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The mixing of cohesive granular materials featuring a large size range in the absence of gravity
Powder Technology 235 (2013) 18–26
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The mixing of cohesive granular materials featuring a large size range in the absence of gravity Lee R. Aarons a,⁎, S. Balachandar a, Yasuyuki Horie b a b
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA Air Force Research Laboratory, Munitions Directorate, Eglin AFB, FL 32542, USA
a r t i c l e
i n f o
Article history: Received 11 April 2012 Received in revised form 28 August 2012 Accepted 23 September 2012 Available online 6 October 2012 Keywords: Mixing Cohesion Shear flow of powders Discrete element simulation
a b s t r a c t We have studied the shear mixing of bidisperse collections of cohesive particles in an effort to develop models that would allow one to predict and control the homogeneity of particle composites. Our focus has been on the effects of interparticle cohesion and shear rate on the microstructure of particle composites. Furthermore, we have focused on particles that have a “large size range,” specifically a 7:1 diameter ratio, such that homogeneous mixtures would include the small particles filling in the gaps formed between big particles, resulting in a correlation between the packing fraction and mixing quality. As a model problem, the cohesion resulting from the van der Waals force acting between particles was considered. Simulations were performed in which initially segregated bidisperse collections of particles were subjected to plane shear under constant applied stress as a method of mixing. Gravity was ignored in these simulations so that the different particles were not driven to different sides of the mixtures and the only hindrance to homogeneous mixing was cohesion. Simulations were performed with a variety of shear rates and particle cohesion strengths for both the large and small particles, and the homogeneity of the resultant mixtures was quantified using two distinct statistics: the estimated mean size of small-particle clusters and the spatial variance in the relative concentrations of the small and large particles. Microstructure images of the mixtures were used to provide additional qualitative measure of homogeneity as well as a measure of the relevance of the order statistics. These data suggested that the cohesiveness (Hamaker constant) of the small particles had the strongest influence on the mixture's homogeneity. When the small particles were not sufficiently cohesive, they did not significantly agglomerate, and so the resulting mixtures were relatively homogeneous over the range of shear rates and Hamaker constants of the big particles explored here. When the small particles were more cohesive, the small particles formed strong agglomerates and the mixtures become significantly inhomogeneous at lower shear rates. Somewhat surprisingly in these cases, better mixing was not achieved by simply making the large particles less cohesive. Rather, reducing the cohesiveness of the large particles far enough caused the large particles to pack more tightly, making the small-particle agglomerates unable to fit in between them, ultimately resulting in worse mixing. As such, the best mixing in these cases was achieved when the big particles were moderately cohesive. A correlation between solid volume fraction and homogeneity was not observed when particle cohesion was varied, as making either the small or large particles more cohesive led to a decrease in solid volume fraction, regardless of the effect on homogeneity. On the other hand, when homogeneity was found to increase with shear rate, so did the solid volume fraction. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The mixing of powders is well-known to be a generally difficult task. When any collection of particles is sufficiently disturbed, particles with similar properties tend to collect with each other and segregate themselves from unlike particles. Segregation generally arises from any differences between the particles that would affect their ⁎ Corresponding author. E-mail address: [email protected] (L.R. Aarons). 0032-5910/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2012.09.044
dynamics, such as size, density, and roughness [1,2]. For example, when flow occurs in a collection of differently sized particles, the bed will dilate, and in the presence of gravity, the small particles will fall through the space yielded by big particles, ending up at the bottom of the collection in a process known as percolation [3]. A further complication to mixing arises when the particles are small enough that the inter-particle attraction (e.g. from the van der Waals force) exceeds their weight. Generally, this occurs when the particles are on the order of 100 μm or smaller [4]. In these cases, the particles will form agglomerates, and in order to achieve good
L.R. Aarons et al. / Powder Technology 235 (2013) 18–26
mixing, these agglomerates would need to be broken up. However, in some cases, particle cohesion can serve to mitigate the natural tendency for the different particles to segregate [5–8]. Homogeneous mixtures are generally of much greater value than inhomogeneous mixtures. Naturally, having a homogeneous mixture means that it features the same chemical and physical properties throughout it, a generally favored characteristic. For example, in the pharmaceutical industry, when pills are created from mixtures of powders, it is desired for each of the pills to contain the same concentration of each of the components. Too much or too little of a particular constituent could result in harm to the consumer or ineffective treatment of the ailment. Another advantage of homogeneous mixing arises when the range of particle sizes is sufficiently large. In these materials, a homogeneous mixture will feature a greater solids' loading than an inhomogeneous mixture, as the smallest particles will fill in the voids between the largest particles. For example, a monodisperse collection of randomly arranged spheres cannot achieve solid volume fractions greater than the random close packed limit of about 0.64 [9,10], but if particles small enough to fit in the voids are added, larger solid volume fractions (approaching 1) can be reached (see, for example, [11]). Li and McCarthy [6–8] analyzed how the homogeneity of bidisperse particle mixtures subjected to shear in the presence of gravity is affected by the addition of water (introducing capillary forces) to the system and surfactants (altering the particle–water–air surface tension) to each group of particles. Using mathematical arguments (which were subjected to some experimental validation), they mapped out what combinations of particle size ratio, particle density ratio, shear rate, and surface tension would result in mixtures that are more homogeneous when the particles are wet than when they are dry. This analysis was extended to cases in which particles are cohesive and attract each other via the van der Waals force rather than capillary forces [5]. In this case the homogeneity of mixtures of cohesive particles was compared to that of mixtures of non-cohesive, but otherwise identical, particles under the same conditions. That being said, these analyses do not allow one to predict if any group of cohesive particles under one set of shearing conditions will mix better or worse if the cohesiveness of the particles or shearing conditions were altered (other than if cohesion were completely removed). Without a model to predict the homogeneity of cohesive particle mixtures, the quest to achieve the best possible mixtures often takes the form of trial and error. This of course leads to wasted time, money, and material, and so the need of such a model is immediately apparent. The primary purpose of the present study is to investigate the influence of cohesion on the mixing of polydisperse particles. To this end, we performed simulations analogous to the aforementioned shearing experiments performed by Li and McCarthy [6], but further explored and quantified the influence of particle cohesion and shear rate on the homogeneity of bidisperse mixtures. We have focused our study on the simulation of bidisperse mixtures that feature a large enough size range for the small particles to fit inside gaps between packed large particles, and as such, could feature a strong packing fraction dependence on homogeneity. As a first step, in this paper we investigate mixing performed in the absence of gravity (or any other body forces). That is, as opposed to the percolation example given earlier, there would be no force that would drive particles of a given size to accumulate in a particular direction. As such, the influence of cohesion on mixing is isolated. Details of these simulations and the metrics used to quantify homogeneity are given in Sections 2 and 3, respectively. Results of the simulations are presented and discussed in Section 4, where it will be shown how particle cohesion and shear rate affect the homogeneity of bidisperse mixtures. Finally, the main results are summarized in Section 5. 2. Details of discrete element simulations This work was approached by modeling the mixing of polydisperse mixtures of cohesive particles via discrete element model
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(DEM) simulations [12] and measuring the homogeneity of the resulting mixtures using multiple techniques. The simulation code used is a modified version of LAMMPS, an open source code developed by Sandia National Laboratories [13]. As simple model systems, we have simulated bidisperse collections of particles, with the large particles having a diameter seven times that of the small particles. This way, the small particles were small enough to fit inside the gaps between packed large particles but were not so small that the simulations required an impractically large number of particles. As typical with DEM, the particles were modeled as soft spheres, such that they were able to overlap, whereupon they would repel each other with forces analogous to those of inelastic springs. In these simulations, the normal force exerted by a pair of colliding particles was given by the linear spring–dashpot (LSD) model, which prescribes a normal force that is the sum of a Hookean spring force, proportional to the extent of overlap (α), and a viscous dashpot force, proportional to the relative normal velocities of the particles (vn), which introduces inelasticity. For colliding particles i and j, this normal force model is given by [14] as F n ¼ keff α−ηeff vn ;
ð1Þ
where for a particle property p, the effective value is given by peff = 2pipj/(pi + pj). For particle i, ki is the normal spring constant, where ki ~ Eidi, and Ei and di are respectively the Young's modulus and diameter of particle i. The damping coefficient of particle i, ηi, can be expressed as ηi ¼ mi η^ i , where mi and η^ i are respectively the mass and the material damping coefficient of i. For simplicity, particle properties without subscripts will refer to those of the small particles. The tangential spring–slider force model gives the force that the surfaces of the particles experience as they rub against each other, which is proportional to the distance the particle surfaces have moved relative to each other since the start of contact (Δs) but is bounded with a Coulombic (frictional) upper limit. It is given by F t ¼ min kt;ef f Δs; μ ij F n ;
ð2Þ
where kt,eff is the effective tangential spring constant and μij is the coefficient of friction between i and j [15–17]. In the simulations presented here, the tangential spring constant for each particle is set equal to its normal spring constant, and the coefficient of friction is set at 0.5 for the interaction between all particles. Cohesion between particles is simulated by having the particles attract each other via the van der Waals force. Particles i and j with radii Ri and Rj, respectively, and with surfaces separated by a distance sij, attract each other with a van der Waals force approximated in these simulations by [18]:
F vdW
3 qffiffiffiffiffiffiffiffiffi 32 Ai Aj Ri Rj sij þ Ri þ Rj ¼ 2 2 : s2ij þ 2Ri sij þ 2Rj sij þ 4Ri Rj 3s2ij sij þ 2Ri þ 2Rj
ð3Þ
The Hamaker constants of particles i and j, given by Ai and Aj, respectively, are material constants which effectively can be altered by coating the particles with surfactants. The influence of particle cohesion on the mixing behavior was examined by varying the Hamaker constants. The expression given by Eq. (3) diverges when particles collide, and so a minimum cutoff separation, smin, is used. That is, FvdW(s b smin) = FvdW(smin). This cutoff separation is generally taken to represent intermolecular distances, typical values of which are 0.4–1 nm [4,18]. When the surface separation becomes much smaller than the particles' radii, then the van der Waals force can be pffiffiffiffiffiffiffiffiffi approximated using the more familiar form, F vdW ¼ Reff Ai Aj =12s2ij . Because the van der Waals force drops off rapidly with increasing separation, to speed up the simulation, when two particles were
