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.     

An intriguing empirical rule for computing the first normal stress difference from steady shear viscosity data for concentrated polymer solutions and melts

The Cox–Merz rule and Laun’s rule are two empirical relations that allow the estimation of steady shear viscosity and first normal stress difference, respectively, using small amplitude oscillatory shear measurements. The validity of the Cox–Merz rule and Laun’s rule imply an agreement between the linear viscoelastic response measured in small amplitude oscillatory shear and the nonlinear response measured in steady shear flow measurements. We show that by using a lesser-known relationship also proposed by Cox and Merz, in conjunction with Laun’s rule, a relationship between the rate-dependent steady shear viscosity and the first normal stress difference can be deduced. The new empirical relation enables a priori estimation of the first normal stress difference using only the steady flow curve (i.e., viscosity vs shear rate data). Comparison of the estimated first normal stress difference with the measured values for six different polymer solutions and melts show that the empirical rule provides values that are in reasonable agreement with measurements over a wide range of shear rates, thus deepening the intriguing connection between linear and nonlinear viscoelastic response of entangled polymeric materials.     

Rheological behaviour of spray-dried egg yolk/xanthan gum aqueous dispersions

This paper provides information on the microstructure of, and reports particle size distributions and rheological results for, aqueous dispersions of spray-dried egg yolk and xanthan gum prepared on a laboratory scale using two types of homogenisers. Laser light scattering results demonstrated that higher energy input during homogenisation yielded a dispersion with a lower average particle size and a wider polydispersity, slightly influencing the linear dynamic viscoelastic functions due to the low concentration of egg yolk particles. These dispersions exhibited weak gel properties at the composition studied. The mechanical spectrum and the corresponding relaxation spectrum were dominated by the xanthan/gum-water matrix which controls the structure of the continuous phase. This fact explained the lack of any wall depletion effects. Several controlled-rate and controlled-stress rotational rheometers and a capillary rheometer were used to obtain information on flow properties. The shear rate dependence of steady state viscosity was determined through twelve decades, and was fitted using the Carreau equation. The kinetics of structural recovery after steady-state shear was studied by start-up at the inception of shear and flow interrupted experiments under controlled shear history. The results were analysed in terms of the ratio of a time-dependent amount of overshoot to the amount of overshoot of the original sample, using the addition of two first order equations. Additionally, combined steady state flow properties at fixed shear stress/oscillatory shear experiments were also used. The increase of the storage modulus with time, checking a linear viscoelastic response, tracked the structural recovery after steady shear. Laser light scattering of sheared samples helped gain a better understanding of the role of egg yolk particles on the rheology of these dispersions. Received: 6 February 2000 Accepted: 5 September 2000     

Thermo-rheological behavior of model protein–polysaccharide mixtures

The use of mixtures of pea protein isolate (PPI) and κ-carrageenan (κ-C) is increasing rapidly with the aim of increasing the stability and viscosity of food products. Recent works have studied their textural and thermal properties but few have studied the influence of the temperature and concentration on the rheological behavior of model systems. In the present work, we study the thermo-rheological properties in the linear and non-linear viscoelastic regimes, in both shear and extensional flows, of mixtures of PPI and κ-C with the aim of obtaining a model for the temperature-induced gelation of complex mixtures of globular vegetable proteins under linear and non-linear deformations. We analyzed the influence of temperature and protein-to-polysaccharide ratios and showed that there are strong changes in the mechanical properties. In shear flows, small-amplitude oscillatory shear was used to study the linear regime and large-amplitude oscillatory shear was used for the non-linear regime. In extensional flows, studies were carried out via the analysis of the dynamics of capillary thinning and breakup process in a filament-thinning rheometer.     

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.     

On the pressure correction of capillary melt rheology data

In this paper, the importance of a pressure correction of viscosity data obtained in capillary melt rheology is demonstrated. A linear polycarbonate has been chosen as a highly pressure-sensitive material for which data obtained by rotational rheometry does not overlap with capillary data. This apparent problem with the Cox–Merz relation is attributed to the existence of a mean pressure inside the capillary which is significantly different from atmospheric conditions. Different methods to determine the pressure coefficient of polycarbonate have been evaluated based on experiments performed with a capillary rheometer equipped with a pressure chamber. It is demonstrated that the pressure coefficient obtained at constant shear stress and the pressure coefficient obtained by the superposition method represent accurate pressure coefficient values. Two approaches are proposed to correct the original capillary data. In the direct methodology, the pressure coefficient is used to rescale the mean pressure inside the capillary to atmospheric conditions. The indirect approach consists of first constructing a mastercurve at a certain reference pressure using capillary data obtained with a pressure chamber. The resulting mastercurve can then be rescaled to atmospheric conditions. It is shown that both methods lead to viscosity curves on which both rotational and capillary data overlap, hence confirming the Cox–Merz relationship for polycarbonate. The indirect method is proven to be advantageous since it opens the possibility to significantly extend the shear rate window in which viscosities can be measured.     

