Noncolloidal Particles Research Articles

A discrete model for the apparent viscosity of polydisperse suspensions including maximum packing fraction

Based on the notion of a construction process consisting of the stepwise addition of particles to the pure fluid, a discrete model for the apparent viscosity as well as for the maximum packing fraction of polydisperse suspensions of spherical, noncolloidal particles is derived. The model connects the approaches by Bruggeman and Farris and is valid for large size ratios of consecutive particle classes during the construction process, appearing to be the first model consistently describing polydisperse volume fractions and maximum packing fraction within a single approach. In that context, the consistent inclusion of the maximum packing fraction into effective medium models is discussed. Furthermore, new generalized forms of the well-known Quemada and Krieger–Dougherty equations allowing for the choice of a second-order Taylor coefficient for the volume fraction (ϕ2-coefficient), found by asymptotic matching, are proposed. The model for the maximum packing fraction as well as the complete viscosity model is compared to experimental data from the literature showing good agreement. As a result, the new model is shown to replace the empirical Sudduth model for large diameter ratios. The extension of the model to the case of small size ratios is left for future work.

The dam-break problem for concentrated suspensions of neutrally buoyant particles

AbstractThis paper addresses the dam-break problem for particle suspensions, that is, the flow of a finite volume of suspension released suddenly down an inclined flume. We were concerned with concentrated suspensions made up of neutrally buoyant non-colloidal particles within a Newtonian fluid. Experiments were conducted over wide ranges of slope, concentration and mass. The major contributions of our experimental study are the simultaneous measurement of local flow properties far from the sidewalls (velocity profile and, with lower accuracy, particle concentration) and macroscopic features (front position, flow depth profile). To that end, the refractive index of the fluid was adapted to closely match that of the particles, enabling data acquisition up to particle volume fractions of 60 %. Particle migration resulted in the blunting of the velocity profile, in contrast to the parabolic profile observed in homogeneous Newtonian fluids. The experimental results were compared with predictions from lubrication theory and particle migration theory. For solids fractions as large as 45 %, the flow behaviour did not differ much from that of a homogeneous Newtonian fluid. More specifically, we observed that the velocity profiles were closely approximated by a parabolic form and there was little evidence of particle migration throughout the depth. For particle concentrations in the 52–56 % range, the flow depth and front position were fairly well predicted by lubrication theory, but taking a closer look at the velocity profiles revealed that particle migration had noticeable effects on the shape of the velocity profile (blunting), but had little impact on its strength, which explained why lubrication theory performed well. Particle migration theories (such as the shear-induced diffusion model) successfully captured the slow evolution of the velocity profiles. For particle concentrations in excess of 56 %, the macroscopic flow features were grossly predicted by lubrication theory (to within 20 % for the flow depth, 50 % for the front position). The flows seemed to reach a steady state, i.e. the shape of the velocity profile showed little time dependence.

Demonstration of Secondary Currents in the Pressure-Driven Flow of a Concentrated Suspension Through a Square Conduit

The existence of secondary flows in the pressure-driven flow of a concentrated suspension of noncolloidal particles through a conduit of square cross section under creeping flow conditions is confirmed experimentally. This Letter lends support to the idea that secondary currents, rather than shear-induced migration, may actually be the dominant mechanism that determines particle distribution in noncolloidal suspension flows through nonaxisymmetric geometries. This work also establishes that coextrusion of two concentrated suspensions through nonaxisymmetric geometries with a stable suspension-suspension interface is not possible, except in special situations.