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separated by a distance greater than 1/4 of the diameter of the larger of the two particles, the van der Waals force was ignored. Rather than simulate complex mixing procedures typical in the preparation of particle composites (e.g. blade mixing), the mixing in these simulations was performed via simple plane shear flow, accomplished by placing particles between two walls and then moving the walls in opposite directions (along the x-axis). To provide a “rough” surface, the walls were made entirely of the small particles. The shearing walls bounded the particles in the vertical (z-) direction. Periodic boundary conditions were employed in the other directions to avoid wall effects in those directions as well as to take into account the effects of more particles than were actually simulated. To avoid constraining the available particle configurations, the length (L) and width (W) of the simulated systems were chosen to be 7 times the diameter of the large particles (i.e. L = W = 49d). To ensure a height (h) of at least about 49d, every simulation featured 367 big particles and 45,769 small particles. This combination of particles gave a total volume of large particles that was about three times the total volume of small particles. A few simulations discussed in the following sections were repeated with larger numbers of particles and boundaries to determine if there are any size effects, but the results showed little difference. To allow for the expansion and contraction of the particle assemblies, the top shearing wall was free to move in the vertical direction. This was done by assigning a finite mass and applying a constant downward force to the top wall. These quantities needed to be large enough so that the top wall did not just slide over the particle bed but small enough to prevent frequent large particle overlaps. Based on trial and error, as well as [19] for some guidance, an applied pressure of 2.87 × 10 −4 k/d and a total wall mass of 6.9 × 10 4ρd 3 were used. Each of the 2542 particles on the top wall was assigned an equal portion of the mass, and at every time step, the total forces exerted by the bulk particles on the top wall was calculated and divided equally among all the wall particles so that they all moved together. Meanwhile, the bottom wall's vertical position was held constant. It should be noted that the overall shear rate (rather than wall velocity) was held constant during each simulation, and so as the top wall moved up or down, the top and bottom walls' streamwise velocities increased or decreased, respectively, by a proportional amount. As will be discussed in Section 4, the shear rate in these simulations was not uniform, i.e. the velocity profile was not linear. This phenomenon is commonly observed during the shear of granular assemblies in simulations (see, for example, [17]) and real systems (such as those used in mixing, [6]) in which the shearing boundaries are sufficiently separated. In simulations, inhomogeneous shearing has been shown to occur even when Lees–Edwards boundary conditions [20] were used in place of solid walls to perform shear [17]. Using the functionality available in LAMMPS, it was easier to use solid shearing walls than Lees–Edwards boundary conditions. Furthermore, as vertically bounding walls are required for shearing in the presence of gravity (which will be covered in a future study), using walls here has made it more convenient to use the same initial conditions and virtually the same input files in both groups of simulations, as well as to compare the different simulations. The initial configurations of the particles featured the large and small particles completely separated side-by-side (see Fig. 1). To investigate and reduce the influence of the initial conditions, three different initial particle configurations were created for each set of Hamaker constants. All of the model equations can be cast in dimensionless form so that all the input parameters for the simulations were dimensionless. qffiffiffiffiffiffiffiffiffiffiffiffiffi 3 A dimensionless material damping coefficient of η^ ρd =k ¼ 0:220 was used in every simulation, giving a coefficient of restitution of 0.7 for collisions between small particles (see [17]). The scaled
Fig. 1. Image of an example of the initial particle configurations used. Large particles (of diameter 7d) are shown in light gray, small particles (of diameter d) are shown in medium gray, and the particles forming the walls are shown in dark gray. 3 2 stiffness, given by k ¼ k=ρd γ_ , where γ_ is the shear rate, is used to express the overall shear rate in dimensionless form when gravity is unimportant or simply nonexistent. This quantity is representative of the ratio of a characteristic potential (spring) energy to a characteristic kinetic energy [21]. The strength of particle attraction between two particles of the same size is expressed in dimensionless form by scaling it by the strength of repulsion of colliding particles, using the scaled Hamaker constant
Ai ¼
500Ai : 3ks2min
ð4Þ
When referring to the strength of cohesive interaction between two small or two large particles, the symbols S and B, respectively, are used in place of Ai*. The cohesive interaction between the small pffiffiffiffiffiffi and big particles can then be expressed as BS. Alternately, the cohesiveness of the particles can be scaled kinetically as
Ai k ¼
500Ai 3ρd3 γ_ 2 s2min :
ð5Þ
This quantity scales the van der Waals potential by the shear energy. This quantity can be seen as a measure of the competition between the particles' tendency to agglomerate (through cohesion) and the shear energy's tendency to break up agglomerates. The cutoff separation distance for the van der Waals force model is expressed in dimensionless form as smin/d. Assuming that the small particles in these simulations represent particles around 20 μm in diameter, a cutoff distance of smin/d = 2 × 10 −5 was used, giving a minimum separation of 0.4 nm. Lastly, the dimensionless mean small-particle cluster size is expressed simply as dcluster/d. ffiffiffiffiffi actual time required to shear for a given strain is proportional pThe to k , and so k⁎ cannot be arbitrarily large (i.e. the shear rate cannot be arbitrarily small) [17]. As will be explained in the next section, simulations were run for a strain of 300. In order to do this within what was considered to be an acceptable time, k⁎ could not exceed 10 9. Assuming particles with diameters on the order of tens of microns and a density on the order of 10 3 kg/m 3, k⁎ ≤ 10 9 would require either shear rates that far exceed those reached in real mixers or particles much softer than real materials [17]. Similarly, the values of S and B used were not necessarily representative of real materials. The multiplicative constant on the right side of Eq. (4) is there to
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give values of S and B of 0.1 to 1 and 0.01 to 10, respectively, when the particles were cohesive in the simulations discussed here. As detailed in [17], these values would be representative of only the softest or most cohesive of particles; S and B would be several orders of magnitude smaller for most materials. By using these values of S and B, as will be shown in the next section, the effects of cohesion became highly visible over the range of k⁎ used. Based on [19], as well as the general expectation that as cohesion is increased, an increase in shear rate is required to offset the effects, we fully expect that the trends observed in these simulations can be extended to more realistic systems that feature both smaller shear rates and less cohesive particles. 3. Homogeneity metrics Two metrics were used to quantify the homogeneity of the mixtures. The first metric used was the mean size of clusters of small particles. Of course, an increasing mean cluster size would be indicative of decreasing homogeneity. A method to calculate the mean cluster size was recently proposed by Gallier [22], summarized as follows. From a given particle configuration, a spherical sample of radius R was taken. Gallier [22] used R/L = 0.3, while for these simulations discussed here, a radius R/L = 15/49≈ 0.306 was used (purely out of convenience due to lengths being expressed in units of a small particle's diameter). Also, whereas Gallier [22] took the sample volume to be centered at the exact center of the mixture, the use of periodic boundaries in the horizontal directions here allowed for multiple sample volumes to be used. Specifically, four sample volumes were used, centered at the center of the mixture (at {x, y, z} = {L/2, L/2, h/2}), at the center of the left edge (at {x, y, z} = {0, L/2, h/2}), at the center of the front edge (at {x, y, z} = {L/2, 0, h/2}), and at the front left corner (at {x, y, z} = {0, 0, h/2}). Using only the N small particles whose centers lie within the sample of volume V = 4πR3/3, the K-Ripley function, K(r), was calculated [23,24]: K ðr Þ ¼
N V X H r−rij ; ∑ 2 N i¼1 j≠i
ð6Þ
where H is the Heaviside function. This quantity is defined as the ratio of the average number of particles within a distance r of each other to the number density, and is essentially equivalent to the radial distribution function integrated over space [22]. Then K(r) was normalized using the Besag L-function [25], rffiffiffiffiffiffiffiffiffiffiffiffi 3 3K ðr Þ Lðr Þ ¼ −r; 4π
ð7Þ
and from this the mean cluster size, dcluster, was estimated using 1=3
dcluster ¼ 2ϕs
L r ag þ r ag ;
ð8Þ
where ϕs is the total volume fraction of the small particles and the aggregate radius, rag, is the value of r at which L(r) achieved its maximum [22]. In the simulations discussed here, the cluster size at a given time was taken as the average of the mean cluster sizes calculated from the four sample volumes. The use of four sample volumes rather than one was found not to affect the time-averaged cluster size estimates, but instead reduced the fluctuations over time. In a homogeneous or simply monodisperse random collection of particles, one would expect the Besag L-function to reach its maximum at the distance between a small particle and its nearest neighbor (i.e. at r = d, neglecting the overlap between contacting particles). When r = d, the K-Ripley function reduces to K ðdÞ ¼
Z s−s ; ns
ð9Þ
21
where Zs − s is the average number of small particles each small particle contacts and ns is the number density of small particles. Using rag = d and Eq. (9), the mean cluster size in a homogeneous mixture can be given simply by dcluster = Zs1/3 − sd. Based on previous simulations of randomly arranged, dense, monodisperse assemblies of particles [10,17], one would expect that for homogeneous systems, an average small particle would be contacting about 5 to 6 other small particles, and so a mean cluster size for homogeneous mixtures would be approximately 1.7d–1.8d. That is, the optimum mean cluster size would be about 1.7d–1.8d, and deviation from this value could serve as a measure of ordering and inhomogeneity. The second metric is based on the spatial variance in the relative volume fractions of the small and large particles, analogous to the sampling technique as described in [26,27]. This was calculated by first dividing the system into a number of smaller sample volumes. As discussed in the next section, shear always eventually became confined to near one of the shearing over the course of a simulation. This, along with the fact that walls constrained the motions and arrangements of near-by particles, the region near one or both walls may be quite different from the bulk. As such, to reduce wall effects, the top and bottom 15.3% of the mixtures were ignored (somewhat similar to how dcluster is calculated here). The fraction of solid volume belonging to the large (or small) particles in each sample volume was found, and then the standard deviation over the sample volumes, σbulk, was calculated. In a bidisperse collection of particles, such as those discussed here, σbulk would not depend on whether it was calculated based on the large particles or the small particles. Of course, the mixtures become more homogeneous as σbulk decreases. The sample volume must be large enough to contain both small and large particles simultaneously, but cannot be so large that the spatial variance is lost [26]. As such, the sample volumes have been chosen to be cubes of length 10d, i.e. sufficiently larger than a large particle (5.6 times the volume of a large particle), while significantly smaller than the entire system (at most 1% of the total volume). To calculate the volume of a group of particles in a sample volume, no smoothing function was used. Rather, if a particle was physically at least partially in a given sample volume, the volume of portion of the particle located within that sample volume was calculated and added to its group's volume. Rather than pick sample volumes at random, they were regularly spaced in all three dimensions. To minimize error, a large number of sample volumes were used; the different sample volumes overlapped. When the number of sample volumes exceeded 1000 (10 by 10 by 10), σbulk negligibly changed with increasing number of sample volumes. The number of sample volumes used had a very weak influence on the overall simulation time, and so, even though it was overkill, a total of 25 3 sample volumes were used. In addition to these two metrics, the total solid volume fraction (or packing fraction), i.e. the ratio of the total particle volume to the system volume, was measured throughout each simulation. As aforementioned, an advantage of achieving homogeneous mixing with particles featuring a large size range is greater solids' loading. As such, homogeneity and solid volume fraction are correlated, and that correlation will be quantified and analyzed. 4. Results and discussion Starting with initially segregated particle collections, shearing led to a general decrease (on average) in σbulk and dcluster and an increase in solid volume fraction, as one would expect and as demonstrated in Fig. 2A–C for a small selection of k⁎, B, and S. It can be observed that after a little while, the rate at which these quantities evolved with strain generally slowed as shear progressed (i.e. decayed exponentially, more or less). Comparisons of the different mixtures needed to be done when the mixtures' evolution with strain had significantly slowed. As such, based on the data partly shown in Fig. 2A–C, it was decided that the mixtures would be compared at strains of 300.
