Powder Technology 314 (2017) 89–101
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DEM simulation of cubical particle packing under mechanical vibration Yongli Wu a,b, Xizhong An a,⁎, A.B. Yu b a b
School of Metallurgy, Northeastern University, Shenyang, 110004, PR China Department of Chemical Engineering, Monash University, Clayton, Victoria 3800, Australia
a r t i c l e
i n f o
Article history: Received 20 May 2016 Received in revised form 9 September 2016 Accepted 11 September 2016 Available online 12 September 2016 Keywords: Particle packing Cubical particles Densification DEM simulation Mechanical vibration
a b s t r a c t Packing densification of cubical particles under mechanical vibrations was dynamically simulated by using discrete element method (DEM). Effects of operating parameters such as vibration amplitude, frequency, vibration intensity, and container size (container wall) on packing densification were comprehensively investigated. The macro property such as packing density and micro properties such as coordination number (CN), radial distribution function (RDF), and particle orientation of the packings were analyzed and compared. It is found that mechanical vibration with proper vibration amplitude and frequency is effective for the densification of cubical particle packing. Packing structures of different packing densities display different properties, based on which random loose packing (RLP) and random close packing (RCP) of cubical particles are identified with the packing density of 0.591 and 0.683, respectively. Two densification mechanisms are discussed as the particle rearrangement is dominant for the transition from RLP to RCP and the crystallization along the container wall is dominant for the transition beyond RCP to ordering. The obtained results are useful for optimizing vibration conditions to generate dense packings and understanding the structural information of some fixed beds with cubical particles. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Particle packing, known as the static assembly of particles, is probably the simplest state of granular matter which are extensively existed both in nature and industries [1,2]. In scientific research, packings of particles can be used to model the structures of liquids [3–5], amorphous metals and alloys [6], solids and crystals [7–9] and their transitions [10–13]. Proper understanding of particle packing is also of primary importance in industrial applications with the following two problems particularly being concerned: i) the packing densification of particles, which is fundamental in powder metallurgy or composite synthesis [14,15]; ii) the quantitative characterization of packing structures, as the permeability and thermal conductivity of granular materials can be quantitatively related to the pore connectivity [16] and particle connectivity [17] of a packing, respectively. Generally, mechanical vibrations or tapping methods are used to densify the packing of particles [12,13,18–20]. Since the work of Scott [21], it is known if uniform hard spheres are poured in a container, a random loose packing (RLP) can be formed with the packing density ρ ≈ 0.60; and this initial RLP structure can be densified to a random close packing (RCP) with ρ ≈ 0.64 by utilizing tapping or vibrations. The obtained RLP and especially RCP of uniform hard spheres exhibit critical features for modelling phase transitions [22–24], thus the mechanism in terms of the transition or densification from RLP to RCP is ⁎ Corresponding author. E-mail address:
[email protected] (X. An).
http://dx.doi.org/10.1016/j.powtec.2016.09.029 0032-5910/© 2016 Elsevier B.V. All rights reserved.
fascinating for researchers. By using mechanical vibrations, packing densification can be realized due to the rearrangement of particles with the elimination of large pores and arching structures initially formed in the loose packings. As a dynamic process, the densification under sinusoidal vibration is affected by vibration time, amplitude A and frequency ω etc. Particularly, A and ω have direct influence on the vibration energy which affects the vibration efficiency greatly. The dimensionless vibration intensity Γ [25,26], defined as Aω2/g (g represents the gravitational acceleration), is thus used to evaluate vibration energy with both A and ω being considered. An et al. [12] reported the densification of spheres from RLP to RCP should be related to Γ with two pore filling mechanisms identified as “pushing filling” when Γ is low and “jumping filling” when Γ is high. It is noting that previous studies were mainly conducted on the packing of spheres, whereas most of the particles that we encounter in reality are of non-spherical shapes. Actually, some non-spherical particles can be beneficial alternatives of spherical particles in many applications. For example, cubical particles are advantageous in forming denser products in powder metallurgy [27] and acting as the drug-delivery media [28] in certain medical field. In addition, it was shown that packing of cubical particles could also be used to study the phase behavior of colloidal particles [29]. But the understanding of cubical particle packing is presently limited. The crystallized packing of cubical particles is easy to be figured out as particles aligned with the neighbors without voids. However, little is known about the RLP and RCP of cubical particles. The studies of cubical particle packing began with physical experiments. Zou and Yu [30] proposed empirical equations to describe
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the relationship between packing density and particle shape for various non-spherical particles. For mono-sized cubical particles, packing densities of RLP and RCP are predicted as about 0.586 and 0.656, respectively. Recently, Barker and Kudrolli [31] reported the RLP and RCP of five Platonic solids by fluidized or mechanically vibrated methods. The packing density of RLP and RCP of cubical particles are 0.54 and 0.67, respectively. They also examined the randomness of packing structures by calculating the variance among the projected areas of images obtained from the top of the packings. Actually, due to the complexity of particle systems, it is difficult and laborious for physical experiments to understand the underlying densification mechanism and the detailed structural information of packing structures. The deficiencies of studies in physical experiments can be readily overcome by numerical simulations. Generally, two kinds of simulation methods are prevalent in modelling particle packings: 1) methods of or related to Monte Carlo (MC) [32]; 2) the Discrete Element Method (DEM) [33]. MC methods are widely used to obtain the maximum packing densities of particles. Zhao and Li [34] reported the packing density of their obtained RCP of cubes using a MC based method is 0.7755. However, they just focused on the relationship between packing density and sphericity without describing further structural information. And it is worth noting that these MC methods actually cannot realize the truly dynamic packing of particles since they are geometrically based without force considerations, which may fail to reproduce the structural information fully comparable to those measured [35]. Based on Newton's second law of motion, DEM [33] can obtain the dynamic information of every particle at each time step, thus can well realize the dynamic packing of particles with inter-particle forces and dynamics being considered. In addition, the experimental vibration conditions performed on particle packing can also be well achieved by DEM [12,20], thus the investigation of densification mechanism is possible. In present work, the dynamic packing of mono-sized cubical particles under mechanical vibration is simulated by DEM with the focuses of reproducing the transition from RLP to RCP and identifying the obtained RCP structure and corresponding densification mechanism. The paper is structured as follows: Section 2 describes the DEM approach used for vibrated packing of cubical particles and the simulation conditions; in Section 3, the effects of vibration condition on packing density, microstructural characterization, and densification mechanisms are systematically discussed and analyzed; final conclusions are summarized in Section 4. 2. Simulation method and conditions 2.1. DEM model
Ip
dvp ¼ F p þ mp g dt
dωp ¼ Tp dt
ns
F p ¼ ∑ F p;s
ð3Þ
s¼1
where ns is the total number of spheres in the particle p; Fp,s is the force of sphere s in particle p, and it is the sum of normal forces Fnp , s and tangential forces Ftp,s acting on its contact points: contacts
F p;s ¼ F np;s þ F tp;s ¼ ∑
C¼1
F nps;c þ F tps;c
ð4Þ
The normal and tangential contact forces Fnps, c and Ftps,c on the contact point c are calculated based on Hertz model [39] and the work of Mindlin and Deresiewicz [40], respectively. The details about the force models can be found in the literature [41,42]. The torque Tp is the sum of three components [43]: ns T p ¼ ∑ T tp;s þ T np;s þ T rp;s s¼1
ð5Þ
where Ttp,s is created by tangential forces, Tnp,s is created by normal forces when the normal force of the element-sphere does not pass through the center of the particle; Trp,s is the rolling friction torque. Ttp,s and Tnp,s are respectively given as contacts
T tp;s ¼ ∑
C¼1
r ps;c F tps;c
T np;s ¼ dp;s F np;s
ð6Þ
ð7Þ
where rps,c is the vector between the contact point c and the center of element sphere s, belonging to particle p; and dp,s represents the relative position vector between the centroid of particle p and the centre of the element sphere s. 2.2. The orientation representation and rotational solution
In DEM model [33], each particle possesses both translational and rotational motions, which are governed by Newton's second law of motion, given by mp
contact detection between non-spherical particles can be simplified to the contact calculation of their element-spheres. Therefore, in this paper the multi-sphere model is adopted to describe cubical particles. By using the multi-sphere model [38], the total force Fp of the particle p is the sum of the forces on its element-spheres.
