven by a strong dependence on the magnitude of the applied strain-rates within an oscillatory cycle rather than the amplitude of the applied strain. A close inspection of the LAOS data reveals strong differences in the viscoelastic nonlinearities of these three different shear-thickening dispersions which can be used to create a nonlinear rheological fingerprint for each and offers valuable new insights into the nonlinear dynamics associated with each of the shear-thickening mechanisms.     

Elongational flow behavior of polymeric fluids according to the Leonov model

The viscoelastic behavior of polymeric systems based upon the Leonov model has been examined for (i) the stress growth at constant strain rate, (ii) the stress growth at constant speed and (iii) the elastic recovery in elongational flow. The model parameters have been determined from the available rheological data obtained either in steady shear flow (shear viscosity and first normal-stress difference as a function of shear rate) or oscillatory flow (storage and loss moduli as a function of frequency in the linear region) or from extensional flow at very small strain rates (time-dependent elongation viscosity in the linear viscoelastic limit). In addition, the effect of the parameter characterizing the strain-hardening of the material during elongation has also been studied. The estimation of this parameter has been based upon the structural characteristics of the polymer chain which include the critical molecular weight and molecular weight of an independent segment. Five different polymer melts have been considered with varying number of modes (maximum four modes). Resulting predictions are in fair agreement with corresponding experimental data in the literature.     

Technical note: correcting for shear strain in an oscillatory squeeze flow rheometer

Currently, rheologists working in the field of oscillatory squeeze flow use extensional strain to characterize the deformations. Due to the shear-dominated flow observed in low Trouton ratio fluids undergoing squeeze flow, it is proposed that an alternate geometry-dependent definition for shear strain in squeeze flow be used instead. Through the use of finite element modelling, it has been shown that this geometry-dependent strain definition allows for better comparison of measurements between both squeeze flow rheometers of different geometric configurations and rotational rheometers. This idea was then explored through laboratory experiments, further supporting this hypothesis. While this definition of strain will only hold true within the bounds of a material’s linear viscoelastic regime, it will help to determine where this boundary is, and thus allow for more accurate material characterization. This type of relationship will become increasingly important with the growing use of squeeze flow rheometers for large-amplitude oscillatory squeezing trials.     

Surface rheology of sorbitan tristearate and β-lactoglobulin: Shear and dilatational behavior

Proteins and surfactants behave very differently under shear and dilatational deformation. In this work, we compare specifically their surface properties by evaluating their rheological response. Oil-soluble surfactant, sorbitan tristearate (Span 65), and globular protein, β-lactoglobulin, were spread and adsorbed onto the surface, respectively. A 2D searle-type measuring geometry with a biconical bob was used for measuring the surface shear rheology, and a pendant drop film balance was used for measuring the dilatational rheology. Both equipments provided the viscoelastic properties (surface shear and dilatational complex moduli) of interfacial layers. Also, the linear and non-linear rheology of these systems was studied by increasing the amplitude of the oscillation. Linear rheology showed that dilatational deformation is mostly affected by the nature of the molecular structure at the interface, whereas shear deformation is affected by the strength of the surface film due to the intermolecular interactions. Furthermore, large-amplitude oscillatory shear rheology indicated that the non-linearity increases with the surface concentration, and is higher for insoluble Span 65 spread films than for soluble protein adsorbed layers. Dilatational and shear deformation provide complementary information about interfacial layers that can be optimized so as to fully characterize the surface depending of the type of film (spread or adsorbed) and the technique used (shear or dilatational rheology under linear or non-linear regimes). This information is useful to correlate the structure and the mechanical properties of interfacial systems.     

Superposition principles for small amplitude oscillatory shearing of nematic mesophases

The linear viscoelasticty of Leslie-Ericksen monodomain liquid crystals subjected to a bend distortion through a small amplitude oscillatory shear flow driven by harmonic wall stress is analyzed, using numerical and asymptotic methods. The viscoelastic material functions were derived using a new scaling approach that extracts the material parameters that control superposition. Small and high frequency superposition schemes for linear viscoleasticity were derived. The schemes were successfully applied to collapse the predicted loss and storage linear viscoelastic moduli of seven experimental data sets. Comparisons between different shear flows (simple shear and capillary Poiseuille) and different director distortion modes (splay and bend) shows that the superposition schemes are applicable to shear flows in any single director distortion mode.     

