Powder Technology 218 (2012) Contents lists available at SciVerse ScienceDirect. Powder Technology

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1 Powder Technology 218 (2012) Contents lists available at SciVerse ScienceDirect Powder Technology journal homepage: Dynamic simulation of granular packing of fine cohesive particles with different size distributions Tao Jia a, Yuwen Zhang a,b,, J.K. Chen a, Y.L. He b a Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA b Key Laboratory of Thermal Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi, , China article info abstract Article history: Received 21 September 2011 Received in revised form 19 November 2011 Accepted 26 November 2011 Available online 3 December 2011 Keywords: Packing Distinct Element Method Size distribution Granular matter A granular packing of fine cohesive particles with three different size distributions (monosize, bimodal and Gaussian) was studied using the Distinct Element Method (DEM). The dissipative forces including viscoelastic and frictional contact forces between the colliding particles cause the energy loss during the packing process, and finally the particles agglomerations is formed. The conservative forces of gravitation and van der Waals forces are also included in the simulations. For the cases of monosized particles packing, the packing structure becomes looser when the particle size becomes smaller. When the particle diameter is less than 100 μm, the type of contact of edge-sharing in-plane equilateral triangle is not dominant in the packing structure. As to the bimodal situations, the change of the particle population ratio does not have much influence on the compactness of the packing structure in the case of small particle size ratio. However, in the cases of large particle size ratio, it is observed that the particle population ratio does affect packing structure compactness. When the smaller particle population ratio increases the packing structure becomes more compact. Regarding the packing of particles with Gaussian distribution, the packing structure becomes more compact as the mean diameter increases. On the other hand, as the particle diameter deviation increases, the packing structure becomes looser. The radial distribution function and force distribution are also analyzed for different packing cases. It is found that the distributed range of contact force is larger than that of van der Waals force in the final granular matters, and the force distribution changes as the particle size distribution changes Elsevier B.V. All rights reserved. 1. Introduction Granular packing of spherical particles has been an important research topic in engineering and physics fields. The forming process of granular matter is usually by agglomeration of particles. Distinct Element Method (DEM) [1 3] provides an effective way to numerically investigate each particle's movement, including the change of its position, orientation, and translational and angular velocities during the agglomeration process. When modeling the collisions of colliding particles, the contact forces [4 11] among them need to be calculated first. The contact force results from the deformation of the colliding particles and can be decomposed into two parts: normal viscoelastic force and tangential frictional force. The former can be calculated by the linear-spring model [12] or by the more accurate nonlinear Hertz model [13 14], and the latter by the Mindlin Deresiewicz theory [15 18]. One important aspect to model the tangential frictional force between contacted particles is to clarify two distinct conditions [19]: static friction and sliding friction. According to the Mindlin Corresponding author at: Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA. Tel.: ; fax: address: zhangyu@missouri.edu (Y. Zhang). Deresiewicz theory, a maximum tangential displacement can be determined once the normal deformation is known. If the tangential displacement is less than or equal to the maximum tangential displacement, the friction is static friction; otherwise, the friction becomes sliding friction. The tangential displacement would stop growing once it reaches the maximum. Particle size plays an important role in forming the agglomeration of the particles. Nowadays a great deal of effort goes to the simulations for mono-sized particles packing [20 24]. However, packing of particles with different size distributions, which is more practical in industry, is rarely studied. Although particles packing with different size distributions was studied using dropping and rolling rules [25 27], these simulations did not take into account the interparticle force and only rooted on the geometry point of view. In these approaches, no any forces, such as contact force, gravity, or van der Waals force, are considered in simulation. To judge whether or not a particle reaches a stable position, this kind of approach only investigates the particles' position, such as whether or not the particle touches the floor, or how the particle contacts neighbor particles. The dynamic simulations [28 29] investigated the packing of millimeterscale particles with different size distributions. In this work, we study micrometer-scale particles packing, ranging from 50 to 100 μm, using the soft-sphere DEM method. The difference /$ see front matter 2011 Elsevier B.V. All rights reserved. doi: /j.powtec

