Rheology of carbon black suspensions. II. Well dispersed system

Linear viscoelastic properties were investigated for the suspensions of carbon black (CB) particles having covalently-fused aggregate structures of an average diameter a=120 nm. The suspending medium, an alkyd resin (AR), had a high affinity toward the CB particles, and the aggregates of these particles were well dispersed to form no higher-order agglomerates. Consequently, the suspensions obeyed the time-temperature superposition and their Arrhenius-type activation energy was identical to that of the medium. From comparison of the zero shear viscosity η₀ for the CB suspensions and hard-sphere silica suspensions, an effective volume fraction φ_eff of the CB particles was found to be 2.7 times larger than the bare volume fraction of the particles. The CB particles exhibited a slow relaxation process, and the terminal relaxation time of this process was close to the Peclet time (Brownian diffusion time) evaluated from the aggregate size a and a high frequency viscosity. Furthermore, the terminal relaxation mode distribution of the CB suspensions was well scaled with an intensity factor H_t that was evaluated from the φ_eff in a way utilized for the hard-sphere silica suspensions. These results demonstrated that the slow relaxation in the CB suspensions was dominated by the Brownian diffusion of the CB aggregates, as similar to the situation for the silica suspensions.     

Rheology of carbon black suspensions. IV. Effect of suspending media on the sol�Cgel transition behavior

Linear viscoelastic properties of carbon black (CB) suspensions with various CB concentrations (c_CB) in two suspending media, mixtures of a rosin-modified phenol resin-type varnish (Varnish-1)/an alkyd resin-type varnish (Varnish-2) and Varnish-1/petroleum solvent (AF) were investigated. The former medium has higher affinity and the latter has poorer affinity toward CB particles than Varnishi-1, although these two media have almost the same viscoelastic properties. Both CB/(Varnish-1/Varnish-2 = 60/40) and CB/(Varnish-1/AF = 80/20) suspensions exhibited a sol?Cgel transition on an increase in c_CB, as similar to the behavior of CB/Varnish-1 suspensions. But, for the CB/(Varnish-1/Varnish-2 = 60/40) suspensions the critical gel concentration (c_gel) was lower and the critical relaxation exponent (n) was higher than that of CB/Varnish-1 suspensions. In contrast, for the CB/(Varnish-1/AF = 80/20) suspensions c_gel and n, respectively, were almost the same value as those of CB/Varnish-1 suspensions. These results strongly suggest that c_gel and n are attributable to the medium affinity, but not to the medium viscosity, and the CB fractal structure changes from plane-like to straight-line structure with increasing medium affinity.     

Flowing colloidal suspensions in non-Newtonian suspending fluids: Decoupling the composite birefringence

The rheology of dilute, colloidal suspensions in polymeric suspending fluids can be studied with simultaneous dichroism and birefringence measurements. The dichroism provides a direct measure of the particle dynamics, but the birefringence is a composite property with independent contributions from the suspended particles and the polymer molecules. For suspensions where the contribution from the particles is significant, the composite birefringence must be decoupled in order to analyze the dynamics of the polymeric suspending fluid. A method to perform the decoupling is derived and then demonstrated through transient shear flow experiments with dilute suspensions ofFeOOH particles in semi-dilute, xanthan gum suspending fluids. The birefringence of the xanthan gum suspending fluid is calculated from experimental measurements of the composite birefringence and the dichroism of the suspension. To gather information on particle/polymer interactions, the calculated birefringence is compared to the birefringence of xanthan gum solutions containing no suspended particles and the dirchoism is compared to that of a suspension in a Newtonian fluid.     

Rheology of carbon black suspensions. V. Heat-induced gelation and its reversibility

Linear viscoelastic properties of carbon black (CB) suspensions in a mixture of a rosin-modified phenol resin-type varnish (Varnish-1)/an alkyd resin-type varnish (Varnish-2), which exhibited a sol?Cgel transition on an increase in CB concentration, were investigated from 30°C to 80°C. The viscoelastic properties were reversible from 30°C to 60°C. In contrast, at temperatures above 60°C, the storage (G??) and loss (G??) moduli were irreversible and increased significantly with increasing temperature. This increase in the moduli is due to a change of the dispersion state to agglomerated state by heating. The agglomerated state was held, when the suspensions were lowered at 30°C. However, the G?? and G?? recovered to the original values upon shearing. This heat-induced gelation should be a universal feature for suspensions of weakly attractive particles. The temperature and shearing histories of the suspensions were discussed in relation to adsorption of polymeric component in the varnish on the CB particles.     