Shear-induced sedimentation in yield stress fluids

Stability of coarse particles against gravity is an important issue in dense suspensions (fresh concrete, foodstuff, etc.). On the one hand, it is known that they are stable at rest when the interstitial paste has a high enough yield stress; on the other hand, it is not yet possible to predict if a given material will remain homogeneous during a flow. Using MRI techniques, we study the time evolution of the particle volume fraction during the flows in a Couette geometry of model density-mismatched suspensions of noncolloidal particles in yield stress fluids. We observe that shear induces sedimentation of the particles in all systems, which are stable at rest. The sedimentation velocity is observed to increase with increasing shear rate and particle diameter, and to decrease with increasing yield stress of the interstitial fluid. At low shear rate (’plastic regime’), we show that this phenomenon can be modelled by considering that the interstitial fluid behaves like a viscous fluid–of viscosity equal to the apparent viscosity of the sheared fluid–in the direction orthogonal to shear. The behavior at higher shear rates, when viscous effects start to be important, is also discussed. We finally study the dependence of the sedimentation velocity on the particle volume fraction, and show that its modelling requires estimating the local shear rate in the interstitial fluid.

Suspensions with a tunable effective viscosity: a numerical study

AbstractIn this paper, we conduct a numerical investigation of sheared suspensions of non-colloidal spherical particles on which a torque is applied. Particles are mono-dispersed and neutrally buoyant. Since the torque modifies particle rotation, we show that it can indeed strongly change the effective viscosity of semi-dilute or even more concentrated suspensions. We perform our calculations up to a volume fraction of 28 %. And we compare our results to data obtained at 40 % by Yeo and Maxey (Phys. Rev.E, vol. 81, 2010, p. 62501) with a totally different numerical method. Depending on the torque orientation, one can increase (decrease) the rotation of the particles. This results in a strong enhancement (reduction) of the effective shear viscosity of the suspension. We construct a dimensionless number$\Theta $which represents the average relative angular velocity of the particles divided by the vorticity of the fluid generated by the shear flow. We show that the contribution of the particles to the effective viscosity can be suppressed for a given and unique value of$\Theta $independently of the volume fraction. In addition, we obtain a universal behaviour (i.e. independent of the volume fraction) when we plot the relative effective viscosity divided by the relative effective viscosity without torque as a function of$\Theta $. Finally, we show that a modified Faxén law can be equivalently established for large concentrations.

Random close packing and relative viscosity of multimodal suspensions

Multimodal suspensions, consisting of non-colloidal spherical particles and a Newtonian matrix, are considered. A new differential (or multi-scale mean field approach) model for the relative viscosity of multimodal suspensions has been developed. The problem of the random close packing fraction of the solid phase is also investigated. We suppose that the multimodal suspension has a dominant large particle composition and that the smaller particles are embedded in the empty space between the larger particles. The relative viscosity model can therefore be based on the theory of monomodal suspensions. Experimental data of Eveson are compared to the predictions given by using three different models of monomodal suspensions respectively. The Maron–Pierce approach appears to give the best agreement with Eveson’s experiments. However, due to experimental uncertainties, we recommend that the Mendoza and Santamaria-Holek (MSH) formula be adopted.

Modeling coupled migration and settling of particulates in curing filled epoxies

AbstractEpoxy resins filled to a high solids loading (40–60% by volume) with noncolloidal particles are used to mitigate stress and vibration in electronic components. We perform continuum‐level finite element method (Schunk et al., A Full‐Newton Finite Element Program for Free and Moving Boundary Problems with Coupled Fluid/Solid Momentum, Energy, Mass, and Chemical Species Transport: User's Guide, Sandia National Laboratories) simulations of filler particle redistribution during the nonisothermal cure of the epoxy under both quiescent and bulk flow conditions. An extent of reaction is used to track the degree of cure. To determine the particle migration, we couple a diffusive flux suspension model (Zhang and Acrivos, Int J Multiphase Flow 1994, 20, 579.) with the curing model. The heat transfer, including the exothermic polymerization reaction, is also modeled. The result is a generalized Newtonian model that has viscosity as a function of temperature, cure and particle volume fraction. With x‐ray computed tomography, we examine settling of the particulate phase in both flowing and quiescent curing systems and compare the experimental results to the model predictions. The model is also validated with temperature measurements. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011

Apparent wall slip velocity measurements in free surface flow of concentrated suspensions

