Search

We synthesized model polyethylene (PE) samples with nearly monodispersed molecular weight distribution and different fractions of short-chain branches to investigate the influences of the branch structure on the rheological properties. The activation energy of the flow process showed linear relations with the weight fractions of comonomer and the short-chain branch. These results suggest that the friction of the chain motion is dominated by the amount of the short-chain branch. The zero-shear viscosity showed a power law to Mw with an exponent of about 3.3 for all linear and branched PEs. The branched PEs showed large entanglement molecular weights compared with the linear PEs because the introduction of the short-chain branches results in the increase of the average molecular weight per backbone unit. Moreover, the introduction of the short-chain branches has no influence on the strain-hardening under an extensional elongation process, indicating that the short-chain branches only affect the friction.

In this work, we numerically investigate the deformation and breakup of a droplet flowing along the centerline of a microfluidic non-orthogonal intersection junction. The relevant boundary data of the velocity field is numerically computed by solving the depth-averaged Brinkman equation via a self-consistent integral equation using the boundary element method. The effect of the capillary number, droplet size, intersection angle, and ratio of outlet channel width to inlet channel width on maximum droplet deformation are studied. We study droplet deformation for the capillary numbers in the range of 0.08-0.3 and find that the maximum droplet deformation scales with the capillary number with power law with an exponent 1.10. We also investigate the effect of droplet size and intersection angle on the maximum droplet deformation and observe that the droplet deformation is proportional to droplet volume and square root of intersection angle, respectively. In continue, we study the droplet breakup phenomenon in an orthogonal intersection junction. By increasing the capillary number, the deformation of a droplet traveling in the cross-junction region becomes larger, until the droplet shape is no longer observed and droplet breakup takes place at a critical value of capillary number. We present a phase diagram for droplet breakup as a function of undeformed droplet radius.

There is an enduring interest in concentrated particulate filled slurries in many industries for a wide variety of products and processes. In previous research, a collection of experimental procedures provided velocity profiles, pressure drops and power law constants for slurries flowing in a tube. A model previously developed for the power law, based on mono-disperse fillers and on percolation theory concepts, is reviewed and extended to bi-modal slurries. The model for the power law constant, n, is shown to be only a function of the volume of non-dissipating clusters in the slurry. Experimental results are used to compare the effects of mono-modal and bi-modal fillers on the viscosity and n in the composite flow of filled Newtonian carrier fluids. The data and analysis demonstrate that n can be related to the ratio of energy dissipated by the slurry divided by the dissipation of a Newtonian fluid flowing under similar conditions. The power law parameters were developed using Bingham model constants employed to approximate dissipation in tube flow. The power law constant was found to be smaller for the bi-modal systems than for mono-modal slurries at the same total filler concentration.

Rheological and microstructural properties of welan gum aqueous solutions were studied as a function of polymer concentration in the 0.2–0.6% (m/m) range at fixed temperature 20 °C. Welan gum is an exopolysaccharide produced by Sphingomonas sp. All the systems exhibited a shear thinning and weak gel–like behavior whose parameters were well adjusted to a power law with the concentration. Furthermore, time-concentration superposition methods were carried out to obtain two master curves, one from the flow curves and the other from the mechanical spectra, which made it possible to extend the accessible experimental range. The non-linear viscoelastic properties were also studied by means of parallel superposition tests. The superimposed shear stress induced a change from weak gel to entangled solution behavior. This latter flow behavior is characterized by a frequency crossover that has been shown to have a linear dependence on the superimposed steady-state shear rate. Finally, Cryo-SEM images revealed a network with numerous junction zones between polymer chains.

A novel approach to the analysis of the electrorheological effect is proposed, based on the expansion of dimensionless relative shear stress as function of electric field strength in the power series $$ {\tau}_{\mathrm{rel}}=\frac{\tau_E}{\tau }=1+\frac{\alpha }{\tau }E+\frac{\beta }{\tau }{E}^n $$ . The application of this approach to investigation of the electrorheological effect in suspensions of isotropic and needle-like CeO2 nanoparticles in polydimethylsiloxane has revealed that the polynomial coefficients can be judged as a measure of the efficiency of transformation of electrical energy into mechanical energy. The values of α and β coefficients depend on the shape and concentration of filler particles, as well as on the shear rate. The value and the sign of these coefficients determine both the magnitude of the electrorheological effect and the type of dependence of the shear stress (linear or power law) on the strength of the electric field. It has been shown that the values of α and β coefficients for the electrorheological fluids with needle-like particles are greater than for fluids with isotropic particles (at the same concentration of suspensions), which is associated with the different polarization of particles in the applied electric field.