2 T. Jia et al. / Powder Technology 218 (2012) between the model of millimeter-scale and the model of micrometerscale is that in millimeter scale, van der Waals force is not considered, but in micro-meter scale van der Waals force needs to be incorporated into the model [30]. In the process of the collision, the coefficient of restitution depends on impact velocity [31 32] and should not be a constant. The use of a constant coefficient of restitution cannot provide us accurate information about the collision process. Therefore, a damping coefficient is employed in the present simulation to consider the collision duration. And the difference between the model of reference [31 32] and the present model is on how to model the tangential contact force. The model of [31 32] is based on the linear-spring method, and the tangential contact force is the product of tangential displacement and a constant coefficient. The present model is based on a nonlinear-spring method which is derived from Mindlin Deresiewicz theory [15 18]; this method is more accurate to model the tangential frictional force. The advantage of the soft-sphere DEM method is that manyparticle simultaneous collisions can be modeled. To investigate the size effects on packing behaviors, three different size distributions are considered: monosize, bimodal and Gaussian distributions. (a) Initial (0 s) (b) Middle (0.05 s) (c) Final (0.1 s) Fig. 1. Monosized particles packing process (d=50 μm). 2. Simulation method There are two kinds of motions for each particle: translation and rotation. The equations describing these motions for each particle i are as follows: m i d 2 r i dt 2 ¼ F i ð1þ (a) Initial (0 s) (b) Middle (0.06 s) (c) Final (0.12 s) Fig. 2. Bimodal particles packing process (d 1 =50 μm, d 2 =100 μm, n 1 :n 2 =8:2). I i dω i dt ¼ T i where m i is the mass of the particle, r i is the position of the particle, ω i is the angular velocity, I i is the inertial moment which equals 0.4m i R i 2, Table 1 Parameters used in simulation. Parameter Value Young's modulus Y 10 7 N/m 2 Poisson's ratio σ 0.3 Particle density ρ 2500kg/m 3 Hamaker constant H a J Sliding friction coefficient μ 0.3 Rolling friction coefficient μ r Damping coefficient γ s ð2þ and R i is the particle radius. F i and T i are respectively the resultant force and torque acting on the particle i: F i ¼ j T i ¼ j F n ij þ Ft ij þ Fv ij þ m i g T t ij þ Tr ij where F n ijis the normal contact force, F t ij is the tangential frictional force, F v ij is the van der Waals force, T t ij is the torque caused by the tangential contact force F t ij, and T r ij is the torque caused by the rolling friction. The subscripts ij denote the force (torque) acting on the particle i by particle j. ð3þ ð4þ Table 2 Particle diameters for different size distributions. (a) Monosized particles' diameters Particle Diameter (μm) (b) Bimodal particles' diameters d 1 (μm) (μm) n 1 : n 2 n 1 :n 2 n 1 :n 2 n 1 :n 2 n 1 : n 2 50:60 2:8 4:6 5:5 6:4 8:2 50:80 2:8 4:6 5:5 6:4 8:2 50:100 2:8 4:6 5:5 6:4 8:2 (c) Gaussian particles' diameters d 1 (μm) d 2 (μm) Deviation (μm) (mean=55) (mean=65) (mean=75) (a) Initial (0 s) (b) Middle (0.05 s) (c) Final (0.1 s) Fig. 3. Gaussian particles packing process (d= μm, d mean =75 μm, d deviation =5 μm).