In this work we have experimentally measured the apparent wall slip velocity in open channel flow of neutrally buoyant suspension of non-colloidal particles. The free surface velocity profile was measured using the tool of particle imaging velocimetry (PIV) for two different channels made of plane and rough walls. The rough walled channel prevents wall slip, whereas the plane wall showed significant wall slip due to formation of slip layer. By comparing the velocity profiles from these two cases we were able to determine the apparent wall slip velocity. This method allows characterization of wall slip in suspension of large sized particles which cannot be performed in conventional rheometers. Experiments were carried out for concentrated suspensions of various particle volume concentrations and for two different sizes of particles. It was observed that wall slip velocity increases with particle size and concentration but decreases with increase in the viscosity of suspending fluid. The apparent wall slip velocity coefficients are in qualitative agreement with the earlier measurements. The effect of wall slip on free surface corrugation was also studied by analyzing the power spectral density (PSD) of the refracted light from the free surface. Our results indicate that free surface corrugation is a bulk flow response and it does not arise from boundary problem such as development of slip layer.

Flows and heterogeneities with a vane tool: Magnetic resonance imaging measurements

We study the local flow properties of various materials in a vane-in-cup geometry. We use magnetic resonance imaging techniques to measure velocities and particle concentrations in flowing Newtonian fluid, yield stress fluid, and in a concentrated suspension of noncolloidal particles in a yield stress fluid. In the Newtonian fluid, we observe that the $\theta$-averaged strain rate component $d_{r,\theta}$ decreases as the inverse squared radius in the gap, in agreement with a Couette analogy. This allows direct comparison (without end-effect corrections) of the resistances to shear in vane and Couette geometries. Here, the mean shear stress in the vane-in-cup geometry is slightly lower than in a Couette cell of same dimensions, and a little higher than when the vane is embedded in an infinite medium. We also observe that the flow enters deeply the region between the blades, leading to significant extensional flow. In the yield stress fluid, in contrast with the usually accepted picture based on simulation results from the literature, we find that the layer of material that is sheared near the blades at low velocity is not cylindrical. There is thus a significant extensional component of shear that should be taken into account in the analysis. Finally and surprisingly, in the suspension, we observe that a thin non-cylindrical slip layer made of the pure interstitial yield stress fluid appears quickly at the interface between the sheared material and the material that moves as a rigid body between the blades. This feature can be attributed to the non-symmetric trajectories of the noncolloidal particles around the edges of the blades. This new important observation is in sharp contradiction with the common belief that the vane tool prevents slippage and may preclude the use of the vane tool for studying the flows of pasty materials with large particles.

Dynamics of bi-dispersed settling suspension of non-colloidal particles in rotating cylinder

Non-neutrally buoyant suspension of bi-dispersed non-colloidal particles in viscous fluid rotating in a horizontal cylinder displays in-homogeneities in particle distribution with alternate bands of high and low particle concentrations along the symmetric axis of the cylinder. Experiments were carried out to characterize the axial segregation in bi-dispersed suspension at various filling fraction and rotation speed of cylinder. The mixture of same particles in absence of any suspending fluid did not show any segregation. However, in case of particles suspended in water it was observed that the rate of segregation increases with increase in filling fraction. Once the particles get segregated along the full length of the cylinder, these bands start to migrate along the tube axis finally merging to give wider bands. For a given filling fraction the rate of segregation increases with the angular speed of the rotating cylinder. When the tube is partially filled the particle segregation is observed at higher angular speed, whereas in fully filled case the segregation starts at much lower rotation speed for the same concentration of particles. The segregation pattern changes as the rotation speed is increased. At higher speed the centrifugal force dominates over gravitational and viscous drag forces and this result into completely different segregation patterns. We have also analyzed the evolution of concentration profile from the image analysis of the particles.