ð1Þ
ð2Þ
where mp, vp, ωp and Ip are respectively the mass, translational velocity, angular velocity, and moment of inertia of particle p; Fp and Tp are the total force and torque acted on the particle p. Due to the geometrical non-uniform characteristics of non-spherical particles, their contact detection is problematic. A number of techniques have been developed in the last decade for solving this problem [36,37]. Especially, the multi-sphere model is widely adopted in DEM simulation. This method is flexible in shape description as a particle of any shape can be constructed by a number of element-spheres; and the
The orientation representation and rotational movement of non-spherical particles are much more complicated than that of spherical particles. This is because the inertia tensor of a nonspherical particle in a global coordinate system will be changed at each time step according to its new orientation [37]. To overcome this problem, two coordinate systems as indicated in Fig. 1 are generally adopted: one is the global coordinate system, which is fixed; and the other is a local coordinate system, which moves with the non-spherical particle together. Therefore, the rotational equation can be solved with the inertial tensor obtained in the local coordinate system. In a local coordinate system, the inertial tensor I of a non-spherical particle can be expressed as: 0
Ixx I ¼ @ I yx Izx
Ixy I yy Izy
1 Ixz I yz A Izz
ð8Þ
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X G ¼ X L M−1
ð15Þ
where M−1 is the inverted matrix of M. And M is expressed with quaternions q (q0, q1, q2, q3) [44]: 0
q20 þ q21 −q22 −q23 @ M¼ 2ðq1 q2 −q0 q3 Þ 2ðq1 q3 þ q0 q2 Þ
2ðq1 q2 þ q0 q3 Þ q20 −q21 þ q22 −q23 2ðq2 q3 −q0 q1 Þ
1 2ðq1 q3 −q0 q2 Þ 2ðq2 q3 þ q0 q1 Þ A q20 −q21 −q22 þ q23
ð16Þ
The quaternions are updated as follows: dq0 1 −q1 ωLpx −q2 ωLpy −q3 ωLpz ¼ 2 dt
ð17Þ
dq1 1 q0 ωLpx −q3 ωLpy þ q2 ωLpz ¼ 2 dt
ð18Þ
dq2 1 q3 ωLpx þ q0 ωLpy þ q1 ωLpz ¼ 2 dt
ð19Þ
dq3 1 −q2 ωLpx þ q1 ωLpy þ q0 ωLpz ¼ 2 dt
ð20Þ
The rotation matrix M of every particle at each time step can thus be updated by the obtained quaternions.
Fig. 1. The local and global coordinate systems for the multi-sphere particle.
For the multi-sphere model, the inertial tensor can be represented
2.3. Simulation conditions
as: 0
ns ns 2 2 B ∑ I p;s þ ∑ mp;s yp;s þ zp;s B s¼1 s¼1 B ns B I¼B − ∑ mp;s xp;s yp;s B s¼1 B ns @ − ∑ mp;s xp;s zp;s s¼1
ns
− ∑ mp;s xp;s yp;s s¼1 ns ns ∑ I p;s þ ∑ mp;s x2p;s þ z2p;s s¼1
s¼1 ns
− ∑ mp;s yp;s zp;s s¼1
1
ns
− ∑ mp;s xp;s zp;s s¼1 ns
− ∑ mp;s yp;s zp;s s¼1 ns ns ∑ I p;s þ ∑ mp;s x2p;s þ y2p;s s¼1
C C C C C C C A
s¼1
ð9Þ where Ip,s and mp,s are the inertial tensor of the main axe and the mass of the element-sphere, respectively; xp,s, yp,s and zp,s are the distances from the center of the element-sphere to the three main axes of the nonspherical particle. It is worth noting that the main axes of the local system are selected to be coincided with the principal axes of inertia of the cubical particle (see Fig. 1). The matrix of inertial tensor can thus be transformed to be a diagonal matrix with the off-diagonal elements becoming zero: 0
Ixx I¼@ 0 0
0 I yy 0
1 0 0A Izz
ð10Þ
where Ixx, Iyy, and Izz are the principal moments of inertia. For cubical particles, the values of Ixx, Iyy, and Izz are actually equivalent, thus the rotational movement solved by Euler's equation can be treated separately in each dimension as follows [37]: Ixx
Iyy
Izz
dωLpx ¼ T Lpx dt dωLpy dt dωLpz dt
ð11Þ
¼ T Lpy
ð12Þ
¼ T Lpz
ð13Þ
As the treatment is in the local coordinate system, a rotation matrix M is used to transfer the global variable XG to the local variables as given by: XL ¼ XGM
ð14Þ