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Fig. 3. The evolution of the streamwise velocity profile over the course of a mixing simulation. Plotted is the streamwise velocity (vx) scaled by the speed of the shearing wall (U) versus the vertical position (z) scaled by the height of the system (H) at strains of 0.5 (solid black line), 6 (dashed black line), and 300 (dash-dot gray line). Data is from a simulation in which S= 0.1, B = 0.01, and k⁎ = 109. The thin dash-dot black line corresponds to uniform shear.
only would have occurred near that wall. Inhomogeneous mixtures would arise when the shear becomes localized before the bulk can be well-mixed. Fig. 4 shows how the dimensionless mean small-particle cluster size, dcluster/d, varied with k⁎ and cohesion after shearing for a strain of 300. Each point represents the mean value obtained from three simulations, only different in their initial particle configurations. The error bar associated with each data point spans the entire range of values obtained from the three simulations. When the particles were not cohesive, dcluster/d was about 1.8, i.e. around the optimal value, regardless of shear rate. Therefore, the mean cluster size estimates would suggest that the mixtures of non-cohesive particles were homogeneous, just as one would likely expect. When the small particles were “slightly cohesive” (S = 0.1), dcluster/d did not differ much from the mixtures of non-cohesive particles, regardless of the cohesiveness of the big particles or shear rate. The small particles attracted each other too weakly to form large clusters that could survive shear at the rates explored, and so they could easily distribute themselves throughout the mixtures before the shear region settled near a wall. It is feasible to assume that as k⁎ becomes sufficiently larger
Fig. 2. The evolution of the (A) solid volume fraction (ϕ), (B) dimensionless mean small-particle cluster size (dcluster/d), and (C) standard deviation in the relative volume fraction of particles (σ) with strain after the start of shear for four different combinations of small-particle cohesion (S), big-particle cohesion (B), and scaled stiffness (k⁎). The evolution of the three quantities in every case slows significantly by a strain of 300. Each curve represents the data produced by a single simulation run.
To explain the behavior observed in Fig. 2A–C, we examine the evolution of the velocity profile over the course of a simulation. Fig. 3 shows an example of how the velocity profile evolved over a strain of 300. In the mixing simulations discussed here, the particles were at rest at the beginning. As shear proceeded, a velocity profile emerged. Eventually, the velocity profile shifted such that almost all the particles flowed as a plug, with a shear region confined to near the top (as shown in Fig. 3) or bottom of the mixture. As mixing would mostly occur where there is shear, mixing would proceed faster when shear is not localized than when shear is confined to a small region. Once the shear region became small and isolated near a wall, mixing in the bulk would have been negligible, and mixing
Fig. 4. The effect of particle cohesion on the mean small-particle cluster size versus shear rate behavior. Plotted is the dimensionless mean small-particle cluster size (d3 2 _ ). Each cluster/d) measured at a strain of 300 versus the scaled stiffness (k ¼ k=ρd γ data point represents an average from three simulations. The error bar associated with each data point covers the entire range of values obtained from the three simulations and is only shown when it extends outside its graph symbol.
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than 109, the mean cluster size would grow when S = 0.1, and so S would need to be even smaller to achieve homogeneous mixing. When the small particles were more cohesive (S = 1), they naturally would have had a greater tendency to attract each other and form large, long-lasting clusters. As such, dcluster was always at least as large as when S was smaller. Furthermore, dcluster exhibited a dependence on k⁎ and B that did not exist when S was smaller. The influence of k⁎ was rather straightforward, as increasing the shear rate and therefore the kinetic energy (decreasing k⁎) helped to shrink the agglomerates of small particles and to give them the inertia to get between the big particles. On the other hand, the influence of B on the mean cluster size was somewhat complex due to competing effects of varying B. Increasing B first of all simply hindered the freedom of the large particles' motion, making it more difficult to get good mixing. However, increasing B also led to the big particles packing more loosely (as observed in the solid volume fraction data discussed later and in [19]) and forming longer-lasting structures, making it easier for the small-particle clusters to fit between the big particles. As such, when k⁎ = 10 9, the smallest dcluster was achieved at an intermediate value of B (i.e. B = 1). The largest dcluster was observed when B was 0.1, as much of the small-particle agglomerates simply could not fit between the big particles. Fig. 4 also shows the weak influence of the cohesive interaction between the large and small particles on mixing. This can be illustrated by looking at the data for S = 0.1 and B = 1 and the data for S = 1 and B = 0.1. The strength of the van der Waals interaction pffiffiffiffiffiffi between the small and large particles is the same in both cases BS ¼ 10−0:5 . However, as just discussed, the corresponding mixtures featured very different degrees of homogeneity. Instead, the homogeneity at a given k⁎ depended primarily on S and secondarily on B. Had the homogeneity depended strongly on the cohesive interaction between the different particles, the influence of B would have been just as strong as that of S. Given that the mean cluster size can be related to the particles' ability to agglomerate, and this ability can be quantified using the kinetically scaled Hamaker constants, we replot the data in Fig. 4 using Sk⁎ and Bk⁎ in Fig. 5 (with error bars removed). This plot shows that multiple mean cluster sizes were achieved for each set of Sk⁎ and Bk⁎; dcluster/d was not a strict function of Sk⁎ and Bk⁎, but rather depended on k⁎ (or similarly, S or B) as well. As such, this figure shows that the transition between homogeneous mixing and inhomogeneous mixing cannot be captured by a single value of Sk⁎ or even a set of Sk⁎ and
Bk⁎ values. Nevertheless, decreasing Sk⁎ while keeping Bk⁎ constant led dcluster/d to decrease until it reached the optimal value of about 1.8, after which it stayed constant. That is, as the small particles became less able to agglomerate, they were able to form more homogeneous mixtures, as previously suggested. The influence of the large particles' ability to form agglomerates on dcluster/d is analogous to the complicated behavior of varying B in Fig. 4. Given that scaling the cohesiveness kinetically did not help to collapse or otherwise simplify the results (and while it will not be shown, this scaling actually would complicate the volume fraction results), this scaling will no longer be used. Fig. 6 shows how the standard deviation in the relative volume fraction of particles in the bulk, σbulk, depended on the scaled stiffness and particle cohesiveness after a strain of 300. Comparing this figure with Fig. 4, it can be seen that for the most part the behaviors of σbulk and dcluster were rather similar to each other (described in the preceding discussion) but featured some differences. First of all, whereas dcluster/d for all cases except S = 1 and B = 0.1 collapsed onto a single value when k⁎ = 10 7, this did not occur for σbulk. Also, when the particles were not cohesive or S = 0.1, σbulk manifested a dependence on k⁎ and B not observed with dcluster/d. For example, when the particles were not cohesive, σbulk decreased with k⁎. All in all, though, the influences of B and k⁎ on σbulk in these cases were much smaller than when S = 1, and so it is questionable if this variation in σbulk was indicative of a significant variation in homogeneity. An explanation for why some of these differences between the behaviors of dcluster/d and σbulk arose (rather than being representative of conflicting data) is that the regions sampled in the calculation of the statistics were different. Because dcluster was calculated based on spherical samples centered at the vertical center of the mixture, it took into account the particles closer to the vertical center more than those farther away; this was not true in the calculation of σbulk. Thus if wall effects extended far enough to creep into the measurements, they would have been captured more by σbulk than dcluster and would have contributed most significantly when σbulk was small. To support and explain the data and conclusions presented so far, images of the mixtures' microstructures were created. Fig. 7A–E show images, generated by POV-Ray [29], of vertical slices of some of the different mixtures generated at a strain of 300 after shear was started. While these images are from one set of simulations, the other two sets of simulations performed with the same groups of parameters produced similar microstructures. Simulations without cohesion and most simulations with S = 0.1 produced visually homogeneous
Fig. 5. The influence of kinetically scaled particle cohesion on the mean small-particle cluster size. Plotted is the dimensionless mean small-particle cluster size (dcluster/d) measured at a strain of 300 versus the kinetically scaled small-particle Hamaker constant (Sk⁎) for different values of the kinetically scaled large-particle Hamaker con3 2 stant (Bk⁎). The symbol color represents the scaled stiffness ( k ¼ k=ρd γ_ ), with black representing k⁎ = 107, dark gray representing k⁎ = 108, and light gray representing k⁎ = 109. Each data point represents an average from three simulations.
Fig. 6. The effect of particle cohesion on the standard deviation in the relative volume fraction of particles versus shear rate behavior based on the particles in the bulk of the mixtures. Plotted is the standard deviation in the relative volume fraction of particles 3 2 (σbulk), measured at a strain of 300, versus the scaled stiffness (k ¼ k=ρd γ_ ). Each data point represents an average from three simulations. The error bar associated with each data point covers the entire range of values obtained from the three simulations and is only shown when it extends outside its corresponding graph symbol.
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3 2 Fig. 7. Images of the microstructure of different mixtures after shearing for a strain of 300 with a scaled stiffness, k ¼ k=ρd γ_ , of 109. Shown are vertical slices of mixtures obtained with (A) no particle cohesion, (B) S= 0.1 and B = 0.01, (C) S = 1 and B = 0.1, (D) S = 1 and B = 1, and (E) S= 1 and B = 10. Large particles (of diameter 7d) are shown in light gray, small particles (of diameter d) are shown in medium gray, and the particles forming the walls are shown in dark gray.
mixtures similar to what is shown in Fig. 7A. When S = 0.1, k⁎ = 10 9, and B = 0.01 or 0.1, the mixtures feature a homogeneous bulk and a less homogeneous wall region, as shown in Fig. 7B, where a void region exists near the center of the bottom wall. This may explain (at least in part) the slight dependence σbulk has on k⁎ and B that
dcluster/d does not have. Fig. 7C–E shows the inhomogeneous mixtures produced when S = 1 and k⁎ = 10 9. These images support the findings that under these conditions, the worst mixing was achieved when B = 0.1 (Fig. 7C), while the best mixing was obtained with B = 1 (Fig. 7D).