Macroscopic behavior of bidisperse suspensions of noncolloidal particles in yield stress fluids

We study both experimentally and theoretically the rheological behavior of isotropic bidisperse suspensions of noncolloidal particles in yield stress fluids. We focus on materials in which noncolloidal particles interact with the suspending fluid only through hydrodynamical interactions. We observe that both the elastic modulus and yield stress of bidisperse suspensions are lower than those of monodisperse suspensions of same solid volume fraction. Moreover, we show that the dimensionless yield stress of such suspensions is linked to their dimensionless elastic modulus and to their solid volume fraction through the simple equation of Chateau et al.[J. rheol. 52, 489-506 (2008)]. We also show that the effect of the particle size heterogeneity can be described by means of a packing model developed to estimate random loose packing of assemblies of dry particles. All these observations finally allow us to propose simple closed form estimates for both the elastic modulus and the yield stress of bidisperse suspensions: while the elastic modulus is a function of the reduced volume fraction $\phi/\phi_m$ only, where $\phi_m$ is the estimated random loose packing, the yield stress is a function of both the volume fraction $\phi$ and the reduced volume fraction.

Shear migration and chaotic mixing of particle suspensions in a time-periodic lid-driven cavity

This work simulates segregation of noncolloidal particle suspensions in a two dimensional time-periodic flow. Two different mixing protocols having alternating moving boundaries in a cavity known to generate chaotic advection while maintaining a constant energy input rate are applied to each suspension. A diffusive flux model is used to capture the essence of shear-induced migration. In this system, fluid deformation drives both mixing and segregation where the local rheology is a function of particle volume fraction. The impact of migration strength, altered by varying the particle size and bulk volume fraction, and topology, altered by breaking symmetry in the flow when varying the period length, are investigated. As a result of the complex interplay between the flow topology and shear migration, the concentration profile ranges from that representing the underlying topology to that of steady flow in a lid-driven cavity and depends on the parameters mentioned above and the structure produced by the two mixing protocols. In this system, increasing the size of chaotic regions does not result in enhancing mixing. These results challenge conventional wisdom in designing small scale flows for mixing and separations in microscale applications.

Dynamics of concentrated suspensions of non-colloidal particles in Couette flow

Fully three-dimensional numerical simulations of concentrated suspensions of O(1000) particles in a Couette flow at zero Reynolds number are performed with the goal of determining the wall effects on concentrated suspensions of non-colloidal particles. The simulations, based on the force-coupling method, are performed for 0.2 ≤ φ ≤ 0.4 and 10 < Ly/a < 30, where φ denotes the volume fraction and Ly and a are, respectively, the channel height and the particle radius. It is shown that the suspensions can be divided into three regions depending on the microstructures; the wall region where a structured particle layering is dominant, the core region in which the suspension field is quasi-homogeneous, and the buffer region which shows the characteristics of both the particle layer and the shear structure. The width of the inhomogeneous region (wall and buffer) is a function of φ and not sensitive to Ly/a, once Ly/a is larger than a threshold. Rheological properties in the inhomogeneous and quasi-homogeneous regions are investigated. The particle stresses are compared with previous rheological models.

Liquid-to-solid transition of concentrated suspensions under complex transient shear histories

Suspensions containing noncolloidal aluminum particles in a Newtonian carrier liquid and the effects of shear history on the rheological properties of suspensions are investigated showing a critical concentration phi(c), which is close to the "freezing" packing fraction of spherical particle suspensions. An apparent liquid-to-solid transition was clearly found for suspensions below phi(c) via transient step shear after long shear history as well as via the large amplitude oscillatory shear flow with controlled shear stress. The microstructures or the local flow characteristics of suspensions with phi<phi(c) and with phi>phi(c) are still different according to the Fourier Transform Rheological analysis. Although suspensions with phi<phi(c) show solidlike behaviors after long shear history, the flow behaviors keep unchanged as shear stress increases, and typically show solidlike, liquid-solid coexistence, liquidlike, and jamming behaviors, successively. However, liquidlike regime of concentrated suspensions with phi>phi(c) disappears after preshear.