The constructed multi-sphere model for representing cubical particles in this work is shown in Fig. 2. Each cubical particle is composed of 64 uniform element-spheres with the diameter of 1.0 cm. To evaluate the shape similarity between the constructed cubical particle and the perfect cube, they are compared in terms of sphericity, which is an important parameter for describing particle shape and defined as the ratio of the surface area of a sphere having the same volume as the particle to the surface area of the particle. For a perfect cube, the sphericity is 0.806. By using AutoCAD, the surface area and volume of the particle is computed as 32.164 cm2 and 13.288 cm3, respectively. Then, the equivalent diameter (dv) and sphericity of the cubical particle are obtained as 2.939 cm and 0.843, respectively. The deviation between the constructed cubical particle and the perfect cube is smaller than 5%, indicating the shape similarity between them is acceptable. It is noted that the cubical particle used in current numerical work is composed of overlapped spheres, which can cause the non-uniform mass distribution within the particle. Therefore, the moment of inertia of the constructed cubical particle is different from that of a perfect cube with the same volume. To decrease this effect, the treatment to obtain a comparable particle density is used as the work of Lu and Mcdowell [45]. The simulation begins with the random generation of equal cubical particles in a cylindrical container. Then, the so-called poured packing [46] is formed with the particles falling down under gravity. During the packing process, interactions among particles and collisions between particles and container walls can cause the energy dissipation of the packing system, thus slowing down the movement of particles. A stable packing structure is formed when all particles reach their stable positions with the kinetic energy becoming zero. Fig. 3 displays the dynamic process of forming a poured initial packing (totally 1121 particles in this case), which generally lasts for about 0.5 s. The mechanical vibration starts from 1.0 s, thus allowing us to get the initial packing structure without vibration (generally adopted at t = 0.98 s) for the comparisons with those vibrated structures in the following structural analyses in Section 3.2. The three dimensional mechanical vibration [13,47], described by R(t) = Asin[ω(t)], is implemented to the container in three inter-perpendicular directions (X, Y, Z), where R represents the displacement in each direction. To simplify the vibration conditions, the values of amplitude A or frequency ω are the same in the X, Y and Z directions. After the vibration process is finished, another 1.0 s is allocated
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Fig. 2. The cubical particle model used in present work.
for the stabilization of the vibrated packing. The simulation parameters in the present study are listed in Table 1.
3. Results and discussion 3.1. Packing densification under 3D vibrations Packing density ρ, as the alternative representation of porosity ε (ε = 1 − ρ), is probably the simplest and most fundamental parameter in particle packing characterization. It is effective to evaluate the loose or dense state of a packing, thus illustrating the densification process and phase transitions. Generally, packing density is affected by a number of variables related to packing conditions and material properties
[35,46]. However, these effects are not dominant towards densification when mechanical vibration is implemented. Therefore, we just focused on the effects of vibration conditions for densification in this study. The effects of material properties and other factors may be studied in future work. Also it needs to note that while studying the effects of each certain vibrated parameter, other parameters are normally being fixed. 3.1.1. Effects of vibration amplitude and frequency It is known the vibration conditions like vibration dimensions (1D or 3D), vibration time, amplitude and frequency all can create effects on the densification process [12,13,20,47,48]. Compared with spherical particles, the rearrangement of cubical particles in the packing is more difficult because of their geometrical angularity. 3D vibration is thus
Fig. 3. Snapshots showing the formation of a cubical particle packing at different time, where: (a) t = 0.02 s, (b) t = 0.20 s, (c) t = 0.40 s, (d) t = 0.45 s.