Orthogonal superposition of small and large amplitude oscillations upon steady shear flow of polymer fluids

The orthogonal superposition of small and large amplitude oscillations upon steady shear flow of elastic fluids has been considered. Theoretical results, obtained by numerical methods, are based on the Leonov viscoelastic constitutive equation. Steady-state components, amplitudes and phase angles of the oscillatory components of the shear stress, the first and second normal stress differences as functions of shear rate, deformation amplitude and frequency have been calculated. These oscillatory components include the first and third harmonic of the shear stresses and the second harmonic of the normal stresses. In the case of small amplitude superposition, the effect of the steady shear flow upon the frequency-dependent storage modulus and dynamic viscosity has been determined and compared with experimental data available in literature for polymeric solutions. The predicted results have been found to be in fair agreement with the experimental data at low shear rates and only in qualitative agreement at high shear rates and low frequencies. A comparison of the present theoretical results has also been made with the predictions of other theories.In the case of large amplitude superposition, the effect of oscillations upon the steady shear flow characteristics has been determined, indicating that the orthogonal superposition has less influence on the steady state shear stresses and the first difference of normal stresses than the parallel superposition. However, in the orthogonal superposition a more pronounced influence has been observed for the second difference of normal stresses.     

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.     

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.     

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.     

Rheological properties of long glass fiber filled polypropylene

Long glass fiber-filled polypropylene (PP) composites are produced by pultrusion, and the extrudate is cut at different lengths producing composites containing long fibers of controlled length. The rheological properties of such composites in the molten state have been studied using different rheometers. A capillary rheometer has been constructed and mounted on a molding-injection machine. The shear viscosity of filled PP determined from the capillary rheometer, after corrections for entrance effects, was found to be very close to that of unfilled PP. However, large excess pressure losses at the capillary entrance were observed and these data have been used to obtain an apparent elongational viscosity. The apparent elongational viscosity was shown to be considerably larger than the shear viscosity for PP and filled PP, and it increased markedly with fiber length and fiber content. Rotational rheometers with a parallel-plate geometry were used to investigate the viscoelastic properties of these composites and their behavior was found to be non-linear, exhibiting a yield stress. A model is proposed to describe the shear viscosity from a solid-like behavior at low stresses to fluid-like behavior at high shear stresses taking into account fiber content and orientation. A modified model, proposed for elongational flow, describes relatively well the apparent elongational data.     

Stress relaxation after oscillatory shear in a thermotropic liquid crystalline polymer

We report investigations of the rheology of a thermotropic liquid crystalline polymer subjected to oscillatory shear. In particular, we have studied its stress relaxation behaviour and found it to be well-represented by the linear viscoelastic model. In our studies, we observed a strong dependence of the relaxation rate on the position in the strain cycle at which the shear is stopped; a result which we show to be a general phenomenon of viscoelastic fluids and not unique to liquid crystalline polymers.     

Pressure-dependent viscosity and free volume of atactic and syndiotactic polystyrene

The effect of pressure on viscosity is an important but often overlooked aspect of the flow properties of polymeric materials. In this work, two polymers (an atactic and a syndiotactic Polystyrene) were characterized to determine the effect of pressure on viscosity. In particular, a device was adopted to increase the exit pressure of a standard capillary rheometer, thus obtaining data of viscosity under high pressure and high shear rates. The Simha-Somcynsky equation of state was applied to the pressure–volume–temperature experimental data of both materials to obtain the dependence of free volume on temperature and pressure. The Doolittle equation was eventually employed to verify the dependence of viscosity on free volume. It was found that, for both materials, a linear relationship holds between the logarithm of zero-shear-rate viscosity (at several temperatures and pressures) and the inverse of free volume.     

Oscillatory squeezing flow of a biological material

Large-amplitude oscillatory squeezing flow data are reported for a complex biological material, which is highly shear-thinning in oscillatory shear flow. This soft tissue has a linear viscoelastic limit at a strain of approximately 0.2%. The oscillatory squeezing flow data at large strain are analyzed using two constitutive models: a bi-viscosity Newtonian model, and a non-linear Maxwell model. It is found that although both models may have the same response in shape, the later matches with our non-linear experimental data better. It is also concluded that the non-linear response of the material in large amplitude oscillatory flow is mainly due to the shear thinning of the material. Received: 9 February 2000/Accepted: 22 February 2000     

Experimental and conceptual problems in the rheological characterization of wheat flour doughs