Rheology of a suspension of particles in viscoelastic fluids

In this paper we study the bulk stress of a suspension of rigid particles in viscoelastic fluids. We first apply the theoretical framework provided by Batchelor [J. Fluid Mech. 41 (1970) 545] to derive an analytical expression for the bulk stress of a suspension of rigid particles in a second-order fluid under the limit of dilute and creeping flow conditions. The application of the suspension balance model using this analytical expression leads to the prediction of the migration of particles towards the centerline of the channel in pressure-driven flows. This is in agreement with experimental observations. We next examine the effects of inertia (or flow Reynolds number) on the rheology of dilute suspensions in Oldroyd-B fluids by two-dimensional direct numerical simulations. Simulation results are verified by comparing them with the analytical expression in the creeping flow limit. It is seen that the particle contribution to the first normal stress difference is positive and increases with the elasticity of the fluid and the Reynolds number. The ratio of the first normal stress coefficient of the suspension and the suspending fluid decreases as the Reynolds number is increased. The effective viscosity of the suspension shows a shear-thinning behavior (in spite of a non-shear-thinning suspending fluid) which becomes more pronounced as the fluid elasticity increases.     

Rheology of SiO 2 /(acrylic polymer/epoxy) suspensions. III. Uniaxial elongational viscosity

Uniaxial elongational viscosity of SiO₂/(acrylic polymer/epoxy (AP/EP)) suspensions with various SiO₂ volume fractions (?) in a blend of acrylic polymer and epoxy was investigated at various temperatures (T). The matrix polymer ((AP/EP) blend) contained 70?vol.% of EP. At ?????35?vol.% at T????80°C, where the suspensions were in sol state, strain-hardening behavior was observed. This strain hardening of the suspensions is attributable to the elongational flow properties of (AP/EP) medium. At critical gel state (??=?35?vol.% and T?=?100°C) and in gel state (?????40?vol.%), the elongational viscosity exhibited the strain-softening behavior. These results strongly suggest that the strain softening results from the strain-induced disruption of the network structure of the SiO₂ particles therein.     

Non-linear viscoelastic behavior of fumed silica suspensions

Suspensions of fumed silica exhibit a wide range of rheological properties depending on the nature and magnitude of the interparticle forces. In a non-polar fluid, the particles interact through hydrogen bonding and can form a three-dimensional network. The microstructure formation is responsible for the non-linear viscoelastic behavior of fumed silica suspensions, even at very small strain. These non-linear rheological properties have been studied in small amplitude oscillatory experiments as a function of particle size, surface treatment of particles, suspending medium polarity and solids concentration. The non-linear viscoelastic behavior is characterized by a non-sinusoidal waveform of the signal response. For suspensions in a non-polar fluid, both the elastic and the loss moduli are shown to be sensitive to the strain amplitude: the elastic modulus is decreasing with increasing strain whereas the loss moduli is initially increasing with strain. We have chosen to examine the dissipated energy which is clearly related to the breakup of the suspension structure. A comparison of model predictions and the experimental data shows the limitations of these models, recently proposed in the literature to describe the behavior of colloidal suspensions. Received: 9 March 1998 Accepted: 17 November 1998     

The effect of pH,ionic strength,and temperature on the rheology and stability of aqueous clay suspensions

The effect of pH level, ionic strength, and temperature on the theology and stability of aqueous suspensions of attapulgite clay was systematically investigated. A Rheometrics Mechanical Spectrometer with cone and plate fixtures was used to measure the steady shear viscosity of the system. The edge charges of the clay particles can be adjusted by changing the pH level of the suspending medium so as to influence the flocculation state and, consequently, the rheological behavior of the suspension. This pH effect may be counteracted by the ionic strength effect at both very high and very low pH levels where the ionic strength is high enough to cause flocculation of the electrostatically stabilized suspension. The temperature effect study indicates that the relative contribution of Brownian motion and shear flow to the viscosity is dependent on the flocculation state of the suspension.     