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Table 1 Parameters used in simulation. Parameters
Values
Sphere diameter, dp Particle density, ρp Number of particles, n Poisson ratio, ν Restitution coefficient, e Shear modulus, G Sliding friction coefficient, μs Rolling friction coefficient, μr Vibration time, t Vibration amplitude, A Vibration frequency, ω Time step, Δt
0.01 m 2.5 × 103 kg/m3 500–1500 0.3 0.6 1.0 × 107 Pa 0.3 0.002 2.0–5.0 s 0.05–0.3 dv (equivalent particle diameter) 50–300 Rad/s 5.0 × 10−5 s
adopted to provide more energy for the packing system to promote the local rearrangement of cubical particles during densification. Among various vibrated parameters, vibration time t is firstly considered, since proper choice of t is the precondition for generating dense packing structures both efficiently and accurately, which has been demonstrated in our physical experiments on vibrated packing densification of spheres [47,48]. By tests, we found vibration time of 3.0 s is appropriate for achieving the dense stable random packing structure of cubical particles. Therefore, unless otherwise specified, the vibration time in the numerical simulations is chosen to be t = 3.0 s. To investigate the effects of amplitude A and frequency ω, A ranging from 0.01 dv to 0.25 dv coupling with ω ranging from 50 to 250 Rad/s are considered. Fig. 4 shows the evolution of packing density ρ with amplitude A when the container diameter D = 0.3 m. It can be seen that under a given ω, ρ initially increases with A to a maximum value and then decreases. And the ρ-A curve with a larger ω shifts to the left, illustrating a larger ω should be coupled with a relatively smaller A to realize dense packings. Similar trend can be found from the effects of ω on ρ with D = 0.3 m as observed in Fig. 5. It is thus indicated that properly controlling A and ω is the necessity for packing densification, and their effects on packing density should be considered together.
Fig. 5. Effects of vibration frequency on packing densities when the container diameter D = 0.30 m.
3.1.2. Combined effects of vibration amplitude and frequency Previous studies have demonstrated that the influences of amplitude A and frequency ω on packing densification of cubical particles can be ascribed to the vibration intensity Γ, which is a function of A and ω (defined as Γ = Aω2/g), and is often used to indicate the combined influence of A and ω [12,20,25,26,47,48]. Fig. 6 gives the relationship between vibration
intensity Γ and packing density ρ. It is found that ρ increases drastically with Γ in the initial stage until to a maximal value, then decreases gradually with the further increase of Γ. This is because the initial increase of Γ can enhance the external energy or forces required to break down the local loose structures such as inter-particle locking or bridges in a packing, and the packing structure is thus transformed from a loose to dense state; but too large Γ may over-excite particles and relax them from a jammed state to a loose packing state. Therefore, an appropriate Γ is needed to realize dense packing structure of high packing density. This is consistent with the pattern in vibrated packing of spheres [20, 47,48]. Fig. 6 also shows that the highest packing density (ρ = 0.663) is achieved with Γ = 5, indicating Γ = 5 is an efficient value for achieving high packing densities. However, Γ = 5 actually corresponds to three combinations of A and ω in Fig. 6, which can lead to three different packing densities. Thus, no one-to-one relationship between vibration intensity and packing density can be identified. The evaluation of vibration intensity can provide certain guidance to choose the proper parameters, but it is not decisive for directly choosing the exact operating parameters. Accordingly, the effects of vibration amplitude and frequency are considered concurrently as shown in Fig. 7. It is indicated high packing densities can be reached when A ranges from 0.5 dv to 0.15 dv and ω ranges from
Fig. 4. Effects of vibration amplitude on packing densities when the container diameter D = 0.30 m.
Fig. 6. Effects of vibration intensity on packing densities at three vibration amplitudes and a range of vibration frequencies when the container diameter D = 0.30 m.
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Fig. 7. Combined effects of both amplitude and frequency on packing densities when the container diameter D = 0.30 m. Fig. 8. Extrapolation of packing densities in different sized containers.