There is a strong interest today in concentrated particulate-filled dispersion and slurries in both polymeric and Newtonian fluids. This paper reviews and extends theoretical approaches using percolation theory concepts to characterize the rheological behavior of fluids filled with particulate solids. First, a previously proposed limiting, zero shear viscosity model based on percolation theory concepts is reviewed. This model has been primarily tested with rigid fillers in a Newtonian carrier and polymeric fluids. Second, all Newtonian fluid-based slurries that have a high concentration of filler become pseudoplastic, shear-thinning slurries at some threshold shear rate. A new theory is reviewed and new data are evaluated that correlate the power law constant, n, to cluster formation of the fillers suspended in the fluids in shear flow. Slurry systems reported here cover a size range from 58 nm to 200 μm. Third, this cluster percolation-based rheological analysis is then extended to a newly proposed model for the calculation of the ratio of infinite shear, η∞, to the zero shear viscosity, η0. Using literature data, it is demonstrated that measurements of the viscosity ratio, η∞/η0, correlate with the power law through the use of an energy dissipation-based model for Bingham rheological fluids.

Rheological measurements were performed to examine the yielding behavior of capillary suspensions prepared by mixing cocoa powder as dispersed phase, vegetable oil as the continuous primary fluid, and water as the secondary fluid. Here, we investigated the yielding behavior of solid-fluid-fluid systems with varying particle volume fraction, ϕ, spanning the regime from a low volume fraction (ϕ = 0.25) to a highly filled regime (ϕ = 0.65) using dynamic oscillatory measurements. While for ϕ ≤ 0.4 with a fixed water volume fraction (ϕ w ) of 0.06 as the secondary fluid, capillary suspensions exhibited a single yield point due to rupturing of aqueous capillary bridges between the particles, while capillary suspensions with ϕ ≥ 0.45 showed a two-step yielding behavior. On plotting elastic stress (G′ γ) as a function of applied strain (γ), two distinct peaks, indicating two yield stresses, were observed. Both the yield stresses and storage modulus at low strains were found to increase with ϕ following a power law dependence. With increasing ϕ w (0 – 0.08) at a fixed ϕ = 0.65, the system shifted to a frustrated, jammed state with particles strongly held together shown by rapidly increasing first and second yield stresses. In particular, the first yield stress was found to increase with ϕ w following a power law dependence, while the second yield stress was found to increase exponentially with ϕ w . Transient steady shear tests were also performed. The single stress overshoot for ϕ ≤ 0.4 with ϕ w = 0.06 reflected one-step yielding behavior. In contrast, for high ϕ (≥ 0.45) values with ϕ w = 0.06, two stress overshoots were observed in agreement with the two-step yielding behavior shown in the dynamic oscillatory measurements. Experiments on the effect of resting time on microstructure recovery demonstrated that aggregates could reform after resting under quiescent conditions.

As one kind of the most important non-Newtonian fluids, Herschel-Bulkley fluids have been applied in practical engineering widely. In order to improve the stability and accuracy in the simulation of Herschel-Bulkley fluids with lattice Boltzmann method (LBM), a modified lattice Boltzmann method was proposed. With Poiseuille flow as the example, the shear-thinning fluids and the shear-thickening fluids were used, respectively, to introduce the method in detail. The comparison between the velocity distributions and the analytical solutions demonstrated the feasibility of the modified method. Also the effect of the power law index, lattice nodes, and the region of the shear rate on the relative error was discussed. Then, the method was applied into the simulation of the cement paste flow in the 3D printing extruder. The streamline figure was obtained and then conducted the flow simulation by the modified method and multiple-relaxation-time lattice Boltzmann method (MRT-LBM), respectively; the comparison further proved the modified method was feasible for Herschel-Bulkley fluids.