Rheology of carbon black suspensions. III. Sol-gel transition system

Linear viscoelastic properties of carbon black (CB) suspensions with various CB volume fractions () in a rosin-modified phenol resin type varnish (Varnish-1) were investigated at various temperatures (T). The CB/Varnish-1 suspensions exhibited a sol-gel transition on an increase in , and the _gel value at the gelation point decreased with increasing T. This T dependence of _gel, being opposite to the dependence seen for usual gelling systems, can be related to a phenol resin type polymeric component included in the Varnish-1. At low T, this polymeric component appeared to be rather well solvated in the Varnish-1 thereby allowing the gelation due to bare attraction between the CB particles at large . In contrast, at high T, the polymeric component appeared to have been less solvated, as evidence from a moderate failure of the time-temperature superposition of pure Varnish-1 and a decrease of its elasticity (in a shifted frequency scale) with increasing T. This less solvated polymeric component would have been adsorbed on the CB particles, thereby allowing the agglomeration of the particles at small _gel at high T.     

The rheology of suspensions of porous zeolite particles in polymer solutions

We present data and predictive models for the shear rheology of suspended zeolite particles in polymer solutions. It was found experimentally that suspensions of zeolite particles in polymer solutions have relative viscosities that dramatically exceed the Krieger–Dougherty predictions for hard sphere suspensions. Our investigations show that the major origin of this discrepancy is due to the selective absorption of solvent molecules from the suspending polymer solution into zeolite pores. The effect raises both the polymer concentration in the suspending medium and the particle volume fraction in the suspension. Consequently, both the viscosity of the polymer solution and the particle contribution to the suspension viscosity are increased. We propose a predictive model for the viscosity of porous zeolite suspensions by incorporating a solvent absorption parameter, α, into the Krieger–Dougherty model. We experimentally determined the solvent absorption parameter by comparing viscosity data for suspensions of porous and nonporous MFI zeolite particles. Our results are in good agreement with the theoretical pore volume of MFI particles.     

An experimental investigation into the kinematics of a concentrated hard-sphere colloidal suspension during Hopkinson bar evaluation at high stresses

The behavior of a concentrated, hard-sphere colloidal suspension is evaluated using the split Hopkinson pressure bar (SHPB) experimental technique. The composition of the suspension is measured using thermograviometric analysis before and after loading. This, combined with recorded pressure distributions, result in the conclusion that the suspension undergoes high rate squeezing flow. Experimental results demonstrate that the suspension exhibits shear thickening consistent with that observed in standard rotational rheometry. At sufficiently high stresses the suspension exhibits a second regime of shear thinning behavior that is consistent with elastohydrodynamic theory that incorporates the shear modulus of the particles themselves. Further increases in stress result in irreversible behavior, i.e., rather than fracturing or crushing, the particles form non-reversible agglomerates during testing. This fact is demonstrated through the use of dynamic light scattering and electron microscopy. This behavior occurs within a regime of viscous material response which is seen to occur at normal strain rates and stresses over 10⁴ s^?1 and 40 MPa, respectively.     

Rheological properties of suspensions of polyethylene-coated aluminum nanoparticles

Ultrafine aluminum powder was identified as very promising fuels for novel energetic materials formulations. However, the large specific surface area of this powder facilitates its oxidation and greatly reduces its shelf life. Therefore, different coating processes were proposed to solve this problem. The rheology of viscous suspensions of nanoparticles still remains poorly understood and the effect of the coating of such particles on the flow behavior is even more difficult to assess. We have studied the rheology of ultrafine aluminum suspensions in three low molecular weight polymers of different viscosities: a hydroxy-terminated polybutadiene, a polypropylene glycol, and a polysiloxane. The nanosize aluminum powder was previously coated by a thin layer of high-density polyethylene using an in situ polymerization process. The rheological characterization of the suspensions was conducted by the means of steady and oscillatory shear flow measurements for noncoated and coated particles. The effect of the coating process on the rheology of the suspensions is discussed in terms of the interactions between the particles and the suspending fluids.     