100 Rad/s to 250 Rad/s. Particularly, A = 0.15 dv and ω = 100 Rad/s can create the highest packing density of 0.663. Thus, vibration intensity Γ = 5 is regarded as the most efficient vibration parameter in this study, since high packing density with relatively low energy consumption can be achieved under this vibration condition. It is noting that this value of Γ = 5 is consistent with the previous experimental work of Baker and Kudrolli [31], who also adopted vibration intensity of 5 to achieve the maximum possible density of particles. 3.1.3. Wall effects The above simulation is carried out in a fixed sized cylindrical container with container diameter D = 0.30 m. It is known that the container size can influence the final packing density due to the wall effects [49]. To eliminate this effect, packing of spheres was generally conducted in different sized containers for extrapolation [21,47,48]. Recently, this method was also used in the packing of non-spherical particles with the obtained results comparable to those results obtained with boundary conditions [43]. It was shown that there was a linear relationship between packing densities and the dv/D ratio. Similar method is used in this work. However, since it is laborious and not meaningful to conduct each of the above vibration conditions in various containers for extrapolation, four cases (C2, C3, C4 and C5) of the packings under vibrations were chosen with the parameters shown in Table 2. One case (C1) of poured packing without vibration is also studied to understand the initial poured packing. The extrapolation of packing densities with different dv/D is shown in Fig. 8. As can be seen, the linear relationship between ρ and dv/D is identified. The R2 values of the trend lines in the inset table of Fig. 8 are around 0.95, indicating the good reliability of the linear relationship. Packing densities increase with the decrease of dv/D. By extrapolation, the interception values on the vertical axis correspond to the packing densities in infinite sized containers. The packing densities of C1 to C5 are 0.591, 0.657, 0.683, 0.694 and 0.727, respectively. However, it still cannot determine which packing is RCP without microscopic structural analyses at present stage. In the following section, the study is thus Table 2 Vibration parameters of the five typical cases of packings. Cases
Packing density (D = 0.30 m)
Vibration amplitude, dv
Vibration frequency, Rad/s
C1 C2 C3 C4 C5
0.562 0.598 0.625 0.650 0.663
0 0.10 0.15 0.10 0.15
0 50 50 100 100
extended to characterize the micro-properties of the 5 packings (C1 to C5) to identify the RCP structure and understand the structural evolution in densification process, which are difficult to be quantitatively studied in physical experiments. 3.2. Micro-structure analyses 3.2.1. Coordination number Coordination number (CN) [23,24,50] is the fundamental microproperty of a packing. It is defined as the number of particles in contact with a considered particle. Based on the multi-sphere model, the contacts between two cubical particles are determined by the contacts between their corresponding element-spheres. In other words, if the distance of two element-spheres in two cubical particles is less than the cut-off distance, then the two particles are considered as in contact. Here, it needs to clarify that for two particles that have multiple contacts of element-spheres in our simulation, the contact number counted for each other is still kept as 1. The cut-off distance, generally ranging from 1.005 d to 1.05 d (d is the diameter of the element-sphere), is very crucial for the results of the CN analysis. In this work, the cut-off distance is set as 1.005 d in order to obtain more precise results. Fig. 9 shows the CN distributions of the five packings in Table 2. The peak of the distribution of curve C1 is around 5.2, corresponding to a loose packing structure. With the implementation of mechanical vibration, the CN distribution curve of C2 shifts to the right, indicating the transition of packing structures from a loose to dense state. It is noting that the peaks of CN curves from C2 to C5 almost keep the same around CN = 6 regardless of the increasing packing densities, which is different from the packing of spheres. This feature illustrates that large rearrangement among cubical particles are difficult due to their angularity compared with spheres, thus further increase of contacts is not easy for most particles. It is also indicated that a high packing density does not require a high CN in the packing of cubes. Actually, even for the perfect ordered packing of cubes with the packing density equals 1.0, the CN of a cube in the packing can still be just 6 with 6 neighbors aligned to its 6 faces. But there are still some changes from the curves of C2 to C5. For example, the part of each curve other than the peak is different, which implies the difference of packing densities. With the increase of the packing density, the probability for those particles with small CN decreases with the increase of particles with large CN, indicating the transition of packing structures from loose to dense state. In addition, Fig. 9 also shows that some of the curves are uniform and symmetrically distributed, whereas others are not so uniform; and some curves are higher and narrower than others. In fact, these
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3.2.2. Radial distribution function Radial distribution function (RDF) [4–6,50] is also frequently used to characterize the packing properties as it contains useful information about inter-particle correlations. It is the probability of finding a particle center at a given distance from a reference one, defined by gðr Þ ¼
Fig. 9. Coordination number distributions for different packing densities.
differences can indicate the uniformity and the transition of the structures. Thus, the standard deviation σCN [24] of each CN curve is adopted to quantify the difference, which defined as:
σ CN
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v" # ffi u 2 u N ¼ t ∑ C ni −C n =N i
ð21Þ
where N is the number of particles, Cni is the CN number of particle i and C n is the averaged CN. Generally, the larger σCN corresponds to the less uniformity of the structure. Fig. 10 plots the results of different packings. With the increase of packing density, σCN decreases initially from C1 to C3 but increases after that. This trend identifies C3 to be a turning point with the most uniform structure. It is known a RCP structure is uniform and can be obtained with the densification from an initial loose structure with less uniformity; but further densification can create local ordered structures which can increase the anisotropy and decrease the uniformity of the whole packing system. The variation of σCN in Fig. 10 is comparable to this pattern. As the turning point, the structure of C3 is thus consistent with the concept of RCP.
