Rheology of physically evolving suspensions

The objective of this work was to investigate the modeling of the whole dynamic rheological behavior of physically evolving suspensions (e.g., polyvinyl chloride plastisols). The evolutions of the complex viscosity with time (isothermal) and with temperature (non-isothermal) were analyzed. To understand the physically involved phenomena, the determination of relationships between the solid volume fraction evolution and the rheological behavior was investigated. Firstly, the evolution of the volume fraction in relation with the variation of radii particle suspensions using a modified Avrami equation was determined. Actually, the rheological study of this physically evolving system is far too complicated due to the many factors involved in the evolving process. Consequently, a phenomenological law using Carreau–Yasuda equation and percolation laws combined with the evolution of the solid volume fraction is investigated to obtain the modeling of the whole dynamic rheological behavior at any frequency and temperature.     

Viscosity of concentrated suspensions of spheres

Difficulties associated with the viscosity measurement of concentrated suspensions of particulate solids in a liquid solvent can effectively be overcome with the falling needle technique reported here. The comparison of the settling (terminal) velocity of a given needle in a Newtonian solvent, with its terminal velocity in a suspension, yields the suspension viscosity ratio directly. The van den Brule and Jongschaap constitutive model describes our high concentration data best. Falling sphere data (diameter of sphere/diameter of suspended particle 10) agree well with the falling needle data over the whole range (up to 40%) of solids concentrations used in our tests.In the opaque suspensions used, the passage of sedimenting needles and spheres was initially observed radiographically. Later tests used a more convenient technique using an inductance coil particle detector driven by a Colpitts oscillator.     

Stability of some shear flows for concentrated suspensions

In this paper we investigate the stability of some viscometric flows for a concentrated suspension model which allows for the effects of shear-induced migration, including plane and circular Couette and Poiseulle flows, and unbounded and bounded torsional flows. In the bounded torsional flow, where its radial outer boundary is assumed frictionless, an exact closeform solution is given. With the exception of torsional flows, we find that a limit point for all the steady-state solutions can exist for certain range in the parameter values. In all cases, disturbances can persist for a long time, O (H ²/a ²), where H is a dimension of the flow field, and a is the particles' radius.     

Viscosity of concentrated noncolloidal bidisperse suspensions

At the same solid volume fraction (Φ) the relative viscosity (η _r ) of a concentrated noncolloidal bidisperse suspension of hard spherical particles is lower than that of a monodisperse suspension. In this paper a semi-analytical viscosity model of noncolloidal bidisperse suspensions is derived using an integration method. In this model the random loose packing density obtained by computer simulation is taken as the limit of solid volume fraction Φ _m which depends upon both the diameter ratio (λ) of large to small particles and the volume fraction of large particles (ξ=Φ _l /Φ). This model shows that at high solid volume fraction, Φ > 0.40, both λ and ξ significantly influence η _r . For example, at Φ=0.5, it predicts that for monodisperse suspensions η _r =70, while for bidisperse suspensions (λ=2 and ξ=0.7) η _r =40. Comparison shows that, at high solid volume fraction (0.4–0.5), the relative viscosity predicted by this model is in good agreement with that predicted by the work of Shapiro and Probstein (1992) and of Patlazhan (1993), but is higher than that predicted by the work of others. Received: 27 February 2001 Accepted: 25 April 2001     