Slip velocity of concentrated suspensions in Couette flow

We present measurement of wall slip velocity in concentrated suspension of non-colloidal particles. The slip in non-colloidal concentrated suspension mainly arises from wall depletion effect since the non-hydrodynamic effects such as those arising from particle-wall interactions can be small. In this work, we provide a simple methodology for the determination of slip velocity, which requires less experimental work compared to other methods available for slip corrections. The experiments were carried out in a cylindrical Couette geometry of a rheometer. The rheological measurements were carried out first with serrated cup and serrated rotor geometry. Next, the serrated rotor was made smooth by a wax coating while the cup remained serrated. The serrated geometry offers no-slip boundary and the measured viscosity is the true viscosity of suspension, whereas smooth rotor showed significant slip at a higher concentration of particles and the measured viscosity was significantly lower. Comparing the wall shear ...

Experimentation on MR fluid using a 2-axis wheel tool

This paper is focused on magnetorheological (MR) fluid assistive polishing of optical aspheric components. MR fluid is a functional mixture of non-colloidal magnetic particle of micrometer size suspended in a host fluid, with the special property that its viscosity can be varied by the application of a magnetic field. This paper introduces the basic principles of the methodology and presents experiment results on MR fluids using a 2-axis wheel-shaped tool supporting dual magnetic fields. Mathematical models taking into account the pressure and the tool velocity are derived. The experiments serve to evaluate the effects of process parameters on material removal and performance using a K9 glass parabolic lens of 60 mm diameter as work-piece. It is shown that surface roughness can be reduced from an initial value of 3.8–1.2 nm after 10 min of polishing. The form errors can also be improved from an initial 2.27 μm rms and 7.89 μm peak-to-valley to become 0.36 μm rms and 2.01 μm peak-to-valley after 60 min of polishing.

Experimental and Field Studies of Type I Settling for Particulate Matter Transported by Urban Runoff

Settling velocity is an important constitutive parameter of particulate matter (PM) transported by runoff. Settling velocity is either explicitly or implicitly utilized when designing or modeling unit operations, and in situ or watershed controls for urban rainfall-runoff. Utilizing two common settling devices, a settling column and an Imhoff cone, settling velocities of discrete noncolloidal particles in source area urban rainfall-runoff were measured. A comparison of settling models applicable to discrete (Type I) PM settling was developed. Models were compared to measured results across the noncohesive silt- and sand-size PM gradation from 2 to 2,000 μm , utilizing measured particle-size distributions (PSDs) and specific gravity. Results indicate that Newton’s Law can reproduce measured settling velocity when measured inputs of PM diameter, specific gravity, and temperature are utilized. Alternative models to Newton’s Law (in the Stokesian regime) did not improve agreement with measured settling veloci...

Effect of coarse particle volume fraction on the yield stress and thixotropy of cementitious materials

In order to help in modelling the yield stress of fresh concrete, we study the behavior of suspensions of coarse particles in a thixotropic cement paste. Our aim is to relate the yield stress of these mixtures to the yield stress of the suspending cement paste, to the time passed at rest, and to the coarse particle volume fraction. We present here procedures that allow for (i) studying a homogeneous and isotropic suspension, (ii) comparing the yield stress of a given cement paste to that of the same cement paste added with particles, and (iii) accounting for the thixotropy of the cement paste. We observe that the yield stress of these suspensions of cement paste with coarse particles follows the very simple Chateau–Ovarlez–Trung model [X. Chateau, G. Ovarlez, K.L. Trung, Homogenization approach to the behavior of suspensions of noncolloidal particles in yield stress fluids, J. Rheol. (2008) 52 489–506.], consistently with the experimental results of Mahaut et al. [F. Mahaut, X. Chateau, P. Coussot, G. Ovarlez, Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids, J. Rheol. (2008) 52 287–313.] obtained with many different particles and suspending yield stress fluids. This consistency between the results obtained in various yield stress fluids shows that the yield stress of the suspension does not depend on the physicochemical properties of the suspending yield stress fluid; it only depends on its yield stress value. This shows that studies of suspensions in model yield stress fluids can be used as a general tool to infer the behavior of fresh concrete. Moreover, we show that the thixotropic structuration rate of the interstitial paste (its static yield stress increase rate in time) is not affected by the presence of the particles. As a consequence, it is sufficient to measure the thixotropic properties of the constitutive cement paste in order to predict the thixotropic structuration rate of a given fresh concrete. This structuration rate is predicted to have the same dependence on the coarse particle volume fraction as the yield stress.