Relaxation effects of slip in shear flow of linear molten polymers

The transient shear response of a linear molten polymer (linear low-density polyethylene) in the nonlinear domain was studied using a true shear (sliding plate) rheometer with different gap spacings to detect slip effects. It was found that nonlinear viscoelasticity is further complicated by wall slip phenomena. Experimental evidence suggested that static slip models coupled with Wagner’s constitutive equation cannot adequately describe the experimental data at large and fast shear deformations. A new dynamic slip model involving multiple slip relaxation times is proposed in this paper, together with a method to assess the model parameters. Significant improvement in predicting the stress response is demonstrated by several examples of start-up of steady shear and large-amplitude oscillatory tests of a linear low-density polyethylene.     

The rheology of layered liquids: lamellar block copolymers and smectic liquid crystals

The frequency-dependence of the viscoelastic shear modulus at low frequencies in a lamellar polystyrene-polyisoprene block copolymer is qualitatively identical to that measured in small-molecule smectics, namely, the rod-like 4-cyano-4-octylbiphenyl and the flexible n-nonyl-1-O--D-glucopyranoside. All three materials were studied after quenching from the isotropic state, and during and after alignment by large-amplitude oscillatory shearing. The kinetics of aligning, as measured by changes in moduli during shearing, are similar, despite great differences in molecular characteristics. These moduli and the aligning process are evidently controlled by smectic fluctuations and defects, the dynamics of which have universal features.     

Low-dimensional intrinsic material functions for nonlinear viscoelasticity

Rheological material functions are used to form our conceptual understanding of a material response. For a nonlinear rheological response, the possible deformation protocols and material measures span a high-dimensional space. Here, we use asymptotic expansions to outline low-dimensional measures for describing leading-order nonlinear responses in large amplitude oscillatory shear (LAOS). This amplitude-intrinsic regime is sometimes called medium amplitude oscillatory shear (MAOS). These intrinsic nonlinear material functions are only a function of oscillatory frequency, and not amplitude. Such measures have been suggested in the past, but here, we clarify what measures exist and give physically meaningful interpretations. Both shear strain control (LAOStrain) and shear stress control (LAOStress) protocols are considered, and nomenclature is introduced to encode the physical interpretations. We report the first experimental measurement of all four intrinsic shear nonlinearities of LAOStrain. For the polymeric hydrogel (polyvinyl alcohol - Borax) we observe typical integer power function asymptotics. The magnitudes and signs of the intrinsic nonlinear fingerprints are used to conceptually model the mechanical response and to infer molecular and microscale features of the material.     

A unified approach to model elasto-viscoplastic thixotropic yield-stress materials and apparent yield-stress fluids

A constitutive model for elasto-viscoplastic thixotropic materials is proposed. It consists of two differential equations, one for the stress and the other for the structure parameter, a scalar quantity that indicates the structuring level of the microstructure. In contrast to previous models of this kind, the structure parameter varies from zero to a positive and typically large number. The lower limit corresponds to a fully unstructured material, whereas the upper limit corresponds to a fully structured material. When the upper limit is finite, the model represents a highly shear-thinning, thixotropic, and viscoelastic liquid that possesses an apparent yield stress. When it tends to infinity, the behavior of a true yield-stress material is achieved. Predictions for rheometric flows such as constant shear rate tests, creep tests, SAOS, and large-amplitude oscillatory shear (LAOS) are presented, and it is shown that, in all cases, the trends observed experimentally are faithfully reproduced by the model. Within the framework of the model, simple explanations are given for the avalanche effect and the shear banding phenomenon. The LAOS results obtained are of particular importance because they provide a piece of information that so far is absent in the literature, namely a quantitative link between the Lissajous–Bowditch curve shapes and rheological effects such as elasticity, thixotropy, and yielding.     

Viscoelastic plastic rheological model for particle filled polymer melts

A viscoelastic plastic model for suspension of small particles in polymer melts has been developed. In this model, the total stress is assumed to be the sum of stress in the polymer matrix and the filler network. A nonlinear viscoelastic model along with a yield criterion were used to represent the stresses in the polymer matrix and the filler network, respectively. The yield function is defined in terms of differential equations with an internal parameter. The internal parameter models the evolution of structure changes during floc rupture and restoration. The theoretical results were obtained for steady and oscillatory shear flow and compared with experimental data for particle filled thermoplastic melt. The experimental data included the steady state shear strress over a wide range of shear rates, the transient stress in a start up shear flow, stress relaxation after cessation of a steady state shear flow, the step shear and the oscillatory shear flow at various amplitudes.     