Rheology of concentrated coagulating suspensions in non-aqueous media

The rheology of concentrated coagulating suspensions is analysed on the basis of the following model: (i) at low shear rates, the shear is not distributed homogeneously but limited to certain shear planes; (ii) the energy dissipation during steady flow is due primarily to the overcoming of viscous drag by the suspended particles during motion caused by encounters of particles in the shear planes. This model is called the giant floc model.With increasing shear rate the distance between successive shear planes diminishes, approaching the suspended particles' diameter at average shear stresses of 88–117 Pa in suspensions of 78 µm particles (glass ballotini coated by a hydrophobic layer) in glycerol — water mixtures, at solid volume fractions between 0.35 and 0.40. Smaller particles form a more persistent coagulation structure. The average force necessary to separate two touching 78 µm particles is too large to be accounted for by London-van der Waels forces; thus coagulation is attributed to bridging connections between polymer chains protruding from the hydrophobic coatings.The frictional ratio of the glass particles in these suspensions is of the order of 10. Coagulation leads to build-up of larger structural units at lower shear rates; on doubling the shear rate the average distance between the shear planes decreases by a factor of 0.81 to 0.88. A inter-shear plane distance - A Hamaker constant - b radius of primary particles - f frictional ratio - F _A attractive force between two particles - g acceleration due to gravity - H distance between the surfaces of two particles - K proportionality constant in power law - l fraction of distance by which a moving particle entrains its neighbours - l effective length of inner cylinder in the rheometer - M torque experienced by inner cylinder during measurements - n exponent in power law - n ₀ ,n ₁ ,n ₂ constants in extended power law - NC _hex number of contacts, per mm², between particles in adjacent layers with an average degree of occupation, assuming a hexagonal arrangement of the particles within the layers - NC _cub asNC _hex, but with a cubical arrangement - p () d increase of slippage probability when the shear stress increases from to + d - q average coordination number of a particle in a coagulate - R _i radius of inner cylinder of rheometer - R _u radius of outer cylinder of rheometer - t _i time during which particlei moves - t ₀ time during which a particle bordering a shear plane moves from its rectilinear course, on meeting another particle - u angle between the direction of motion, and the line connecting the centers of two successive particles bordering a shear plane - V _A attractive energy between two particles - x, y, z Cartesian coordinates:x — the direction of motion;y — the direction of the velocity gradient - y ₀ ,z ₀ y, z value of a particle meeting another particle, when both are far removed from each other - y ₀ spread iny ₀ values - —2/n - ₀ capture efficiency - shear rate - average shear rate calculated for a Newtonian liquid - _i distance by which particlei moves - ₀ distance by which a particle bordering a shear plane moves from its rectilinear course, when it encounters another particle - square root of area occupied by a particle bordering a shear plane, in this plane - _c energy dissipated during one encounter of two particles bordering a shear plane - _p energy dissipated by one particle - energy dissipated per unit of volume and time during steady flow - viscosity - _app calculated as if the liquid is Newtonian - ₀ viscosity of suspension medium - _PL lim - [] intrinsic viscosity - _diff - _{diff, rel diff}/ ₀ - standard deviation of distribution ofy ₀ values - shear stress - _n average shear stress at the highest values applied - mass average particle diameter - _n number average particle diameter - solid volume fraction - _eff effective solid volume fraction in Dougherty-Krieger relation - _max maximum solid volume fraction permitting flow - _i angular velocity of inner cylinder in rheometer during measurements     

On the behaviour of gelled fibre suspensions in steady shear

The shear rheological properties of suspensions of gelled agar fibres in a low viscosity Newtonian matrix fluid were investigated. Two classes of fibres, low aspect ratio fibres and high aspect ratio fibres with an aspect ratio of the order of 10 and 100 respectively were included in the investigations. For all fibre phase volumes investigated, from as low as 0.01 upwards, the flow curves are characterised by an apparent yield stress followed by shear-thinning which was independent of the fibre aspect ratio. Based on our analysis of the flow curves, we conclude that the high aspect ratio fibres behave like flexible threads in contrast to the low aspect ratio fibres whose high shear relative viscosity is successfully described by a relation for long rigid rods. These findings are supported by flow visualisation using an optical shearing stage coupled to a light microscope.     

A new method for direct simulations of flexible filament suspensions in non‐zero Reynolds number flows

A new method for direct simulations of flexible filament suspensions in a non‐zero Reynolds number flow is developed. For fluid domain, simulations are based on a lattice Boltzmann equation. For solid domain, a slender solid body is discretized into a chain of consecutive spherical segments contacting each other. A constraint force algorithm is proposed to warrant constant bonding distance between two neighbouring segments and non‐slip velocity conditions at the contacting points so that the flexible filament moves and rotates as a whole body. The fibre could be bent and twisted in the model. Non‐linear inertial interactions between fluid and flexible filament can be naturally studied by using this model embedded in the lattice Boltzmann scheme. The present flexible fibre method is tested by using a rigid particle method when the fibre stiffness is very large and by comparing the present results with theoretical and experimental results. It is demonstrated that the present results have a reasonable accuracy and that the computational results are consistent with the existed experimental results at non‐zero Reynolds number flows. Copyright © 2006 John Wiley & Sons, Ltd.     

Rheology of molecular liquids and concentrated suspensions by microscopic dynamical simulations

The rheology and the associated changes that arise in sheared molecular and colloidal liquids are investigated by Molecular and Brownian Dynamics Computer Simulation. Significant shear thinning and normal pressure effects occur in all liquids when the shear rate approximately equals an inverse characteristic relaxation time for the material. The shear and bulk moduli, and self-diffusion coefficients increase with shear rate for all liquids and stable dispersions. The importance for rheology of hydrodynamic coupling between macromolecule trajectories at high packing fractions is demonstrated. The infinite frequency moduli depend on the packing fraction to a power which is effectively the same for all materials, i.e. ca. 3.5, above a percolation transition at a packing fraction 0.25. The suspending fluid enhances the degree of shear thinning above that of the corresponding single component fluid consisting of pure macroparticles.     