Shear and extensional rheology of carbon nanofiber suspensions

The morphology and rheology of carbon nanofiber (CNF) suspensions were studied. The CNFs, produced by decomposing organic vapors at elevated temperature in the presence of metal catalysts, have characteristic diameter and length of 100 nm and 20–100 m, respectively. The CNFs, as delivered, have a strong tendency to clump into mm-sized agglomerates. The efficacy of CNF/glycerol-water suspensions was studied vs. two processing parameters: mechanical sonication and chemical treatment. Experimental measurements revealed that sonication alone reduces the size of CNF clumps from millimeter to micrometer scale, but cannot achieve uniform dispersion. The chemically untreated sonicated suspensions contain clumps of nanofibers with a characteristic size of 20×50 m, together with smaller aggregations of partially dispersed nanofibers. In response to this unsatisfactory dispersion, the effect of acid treatment before dispersion was investigated. This acid treatment, which makes the surface of the CNFs more hydrophilic, greatly improves dispersion in the aqueous solution: treatment followed by sonication results in a uniform dispersion of individual nanofibers. At the same time, however, we observed that surface treatment and subsequent sonication greatly shorten the nanofibers.The rheology of CNF/glycerol-water suspensions is highly non-Newtonian both in shear and extensional flows, with strong dependence on the dispersion, particle length, and concentration of the CNFs. As the solvent is Newtonian, all of the elastic and strain-rate dependent behavior in the CNF/aqueous suspensions derives from the addition of nanofibers. The steady shear viscosity of the untreated-sonicated (poorly dispersed, with longer fibers) suspensions is highly shear thinning with a viscosity that increases three orders of magnitude as concentration varies from 0.5 wt% to 5 wt%. Beyond 5 wt% the suspensions are too viscous to be effectively mixed by sonication. When the CNFs are chemically treated and then sonicated (resulting in much better dispersion but shorter fibers), the viscosity exhibits little shear thinning, and only varies by a factor of two from pure solvent to 5 wt%. In small amplitude oscillatory shear measurements, we found strong indications of elastic behavior in both the treated and untreated suspensions, with elastic modulus G greater than loss modulus G. In particular, for both systems G exhibits a low-frequency plateau when nanofiber concentration is 3 wt% or above, a characteristic of elastic solidlike response. Again, there is a strong dependence on CNF dispersion and fiber length: At low frequencies, the elastic modulus of the 5 wt% untreated suspension (with agglomerates and longer fibers) is four orders of magnitude larger than that of the 5 wt% treated suspension (with uniformly dispersed, shorter fibers). In addition, G of untreated suspensions is a much stronger function of concentration than that of treated suspensions, indicative of network formation.The rheology and morphology of nanofiber suspensions were related by identifying morphology of the suspensions with the assumptions of the kinetic theory-based elastic and rigid dumbbell constitutive models; the approach is to specify the parameters in the kinetic theory models in terms of microscale morphological features measured in the SEM. Of those investigated, the elastic dumbbell model with anisotropic hydrodynamic drag is the most successful, effectively modeling the small amplitude oscillatory shear and steady shear behavior of the treated sonicated suspensions. As for the treated unsonicated and untreated sonicated suspensions, which contain mesoscale agglomerates not present in the underlying assumptions of the dumbbell models, it is discovered that the elastic dumbbell with parameters assigned from morphological measurements predicts the correct trends in the steady shear experiments, but fails to accurately predict the small amplitude oscillatory shear experiments.     