We report on the interplay between creep and residual stresses in a carbopol microgel. When a constant shear stress $\sigma$ is applied below the yield stress $\sigma_\text{y}$, the strain is shown to increase as a power law of time, $\gamma(t)=\gamma_0 + (t/\tau)^\alpha$, with an exponent $\alpha=0.39\pm 0.04$ that is strongly reminiscent of Andrade creep in hard solids. For applied shear stresses lower than some typical value $\sigma_\text{c}\simeq 0.2 \sigma_\text{y}$, the microgel experiences a more complex, anomalous creep behaviour, characterized by an initial decrease of the strain, that we attribute to the existence of residual stresses of the order of $\sigma_\text{c}$ that persist after a rest time under a zero shear rate following preshear. The influence of gel concentration on creep and residual stresses are investigated as well as possible aging effects. We discuss our results in light of previous works on colloidal glasses and other soft glassy systems., Comment: 17 pages, 14 figures, Rheologica Acta (2016)

We propose a micromechanical model for the behavior of dilute magnetorheological fluids under unidirectional slow-compression, constant-volume squeeze flow mode. In the linear magnetization regime, the model predicts a power law scaling of the normal stress with the particle volume fraction and magnetic field strength squared at low fields. The predictions are satisfactorily compared with experimental measurements for different particle loadings, sample volume, surface roughness, and initial gap distance.

The stick-slip motion of magnetorheological (MR) fluids under a simple linear shear mode has been experimentally reported. The critical shear strain for the onset of the stick-slip motion decreases logarithmically with the magnetic flux intensity applied. The rate of energy accumulation and releasing during stick-slip motion changed in a power law of the viscosity of the continuous phase in a range of 0.01~1000 mPa S. Different carrier fluids led to different stick-slip characteristics. The stick-slip motion can be fully suppressed with certain carrier fluids. When the MR fluids are in a shear-thickening state, the critical shear strain of the onset of the stick-slip motion is larger than a normal state. But, the relative spike amplitude of the stick-slip remains similar.

In this study, an inverse method based on the Levenberg–Marquardt algorithm was evaluated in a numerical experiment to determine the large strain viscoelastic properties from the bubble inflation test. The properties were determined by iteratively matching the calculated bubble pressure–piston displacement data from finite element simulations to a single set of bubble pressure–piston displacement data. The strain-dependent behaviour was characterised by a two-parameter Mooney–Rivlin hyperelastic model, while the time-dependent behaviour was characterised by a three-parameter power law equation. Different initial guesses were used to evaluate the inverse method, and transformation functions were applied to constrain the intermediate guesses to be within bounds. It was found that estimates of the viscoelastic properties could be obtained reasonably using only one set of bubble pressure–piston displacement data. Estimates of the properties were likely affected by the limited time duration of the test, as the behaviour at shorter and particularly larger time scales was less accurately predicted.

This work is concerned with an experimental and theoretical study on compression properties of magnetorheological fluids under the nonuniform field. Experimental tests of unidirectional monotonic compression were firstly carried out under constant area operation using a commercial plate–plate magneto-rheometer where the magnetic field radial distribution was nonuniform. Normal forces increased with decreasing of the gap distance, and two regions were found through the normal force versus gap distance curves: elastic deformation and plastic flow. High normal forces could be obtained in the case of high magnetic field, high compression velocity, low initial gap distance, high volume fraction, and high medium viscosity. In the plastic flow region, the normal force with the gap distance could be fitted with a power law relation $F_{\textrm {N}} \propto h^n$ , and the index n was around well in the range (−3, −2). Taking nonuniform magnetic field into account, the theoretical modeling in the plastic flow was then developed to calculate the normal force under compression based on the continuum media theory. Compared to the uniform field, there existed a magnetic field gradient-induced normal force under nonuniform field. Considering the sealing and squeeze strengthening effect, the gap distance-dependent shear yield stress was proposed, and a good correspondence between the theoretical and experimental results was obtained.

ZrO2 nanoparticle dispersions containing associative polymers exhibit two relaxation modes: Maxwellian behavior at high frequency imparted by the associating polymers and a power law spectrum at low frequency generated by the particle dynamics. The timescales and volume fraction dependence of the dispersions reflect weak attractions between particles with adsorbed polymer layers dispersed in a percolated network of associative polymer. The Baxter stickiness parameter extracted from the high frequency viscosity data indicates strong attractions, whereas the high frequency modulus reveals three sources of elasticity: micelle–micelle associations in the solution, rigidity of the particles and adsorbed layer, and adsorbed layer–adsorbed layer interactions. The sol-gel transition of the dispersions occurs around 12–14% particle loading. Comparison with latex dispersions suggests a slower relaxation mode and greater rigidity with the ZrO2 particles.