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.     

Reological behaviour of concentrated monodisperse suspensions as a function of preshear conditions and temperature: an experimental study

 The influence of preshearing on the rheological behaviour of model suspensions was investigated with a stress-controlled cone-and-plate rheometer. The used matrix fluids showed Newtonian behaviour over the whole range of applied shear stresses. Highly monodisperse spherical glass spheres with various particle diameters were used as fillers. By applying steady preshearing at a low preshear stress, where a diffusion of particles can be expected, it was found for all model suspensions investigated at volume fractions ranging from 0.20 to 0.35 that the time-temperature superposition in the steady shear and in the dynamic mode holds within the chosen temperature range. Furthermore, all presheared model suspensions displayed a high and a low frequency range which are either separated by a shoulder or by a plateau value of G′ at intermediate frequencies. It could clearly be demonstrated that the low frequency range strongly depends on the preshear conditions. Hence, the features observed in the low frequency range can be attributed to a structure formation of a particulate network. In the high frequency range a frequency-dependent behaviour was observed which obeys the classical behaviour of Newtonian fluids (G′∝ω², G′′∝ω). The resulting temperature shift factors from the dynamic and the steady shear mode are identical and independent of the volume fraction and the particle size of the filler. Received: 29 November 2000 Accepted: 12 July 2001     

Steady light diffusion application to rheology: a new tool for the characterization of concentrated suspensions

We present new developments of steady light diffusion applied to rheology. Though many techniques allow the structural investigation of diluted or transparent media, very few give direct information on optically dense systems. The technique proposed here is based on the diffusion approximation and is thus valid for sheared, time-dependent flows.After recalling important theoretical results, we show the techniques ability to determine typical sizes and orientation of structures in shear flows for various concentrated suspensions (emulsions, and an industrial softener). In particular, it is able to demonstrate the effect of shear on the orientation of anisotropic objects. Moreover, the use of simple structural models incorporating the measured anisotropy allows good predictions of experimental rheological measurements. This new technique, applicable to a wide range of colloidal systems, is very helpful to characterize the shear induced structural organization of optically dense materials.This paper was presented at the first Annual European Rheology Conference (AERC) held in Guimarães, Portugal, September 11-13, 2003.     

Recoverable strains and retardation times of monodisperse suspensions of silicon dioxide spheres in poly(dimethylsiloxane)

Steady-state viscosities η, steady-state recoverable strains γ _rs and characteristic retardation time τ _1/2 were measured for suspensions of monodisperse silicon dioxide (SiO₂) spheres in poly(dimethylsiloxane) (PDMS) with various volume fractions Φ of the suspended spheres at various creep stresses σ ₀. Two different regions are found in plots of η/η _m vs γ _rs, where η/η _m denotes the relative viscosity of the suspensions. In one region, η/η _m is proportional to γ _rs, while γ _rs is independent of η/η _m in the other region. In both regions, τ _1/2 is the functions of the shear strain rate in the steady-state of creep test independently of Φ. The origin of the elasticity is related to the ‘maximally distorted’ cages recovered owing to the repulsive interaction between the SiO₂ spheres and recovery of the cages in the shear-induced clusters of the suspended spheres.     

Fragile networks and rheology of concentrated suspensions

Suspensions consisting of particles of colloidal dimensions have been reported to form connected structures. When attractive forces act between particles in suspension they may flocculate and, depending on particle concentration, shear history and other parameters, flocs may build-up in a three-dimensional network which spans the suspension sample. In this paper a floc network model is introduced to interpret the elastic behavior of flocculated suspensions at small deformations. Elastic percolation concepts are used to explain the variation of the elastic modulus with concentration. Data taken from the suspension rheology literature, and new results with suspensions of magnetic -Fe₂O₃ and non-magnetic -Fe₂O₃ particles in mineral oil are interpreted with the model proposed.Non-zero elastic modulus appeared at threshold particle concentrations of about 0.7 vol.% and 0.4 vol.% of the magnetic and non-magnetic suspensions, respectively. The difference is attributed to the denser flocs formed by magnetic suspensions. The volume fraction of particles in the flocs was estimated from the threshold particle concentration by transforming this concentration into a critical volume concentration of flocs, and identifying this critical concentration with the theoretical percolation threshold of three-dimensional networks of different coordination numbers. The results obtained indicate that the flocs are low-density structures, in agreement with cryo-scanning electron micrographs. Above the critical concentration the dynamic elastic modulus G was found to follow a scaling law of the type G ( _f - _f ^c ) ^f , where _f is the volume fraction of flocs in suspension, and _f ^c is its threshold value. For magnetic suspensions the exponent f was found to rise from a low value of about 1.0 to a value of 2.26 as particle concentration was increased. For the non-magnetic a similar change in f was observed; f changed from 0.95 to 3.6. Two other flocculated suspension systems taken from the literature showed a similar change in exponent. This suggests the possibility of a change in the mechanism of stress transport in the suspension as concentration increases, i.e., from a floc-floc bond-bending force mechanism to a rigidity percolation mechanism.     