Non-linear rheology of a face-centred cubic phase in a diblock copolymer gel

The viscoelastic behaviour of a poly(oxyethylene)-poly(oxybutylene) diblock copolymer in aqueous solution forming a face-centred cubic (fcc) micellar phase has been investigated using oscillatory shear rheometry. With increasing strain amplitude, the micellar solution was observed to undergo a transition from linear to non-linear behaviour, characterized by strong shear thinning. The non-linear behaviour observed in the stress response was analyzed by Fourier transformation of the waveform. Fourier analysis revealed that the high harmonic contributions to the shear stress response increased with strain amplitude and up to the 81st harmonic was observed for very large amplitudes. The onset of non-linear response as defined from the dependence of isochronal dynamic shear moduli on strain amplitude was found to be in good agreement with that defined by the appearance of a higher harmonic in the stress waveform. The amplitudes of the harmonic coefficients are compared to the predictions of a model for the nonlinear rheological response of a lyotropic cubic mesophase based on the stress response to a periodic lattice potential (Jones and McLeish 1995). It is found that the model is able to account for qualitative trends in the data such as the development of finite higher harmonics with increasing strain, but it does not describe the full frequency and strain dependence of these coefficients. Received: 31 May 2000 Accepted: 21 August 2000     

A “quasi-elastic” affine formulation for the homogenised behaviour of nonlinear viscoelastic polycrystals and composites

The derivation of the overall behaviour of nonlinear viscoelastic (or rate-dependent elastoplastic) heterogeneous materials requires a linearisation of the constitutive equations around uniform per phase stress (or strain) histories. The resulting Linear Comparison Material (LCM) has to be linear thermoviscoelastic to fully retain the viscoelastic nature of phase interactions. Instead of the exact treatment of this LCM (i.e., correspondence principle and inverse Laplace transforms) as proposed by the “classical” affine formulation, an approximate treatment is proposed here. First considering Maxwellian behaviour, comparisons for a single phase as well as for two-phase materials (with “parallel” and disordered morphologies) show that the “direct inversion method” of Laplace transforms, initially proposed by Schapery (1962), has to be adapted to fit correctly exact responses to creep loading while a more general method is proposed for other loading paths. When applied to nonlinear viscoelastic heterogeneous materials, this approximate inversion method gives rise to a new formulation which is consistent with the classical affine one for the steady-state regimes. In the transient regime, it leads to a significantly more efficient numerical resolution, the LCM associated to the step by step procedure being no more thermoviscoelastic but thermoelastic. Various comparisons for nonlinear viscoelastic polycrystals responses to creep as well as relaxation loadings show that this “quasi-elastic” formulation yields results very close to classical affine ones, even for high contrasts.     

Differences between stress and strain control in the non-linear behavior of complex fluids

Various techniques have been proposed to characterize the behavior in the non-linear regime. A new theoretical framework, as proposed recently by Ewoldt et al. (J Rheol 52(6):1427–1458, 2008), provides a quantitative analysis of Lissajous figures during large-amplitude oscillatory shear (LAOS). Intra- and intercycle non-linearities, strain stiffening and softening, and shear thinning and thickening are described and can be distinguished. The new LAOS framework from Ewoldt et al. has been extended to a sinusoidal stress input. Measurements on two different samples reveal significant different results for sinusoidal strain or sinusoidal stress input. For both sinusoidal inputs, the results have been verified by cyclic stress and strain loading tests. The sinusoidal input tests are analyzed as an oscillatory test by the rheometer software and firmware, whereas the cyclic loading tests are purely rotational tests. Since both types of testing give the same results, any instrumental artifacts can be excluded. This implies that complex fluids can behave differently whether periodic stress or strain input functions outside the linear visco-elastic range are applied. All tests in controlled strain and stress in rotational and oscillatory modes have been performed with the same rheometer based on an air bearing-supported electrically commutated synchronous motor.     