Flows involving different types of chain branches have been modelled as functions of the uniaxial elongation using the recently generated constitutive model and molecular dynamics for linear viscoelasticity of polymers. Previously control theory was applied to model the relationship between the relaxation modulus, dynamic and shear viscosity, transient flow effects, power law and Cox–Merz rule related to the molecular weight distribution (MWD) by melt calibration. Temperature dependences and dimensions of statistical chain tubes were also modelled. The present study investigated the elongational viscosity. We introduced earlier the rheologically effective distribution (RED), which relates very accurately and linearly to the viscoelastic properties. The newly introduced effective strain-hardening distribution (REDH) is related to long-chain branching. This REDH is converted to real long-chain branching distribution by melt calibration and a simple relation formula. The presented procedure is very effective at characterizing long-chain branches, and also provides information on their structure and distribution. Accurate simulations of the elongational viscosities of low-density polyethylene, linear low-density polyethylene and polypropylene, and new types of MWDs are presented. Models are presented for strain-hardening that includes the monotonic increase and overshoot effects. Since the correct behaviour at large Hencky strains is still unclear, these theoretical models may aid further research and measurements.

Lubricated squeezing flow experiments on wheat flour dough have, until now, mostly been performed in constant plate speed mode (CPS), i.e. at a permanently increasing extension rate. We have compared the results obtained under the CPS and constant extension rate (CER) modes using one of the very few commercial rheometers that allow operation in the CER mode. In both cases, and at any constant biaxial strain, a power law could be fitted to the stress versus extension rate data, the "consistency index" (K) increasing continuously with the strain and the "flow behaviour index" (n) being constant only up to a low strain value (a parts per thousand 0.25) and then decreasing. When compared to the CER mode, the CPS mode produced higher K and n values. For wheat flour doughs, an increase in K with extension may be associated to a strain-hardening phenomenon but the roles of viscoelasticity and lubricant thinning are discussed.

We present an experimental investigation of the solid–fluid transition in a yield stress shear thinning physical gel (Carbopol® 940) under shear. Upon a gradual increase of the external forcing, we observe three distinct deformation regimes: an elastic solid-like regime (characterized by a linear stress–strain dependence), a solid–fluid phase coexistence regime (characterized by a competition between destruction and reformation of the gel), and a purely viscous regime (characterized by a power law stress-rate of strain dependence). The competition between destruction and reformation of the gel is investigated via both systematic measurements of the dynamic elastic moduli (as a function of stress, the amplitude, and temperature) and unsteady flow ramps. The transition from solid behavior to fluid behavior displays a clear hysteresis upon increasing and decreasing values of the external forcing. We find that the deformation power corresponding to the hysteresis region scales linearly with the rate at which the material is being forced (the degree of flow unsteadiness). In the asymptotic limit of small forcing rates, our results agree well with previous steady state investigations of the yielding transition. Based on these experimental findings, we suggest an analogy between the solid–fluid transition and a first-order phase transition, e.g., the magnetization of a ferro-magnet where irreversibility and hysteresis emerge as a consequence of a phase coexistence regime. In order to get further insight into the solid–fluid transition, our experimental findings are complemented by a simple kinetic model that qualitatively describes the structural hysteresis observed in our rheological experiments. The model is fairly well validated against oscillatory flow data by a partial reconstruction of the Pipkin space of the material’s response and its nonlinear spectral behavior.

We propose to analyse power law shear stress relaxation modulus observed at the sol–gel transition (SGT) in many gelling systems in terms of fractional calculus. We show that the critical gel (gel at SGT) can be associated to a single fractional element and the gel in the post-SGT state to a fractional Kelvin–Voigt model. In this case, it is possible to give a physical interpretation to the fractional derivative order. It is associated to the power law exponent of the shear modulus related to the fractal dimension of the critical gel. A preliminary experimental application to silica alkoxide-based systems is given.