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Fig. 8 shows how the solid volume fraction, ϕ, measured at a strain of 300, varied with the scaled stiffness and particle cohesiveness. If the particles were monodisperse, one would expect that as k⁎ decreases, the solid volume fraction either would remain virtually constant or decrease slightly, indicative of elastic–quasistatic flow, or would decrease significantly, indicative of inertial or elastic–inertial flow [19,28]. Thus when the particles are polydisperse and feature a large range in size, an increase in solid volume fraction with a decrease in k⁎ would likely be attributed to more small particles fitting in between the big particles, i.e. an increase in homogeneity. This is most evident when S = 1, as decreasing k⁎ increased homogeneity and solid volume fraction. However, when S = 0.1 and B =1 and k⁎ was decreased from 109 to 108, the solid volume fraction increased noticeably despite a negligible change in homogeneity. When k⁎ =109, the small particles would not have agglomerated with each other much but instead (at least partially) would have coated the more cohesive big particles, forming a homogeneous mixture. At the same time, the particles formed structures that limited the solid volume fraction. Decreasing k⁎ to 108 gave the small particles enough energy to escape the pull of the large particles, so they could pack more tightly while still achieving a homogeneous mixture. The influence of cohesion on solid volume fraction would be two-fold, as it would directly affect solid volume fraction by causing the small and large particles to pack less tightly on their own and would indirectly affect solid volume fraction by influencing homogeneity. When S was increased from 0 to 0.1, the small particles still did not agglomerate much and so they had little effect on the solid volume fraction. As such, at a given shear rate, the solid volume fraction was dictated by how efficiently the big particles packed. So as B increased, the big particles of course packed less densely, and a decrease in solid volume fraction was observed. When S was increased to 1, the small particles agglomerated enough to significantly lower the solid volume fraction, both because of a reduction in homogeneity and because they packed less densely. As B increased and the big particles packed less densely, the solid volume fraction decreased monotonically, unaffected by the improvement in homogeneity as B went from 0.1 to 1. This behavior likely arose because the large particles constitute 73% of the solid volume and form the “backbone” of the mixtures. Before closing this section, it should be emphasized that the results presented here are from simulations performed under very specific conditions, most notably with a bidisperse mixture featuring a 7:1 size ratio and in the absence of gravity. That being said, these
Fig. 8. The effect of particle cohesion on the solid volume fraction versus shear rate behavior. Plotted is fraction (ϕ), measured at a strain of 300, versus the the solid volume 3 2 scaled stiffness k ¼ k=ρd γ_ . Each data point represents an average from three simulations. The error bar associated with each data point covers the entire range of values obtained from the three simulations and is only shown when it extends outside its corresponding graph symbol.
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results may provide insight into mixing under some other conditions. Of course, in the presence of gravity, there will be gravity-driven size segregation, and so the conclusions presented in this paper likely cannot be readily extended to those situations. Otherwise, should the particle size range be expanded, the mixing mechanics remain essentially the same. If anything, it would make it easier for the small particles fit between the large particles, and so the small particles would need to be more cohesive for the mixtures to become noticeably inhomogeneous. On the other hand, when the size ratio becomes smaller (less than 6.46:1), the mixing mechanics are a bit different, as the small particles cannot fit between the gaps formed by packed large particles. Rather, bed dilation would need to occur to give the smaller particles room to intermingle with the large particles. As such, we would not expect the homogeneity and solid volume fraction to qualitatively depend on cohesion and shear rate in the same way as described in this paper. 5. Summary In this study, we have investigated the influence of cohesion on the homogeneity and solid volume fraction of bidisperse collections of particles subjected to plane shear between two walls as a means of mixing in the absence of gravity. Attention was focused on particles featuring a 7:1 diameter ratio, such that mixing would involve the small particles filling in the gaps between the large particles. The homogeneity of the mixtures was quantified by measuring the mean size of the small-particle clusters, dcluster, as defined by [22], and the standard deviation in the relative concentration of the different particles in the bulk, σbulk, at a strain of 300 after starting with a completely segregated arrangement. Images of the mixtures' microstructure were also used to capture homogeneity. In short, a mixture's homogeneity depended on how easily the small particles were able to fit inside the gaps between the big particles. Inhomogeneous mixtures were observed only when the small particles were able enough to form and maintain agglomerates. In these cases, the agglomerates, rather than individual particles, had to find their way between the big particles, and homogeneity, as measured by dcluster and σbulk, was greatly improved with decreasing scaled stiffness 3 2 k ¼ k=ρd γ_ and was significantly influenced by the cohesiveness of the big particles. If the big particles were not sufficiently cohesive, they packed too tightly for the small-particle agglomerates to get between them, resulting in the most inhomogeneous mixtures. As the big particles became more cohesive, the small-particle agglomerates had more room into which they can fit, improving homogeneity until the cohesion restricted particle motion so much that mixing was prevented and homogeneity worsened. Furthermore, as k⁎ decreased, the particles had more kinetic energy to mix and break up agglomerates, naturally improving homogeneity. On the other hand, when the small particles were less cohesive or the particles were not cohesive at all, the mixtures appeared rather homogeneous, especially in the bulk. In these cases, dcluster/d was measured to be about 1.8, close to the lowest possible value, for all values of k⁎ and large-particle cohesiveness explored. As long as the particles were not negligibly cohesive at a given shear rate, the mixture's solid volume fraction increased with shear rate, generally as a result of improving homogeneity. Otherwise, like monodisperse materials in the elastic–quasistatic regime, the volume fraction decreased slightly with increasing shear rate. Meanwhile, as the particles became more cohesive, the solid volume fraction decreased, regardless if homogeneity was improved or not. Acknowledgments This work was funded by the U.S. Air Force Research Laboratory under contract number FA8651-08-D-0108, as part of the Florida
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Institute for Research in Energetics (FIRE). Some computational resources and work space were provided by the Eglin Air Force Base. References [1] G. Plantard, H. Saadaoui, P. Snabre, B. Pouligny, Surface-roughness-driven segregation in a granular slurry under shear, Europhysics Letters 75 (2) (2006) 335–341. [2] J.C. Williams, The segregation of particulate materials, Powder Technology 15 (1976) 245–251. [3] In: M. Rhodes (Ed.), Introduction to Particle Technology, John Wiley and Sons, New York, 1998. [4] J.P.K. Seville, C.D. Willett, P.C. Knight, Interparticle forces in fluidisation: a review, Powder Technology 113 (3) (2000) 261–268. [5] H. Li, Impact of cohesion forces on particle mixing and segregation, PhD Thesis, University of Pittsburgh, Pittsburgh, PA, 2005. [6] H. Li, J.J. McCarthy, Cohesive particle mixing and segregation under shear, Powder Technology 164 (2006) 58–64. [7] H. Li, J.J. McCarthy, Phase diagrams for cohesive particle mixing and segregation, Physical Review E 71 (2) (2005). [8] H. Li, J.J. McCarthy, Controlling cohesive particle mixing and segregation, Physical Review Letters 90 (18) (2003). [9] S. Torquato, T.M. Truskett, P.G. Debenedetti, Is random close packing of spheres well defined? Physical Review Letters 84 (10) (2000) 2064. [10] A.Z. Zinchenko, Algorithm for random close packing of spheres with periodic boundary conditions, Journal of Computational Physics 114 (1994) 298–307. [11] H.Y. Sohn, C. Moreland, The effect of particle size distribution on packing density, Canadian Journal of Chemical Engineering 46 (1968) 162–167. [12] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1) (1979) 47–65. [13] S.J. Plimpton, Fast parallel algorithms for short-range molecular dynamics, Journal of Comparative Psychology 117 (1995) 1–19. [14] L.E. Silbert, D. Ertas, G.S. Grest, T.C. Halsey, D. Levine, S.J. Plimpton, Granular flow down an inclined plane: Bagnold scaling and rheology, Physical Review E 64 (5) (2001) 051302.
[15] L. Vu-Quoc, X. Zhang, An accurate and efficient tangential force-displacement model for elastic frictional contact in particle-flow simulations, Mechanics of Materials 31 (4) (1999) 235–269. [16] O.R. Walton, R.L. Braun, Viscosity, granular-temperature, and stress calculations for shearing assemblies of inelastic, frictional disks, Journal of Rheology 30 (5) (1986) 949–980. [17] L. Aarons, S. Sundaresan, Shear flow of assemblies of cohesive and non-cohesive granular materials, Powder Technology 169 (1) (2006) 10–21. [18] R.Y. Yang, R.P. Zou, A.B. Yu, Computer simulation of the packing of fine particles, Physical Review E 62 (3) (2000) 3900–3908. [19] L. Aarons, S. Sundaresan, Shear flow of assemblies of cohesive granular materials under constant applied normal stress, Powder Technology 183 (2008) 340–355. [20] A.W. Lees, S.F. Edwards, The computer study of transport processes under extreme conditions, Journal of Physics C: Solid State Physics 5 (1972) 1921–1929. [21] C.S. Campbell, Granular shear flows at the elastic limit, Journal of Fluid Mechanics 465 (2002) 261–291. [22] S. Gallier, A stochastic pocket model for aluminum agglomeration in solid propellants, Propellants, Explos, Pyrotechnics 34 (2009) 97–105. [23] B.D. Ripley, The second-order analysis of stationary point process, Journal of Applied Probability 13 (1976) 255. [24] B.D. Ripley, Modelling spatial patterns, Journal of the Royal Statistical Society B39 (1977) 172. [25] J. Besag, Contribution to the discussion of Dr. Ripley's paper, Journal of the Royal Statistical Society B39 (1977) 193. [26] L.T. Fan, S.J. Chen, C.A. Watson, Solids mixing, Industrial and Engineering Chemistry 62 (7) (1970) 52–69. [27] L.T. Fan, J.R. Too, R.M. Rubison, F.S. Lai, Studies on multicomponent solids mixing and mixtures part III: mixing indices, Powder Technology 24 (1979) 73–89. [28] C.S. Campbell, Stress-controlled elastic granular shear flows, Journal of Fluid Mechanics 539 (2005) 273–297. [29] Persistence of Vision Pty. Ltd., Persistence of Vision™ Raytracer, Persistence of Vision Pty. Ltd., Williamstown, Victoria, Australia, 2004. http://www.povray.org/.