Fig. 10. The standard deviation of coordination number for different packing densities.
△Nðr Þ 4πr 2 Δrρ0
ð22Þ
where ΔN(r) is the number of particle centers situated at a distance between r and r + Δr from the center of a given particle and ρ0 is the average number of particle per unit volume in the packing. For the packing of uniform spheres, the splitting second peak in the pffiffiffi RDF with the first component at 3 d (diameter of the sphere) and the second component at 2.0 d can indicate the RCP feature of the packing [5,50]. And the two peaks respectively correspond to two types of particle connection as the so-called edge sharing in-plane equilateral triangles and three spheres along a line [5,50]. Similarly, we tried to identify such local structures in cubical particle packings. Fig. 11 shows the RDF variation of the five packings. It can be seen from Fig. 11(a) that the first peak of RDF changes gradually from C1 to C5. In parpffiffiffi pffiffiffi ticular, three peaks exists obviously at r1 = ð1 þ 2Þ=2, r2 = 5=2 and r3 = 1.0, which correspond to three types of particle connection as shown in Fig. 11(b). As can be seen, the first possible particle connection pffiffiffi (r1 = ð1 þ 2Þ=2) is between a cube i and its neighbor j1 with the interpffiffiffi particle angle of 45°; the second type of particle connection (r2 = 5=2) is between the cube i and the cube j2 sharing a half of a face; and the third type of particle connection (r3 = 1.0) is between the cube i and the cube j3 which are aligned to each other along a line. In the RDF of C1, the first peak is splitting as two components located at r1 and r2 respectively. The particle connection corresponding to the peak at r1 is actually loose and not mechanically stable as the particles just contact with each other by a single edge. With the mechanical vibration, the peak at r1 disappears in C2, and only the peak at r2 exists obviously. It is indicated that the weak particle connection (r1 = pffiffiffi ð1 þ 2Þ=2) is likely to be eliminated in the vibration process. In the RDF of C3, the first peak becomes splitting again with two components existing at r2 and r3, respectively. In RDF of C4 and C5, the first peak becomes much sharper at just r3 corresponding to cubes aligned with each other along a line, which is actually a local structure of high ordering. Therefore, the RDF of C3 is the final packing to keep the splitting feature of the first peak. Like the splitting second peak in the RDF of RCP for uniform spheres, the splitting first peak at r2 and r3 in RDF of the packing of C3 may also be considered as the RCP feature of uniform cubes. The packing of C3 as the critical state of structural transition is also consistent with the previous CN analyses in Section 3.2.1. Besides, the structure of C1 formed by poured packing is loose and uniform in CN analyses, and also keeps loose local structures corresponding to the pffiffiffi peak at r1 = ð1 þ 2Þ=2 in RDF. Therefore, the packing of C1 is consistent with the concept of RLP as a loose state of poured packing without tapping or vibration [21]. 3.2.3. Particle orientation In addition to the importance of determining the movement of nonspherical particles, orientation is also of primary consideration for packing structures in terms of particle connectivity and pore shapes. The method that we use to investigate the orientation of cubical particles is similar as that used by Zhou et al. [51] in describing the orientation variation of ellipsoids. Since the three axes of cubical particles are equivalent, three vectors in the direction of three axes are traced respectively for orientation determination as shown in Fig. 12. The distributions of the orientation angles in the horizontal X-O-Y plane, the vertical X-OZ plane or Y-O-Z plane are computed respectively. Then, the three distributions are averaged as the orientation distribution of the cubical particles.
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Fig. 11. Radial distribution function for different packing densities: (a) the variation of RDF curves; (b) three typical local structures (particle connection) at the peaks in RDF.