Experimentally, it is well-known that the Rouse model gives a superb description of the concentration dependence of terminal relaxation time, terminal modulus, zero shear-rate viscosity, and diffusion coefficient of semidilute unentangled polyelectrolyte solutions. However, such solutions exhibit shear thinning of the apparent viscosity when the shear rate exceeds the reciprocal of the terminal relaxation time, which is not immediately anticipated by the Rouse model. We present a simple calculation based on the Rouse model for the dependence of the apparent viscosity η on shear rate $\dot{\gamma}$ in steady shear. The derived power law $\eta \sim \dot{\gamma}^{-1/2}$ applies to nearly mono- disperse unentangled polymer melts and polymer solutions that have a high enough concentration so that chains overlap, but have low enough concentration that they are not entangled. We find that the predicted power law agrees nicely with data on unentangled polymer melts and semidilute unentangled solutions of polyelectrolytes. The exponent 1/2 means the empirical Cox-Merz rule applies to Rouse chains. This potentially has far-reaching consequences for entangled polymer melts, for which motion of a Rouse chain confined to a tube describes dynamics.

The drag force on a sphere moving through an aqueous foam is measured as the foam ages. After an initial period, the steady-state drag decreases with age T as T−0.54±0.14. As the mean bubble size R in the foam coarsens as T0.5, this implies that the drag force scales as \(1/\overline{R} .\) The transient buildup of the force when the sphere starts to move is described by a single exponential approach to the steady-state drag while its relaxation when the motion stops is described by the sum of three exponential relaxations. This is as for fresh foam, but the coefficients and time constants vary systematically with age. For the most part, these quantities also show a power law scaling with T. The age dependence of the quantities determined in this study is discussed in terms of the mean bubble size.

The Landau-de Gennes tensor order parameter equations of nemato-dynamics are formulated, solved and used to find the impact of textural transformation on stress growth in thermotropic liquid crystalline polymers subjected to shear start-up flow. The simulated textural transformations include nucleation and annihilation of twist inversion walls. Coarsening processes include wall-wall annihilation, wall pinching and wall-bounding surface reactions. In the absence of defect-related effects, the stress growth is characterized by an early stress plateau, intermediate power law growth, and a late stage stress plateau. As the Deborah number (De) increases, flow-induced textural transformations affect the late stage and then the intermediate stress growth stage. Defects are found to be stress sinks, and so removal of defects increases stress. At lower Deborah numbers, few defects arise and coarsening rates are low, so the main texture effect in this regime is in the late stage plateau region, causing localized step increases. At Deborah numbers close to one, nucleation and coarsening rates increase, and textural effects appear closer and closer to the intermediate stress growth regime. As De increases further, coarsening by pinching processes overcomes nucleation, and all defects disappear in the intermediate stress growth regime, causing the stress growth to exhibit a smooth staircase shape. Strain and amplitude scaling is not observed. Simulated textural transformations show that smooth staircase stress growth is the result of defect annihilation processes. The non-monotonic stress growth is consistent with experimental observations. Simulated textures provide specific knowledge important to the eventual understanding of the rheologies of textured liquid crystal polymers.

The evolution of native egg yolk undergoing a thermal-induced sol-gel transition was studied by using temperature controlled small amplitude oscillatory shear measurements. The critical gel point was determined according to Winter’s criterion: 1) from the measurements of storage and loss moduli as a function of heating time at different frequencies, and 2) from the exponents of the power law mechanical spectra obtained after cure experiments performed up to a maximum temperature (60–90 °C) followed by a sudden decrease in temperature up to 20 °C. Differential Scanning Calorimetry (DSC) was performed in order to investigate thermal transitions in egg yolk. Microstructure of gels was evaluated by Transmission and Scanning Electron Microscopy. The results obtained were discussed in terms of the processes involved in protein gelation: change in the protein system, aggregation of partially denaturated protein molecules and association of aggregates. As a result, an elastic gel network was always obtained. The influence of frequency, heating rate, solids concentration and maximum temperature of processing, was analysed. Most of the transformations found during thermal processing were found to be basically irreversible, even at the sol state and gel point. However, some reversible phenomena were detected during constant temperature processing depending on the maximum temperature performed.