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
The mixing of cohesive granular materials featuring a large size range in the absence of gravity Lee R. Aarons a,⁎, S. Balachandar a, Yasuyuki Horie b a b
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA Air Force Research Laboratory, Munitions Directorate, Eglin AFB, FL 32542, USA
a r t i c l e
i n f o
Article history: Received 11 April 2012 Received in revised form 28 August 2012 Accepted 23 September 2012 Available online 6 October 2012 Keywords: Mixing Cohesion Shear flow of powders Discrete element simulation
a b s t r a c t We have studied the shear mixing of bidisperse collections of cohesive particles in an effort to develop models that would allow one to predict and control the homogeneity of particle composites. Our focus has been on the effects of interparticle cohesion and shear rate on the microstructure of particle composites. Furthermore, we have focused on particles that have a “large size range,” specifically a 7:1 diameter ratio, such that homogeneous mixtures would include the small particles filling in the gaps formed between big particles, resulting in a correlation between the packing fraction and mixing quality. As a model problem, the cohesion resulting from the van der Waals force acting between particles was considered. Simulations were performed in which initially segregated bidisperse collections of particles were subjected to plane shear under constant applied stress as a method of mixing. Gravity was ignored in these simulations so that the different particles were not driven to different sides of the mixtures and the only hindrance to homogeneous mixing was cohesion. Simulations were performed with a variety of shear rates and particle cohesion strengths for both the large and small particles, and the homogeneity of the resultant mixtures was quantified using two distinct statistics: the estimated mean size of small-particle clusters and the spatial variance in the relative concentrations of the small and large particles. Microstructure images of the mixtures were used to provide additional qualitative measure of homogeneity as well as a measure of the relevance of the order statistics. These data suggested that the cohesiveness (Hamaker constant) of the small particles had the strongest influence on the mixture's homogeneity. When the small particles were not sufficiently cohesive, they did not significantly agglomerate, and so the resulting mixtures were relatively homogeneous over the range of shear rates and Hamaker constants of the big particles explored here. When the small particles were more cohesive, the small particles formed strong agglomerates and the mixtures become significantly inhomogeneous at lower shear rates. Somewhat surprisingly in these cases, better mixing was not achieved by simply making the large particles less cohesive. Rather, reducing the cohesiveness of the large particles far enough caused the large particles to pack more tightly, making the small-particle agglomerates unable to fit in between them, ultimately resulting in worse mixing. As such, the best mixing in these cases was achieved when the big particles were moderately cohesive. A correlation between solid volume fraction and homogeneity was not observed when particle cohesion was varied, as making either the small or large particles more cohesive led to a decrease in solid volume fraction, regardless of the effect on homogeneity. On the other hand, when homogeneity was found to increase with shear rate, so did the solid volume fraction. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The mixing of powders is well-known to be a generally difficult task. When any collection of particles is sufficiently disturbed, particles with similar properties tend to collect with each other and segregate themselves from unlike particles. Segregation generally arises from any differences between the particles that would affect their ⁎ Corresponding author. E-mail address: [email protected] (L.R. Aarons). 0032-5910/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2012.09.044
dynamics, such as size, density, and roughness [1,2]. For example, when flow occurs in a collection of differently sized particles, the bed will dilate, and in the presence of gravity, the small particles will fall through the space yielded by big particles, ending up at the bottom of the collection in a process known as percolation [3]. A further complication to mixing arises when the particles are small enough that the inter-particle attraction (e.g. from the van der Waals force) exceeds their weight. Generally, this occurs when the particles are on the order of 100 μm or smaller [4]. In these cases, the particles will form agglomerates, and in order to achieve good
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mixing, these agglomerates would need to be broken up. However, in some cases, particle cohesion can serve to mitigate the natural tendency for the different particles to segregate [5–8]. Homogeneous mixtures are generally of much greater value than inhomogeneous mixtures. Naturally, having a homogeneous mixture means that it features the same chemical and physical properties throughout it, a generally favored characteristic. For example, in the pharmaceutical industry, when pills are created from mixtures of powders, it is desired for each of the pills to contain the same concentration of each of the components. Too much or too little of a particular constituent could result in harm to the consumer or ineffective treatment of the ailment. Another advantage of homogeneous mixing arises when the range of particle sizes is sufficiently large. In these materials, a homogeneous mixture will feature a greater solids' loading than an inhomogeneous mixture, as the smallest particles will fill in the voids between the largest particles. For example, a monodisperse collection of randomly arranged spheres cannot achieve solid volume fractions greater than the random close packed limit of about 0.64 [9,10], but if particles small enough to fit in the voids are added, larger solid volume fractions (approaching 1) can be reached (see, for example, [11]). Li and McCarthy [6–8] analyzed how the homogeneity of bidisperse particle mixtures subjected to shear in the presence of gravity is affected by the addition of water (introducing capillary forces) to the system and surfactants (altering the particle–water–air surface tension) to each group of particles. Using mathematical arguments (which were subjected to some experimental validation), they mapped out what combinations of particle size ratio, particle density ratio, shear rate, and surface tension would result in mixtures that are more homogeneous when the particles are wet than when they are dry. This analysis was extended to cases in which particles are cohesive and attract each other via the van der Waals force rather than capillary forces [5]. In this case the homogeneity of mixtures of cohesive particles was compared to that of mixtures of non-cohesive, but otherwise identical, particles under the same conditions. That being said, these analyses do not allow one to predict if any group of cohesive particles under one set of shearing conditions will mix better or worse if the cohesiveness of the particles or shearing conditions were altered (other than if cohesion were completely removed). Without a model to predict the homogeneity of cohesive particle mixtures, the quest to achieve the best possible mixtures often takes the form of trial and error. This of course leads to wasted time, money, and material, and so the need of such a model is immediately apparent. The primary purpose of the present study is to investigate the influence of cohesion on the mixing of polydisperse particles. To this end, we performed simulations analogous to the aforementioned shearing experiments performed by Li and McCarthy [6], but further explored and quantified the influence of particle cohesion and shear rate on the homogeneity of bidisperse mixtures. We have focused our study on the simulation of bidisperse mixtures that feature a large enough size range for the small particles to fit inside gaps between packed large particles, and as such, could feature a strong packing fraction dependence on homogeneity. As a first step, in this paper we investigate mixing performed in the absence of gravity (or any other body forces). That is, as opposed to the percolation example given earlier, there would be no force that would drive particles of a given size to accumulate in a particular direction. As such, the influence of cohesion on mixing is isolated. Details of these simulations and the metrics used to quantify homogeneity are given in Sections 2 and 3, respectively. Results of the simulations are presented and discussed in Section 4, where it will be shown how particle cohesion and shear rate affect the homogeneity of bidisperse mixtures. Finally, the main results are summarized in Section 5. 2. Details of discrete element simulations This work was approached by modeling the mixing of polydisperse mixtures of cohesive particles via discrete element model
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(DEM) simulations [12] and measuring the homogeneity of the resulting mixtures using multiple techniques. The simulation code used is a modified version of LAMMPS, an open source code developed by Sandia National Laboratories [13]. As simple model systems, we have simulated bidisperse collections of particles, with the large particles having a diameter seven times that of the small particles. This way, the small particles were small enough to fit inside the gaps between packed large particles but were not so small that the simulations required an impractically large number of particles. As typical with DEM, the particles were modeled as soft spheres, such that they were able to overlap, whereupon they would repel each other with forces analogous to those of inelastic springs. In these simulations, the normal force exerted by a pair of colliding particles was given by the linear spring–dashpot (LSD) model, which prescribes a normal force that is the sum of a Hookean spring force, proportional to the extent of overlap (α), and a viscous dashpot force, proportional to the relative normal velocities of the particles (vn), which introduces inelasticity. For colliding particles i and j, this normal force model is given by [14] as F n ¼ keff α−ηeff vn ;
ð1Þ
where for a particle property p, the effective value is given by peff = 2pipj/(pi + pj). For particle i, ki is the normal spring constant, where ki ~ Eidi, and Ei and di are respectively the Young's modulus and diameter of particle i. The damping coefficient of particle i, ηi, can be expressed as ηi ¼ mi η^ i , where mi and η^ i are respectively the mass and the material damping coefficient of i. For simplicity, particle properties without subscripts will refer to those of the small particles. The tangential spring–slider force model gives the force that the surfaces of the particles experience as they rub against each other, which is proportional to the distance the particle surfaces have moved relative to each other since the start of contact (Δs) but is bounded with a Coulombic (frictional) upper limit. It is given by F t ¼ min kt;ef f Δs; μ ij F n ;
ð2Þ
where kt,eff is the effective tangential spring constant and μij is the coefficient of friction between i and j [15–17]. In the simulations presented here, the tangential spring constant for each particle is set equal to its normal spring constant, and the coefficient of friction is set at 0.5 for the interaction between all particles. Cohesion between particles is simulated by having the particles attract each other via the van der Waals force. Particles i and j with radii Ri and Rj, respectively, and with surfaces separated by a distance sij, attract each other with a van der Waals force approximated in these simulations by [18]:
F vdW
3 qffiffiffiffiffiffiffiffiffi 32 Ai Aj Ri Rj sij þ Ri þ Rj ¼ 2 2 : s2ij þ 2Ri sij þ 2Rj sij þ 4Ri Rj 3s2ij sij þ 2Ri þ 2Rj
ð3Þ
The Hamaker constants of particles i and j, given by Ai and Aj, respectively, are material constants which effectively can be altered by coating the particles with surfactants. The influence of particle cohesion on the mixing behavior was examined by varying the Hamaker constants. The expression given by Eq. (3) diverges when particles collide, and so a minimum cutoff separation, smin, is used. That is, FvdW(s b smin) = FvdW(smin). This cutoff separation is generally taken to represent intermolecular distances, typical values of which are 0.4–1 nm [4,18]. When the surface separation becomes much smaller than the particles' radii, then the van der Waals force can be pffiffiffiffiffiffiffiffiffi approximated using the more familiar form, F vdW ¼ Reff Ai Aj =12s2ij . Because the van der Waals force drops off rapidly with increasing separation, to speed up the simulation, when two particles were