Rheological behavior of concentrated suspensions as affected by the dynamics of the mixing process

The rheological characterizations of concentrated suspensions are generally carried out assuming “well-mixed” suspensions. However, the variation of the concentration distributions of the ingredients of the formulation, i.e., the “goodness of mixing”, the size and shape distributions of the particle clusters and the rheological behavior of the suspension all depend on the thermo-mechanical history that the suspension is exposed to during the mixing process. Here, various experimental tools are used for the characterization of the degree of mixedness (concentration distributions) of various ingredients along with the characterization of rheological material functions, wall slip behavior and the maximum packing fraction of a graphite/elastomer suspension. The degree of mixedness values of the ingredients of the suspensions processed using batch and continuous processes and under differing operating conditions were characterized quantitatively using wide-angle X-ray diffraction and thermo gravimetric analysis and were elucidated under the light of the electrical properties of the suspension as affected by the mixing process. Upon achieving better homogeneity of the graphite particles and the binder and decreases in the size and breadth of the size distributions of particle clusters (as inferred from electrical measurements and maximum packing fraction values), the elasticity (storage modulus) and the shear viscosity (magnitude of the complex viscosity from small-amplitude oscillatory shear and shear viscosity from steady torsional and capillary rheometry) of the suspension decreased significantly and the wall slip velocity values increased. These findings demonstrate the intimate relationships that exist between the rheological behavior of concentrated suspensions and the thermo-mechanical history that they are exposed to during the processing stage and suggest that the preparation conditions for suspensions should be carefully selected and well documented to achieve reproducible characterization of rheological material functions.     

L1 Continuous Dependence Property¶for Systems of Conservation Laws

This paper is concerned with the uniqueness and L¹ continuous dependence of entropy solutions for nonlinear hyperbolic systems of conservation laws. We study first a class of linear hyperbolic systems with discontinuous coefficients: Each propagating shock wave may be a Lax shock, or a slow or fast undercompressive shock, or else a rarefaction shock. We establish a result of L¹ continuous dependence upon initial data in the case where the system does not contain rarefaction shocks. In the general case our estimate takes into account the total strength of rarefaction shocks. In the proof, a new time-decreasing, weighted L¹ functional is obtained via a step-by-step algorithm. To treat nonlinear systems, we introduce the concept of admissible averaging matrices which are shown to exist for solutions with small amplitude of genuinely nonlinear systems. Interestingly, for many systems of continuum mechanics, they also exist for solutions with arbitrary large amplitude. The key point is that an admissible averaging matrix does not exhibit rarefaction shocks. As a consequence, the L¹ continuous dependence estimate for linear systems can be extended to nonlinear hyperbolic systems using a wave-front tracking technique.     

Combined natural convection and radiation in a vertical annulus

This paper presents a numerical study of the effects of thermal radiation on the heat transfer of steady natural convection in a participating fluid contained inside a cylindrical annulus. The fluid absorbs and emits radiation energy. Uniform temperature difference is imposed across the inner and outer cylinder walls. The radiation part of the problem is treated by using the two-dimensional P-1 approximation. The numerical calculations were executed using the PROJECTION algorithm with a power law scheme. The analysis are performed for Rayleigh number ranging from 10⁴ to 10⁶, aspect ratio varying between 2 and 10, and the radius ratio ranging from 1.25 to 10 for N=0.1, 1, and X where N is the conduction-to-radiation parameter. The thermal radiation greatly influences the flow field and temperature distribution inside the enclosure for different aspect ratio and radius ratios. Those results are presented vividly by the streamlines and isotherms. Based on the results, the average Nusselt number is correlated by a relation of the form N¥ˆ=a+b ln A and N¥ˆ=c+d exp(m-/e). The coefficients for various N are tabulated. Zusammenfassung Ziel der numerischen Studie ist es, die Einflüsse der Wärmestrahlung auf den Wärmeübergang durch stationäre, natürliche Konvektion an ein, in einem zylindrischen Ringraum befindlichen Fluid zu bestimmen, das Strahlungsenergie absorbiert und emittiert. Innere und äußere Zylinderwand werden auf unterschiedlichen, aber konstanten Temperaturen gehalten. Der Strahlungstransport wird mittels der zweidimensionalen P-1-Approximation erfaßt; die numerischen Berechnungen erfolgten unter Verwendung des PROJECTION-Algorithms nach einem Potenzgesetzschema. Die Untersuchung erstreckt sich auf Rayleigh-Zahlen von 10⁴ bis 10⁶, auf Seitenverhältnisse von 2 bis 10 und auf Radienverhältnisse von 1,25 bis 10. Der Leitungs/Strahlungs-Parameter N hatte die Werte 0,1-1,0 und X. Die Wärmestrahlung hat im Variationsbereich des Seiten-und des Radienverhältnisses großen Einfluß auf das Strömungsverhalten und die Temperaturverteilung im Ringspalt, was sich sehr gut durch Stromlinien-und Isothermenfeld-Darstellungen belegen läßt. Die Resultate stützen Berechnungsgleichungen für die gemittelte Nusselt-Zahl der Form N¥ˆ=a+b ln A und N¥ˆ=e+d exp(-/e). Die einzelnen Koeffizienten sind für verschiedene N-Werte in Tabellen zusammengestellt.     