The fundamental assumption of the paper is that the extra stress tensor τ of an electrorheological fluid is an isotropic tensor valued function of the rate of strain tensor D and the vector n (which characterizes the orientation \({\mathbf{\hat{n}}}\) and length N of the fibers formed by application of an electric field). The resulting constitutive equation for τ is supplemented by the solution of the previously studied time evolution equation for n. Plastic behavior for the shear and normal stresses is predicted. Anticipating that the action of increasing shear rate \( \dot{\gamma } \) is i) to orient the fibers more and more in the direction of flow and ii) simultaneously to break up the fibers leads to the conclusion that for \( \dot{\gamma } \to \infty \) the same behavior is encountered as without an electric field. Using realistically possible approximation formulas for the dependence of \({\mathbf{\hat{n}}}\) and N on \( \dot{\gamma } \) leads to the Bingham behavior for \( \dot{\gamma } \to 0 \) and power law behavior for large shear rates.

For the most common types of viscometers the apparent flow curve of plastic fluids is studied. For torsional flow, where the shear rate is the natural variable, the apparent yield stress exceeds the true yield stress $$\tau _c $$ by more than 33%. If, on the other hand, $$\tau $$ is the natural variable (like in capillary flow, slit flow, and concentric cylinder flow) the yield stress is correctly predicted, but the behavior close to $$\tau _c $$ differs fundamentally. If the apparent shear rate $$\dot \gamma _a $$ goes to zero like $$\left( {\tau - \tau _c } \right)^{1/n} $$ (where the power law index n could be the power law index of a Herschel-Bulkley fluid), the true shear rate has to be proportional to $$ \left( {\tau - \tau _c } \right)^{\left[ {1/n} \right] - 1} $$ for $$\tau \to \tau _c $$ . For n=1 this implies a discontinuity of $$\dot \gamma $$ at $$\tau _c $$ ( $$\dot \gamma = 0$$ for $$\tau < \tau _c $$ ). For tangential annular flow between concentric cylinders the ratio of radii $$\left( \kappa \right) $$ enters. Using an exact relation between $$\dot \gamma $$ and $$\dot \gamma _a $$ reveals that no single (κ-dependent) expression for the apparent flow curve can exist, which would for plastic fluids cover the entire flow regime ( $$ \tau > \tau _c $$ ). Irrespective as to what viscometer is used the far field behavior of the apparent flow curve and the true flow curve will, in general, differ too, though only quantitatively.

Samples of commercial tomato paste, low fat mayonnaise and mustard about 6–8 mm thick were squeezed to 0.8 mm at various speeds between 5 mm min−1 and 25 mm min−1 between Teflon-coated parallel plates 127 mm in diameter using an Instron UTM Model 5542. All the log force vs log height relationships had a clearly identified linear region. This indicated that a dominant squeezing flow regime was achieved at about 3 mm height, and that the machine has the proper stiffness to perform the tests. The stress level at a pre-selected height in this region is a measure of consistency, sensitive enough to distinguish between products of different brands. The residual stress after relaxation for about 2 min was on order of 10–50% of the initial stress, an indication that all three foods have a considerable structural integrity. In all three products there was a considerable discrepancy between the observed rate effects and predictions based on a pseudoplastic (power law) model. It could be described by the empirical relation (Fv1 − FR)/(Fv2 − FR)=(V1/V2)m where Fv1 and Fv2 are the forces at the given displacement reached at speeds v1 and v2 respectively, FR is the residual force after relaxation (found to be practically rate independent), and m is a constant of the order of 0.15–0.33, independent of the compression velocities ratio but characteristic of the food and brand. The calculated elongational viscosity was not a unique function of biaxial strain rate. To a certain extent, this was probably due to imperfect lubrication. But it was also a manifestation of these products considerable structural integrity which cannot be accounted for by models developed for ideal liquids.

Recent computational analysis of entrance flows (Mitsoulis et al. 1998) suggests that the entrance pressure drop is insensitive to large changes in steady extensional viscosity-a result that directly contradicts a large body of experimental work in this area. A re-examination of entrance flows using numerical simulations is presented in this work which shows that entrance pressure drops do depend on the steady extensional viscosity, provided the extension rate in the entrance flow is large enough. Numerical simulations are presented using both the strain thinning and thickening versions of the Phan-Thien–Tanner (PTT) constitutive model. Several techniques for extracting extensional viscosity from entrance pressure are applied to the results of these simulations. The resulting predictions of extensional viscosity are compared to the steady extensional viscosity curves predicted by the PTT constitutive model used to generate the simulated pressure drop curves. The analytical techniques examined here are shown to provide reasonably accurate estimates of the steady extensional viscosity. This work also clearly demonstrates the advantage of using variable power-law coefficients for the rheological properties, used as inputs to the analyses, to capture the extensional behavior at deformation rates below the power law region more accurately.