The influence of secondary flows induced by normal stress differences on the shear-induced migration of particles in concentrated suspensions

It was first demonstrated experimentally by H. Giesekus in 1965 that the second normal stress difference in polymers can induce a secondary flow within the cross-section of a non-axisymmetric conduit. In this paper, we show through simulations that the same may be true for suspensions of rigid non-colloidal particles that are known to exhibit a strong negative second normal stress difference. Typically, the magnitudes of the transverse velocity components are small compared to the average axial velocity of the suspension; but the ratio of this transverse convective velocity to the shear-induced migration velocity is characterized by the shear-induced migration Péclet number χ which scales as B2/a2, B being the characteristic length scale of the cross-section and a being the particle radius. Since this Péclet number is kept high in suspension experiments (typically 100 to 2500), the influence of the weak circulation currents on the concentration profile can be very strong, a result that has not been appreciated in previous work. The principal effect of secondary flows on the concentration distribution as determined from simulations using the suspension balance model of Nott &amp; Brady (J. Fluid Mech. vol. 275, 1994, p. 157) and the constitutive equations of Zarraga et al. (J. Rheol. vol. 44, 2000, p. 185) is three-fold. First, the steady-state particle concentration distribution is no longer independent of particle size; rather, it depends on the aspect ratio B/a. Secondly, the direction of the secondary flow is such that particles are swept out of regions of high streamsurface curvature, e.g. particle concentrations in corners reach a minimum rather than the local maximum predicted in the absence of such flows. Finally, the second normal stress differences lead to instabilities even in such simple geometries as plane-Poiseuille flow.

Studies on the migration in PA6‐OMMT nanocomposites: effect of annealing on migration as evidenced by ARXPS (angle resolved X‐ray photoelectron spectroscopy)

AbstractThe present paper deals with experiments in which the angle resolved X‐ray photoelectron spectroscopy (ARXPS) technique was applied to investigate the phenomenon of migration in polyamide 6 with organically modified montmorillonite (PA6‐OMMT) nanocomposites. This is the first time ARXPS was used, to gain a more detailed insight into the migration process. Curve‐fitting analyses are reported which enable the differentiation between a manual mixture of PA6 with OMMT at room temperature from a nanocomposite structure. The ARXPS technique was applied to annealed samples at 250, 275, and 285°C, for 2 hr, and with three take of angles, α, of 90, 60, and 30°. The depth of the layer investigated is 9 nm in case the sample surface is well defined. By tilting the take‐off angle of the beam‐out electrons one can determine the intensity of the signals at various distances from the surface within the investigated layer. The concentration gradient of the Si signals is observed. The rate of migration is found to be more rapid in the layer closest to the surface. The intensity of the Si signals in the investigated layer is found to be 1.8 to 9.1 times that of the same composite sample at room temperature. This ratio determines the extent of migration. At temperatures higher than 250°C and times of annealing greater than 30 min a decrease in the extent of migration is observed. This decrease is explained by the gradual decomposition of the surfactant and the consequent removal of the polymeric matrix molecules from the migrating exfoliated units, culminating in denuded alumino‐silicate layers. These layers aggregate to noncolloidal microcomposite particles which do not migrate. The concentration of the exfoliated units decreases with consequent decrease in migration. A gradient of decreasing concentration of Si2p signals was observed after various times of annealing, where the gradient becomes more uniform with increase in time. Copyright © 2008 John Wiley &amp; Sons, Ltd.