3 78 T. Jia et al. / Powder Technology 218 (2012) (a) Coordination number (a) Coordination number Coordination number d 1 = 1 : 1.2 d 1 = 1 : 1.6 d 1 = 1 : (b) Porosity (b) Porosity Particle population ratio( n 1 : n 2 ) 0.58 d 1 = 1 : 1.2 Fig. 4. Coordination number and porosity of monosized particle granular structures. Porosity d 1 = 1 : 1.6 d 1 = 1 : Particle population ratio( n 1 : n 2 ) Fig. 5. Coordination number and porosity of bimodal particles granular structures. According to the nonlinear Hertz theory [22], the normal contact force acting on particle i by particle j is calculated by: F n ij ¼ 2 pffiffiffi pffiffiffiqffiffiffiffiffi 3 E R ξ 3=2 n γe R ξ n v ij :n ij n ij ð5þ v t ¼ v i v j t ij þ ω i R i ω j R j ð8þ where E=Y/(1 σ 2 ) with Y denoting Young's modulus and σ Poisson's ratio, R is effective radius that equals R i R j /(R i +R j ) with R i and R j being the vectors running from the center of particle i and j to the contact point of the two particles respectively, γ is damping coefficient, v ij is the velocity of particle i relative to the velocity of particle j, ξ n =R i +R j R ij is the deformation between the two particles with R ij being the distance between the spherical center of particle iand that of particle j, and n ij is the unit vector whose direction is from the center of particle j to that of particle i. Based on the Mindlin Deresiewicz theory [15 18], the tangential contact force F t ij is determined as: " F t ij ¼ μf n j ij 1 1 ξ # tj 1:5 t jξ max j ij ð6þ where μ is sliding friction coefficient, F n ij is the magnitude of the normal contact force, ξ t is the tangential displacement which is determined as: where v t is the tangential relatively velocity, t ij is the unit vector along the tangential direction, t 0 is the time when the two particles just touch without deformation, and ω i and ω j are angular velocities of particles iand j respectively. If the tangential displacement is larger than the magnitude of the maximum tangential displacement jξ max j, sliding friction occurs between the two particles. The maximum tangential displacement ξ max is calculated as: ξ max ¼ μ 2 σ 2 2σ ξ n The van der Waals force is expressed in the following form: F v ij ¼ H 3 64R a 6 i R 3 j h þ R i þ R j 2 2 n ij ð10þ h 2 þ 2R i h þ 2R j h h 2 þ 2R i h þ 2R j h þ 4R i R j ð9þ t ξ t ¼ v t dt t 0 ð7þ where H a is Hamaker constant that depends on particle material property, and h is the distance between the centers of particles i and j minus the sum of the radius of the two particles.

4 T. Jia et al. / Powder Technology 218 (2012) (a) Coordination number the smallest particle diameter. Due to the gravity the particles start to fall down and collide to each other. Periodical boundary conditions are applied to the horizontal directions. For monosize particles, six cases with particle diameters of 50, 60, 70, 80, 90, and 100 μm are studied. For the bimodal particles cases, 15 cases are simulated for three particle diameter ratios: 1:1.2 (50 μm:60 μm), 1:1.6 (50 μm:80 μm), and 1:2 (50 μm:100 μm), and five particle population ratios: 2:8, 4:6, 5:5, 6:4 and 8:2. As to the Gaussian distributions, 15 cases are simulated for three ranges of particle diameters: μm (the diameter mean value is 55 μm), μm (the diameter mean value is 65 μm), and μm (the diameter mean value is 75 μm) and five deviations: 1, 2, 3, 4 and 5 μm. The probability density function for the particle diameter is determined by [35]: " fðdþ ¼ pffiffiffiffiffiffi 1 exp d 2 # d : ð13þ 2π S 2S 2 (b) Porosity Fig. 6. Coordination number and porosity of Gaussian particles granular structures. The torque acted on particle i due to the rolling friction between particle i and j is: T r ij ¼ μ r R i F n ijω i ð11þ where μ r is rolling friction coefficient. The torque T t ij caused by the tangential contact force F t ij is gven by: T t ij ¼ R i Ft ij : ð12þ The Verlet method is employed in the present simulations due to its simplicity and effectiveness. The details of the Verlet method can be found in [33 34] and are not given here for brevity. The computer code is written in FORTRAN and simulations are carried out on a PC. 3. Result and discussion Initially there are 4500 ( ) particles assigned in a rectangular space, i.e., 15 cell (layers) lengths in each of the x- and y- directions and 20-cell lengths in the z-direction. To make sure that there is no overlap exists, the cell lengths are slightly larger than Table 1 shows the particle material properties [5 6] including Young's modulus, Poisson's ratio, density, Hamaker constant, friction coefficients between the particles, and the damping coefficient that is used to calculate the normal contact force between colliding particles. The time step for the cases of monosized particles simulations is 10 7 s. As to the bimodal and Gaussian particles cases, the time-steps are s and s, respectively. In the DEM simulations, the time-step must be small enough to avoid unrealistic large deformation of colliding particles. If the deformation is unrealistically large, the contact force acting on the colliding particles will be unrealistically large. The contact force also depends on particle size. For the cases of monosized particles, the smaller the particle is, the smaller the time-step should be. For the situation of bimodal and Gaussian particles, the time-step not only depends on the smallest particle, but also depends on the size difference of the colliding particles. The larger the size difference is, the smaller the time-step should be Table 2 summarized particle diameters for different size distributions. Fig. 1 illustrates the dynamic process of monosized particles packing with the particle diameter of 50 μm. Fig. 2 shows the bimodal particles packing for the 50 μm and 100 μm particles with a particle population ratio of 8:2. Fig. 3 shows the Gaussian particles packing process for the particle diameters ranging from 50 μm to 100 μm with the particle diameter mean value of 75 μm and the diameter deviation of 5 μm Coordination number and porosity The coordination number [25] for a particle is defined as the number of particles contacting it. The coordination number of the final granular structure is the mean value of all the individual particle coordination numbers. In the monosized particles packing cases, two particles are considered to contact each other if the distance between the centers of them is less than the critical distance of 1.01d where d is the particle diameter. As to the bimodal and Gaussian particles packing, the critical distance is 1.01(d 1 +d 2 )/2 where d 1 and d 2 are the two particles' diameters. Porosity is another measure of the packing structure [25]. Itisdefined as the volume of void to the total volume of the rectangular space that just covers the final granular matter. It can be seen from Fig. 4 that for the monosized particle packing cases, the coordination number decreases and the porosity increases as the particle size decreases. This is different from the situation of coarse monosized particles packing [31 32] in which the coordination number and porosity are independent of particle size. Fig. 5 shows the coordination number and porosity of the bimodal particles granular structures. It can be seen that for the cases of larger particle size ratio (d 1 :d 2 =1:1.6 and d 1 :d 2 =1:2.0), the coordination number increases and the porosity decreases with increasing the smaller