Yield stress and maximum packing fraction of concentrated suspensions

The yield stress and features of the structure of concentrated suspensions based on silica flour, with particles of average diameter around 4 m, were investigated in terms of a phenomenological model. The yield stress of a concentrated suspension of known solid volume concentration is estimated by employing a shear-dependent maximum packing fraction _m which is obtained by model fitting equilibrium viscosity data, and by incorporating a first-order kinetic equation. The model proposed was examined by using several mineral suspensions in which silica flour was mixed with metal oxide particles so that microstructural features of the suspensions could be adjusted. A cocoa fat suspension was also used as a test sample having radically different chemistry. The agreement between the model prediction and independently obtained experimental evidence is acceptable. Furthermore, a qualitative explanation is obtained by a scaling analysis in an effort to relate the model parameters with the suspension structure that stems from interactions among the suspension constituents.     

Processing,characterisation and rheology of transparent concentrated fibre-bundle suspensions

Highly concentrated planar fibre-bundle suspensions with a transparent PMMA matrix were processed with various initial bundle contents and orientations. They were submitted to simple compression and plane strain compression deformation modes. First rheological measurements are presented. They highlight the role of the bundle content and orientation on recorded stress levels. The transparent matrix allows the observation of fibrous microstructures before and after compressions: The in-plane deformation of bundles (flattening and bending) as well as the evolution of their orientation are analysed and discussed. This paper was presented at the 3rd Annual European Rheology Conference (AERC) 2006, held in Hersonisos, Crete, Greece, April 27–29, 2006.     

Orientation tensors in simple flows of dilute suspensions of non-Brownian rigid ellipsoids,comparison of analytical and approximate solutions

General analytical solutions are obtained for the planar orientation structure of rigid ellipsoid of revolutions subjected to an arbitrary homogeneous flow in a Newtonian fluid. Both finite and infinite aspect ratio particles are considered. The orientation structure is described in terms of two-dimensional, time-dependent tensors that are commonly employed in constitutive equations for anisotropic fluids such as fiber suspensions. The effect of particle aspect ratio on the evolution of orientation structure is studied in simple shear and planar elongational flows. With the availability of analytical solutions, accuracies of quadratic closure approximations used for nonhomogeneous flows are analyzed, avoiding numerical integration of orientation distribution function. In general, fourth-order orientation evolution equations with sixth-order quadratic closure approximations yield more accurate representations compared to the commonly used second-order evolution equations with fourth-order quadratic closure approximations. However, quadratic closure approximations of any order are found to give correct maximum orientation angle (i.e., preferred direction) results for all particle aspect ratios and flow cases.     

Shear behaviour of biopolymer suspensions with spheroidal and cylindrical particles

The rheological behaviour of suspensions is influenced by many parameters, one of which is the particle shape. For rigid particle suspensions a number of studies demonstrate the effects of the particle aspect ratio. Indeed, fibres are widely used as rheology modifiers in different materials such as synthetic polymers. This work is concerned with testing the hypothesis that regularly shaped particles with aspect ratios larger than one that are made of gelled biopolymers could be used as rheology modifiers for biopolymer solutions. Biopolymers, and mixtures thereof are a widely used ingredient in foods and other products with structure functionality. Tailoring rheology modifiers by morphology offers an alternative to using different biopolymers. It is demonstrated how biopolymer suspensions with regular spheroidal, or cylindrical particle shapes can be produced by gelling the droplet phase of a liquid two phase biopolymer mixture in a shear field. Biopolymers were chosen such that gelation is initiated by cooling. Shear-cooling at constant stresses leads to the formation of ellipsoidal particles. Cylindrical particles can be generated by stepping up the shear stress prior to gelation, i.e., stretching the droplet phase into fibrils, and trapping the shape prior to break-up through gelation. Morphologies and steady shear rheological data for suspensions of the two biopolymers gellan and κ-carrageenan with an internal phase volume of 0.2 are reported. The influence of particle shape on relative viscosity is pronounced. At high shear stresses particle orientation leads to decreased viscosity with increasing particle aspect ratio. In the low shear region, higher aspect ratio suspensions show higher viscosities. Additionally, the material properties, including the interfacial tension, which influence the suspension morphology are reported. Received: 3 March 2000 Accepted: 22 August 2000     

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.