Stress communication and filtering of viscoelastic layers in oscillatory shear

We revisit a classical topic: response functions of viscoelastic layers in large amplitude oscillatory shear. Motivated by questions concerning protective biological layers, we focus on boundary stresses in a parallel plate geometry with imposed oscillatory strain or stress. These features are gleaned from resolution and analysis of coupled standing waves of deformation and stress. We identify a robust non-monotone variation in boundary stress signals with respect to all experimental controls: viscoelastic moduli of the layer, layer thickness, and driving frequency. This structure of peaks and valleys in boundary values of shear and normal stress indicates redundant mechanisms for stress communication (by tuning to the peaks) and stress filtering (by tuning to the valleys). In this paper, we first restrict to a single-mode non-linear Maxwell model for the viscoelastic layer, where analysis renders a transparent explanation of the phenomena. We then consider a Giesekus constitutive model of the layer, where analysis is supplanted by numerical simulations of coupled non-linear partial differential equations. Parametric studies of wall stress values from standing waves confirm persistence of the Maxwell model phenomena. The analysis and simulations rely on and extend our recent studies of shear waves in a micro parallel plate rheometer [S.M. Mitran, M.G. Forest, L. Yao, B. Lindley, D. Hill, Extenstions of the Ferry shear wave model for active linear and nonlinear microrheology, J. Non-Newtonian Fluid Mech. 154 (2008) 120–135; D.B. Hill, B. Lindley, M.G. Forest, S.M. Mitran, R. Superfine, Experimental and modeling protocols from a micro-parallel plate rheometer, UNC Preprint, 2008].     

On secondary loops in LAOS via self-intersection of Lissajous–Bowditch curves

When the shear stress measured in large amplitude oscillatory shear (LAOS) deformation is represented as a 2-D Lissajous–Bowditch curve, the corresponding trajectory can appear to self-intersect and form secondary loops. This self-intersection is a general consequence of a strongly nonlinear material response to the imposed oscillatory forcing and can be observed for various material systems and constitutive models. We derive the mathematical criteria for the formation of secondary loops, quantify the location of the apparent intersection, and furthermore suggest a qualitative physical understanding for the associated nonlinear material behavior. We show that when secondary loops appear in the viscous projection of the stress response (the 2-D plot of stress vs. strain rate), they are best interpreted by understanding the corresponding elastic response (the 2-D projection of stress vs. strain). The analysis shows clearly that sufficiently strong elastic nonlinearity is required to observe secondary loops on the conjugate viscous projection. Such a strong elastic nonlinearity physically corresponds to a nonlinear viscoelastic shear stress overshoot in which existing stress is unloaded more quickly than new deformation is accumulated. This general understanding of secondary loops in LAOS flows can be applied to various molecular configurations and microstructures such as polymer solutions, polymer melts, soft glassy materials, and other structured fluids.     

Non-linear viscoelastic response of magnetic fiber suspensions in oscillatory shear

This paper reports the first study on the large amplitude oscillatory shear flow for magnetic fiber suspensions subject to a magnetic field perpendicular to the flow. The suspensions used in our experiments consisted of cobalt microfibers of the average length of 37 μm and diameter of 4.9 μm, dispersed in a silicon oil. Rheological measurements have been carried out at imposed stress using a controlled stress magnetorheometer. The stress dependence of the shear moduli presented a staircase-like decrease with, at least, two viscoelastic quasi-plateaus corresponding to the onset of microscopic and macroscopic scale rearrangement of the suspension structure, respectively. The frequency behavior of the shear moduli followed a power-law trend at low frequencies and the storage modulus showed a high-frequency plateau, typical for Maxwell behavior. Our simple single relaxation time model fitted reasonably well the rheological data. To explain a relatively high viscous response of the fiber suspension, we supposed a coexistence of percolating and pivoting aggregates. Our simulations revealed that the former became unstable beyond some critical stress and broke in their middle part. At high stresses, the free aggregates were progressively destroyed by shear forces that contributed to a drastic decrease of the moduli. We have also measured and predicted the output strain waveforms and stress–strain hysteresis loops. With the growing stress, the shape of the stress–strain loops changed progressively from near-ellipsoidal one to the rounded-end rectangular one due to a progressive transition from a linear viscoelastic to a viscoplastic Bingham-like behavior.