Wheat flour dough is an industrially important material and a better understanding of its rheological behavior could have long ranging impact on the agricultural and the food processing industries. However, rheological characterization of dough is proving to be difficult due to a range of testing issues and anomalies in flow behavior. In a cone-and-plate rheometer wheat flour doughs “roll-out” of the gap before steady state viscosities can be established, as discussed by Bloksma and Nieman (1975). However, the mirror image of the transient viscosity-time plot obtained using a cone-and-plate viscometer has been used to obtain an estimate of steady shear viscosity behavior (Gleissle, 1975). To check this transient methodology for doughs, a second method, in addition to cone-and-plate transient flow, for determination of the shear viscosity, was needed. For this, capillary extrusion was chosen. Both a piston-driven and pressure driven capillary rheometer were employed. End corrections were determined to provide information on both the shear viscosity and, following Binding (1988), the extensional viscosity of the doughs. There are few data available on end corrections for doughs, though published data by Kieffer indicate that the corrections are unexpectedly very high. In this present work it was found that the end correction experiments were very difficult and imprecise in part due to the time-dependent nature of the doughs and difficulties in preparing replicate batches required to compare dies of differing L/R values. Further it was unexpectedly found that the samples, though prepared by normal mixing procedures to the “optimum” level, were so heterogeneous that large fluctuations in the pressure at constant output rate (in the piston-driven rheometer) and in output rate at constant pressure (in the pressure-driven instrument) were observed. These fluctuations could be eliminated by overmixing of the doughs, but overmixed doughs are of little practical interest. Although the problems encountered in this work were significant, it was encouraging that even these preliminary studies indicate that rheological measurements are effective in differentiating between spring and winter wheats. Defining a constitutive model for dough rheology still remains a major challenge, as results from one type of testing do not corroborate the findings from a different type of testing. Received: 19 May 1998 Accepted: 27 July 1998     

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.     

Evaluating rheological models for human blood using steady state,transient, and oscillatory shear predictions

The rheological characterization of human blood, through modeling and analysis of steady state, transient, and oscillatory shear flows, has made tremendous progress recently. Due to the aggregation of red blood cells at low shear rates, many recent models for blood rheology include a structural, thixotropic component with one of the most recent attempts unifying this approach with a viscoelastic formulation. We will show how these models, along with proposed modifications to another recent structural, kinetic thixotropy model, can improve modeling predictions. Results are compared to the Maxwell-like Bautista-Manero-Puig model, the Oldroyd-8 inspired viscoelastic Anand-Kwack-Masud model, a viscoelastic-thixotropic model from Blackwell and Ewoldt, and the Herschel-Bulkley model. We explore the weaknesses of the legacy blood models and then demonstrate the efficacy of the newly improved models for modeling human blood steady state and transient shear rheology to predict oscillatory shear flow.     

Comparison of stress-controlled and strain-controlled rheometers for large amplitude oscillatory shear

In linear viscoelastic region, it is well known that dynamic modulus and dynamic compliance can be converted to each other. However, it is questionable whether there exists an interconversion between large amplitude oscillatory shear (LAOS) data measured from different types of rheometers—stress-controlled and strain-controlled rheometers. Hence, we tried to prove the existence by use of polyethylene oxide (PEO) aqueous solutions with well-developed entanglements. From this experiment, we can conclude that a stress-controlled rheometer can simulate LAOS behavior measured from a strain-controlled rheometer under the conditions where inertia effect is not significant. Furthermore, it is investigated whether the LAOS data of the stress-controlled rheometer obey stress–frequency superposition as the strain–frequency superposition found by Cho et al. (J Rheol 54:27–63, 2010) from LAOS data measured by the strain-controlled rheometer. This scaling relation shows that the dimensionless stress amplitude is a function of zeta which is the product of the stress amplitude and linear viscoelastic function J′(ω). The plot shows that all of the data are superposed in a single curve without regard to frequency, molecular weight, and concentration of PEO aqueous solutions.     

A new look at extensional rheology of low-density polyethylene

The nonlinear rheology of three selected commercial low-density polyethylenes (LDPE) is measured in uniaxial extensional flow. The measurements are performed using three different devices including an extensional viscosity fixture (EVF), a homemade filament stretching rheometer (DTU-FSR) and a commercial filament stretching rheometer (VADER-1000). We show that the measurements from the EVF are limited by a maximum Hencky strain of 4, while the two filament stretching rheometers are able to probe the nonlinear behavior at larger Hencky strain values where the steady state is reached. With the capability of the filament stretching rheometers, we show that LDPEs with quite different linear viscoelastic properties can have very similar steady extensional viscosity. This points to the potential for independently controlling shear and extensional rheology in certain rate ranges.