Phase separation processes in mixtures of poly-α-methylstyrene-co-acrylonitrile (PαMSAN) and poly-methylmethacrylate (PMMA) with lower critical solution temperature (LCST) behavior have been studied, focusing on the manifestation of the interface in oscillatory shear measurements. By using blends of different composition, systems with a droplet-matrix morphology or a co-continuous structure are generated during the phase separation process. The feasibility of probing this morphology development by rheological measurements has been investigated. The development of a disperse droplet phase leads to an increase in the elasticity of the blend at low frequency, showing up as a shoulder in the plot of storage modulus versus frequency. Here, the droplet growth is unaffected by the shear amplitude up to strains of 0.2; therefore the resulting dynamic data are suitable for quantitative analysis. In contrast, for blends in which phase separation leads to a co-continuous structure, the storage modulus shows a power law behavior at low frequency and its value decreases as time proceeds. For the latter systems, effects of the dynamic measurement on the morphology development have been observed, even for strain amplitudes as low as 0.01. To probe the kinetics of morphology evolution in droplet-matrix systems, measurements of the time dependence of the dynamic moduli at fixed frequency should be performed (for a whole series of frequencies). Only from such measurements, curves of the frequency dependence of the moduli at a well defined residence time can be constructed. From fitting these curves to the emulsion model of Palierne, the droplet diameter distribution at that particular stage in the phase separation and growth process can be obtained. It is not appropriate to use a simplified version of the Palierne model containing only the average droplet size, because a morphology with too broad a size distribution is generated.

The empirical Cole-Cole distribution is an analytical three parameter model of the relaxation spectra that provides accurate fits to experimental dynamic viscosity data for many systems of commercial linear backbone flexible polymers. We demonstrate that for disparate systems of polyethylenes, the three Cole-Cole model parameters have simple power law relationships to moments of the molecular weight distribution enabling direct molecular interpretation of the mechanical relaxation spectrum. A simple relationship between the Cole-Cole distribution and the Cross model for the non-linear flow curve can be deduced utilizing the empirical Cox-Merz rule. Accurately representing the linear viscoelastic material functions with empirical analytical relaxation spectra containing relatively few fitting parameters that can be readily interpreted is a major advance in polymer characterization. The three Cole-Cole parameters effectively replace single point material characterizations such as melt flow index. Development of higher resolution polymer characterization methods is imperative with the advent of metallocene catalyst technology, which enables the molecular weight and backbone architecture to be carefully controlled.

We investigated the stress relaxation behavior of critical gels originating from six nearly monodisperse, highly entangled polybutadiene melts of different molecular weight from 18000 to 97 000 g/mole. The polymers were vulcanized by a hydrosilation reaction which takes place nearly exclusively at the pendant 1,2-vinyl sites distributed randomly along the polybutadiene chain. The BSW spectrum represents the relaxation of the initial uncrosslinked precursor. A characteristic parameter is the longest relaxation time of the precursor. Crosslinking increases this longest time even further. Surprisingly, the relaxation spectrum of the precursor is not altered much by the crosslinking except for an additional long time behavior. At the gel point (critical gel), this long time behavior is self-similar. It follows the typical power law as described by the Chambon-Winter gel equation, G(t) = St −n , in the terminal zone. The critical relaxation exponent was found to be close to n = 0.5 over a range of stoichiometric ratios and for all precursor molecular weights analyzed. A new scaling relationship was found between the gel stiffness, S, and the precursor molecular weight of the form: S ∼ M , where exponent z from the zero shear viscosity-molecular weight relationship, η0 ∼ M , is commonly found to be z = 3.3 – 3.6.

The phenomenon of wall slip during flow of rubber compounds through capillaries is investigated for a typical styrene-butadiene elastomer with carbon black. It was found that at low temperature (110°C) the dependencies of slip velocity V c on shear stress are described by the power law but, additionally, V c depends on radius of a channel. At high temperatures there is a critical shear stress below which sliding is absent. Sliding appears only at higher shear stresses where, again, V c depends on shear stress and the radius of a channel.