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separated by a distance greater than 1/4 of the diameter of the larger of the two particles, the van der Waals force was ignored. Rather than simulate complex mixing procedures typical in the preparation of particle composites (e.g. blade mixing), the mixing in these simulations was performed via simple plane shear flow, accomplished by placing particles between two walls and then moving the walls in opposite directions (along the x-axis). To provide a “rough” surface, the walls were made entirely of the small particles. The shearing walls bounded the particles in the vertical (z-) direction. Periodic boundary conditions were employed in the other directions to avoid wall effects in those directions as well as to take into account the effects of more particles than were actually simulated. To avoid constraining the available particle configurations, the length (L) and width (W) of the simulated systems were chosen to be 7 times the diameter of the large particles (i.e. L = W = 49d). To ensure a height (h) of at least about 49d, every simulation featured 367 big particles and 45,769 small particles. This combination of particles gave a total volume of large particles that was about three times the total volume of small particles. A few simulations discussed in the following sections were repeated with larger numbers of particles and boundaries to determine if there are any size effects, but the results showed little difference. To allow for the expansion and contraction of the particle assemblies, the top shearing wall was free to move in the vertical direction. This was done by assigning a finite mass and applying a constant downward force to the top wall. These quantities needed to be large enough so that the top wall did not just slide over the particle bed but small enough to prevent frequent large particle overlaps. Based on trial and error, as well as [19] for some guidance, an applied pressure of 2.87 × 10 −4 k/d and a total wall mass of 6.9 × 10 4ρd 3 were used. Each of the 2542 particles on the top wall was assigned an equal portion of the mass, and at every time step, the total forces exerted by the bulk particles on the top wall was calculated and divided equally among all the wall particles so that they all moved together. Meanwhile, the bottom wall's vertical position was held constant. It should be noted that the overall shear rate (rather than wall velocity) was held constant during each simulation, and so as the top wall moved up or down, the top and bottom walls' streamwise velocities increased or decreased, respectively, by a proportional amount. As will be discussed in Section 4, the shear rate in these simulations was not uniform, i.e. the velocity profile was not linear. This phenomenon is commonly observed during the shear of granular assemblies in simulations (see, for example, [17]) and real systems (such as those used in mixing, [6]) in which the shearing boundaries are sufficiently separated. In simulations, inhomogeneous shearing has been shown to occur even when Lees–Edwards boundary conditions [20] were used in place of solid walls to perform shear [17]. Using the functionality available in LAMMPS, it was easier to use solid shearing walls than Lees–Edwards boundary conditions. Furthermore, as vertically bounding walls are required for shearing in the presence of gravity (which will be covered in a future study), using walls here has made it more convenient to use the same initial conditions and virtually the same input files in both groups of simulations, as well as to compare the different simulations. The initial configurations of the particles featured the large and small particles completely separated side-by-side (see Fig. 1). To investigate and reduce the influence of the initial conditions, three different initial particle configurations were created for each set of Hamaker constants. All of the model equations can be cast in dimensionless form so that all the input parameters for the simulations were dimensionless. qffiffiffiffiffiffiffiffiffiffiffiffiffi 3 A dimensionless material damping coefficient of η^ ρd =k ¼ 0:220 was used in every simulation, giving a coefficient of restitution of 0.7 for collisions between small particles (see [17]). The scaled
Fig. 1. Image of an example of the initial particle configurations used. Large particles (of diameter 7d) are shown in light gray, small particles (of diameter d) are shown in medium gray, and the particles forming the walls are shown in dark gray. 3 2 stiffness, given by k ¼ k=ρd γ_ , where γ_ is the shear rate, is used to express the overall shear rate in dimensionless form when gravity is unimportant or simply nonexistent. This quantity is representative of the ratio of a characteristic potential (spring) energy to a characteristic kinetic energy [21]. The strength of particle attraction between two particles of the same size is expressed in dimensionless form by scaling it by the strength of repulsion of colliding particles, using the scaled Hamaker constant
Ai ¼
500Ai : 3ks2min
ð4Þ
When referring to the strength of cohesive interaction between two small or two large particles, the symbols S and B, respectively, are used in place of Ai*. The cohesive interaction between the small pffiffiffiffiffiffi and big particles can then be expressed as BS. Alternately, the cohesiveness of the particles can be scaled kinetically as
Ai k ¼
500Ai 3ρd3 γ_ 2 s2min :
ð5Þ
This quantity scales the van der Waals potential by the shear energy. This quantity can be seen as a measure of the competition between the particles' tendency to agglomerate (through cohesion) and the shear energy's tendency to break up agglomerates. The cutoff separation distance for the van der Waals force model is expressed in dimensionless form as smin/d. Assuming that the small particles in these simulations represent particles around 20 μm in diameter, a cutoff distance of smin/d = 2 × 10 −5 was used, giving a minimum separation of 0.4 nm. Lastly, the dimensionless mean small-particle cluster size is expressed simply as dcluster/d. ffiffiffiffiffi actual time required to shear for a given strain is proportional pThe to k , and so k⁎ cannot be arbitrarily large (i.e. the shear rate cannot be arbitrarily small) [17]. As will be explained in the next section, simulations were run for a strain of 300. In order to do this within what was considered to be an acceptable time, k⁎ could not exceed 10 9. Assuming particles with diameters on the order of tens of microns and a density on the order of 10 3 kg/m 3, k⁎ ≤ 10 9 would require either shear rates that far exceed those reached in real mixers or particles much softer than real materials [17]. Similarly, the values of S and B used were not necessarily representative of real materials. The multiplicative constant on the right side of Eq. (4) is there to
L.R. Aarons et al. / Powder Technology 235 (2013) 18–26
give values of S and B of 0.1 to 1 and 0.01 to 10, respectively, when the particles were cohesive in the simulations discussed here. As detailed in [17], these values would be representative of only the softest or most cohesive of particles; S and B would be several orders of magnitude smaller for most materials. By using these values of S and B, as will be shown in the next section, the effects of cohesion became highly visible over the range of k⁎ used. Based on [19], as well as the general expectation that as cohesion is increased, an increase in shear rate is required to offset the effects, we fully expect that the trends observed in these simulations can be extended to more realistic systems that feature both smaller shear rates and less cohesive particles. 3. Homogeneity metrics Two metrics were used to quantify the homogeneity of the mixtures. The first metric used was the mean size of clusters of small particles. Of course, an increasing mean cluster size would be indicative of decreasing homogeneity. A method to calculate the mean cluster size was recently proposed by Gallier [22], summarized as follows. From a given particle configuration, a spherical sample of radius R was taken. Gallier [22] used R/L = 0.3, while for these simulations discussed here, a radius R/L = 15/49≈ 0.306 was used (purely out of convenience due to lengths being expressed in units of a small particle's diameter). Also, whereas Gallier [22] took the sample volume to be centered at the exact center of the mixture, the use of periodic boundaries in the horizontal directions here allowed for multiple sample volumes to be used. Specifically, four sample volumes were used, centered at the center of the mixture (at {x, y, z} = {L/2, L/2, h/2}), at the center of the left edge (at {x, y, z} = {0, L/2, h/2}), at the center of the front edge (at {x, y, z} = {L/2, 0, h/2}), and at the front left corner (at {x, y, z} = {0, 0, h/2}). Using only the N small particles whose centers lie within the sample of volume V = 4πR3/3, the K-Ripley function, K(r), was calculated [23,24]: K ðr Þ ¼
N V X H r−rij ; ∑ 2 N i¼1 j≠i
ð6Þ
where H is the Heaviside function. This quantity is defined as the ratio of the average number of particles within a distance r of each other to the number density, and is essentially equivalent to the radial distribution function integrated over space [22]. Then K(r) was normalized using the Besag L-function [25], rffiffiffiffiffiffiffiffiffiffiffiffi 3 3K ðr Þ Lðr Þ ¼ −r; 4π
ð7Þ
and from this the mean cluster size, dcluster, was estimated using 1=3
dcluster ¼ 2ϕs
L r ag þ r ag ;
ð8Þ
where ϕs is the total volume fraction of the small particles and the aggregate radius, rag, is the value of r at which L(r) achieved its maximum [22]. In the simulations discussed here, the cluster size at a given time was taken as the average of the mean cluster sizes calculated from the four sample volumes. The use of four sample volumes rather than one was found not to affect the time-averaged cluster size estimates, but instead reduced the fluctuations over time. In a homogeneous or simply monodisperse random collection of particles, one would expect the Besag L-function to reach its maximum at the distance between a small particle and its nearest neighbor (i.e. at r = d, neglecting the overlap between contacting particles). When r = d, the K-Ripley function reduces to K ðdÞ ¼
Z s−s ; ns
ð9Þ
21
where Zs − s is the average number of small particles each small particle contacts and ns is the number density of small particles. Using rag = d and Eq. (9), the mean cluster size in a homogeneous mixture can be given simply by dcluster = Zs1/3 − sd. Based on previous simulations of randomly arranged, dense, monodisperse assemblies of particles [10,17], one would expect that for homogeneous systems, an average small particle would be contacting about 5 to 6 other small particles, and so a mean cluster size for homogeneous mixtures would be approximately 1.7d–1.8d. That is, the optimum mean cluster size would be about 1.7d–1.8d, and deviation from this value could serve as a measure of ordering and inhomogeneity. The second metric is based on the spatial variance in the relative volume fractions of the small and large particles, analogous to the sampling technique as described in [26,27]. This was calculated by first dividing the system into a number of smaller sample volumes. As discussed in the next section, shear always eventually became confined to near one of the shearing over the course of a simulation. This, along with the fact that walls constrained the motions and arrangements of near-by particles, the region near one or both walls may be quite different from the bulk. As such, to reduce wall effects, the top and bottom 15.3% of the mixtures were ignored (somewhat similar to how dcluster is calculated here). The fraction of solid volume belonging to the large (or small) particles in each sample volume was found, and then the standard deviation over the sample volumes, σbulk, was calculated. In a bidisperse collection of particles, such as those discussed here, σbulk would not depend on whether it was calculated based on the large particles or the small particles. Of course, the mixtures become more homogeneous as σbulk decreases. The sample volume must be large enough to contain both small and large particles simultaneously, but cannot be so large that the spatial variance is lost [26]. As such, the sample volumes have been chosen to be cubes of length 10d, i.e. sufficiently larger than a large particle (5.6 times the volume of a large particle), while significantly smaller than the entire system (at most 1% of the total volume). To calculate the volume of a group of particles in a sample volume, no smoothing function was used. Rather, if a particle was physically at least partially in a given sample volume, the volume of portion of the particle located within that sample volume was calculated and added to its group's volume. Rather than pick sample volumes at random, they were regularly spaced in all three dimensions. To minimize error, a large number of sample volumes were used; the different sample volumes overlapped. When the number of sample volumes exceeded 1000 (10 by 10 by 10), σbulk negligibly changed with increasing number of sample volumes. The number of sample volumes used had a very weak influence on the overall simulation time, and so, even though it was overkill, a total of 25 3 sample volumes were used. In addition to these two metrics, the total solid volume fraction (or packing fraction), i.e. the ratio of the total particle volume to the system volume, was measured throughout each simulation. As aforementioned, an advantage of achieving homogeneous mixing with particles featuring a large size range is greater solids' loading. As such, homogeneity and solid volume fraction are correlated, and that correlation will be quantified and analyzed. 4. Results and discussion Starting with initially segregated particle collections, shearing led to a general decrease (on average) in σbulk and dcluster and an increase in solid volume fraction, as one would expect and as demonstrated in Fig. 2A–C for a small selection of k⁎, B, and S. It can be observed that after a little while, the rate at which these quantities evolved with strain generally slowed as shear progressed (i.e. decayed exponentially, more or less). Comparisons of the different mixtures needed to be done when the mixtures' evolution with strain had significantly slowed. As such, based on the data partly shown in Fig. 2A–C, it was decided that the mixtures would be compared at strains of 300.