5 80 T. Jia et al. / Powder Technology 218 (2012) Fig. 7. Radial distribution function of monosized particles packing structure. particles population ratio. This is because more smaller particles can enter the void space among larger particles to make the final granular structure more compact. For the cases of the smallest particle size ratio (d 1 :d 2 =1:1.2), on the other hand, the coordination number and porosity does not change much as the particles population ratio changes. Thus, to achieve a more compact packing structure in bimodal cases, a larger particle size ratio with higher smaller particle population ratio is recommended. Regarding the Gaussian particles cases, it can be observed from Fig. 6 that as the particle diameter mean value decreases, the coordination number decreases and the Fig. 8. Radial distribution function of bimodal particles packing structure (d 1 =50 μm, d 2 =100 μm).

6 T. Jia et al. / Powder Technology 218 (2012) porosity increases. On the other hand, the trends are opposite as the particle diameter deviation increases. The reason is that as the mean particle diameter decreases, the mean magnitude of van der Waals force increases, making the packing structure looser. As the particle diameter deviation increases, the variety of particle size increases, making the packing structure looser Radial distribution function Radial distribution function (RDF), g(r), is a measure of how particle number density changes at a given distance from a reference particle: gr ðþ¼ dnðþ r 4πr 2 drρ ð14þ where N(r)is the number of the particles in the spherical space with a radius of r around the reference point, and ρ the number of particles per volume. Fig. 7 shows the RDFs for the monosized particles packings where the unit of the distance ris the particle diameter. It can be seen that all RDFs have a peak value at the distance of 2, which corresponds to the contact type of three particles centers lying on a line [31]. One exception is that the packing of p 100-μm particles also have a peak at the distance of 1.73 (close to ffiffiffi 3 ), which corresponds to the contact type of edge-sharing in-plane equilateral triangle [31]. This means that when particle diameter is less than 100 μm, the edge-sharing inplane equilateral triangle contact is not dominant in the packing structure. Fig. 8 shows the RDF changes with particle population ratio for the bimodal particles packing structure (d 1 =50μm and d 2 =100 μm). It can be seen that RDF has peak values at the distance of 2, 3, and 4. These three numbers correspond to three different kinds of contact: smaller particle to smaller particle, smaller particle to larger particle, and larger particle to larger particle. A clear increasing trend of RDF at the distance 2 appears in Fig. 8. This can be explained that when the population ratio of the smaller particle increases, the contacts between smaller particles increase in the final packing structure. The decreasing trend of the RDF at the distance 4 (this corresponds to the contact between larger particle and larger particle) is due to the fact that when the smaller particle population ratio increases there are fewer contacts between larger particles. In the cases of medium particle population ratios (n 1 :n 2 =4:6,n 1 :n 2 =5:5, and n 1 : n 2 =6:4), the RDF at the distance of 3 is larger than the values at the distances of 2 and 4. This indicates that in these three cases there are more contacts between smaller particles and larger particles than the other two kinds of contact. Fig. 9 shows the RDFs for the Gaussian particles packing cases (d= μm). The units of the distance r are the radius of the smallest particles. Here we do not see the discrete peak values of RDF as the bimodal RDFs shown in Fig. 8. The reason is that the particle diameters change continuously in a range, not the same as the bimodal cases where the particle diameters have distinct value. It is also observed that the RDF peak value is larger in the particle diameter deviation equal to 1 μm than those in the other four cases. And as the deviation increases, the radial distance at which RDF reaches the peak value increases. The physical meaning of this is that as the deviation increases, there are more contact between small particles and large particles in the final granular packing structures Force distribution The Distinct Element Method offers us the opportunity to explore the force distribution in the final granular packing structure. In the present simulations, there are three kinds of forces acting on each particle: contact force F c ij, van der Waals force Fv ij,andgravitational force m i g. The contact and van der Waals forces are different from gravitational force in the way that they do not have preferential direction and are randomly oriented in the granular packing structure. Two force ratios are considered in the present study: z c ¼ j F c ij = j m igj ð15þ Fig. 9. Radial distribution function of Gaussian particles packing structure (d=50 μm 100 μm).

7 82 T. Jia et al. / Powder Technology 218 (2012) (a) Contact force distribution (b) Van der Waals force distribution Fig. 10. Force distribution in monosized particles packing structure. z v ¼ j F v ij = j m igj ð16þ where z c (z v ) are the ratios of the magnitude of contact force (van der Waals force) to the magnitude of gravitational force. Figs show the force probability distributions in the granular packing structures for the three kinds of particle size distributions: monosize, bimodal, and Gaussian. The contact force probability distribution P(z c ) is exponential tail-like and the van der Waals force probability distribution P(z v ) is bell-shape-like. In the monosized particles packing structures, the contact force mainly covers the range of z c =0 100 while the van der Waals force range decreases as the particle size increases. The range of z v is 0 5 in the case of 100- μm particles and 0 20 in the 50-μm particles. As to the bimodal particles packing structures shown in Fig. 11, the contact force is mainly distributed in the range of z c =0 100 and the van der Waals force z v =0 20. For the Gaussian distribution cases, the contact force is distributed in the range ofz c = in the case of particle diameter range of μm, the van der Waals force is mainly distributed in the range of z v =0 20, in the other two cases of the particle diameter range of μm and μm, the van der Waals force is dispensed in the range of z v =0 10.

8 T. Jia et al. / Powder Technology 218 (2012) (a) Contact force distribution (b) Van der Waals force distribution Fig. 11. Force distribution in bimodal particles packing structure (d 1 =50 μm, d 2 =100 μm). 4. Conclusion Three-dimensional dynamic simulations of granular packing of fine cohesive particles with different size distributions (monosize, bimodal, and Gaussian) are conducted based on Distinct Element Method. Four kinds of forces are considered in the simulations, including two dissipative forces: viscoelastic and frictional forces and two conservative forces: gravity and van der Waals forces. The dissipative forces cause the energy loss during the particles packing process; finally forming the particles agglomeration. The structures of the final granular matters composed of particles with different size distributions are quantified by the parameters of coordination number, porosity, radial distribution function. The force distributions in the granular matters are also analyzed. The distributed range of contact force is larger than that of van der Waals force, and the contact force probability distributions show exponential-like tail and the van der Waals forces bellshape. Acknowledgment Support for this work by the U.S. National Science Foundation under grant number CBET is gratefully acknowledged.

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