Traversing field of view and AR-PIV for mid-field wake vortex investigation in a towing tank

Wake vortex flow experiments are performed in a water tank where a 1:48 scaled model of a large transport aircraft A340-300 is towed at the speed of 3 and 5 ms^-1 with values of the angle of attack !={2°, 4°, 8°}. Particle image velocimetry (PIV) measurements are performed in a plane perpendicular to the towing direction describing the streamwise component of the wake vorticity. The instantaneous field of view (I-FOV) is traversed vertically with an underwater moving-camera device tracking the vortex core during the downward motion. An adaptive resolution (AR) image-processing technique is introduced that enhances the PIV interrogation in terms of spatial resolution and accuracy. The main objectives of the investigation are to demonstrate the applicability of PIV diagnostics in wake vortex research with towing-tank facilities. The specific implementation of the traversing field-of-view (T-FOV) technique and the AR image processing are driven by the need to characterize the vortex wake global properties as well as the vortex decay phenomenon in the mid- and far-field. Relevant aerodynamic information is obtained in the mid-field where the time evolution of the vortex structure (core radius and tangential velocity) and of the overall vortex wake (vortex trajectory, descent velocity, circulation) are discussed.     

Study of the flow around surface-mounted pyramids

The turbulent flow around square-based wall-mounted pyramids in thin and thick boundary layers was experimentally investigated as a function of the pyramid apex angle, ‘, and angle of attack, f, based on mean flow surface patterns, pressure and velocity measurements. For thin boundary layers, wake periodicity is observed. For slender pyramids (15°<‘<75°), the periodic formation and shedding of vortices is observed. The shedding frequency scales with the length scale L=w'(w'/h)^-1/4, where h is the pyramid height and w' is the frontal (projected) base width. For broad pyramids, wake periodicity exists but cannot be related to vortex shedding. Vortex shedding appears related to the existence of a double vortex structure along the side faces of the slender pyramids. The location of the separation point upstream and the attachment point downstream of the obstacle also scales with L. For thick boundary layers, no wake periodicity is observed and the mean flow structure in the wake differs from that seen for thin boundary layers. The location of the separation and attachment points scales only approximately with L.     

Extensional rheology of a shear-thickening cornstarch and water suspension

A filament-stretching rheometer is used to measure the extensional viscosity of a shear-thickening suspension of cornstarch in water. The experiments are performed at a concentration of 55 wt.%. The shear rheology of these suspensions demonstrates a strong shear-thickening behavior. The extensional rheology of the suspensions demonstrates a Newtonian response at low extension rates. At moderate strain rates, the fluid strain hardens. The speed of the strain hardening and the extensional viscosity achieved increase quickly with increasing extension rate. Above a critical extension rate, the extensional viscosity goes through a maximum and the fluid filaments fail through a brittle fracture at a constant tensile stress. The glassy response of the suspension is likely the result of jamming of particles or clusters of particles at these high extension rates. This same mechanism is responsible for the shear thickening of these suspensions. In capillary breakup extensional rheometry, measurement of these suspensions demonstrates a divergence in the extensional viscosity as the fluid stops draining after a modest strain is accumulated.