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Fig. 3. The evolution of the streamwise velocity profile over the course of a mixing simulation. Plotted is the streamwise velocity (vx) scaled by the speed of the shearing wall (U) versus the vertical position (z) scaled by the height of the system (H) at strains of 0.5 (solid black line), 6 (dashed black line), and 300 (dash-dot gray line). Data is from a simulation in which S= 0.1, B = 0.01, and k⁎ = 109. The thin dash-dot black line corresponds to uniform shear.
only would have occurred near that wall. Inhomogeneous mixtures would arise when the shear becomes localized before the bulk can be well-mixed. Fig. 4 shows how the dimensionless mean small-particle cluster size, dcluster/d, varied with k⁎ and cohesion after shearing for a strain of 300. Each point represents the mean value obtained from three simulations, only different in their initial particle configurations. The error bar associated with each data point spans the entire range of values obtained from the three simulations. When the particles were not cohesive, dcluster/d was about 1.8, i.e. around the optimal value, regardless of shear rate. Therefore, the mean cluster size estimates would suggest that the mixtures of non-cohesive particles were homogeneous, just as one would likely expect. When the small particles were “slightly cohesive” (S = 0.1), dcluster/d did not differ much from the mixtures of non-cohesive particles, regardless of the cohesiveness of the big particles or shear rate. The small particles attracted each other too weakly to form large clusters that could survive shear at the rates explored, and so they could easily distribute themselves throughout the mixtures before the shear region settled near a wall. It is feasible to assume that as k⁎ becomes sufficiently larger
Fig. 2. The evolution of the (A) solid volume fraction (ϕ), (B) dimensionless mean small-particle cluster size (dcluster/d), and (C) standard deviation in the relative volume fraction of particles (σ) with strain after the start of shear for four different combinations of small-particle cohesion (S), big-particle cohesion (B), and scaled stiffness (k⁎). The evolution of the three quantities in every case slows significantly by a strain of 300. Each curve represents the data produced by a single simulation run.
To explain the behavior observed in Fig. 2A–C, we examine the evolution of the velocity profile over the course of a simulation. Fig. 3 shows an example of how the velocity profile evolved over a strain of 300. In the mixing simulations discussed here, the particles were at rest at the beginning. As shear proceeded, a velocity profile emerged. Eventually, the velocity profile shifted such that almost all the particles flowed as a plug, with a shear region confined to near the top (as shown in Fig. 3) or bottom of the mixture. As mixing would mostly occur where there is shear, mixing would proceed faster when shear is not localized than when shear is confined to a small region. Once the shear region became small and isolated near a wall, mixing in the bulk would have been negligible, and mixing
Fig. 4. The effect of particle cohesion on the mean small-particle cluster size versus shear rate behavior. Plotted is the dimensionless mean small-particle cluster size (d3 2 _ ). Each cluster/d) measured at a strain of 300 versus the scaled stiffness (k ¼ k=ρd γ data point represents an average from three simulations. The error bar associated with each data point covers the entire range of values obtained from the three simulations and is only shown when it extends outside its graph symbol.
L.R. Aarons et al. / Powder Technology 235 (2013) 18–26
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than 109, the mean cluster size would grow when S = 0.1, and so S would need to be even smaller to achieve homogeneous mixing. When the small particles were more cohesive (S = 1), they naturally would have had a greater tendency to attract each other and form large, long-lasting clusters. As such, dcluster was always at least as large as when S was smaller. Furthermore, dcluster exhibited a dependence on k⁎ and B that did not exist when S was smaller. The influence of k⁎ was rather straightforward, as increasing the shear rate and therefore the kinetic energy (decreasing k⁎) helped to shrink the agglomerates of small particles and to give them the inertia to get between the big particles. On the other hand, the influence of B on the mean cluster size was somewhat complex due to competing effects of varying B. Increasing B first of all simply hindered the freedom of the large particles' motion, making it more difficult to get good mixing. However, increasing B also led to the big particles packing more loosely (as observed in the solid volume fraction data discussed later and in [19]) and forming longer-lasting structures, making it easier for the small-particle clusters to fit between the big particles. As such, when k⁎ = 10 9, the smallest dcluster was achieved at an intermediate value of B (i.e. B = 1). The largest dcluster was observed when B was 0.1, as much of the small-particle agglomerates simply could not fit between the big particles. Fig. 4 also shows the weak influence of the cohesive interaction between the large and small particles on mixing. This can be illustrated by looking at the data for S = 0.1 and B = 1 and the data for S = 1 and B = 0.1. The strength of the van der Waals interaction pffiffiffiffiffiffi between the small and large particles is the same in both cases BS ¼ 10−0:5 . However, as just discussed, the corresponding mixtures featured very different degrees of homogeneity. Instead, the homogeneity at a given k⁎ depended primarily on S and secondarily on B. Had the homogeneity depended strongly on the cohesive interaction between the different particles, the influence of B would have been just as strong as that of S. Given that the mean cluster size can be related to the particles' ability to agglomerate, and this ability can be quantified using the kinetically scaled Hamaker constants, we replot the data in Fig. 4 using Sk⁎ and Bk⁎ in Fig. 5 (with error bars removed). This plot shows that multiple mean cluster sizes were achieved for each set of Sk⁎ and Bk⁎; dcluster/d was not a strict function of Sk⁎ and Bk⁎, but rather depended on k⁎ (or similarly, S or B) as well. As such, this figure shows that the transition between homogeneous mixing and inhomogeneous mixing cannot be captured by a single value of Sk⁎ or even a set of Sk⁎ and
Bk⁎ values. Nevertheless, decreasing Sk⁎ while keeping Bk⁎ constant led dcluster/d to decrease until it reached the optimal value of about 1.8, after which it stayed constant. That is, as the small particles became less able to agglomerate, they were able to form more homogeneous mixtures, as previously suggested. The influence of the large particles' ability to form agglomerates on dcluster/d is analogous to the complicated behavior of varying B in Fig. 4. Given that scaling the cohesiveness kinetically did not help to collapse or otherwise simplify the results (and while it will not be shown, this scaling actually would complicate the volume fraction results), this scaling will no longer be used. Fig. 6 shows how the standard deviation in the relative volume fraction of particles in the bulk, σbulk, depended on the scaled stiffness and particle cohesiveness after a strain of 300. Comparing this figure with Fig. 4, it can be seen that for the most part the behaviors of σbulk and dcluster were rather similar to each other (described in the preceding discussion) but featured some differences. First of all, whereas dcluster/d for all cases except S = 1 and B = 0.1 collapsed onto a single value when k⁎ = 10 7, this did not occur for σbulk. Also, when the particles were not cohesive or S = 0.1, σbulk manifested a dependence on k⁎ and B not observed with dcluster/d. For example, when the particles were not cohesive, σbulk decreased with k⁎. All in all, though, the influences of B and k⁎ on σbulk in these cases were much smaller than when S = 1, and so it is questionable if this variation in σbulk was indicative of a significant variation in homogeneity. An explanation for why some of these differences between the behaviors of dcluster/d and σbulk arose (rather than being representative of conflicting data) is that the regions sampled in the calculation of the statistics were different. Because dcluster was calculated based on spherical samples centered at the vertical center of the mixture, it took into account the particles closer to the vertical center more than those farther away; this was not true in the calculation of σbulk. Thus if wall effects extended far enough to creep into the measurements, they would have been captured more by σbulk than dcluster and would have contributed most significantly when σbulk was small. To support and explain the data and conclusions presented so far, images of the mixtures' microstructures were created. Fig. 7A–E show images, generated by POV-Ray [29], of vertical slices of some of the different mixtures generated at a strain of 300 after shear was started. While these images are from one set of simulations, the other two sets of simulations performed with the same groups of parameters produced similar microstructures. Simulations without cohesion and most simulations with S = 0.1 produced visually homogeneous
Fig. 5. The influence of kinetically scaled particle cohesion on the mean small-particle cluster size. Plotted is the dimensionless mean small-particle cluster size (dcluster/d) measured at a strain of 300 versus the kinetically scaled small-particle Hamaker constant (Sk⁎) for different values of the kinetically scaled large-particle Hamaker con3 2 stant (Bk⁎). The symbol color represents the scaled stiffness ( k ¼ k=ρd γ_ ), with black representing k⁎ = 107, dark gray representing k⁎ = 108, and light gray representing k⁎ = 109. Each data point represents an average from three simulations.
Fig. 6. The effect of particle cohesion on the standard deviation in the relative volume fraction of particles versus shear rate behavior based on the particles in the bulk of the mixtures. Plotted is the standard deviation in the relative volume fraction of particles 3 2 (σbulk), measured at a strain of 300, versus the scaled stiffness (k ¼ k=ρd γ_ ). Each data point represents an average from three simulations. The error bar associated with each data point covers the entire range of values obtained from the three simulations and is only shown when it extends outside its corresponding graph symbol.
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3 2 Fig. 7. Images of the microstructure of different mixtures after shearing for a strain of 300 with a scaled stiffness, k ¼ k=ρd γ_ , of 109. Shown are vertical slices of mixtures obtained with (A) no particle cohesion, (B) S= 0.1 and B = 0.01, (C) S = 1 and B = 0.1, (D) S = 1 and B = 1, and (E) S= 1 and B = 10. Large particles (of diameter 7d) are shown in light gray, small particles (of diameter d) are shown in medium gray, and the particles forming the walls are shown in dark gray.
mixtures similar to what is shown in Fig. 7A. When S = 0.1, k⁎ = 10 9, and B = 0.01 or 0.1, the mixtures feature a homogeneous bulk and a less homogeneous wall region, as shown in Fig. 7B, where a void region exists near the center of the bottom wall. This may explain (at least in part) the slight dependence σbulk has on k⁎ and B that
dcluster/d does not have. Fig. 7C–E shows the inhomogeneous mixtures produced when S = 1 and k⁎ = 10 9. These images support the findings that under these conditions, the worst mixing was achieved when B = 0.1 (Fig. 7C), while the best mixing was obtained with B = 1 (Fig. 7D).