A four-parameter Maxwell model is formulated with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville definition. This model is used to determine the relaxation and retardation functions. The relaxation function was found in the time domain with the help of a power law series; a direct solution was used in the Laplace domain. The solution can be presented as a product of a power law term and the Mittag-Leffler function. The retardation function is determined via Laplace transformation and is solely a power law type. The investigation of the relaxation function shows that it is strongly monotonic. This explains why the model with fractional derivatives is consistent with thermodynamic principles. This type of rheological constitutive equation shows fluid behavior only in the case of a fractional derivative of the stress and a first order derivative of the strain. In all other cases the viscosity does not reach a stationary value. In a comparison with other relaxation functions like the exponential function or the Kohlrausch-Williams-Watts function, the investigated model has no terminal relaxation time. The time parameter of the fractional Maxwell model is determined by the intersection point of the short- and long-rime asymptotes of the relaxation function.

The evolution of linear viscoelasticity during the vulcanization of polyethylene is studied through the gel point. The material in the vicinity of the gel point is described by two scaling laws: one characterizes the viscoelasticity at the critical point and a second characterizes the evolution of viscoelasticity near the gel point. Time Resolved Mechanical Spectroscopy is used to observe both scaling phenomena. The material at the gel point displays power law relaxation over five decades of time with a power-law relaxation exponent equal to 0.32. This study conforms with previous findings that this exponent is composition-dependent. The longest relaxation time diverges in the vicinity of the gel point as λmax ∼ |pc - p| −1/κ, and we find κ = 0.2. This result conforms with previous reports that this exponent may be independent of composition. The Arrhenius flow activation energy for this material undergoes three-fold changes during crosslinking up to the gel point. A single-adjustable-parameter stretched-exponential-power law relaxation function is an inadequate representation of crosslinked materials over any significant range of extent of the reaction up to the gel point.

A radiation crosslinked model linear low-density polyethylene (LLDPE) exhibits power-law relaxation,G(t) =St−n at its gel point (GP). The relaxation exponent has a value of about 0.46. The relaxation behavior is dominated by power laws, not only directly at GP, but in a very broad vicinity of GP and in a frequency window, which narrows with distance from the gel point. The power law exponent decreases with increasing radiation dose (increasing extent of crosslinking). Independent measurements of the gel fraction and the molecular-weight distribution of the radiated samples' soluble fraction support the rheological observations.

Force-time relationships of a double-layered array of two power law liquids in squeezing flow at a constant displacement rate were generated with a computer. As in the case of a single layer, lubrication, or lack of it, has the strongest influence on the magnitude of the forces and the flow pattern. Transient flow regimes that were prominent in the behavior of Newtonian liquid arrays in lubricated squeezing flow were also found in the behavior of the power law liquids. Their prominence was influenced by the liquid's flow index and it was drastically magnified as the differences in the liquid's consistency increased.

The present work offers an analysis of the regularities of the flow of plastic disperse systems (PDS) as applied to the Bulkley-Herschel three-constant rheological equation. Theoretical data on the velocity profile distribution have been confirmed by experiments on the device, in which the flow structure was studied by the method of the frozen flow. By introducing expressions for the generalized Bingham and Reynolds numbers for non-linear viscoplastic bodies, formulas have been obtained for the calculation of the resistance coefficient of a hydrodynamically stabilized flow; the formulas for power law liquids and the Schwedoff-Bingham model follow from it as special cases. The results of the calculations are in good accord with experimental data for various PDS. A nomogram is given for the determination of the resistance coefficient in engineering calculations. Finally the present work considers the flow in the initial hydrodynamic zone of the tube. An analysis is given of the existing methods used for the solution of the problems and it is shown that the method used in this work for numerical calculation of simplified equations of continuity and motion provides sufficient accuracy. The results of numerical calculations are compared with the literature data, and these results are used for the analysis of the development of the flow of nonlinearly viscoplastic bodies in the initial zone. Relations are given for the determination of the length of the initial zone and of the increase of the resistance to flow due to the formation of the velocity profile. With account taken of this resistance, formulas have been derived for the calculation of pressure losses during the flow of PDS in tubes, including the initial zone.

An expression is developed that predicts the concentration dependence of the apparent viscosity of a concentrated pseudoplastic suspension. The result, which is an extension of the Frankel and Acrivos Newtonian suspension analysis, shows that the influence of particles concentration on the effective viscosity of pseudoplastic suspensions increases as the power law index of the suspension increases. Experimental data shows good agreement with the theoretical predictions.