L.R. Aarons et al. / Powder Technology 235 (2013) 18–26
Fig. 8 shows how the solid volume fraction, ϕ, measured at a strain of 300, varied with the scaled stiffness and particle cohesiveness. If the particles were monodisperse, one would expect that as k⁎ decreases, the solid volume fraction either would remain virtually constant or decrease slightly, indicative of elastic–quasistatic flow, or would decrease significantly, indicative of inertial or elastic–inertial flow [19,28]. Thus when the particles are polydisperse and feature a large range in size, an increase in solid volume fraction with a decrease in k⁎ would likely be attributed to more small particles fitting in between the big particles, i.e. an increase in homogeneity. This is most evident when S = 1, as decreasing k⁎ increased homogeneity and solid volume fraction. However, when S = 0.1 and B =1 and k⁎ was decreased from 109 to 108, the solid volume fraction increased noticeably despite a negligible change in homogeneity. When k⁎ =109, the small particles would not have agglomerated with each other much but instead (at least partially) would have coated the more cohesive big particles, forming a homogeneous mixture. At the same time, the particles formed structures that limited the solid volume fraction. Decreasing k⁎ to 108 gave the small particles enough energy to escape the pull of the large particles, so they could pack more tightly while still achieving a homogeneous mixture. The influence of cohesion on solid volume fraction would be two-fold, as it would directly affect solid volume fraction by causing the small and large particles to pack less tightly on their own and would indirectly affect solid volume fraction by influencing homogeneity. When S was increased from 0 to 0.1, the small particles still did not agglomerate much and so they had little effect on the solid volume fraction. As such, at a given shear rate, the solid volume fraction was dictated by how efficiently the big particles packed. So as B increased, the big particles of course packed less densely, and a decrease in solid volume fraction was observed. When S was increased to 1, the small particles agglomerated enough to significantly lower the solid volume fraction, both because of a reduction in homogeneity and because they packed less densely. As B increased and the big particles packed less densely, the solid volume fraction decreased monotonically, unaffected by the improvement in homogeneity as B went from 0.1 to 1. This behavior likely arose because the large particles constitute 73% of the solid volume and form the “backbone” of the mixtures. Before closing this section, it should be emphasized that the results presented here are from simulations performed under very specific conditions, most notably with a bidisperse mixture featuring a 7:1 size ratio and in the absence of gravity. That being said, these
Fig. 8. The effect of particle cohesion on the solid volume fraction versus shear rate behavior. Plotted is fraction (ϕ), measured at a strain of 300, versus the the solid volume 3 2 scaled stiffness k ¼ k=ρd γ_ . Each data point represents an average from three simulations. The error bar associated with each data point covers the entire range of values obtained from the three simulations and is only shown when it extends outside its corresponding graph symbol.
25
results may provide insight into mixing under some other conditions. Of course, in the presence of gravity, there will be gravity-driven size segregation, and so the conclusions presented in this paper likely cannot be readily extended to those situations. Otherwise, should the particle size range be expanded, the mixing mechanics remain essentially the same. If anything, it would make it easier for the small particles fit between the large particles, and so the small particles would need to be more cohesive for the mixtures to become noticeably inhomogeneous. On the other hand, when the size ratio becomes smaller (less than 6.46:1), the mixing mechanics are a bit different, as the small particles cannot fit between the gaps formed by packed large particles. Rather, bed dilation would need to occur to give the smaller particles room to intermingle with the large particles. As such, we would not expect the homogeneity and solid volume fraction to qualitatively depend on cohesion and shear rate in the same way as described in this paper. 5. Summary In this study, we have investigated the influence of cohesion on the homogeneity and solid volume fraction of bidisperse collections of particles subjected to plane shear between two walls as a means of mixing in the absence of gravity. Attention was focused on particles featuring a 7:1 diameter ratio, such that mixing would involve the small particles filling in the gaps between the large particles. The homogeneity of the mixtures was quantified by measuring the mean size of the small-particle clusters, dcluster, as defined by [22], and the standard deviation in the relative concentration of the different particles in the bulk, σbulk, at a strain of 300 after starting with a completely segregated arrangement. Images of the mixtures' microstructure were also used to capture homogeneity. In short, a mixture's homogeneity depended on how easily the small particles were able to fit inside the gaps between the big particles. Inhomogeneous mixtures were observed only when the small particles were able enough to form and maintain agglomerates. In these cases, the agglomerates, rather than individual particles, had to find their way between the big particles, and homogeneity, as measured by dcluster and σbulk, was greatly improved with decreasing scaled stiffness 3 2 k ¼ k=ρd γ_ and was significantly influenced by the cohesiveness of the big particles. If the big particles were not sufficiently cohesive, they packed too tightly for the small-particle agglomerates to get between them, resulting in the most inhomogeneous mixtures. As the big particles became more cohesive, the small-particle agglomerates had more room into which they can fit, improving homogeneity until the cohesion restricted particle motion so much that mixing was prevented and homogeneity worsened. Furthermore, as k⁎ decreased, the particles had more kinetic energy to mix and break up agglomerates, naturally improving homogeneity. On the other hand, when the small particles were less cohesive or the particles were not cohesive at all, the mixtures appeared rather homogeneous, especially in the bulk. In these cases, dcluster/d was measured to be about 1.8, close to the lowest possible value, for all values of k⁎ and large-particle cohesiveness explored. As long as the particles were not negligibly cohesive at a given shear rate, the mixture's solid volume fraction increased with shear rate, generally as a result of improving homogeneity. Otherwise, like monodisperse materials in the elastic–quasistatic regime, the volume fraction decreased slightly with increasing shear rate. Meanwhile, as the particles became more cohesive, the solid volume fraction decreased, regardless if homogeneity was improved or not. Acknowledgments This work was funded by the U.S. Air Force Research Laboratory under contract number FA8651-08-D-0108, as part of the Florida
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Institute for Research in Energetics (FIRE). Some computational resources and work space were provided by the Eglin Air Force Base. References [1] G. Plantard, H. Saadaoui, P. Snabre, B. Pouligny, Surface-roughness-driven segregation in a granular slurry under shear, Europhysics Letters 75 (2) (2006) 335–341. [2] J.C. Williams, The segregation of particulate materials, Powder Technology 15 (1976) 245–251. [3] In: M. Rhodes (Ed.), Introduction to Particle Technology, John Wiley and Sons, New York, 1998. [4] J.P.K. Seville, C.D. Willett, P.C. Knight, Interparticle forces in fluidisation: a review, Powder Technology 113 (3) (2000) 261–268. [5] H. Li, Impact of cohesion forces on particle mixing and segregation, PhD Thesis, University of Pittsburgh, Pittsburgh, PA, 2005. [6] H. Li, J.J. McCarthy, Cohesive particle mixing and segregation under shear, Powder Technology 164 (2006) 58–64. [7] H. Li, J.J. McCarthy, Phase diagrams for cohesive particle mixing and segregation, Physical Review E 71 (2) (2005). [8] H. Li, J.J. McCarthy, Controlling cohesive particle mixing and segregation, Physical Review Letters 90 (18) (2003). [9] S. Torquato, T.M. Truskett, P.G. Debenedetti, Is random close packing of spheres well defined? Physical Review Letters 84 (10) (2000) 2064. [10] A.Z. Zinchenko, Algorithm for random close packing of spheres with periodic boundary conditions, Journal of Computational Physics 114 (1994) 298–307. [11] H.Y. Sohn, C. Moreland, The effect of particle size distribution on packing density, Canadian Journal of Chemical Engineering 46 (1968) 162–167. [12] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1) (1979) 47–65. [13] S.J. Plimpton, Fast parallel algorithms for short-range molecular dynamics, Journal of Comparative Psychology 117 (1995) 1–19. [14] L.E. Silbert, D. Ertas, G.S. Grest, T.C. Halsey, D. Levine, S.J. Plimpton, Granular flow down an inclined plane: Bagnold scaling and rheology, Physical Review E 64 (5) (2001) 051302.
[15] L. Vu-Quoc, X. Zhang, An accurate and efficient tangential force-displacement model for elastic frictional contact in particle-flow simulations, Mechanics of Materials 31 (4) (1999) 235–269. [16] O.R. Walton, R.L. Braun, Viscosity, granular-temperature, and stress calculations for shearing assemblies of inelastic, frictional disks, Journal of Rheology 30 (5) (1986) 949–980. [17] L. Aarons, S. Sundaresan, Shear flow of assemblies of cohesive and non-cohesive granular materials, Powder Technology 169 (1) (2006) 10–21. [18] R.Y. Yang, R.P. Zou, A.B. Yu, Computer simulation of the packing of fine particles, Physical Review E 62 (3) (2000) 3900–3908. [19] L. Aarons, S. Sundaresan, Shear flow of assemblies of cohesive granular materials under constant applied normal stress, Powder Technology 183 (2008) 340–355. [20] A.W. Lees, S.F. Edwards, The computer study of transport processes under extreme conditions, Journal of Physics C: Solid State Physics 5 (1972) 1921–1929. [21] C.S. Campbell, Granular shear flows at the elastic limit, Journal of Fluid Mechanics 465 (2002) 261–291. [22] S. Gallier, A stochastic pocket model for aluminum agglomeration in solid propellants, Propellants, Explos, Pyrotechnics 34 (2009) 97–105. [23] B.D. Ripley, The second-order analysis of stationary point process, Journal of Applied Probability 13 (1976) 255. [24] B.D. Ripley, Modelling spatial patterns, Journal of the Royal Statistical Society B39 (1977) 172. [25] J. Besag, Contribution to the discussion of Dr. Ripley's paper, Journal of the Royal Statistical Society B39 (1977) 193. [26] L.T. Fan, S.J. Chen, C.A. Watson, Solids mixing, Industrial and Engineering Chemistry 62 (7) (1970) 52–69. [27] L.T. Fan, J.R. Too, R.M. Rubison, F.S. Lai, Studies on multicomponent solids mixing and mixtures part III: mixing indices, Powder Technology 24 (1979) 73–89. [28] C.S. Campbell, Stress-controlled elastic granular shear flows, Journal of Fluid Mechanics 539 (2005) 273–297. [29] Persistence of Vision Pty. Ltd., Persistence of Vision™ Raytracer, Persistence of Vision Pty. Ltd., Williamstown, Victoria, Australia, 2004. http://www.povray.org/.