Fig. 13 shows the orientation information of cubical particles in cases of C1, C3 and C5. For the poured packing C1, the orientation distribution is relatively uniform, indicating the orientation of the cubical particles is random at the initial loose state. In previous study, Delaney et al. [52] argued that the RLP of non-spherical particles cannot be defined like that of spheres obtained by pouring particles, since the orientational ordering increases with the aspect ratio of ellipsoids. From the present study, it is shown that the orientational ordering of RLP can be limited
for cubes. And the present orientational ordering of cubes is likely to be even lower by settling particles in liquids or using particles of large friction coefficients, which can be explored in the future. The orientation distribution of C3 is not so uniform with the implementation of vibration. But no obvious preferred orientation appears in C3, thus the orientational randomness of C3 can still be accepted as the RCP structure. However, the orientational ordering emerges with further vibration. In C5, obvious preferred orientations appears particularly at 0° and 180°
Fig. 12. The sketch of orientation angle for cubical particles, where: (a) represents the orientation vectors; (b) describes the angle in the X-O-Y plane.
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Fig. 13. Distributions of orientation angles for packings with different packing densities.
in X-O-Z plane and Y-O-Z plane, which indicates cubical particles tend to face upward or downward direction in the vibration process. This is because cubical particles of horizontal state possess less potential energy, thus are more stable under gravity [51]. The particle orientation of cubes affected by gravity can be further understood by analyzing the force orientation distribution [53]. Fig. 14 shows the orientation distribution of the normal contact force P(θ) of C1, C3 and C5, where θ is the angle between the force and an upward (vertical) vector with θ ∈ (0°, 90°]. The peaks of the curves at θ = 90° indicates more forces tend to be horizontal under gravity [53]. And this trend is more obvious with the increase of packing density (the higher points at θ = 90°). But compared to the large increase of C5 at θ = 90°, the increase of C3 is much smaller, illustrating the increase of anisotropy effect caused by gravity is limited from C1 to C3. Besides, it can be seen in the P(θ) of C1 there is another peak at θ = 45°, which disappears in the P(θ) of C3. This variation illustrates the initial anisotropic structure can be more uniform under proper vibration. And in C3, the distribution of P(θ) is uniform in most of the scope without obvious peaks, indicating the randomness of the packing structure. It is thus more reasonable for considering C3 as RCP without large structural anisotropy. In addition, the further increase of packing density displays anisotropy again as shown in C5 with the appearance of peaks at 15° and 40°, which is consistent with the anisotropy of particle orientations in Fig. 13(c). It is thus shown the structural properties are connected to the interparticle forces in the packing system. From the above structural analyses, RLP and RCP are identified as C1 with ρ = 0.591 and C3 with ρ = 0.683, respectively. Although the two packing structures that we obtained still have some deficiencies as they are not absolutely random and uniform under the effect of gravity, but
Fig. 14. Distributions of force orientation for the packings with different packing densities.
the ordering and anisotropy are not obvious for them. Compared with the present literature with little structural information of the packing of cubes, our work with the analyses of CN, RDF and orientation could give some new sights for the RLP and RCP of cubes. Actually, the obtained RCP and RLP in our work are also close to the previous experimental results. For example, the packing densities for RLP and RCP of cubes are predicted as 0.586 and 0.656 respectively in the empirical equations proposed by Zou and Yu [29]. And the obtained RCP is also consistent with the result of Barker and Kudrolli [30] who obtained the RCP packing density of cubes as 0.67. However, the RLP packing density they obtained is 0.54 which is much lower than 0.591. The reason is probably due to the difference of packing methods as they obtained the loose structure by settling cubes in liquids. 3.3. Densification mechanism The densification mechanism is of great importance for both the understanding of granular dynamics as well as the industrial applications. From the analyses of the micro-structures, it is known that the densification from C1 to C3 and C3 to C5 are not the same. Densification develops limited ordering occurrence from C1 to C3; but the densification from C3 to C5 can lead to more obvious ordering structures. Accordingly, we investigated the densification mechanisms behind the two densification processes with two stages identified. 3.3.1. Particle rearrangement Fig. 15 plots the distribution of particle centers of C1 and C3, where the spheres represent the centers of cubical particles with the color representing its coordination number (CN). It is clear that particles are randomly distributed in both C1 and C3 without obvious crystallization. The outer ring of C3 from the top view is of slight ordering, but the ordering region is limited and does not propagate to inner zones. However, particles of C3 are much darker than particles of C1, indicating the contacts of particles increase in this densification process. In previous studies, it is generally accepted that there are macro-pores or arching structures [3,12,54] in RLP of spherical particles. And densification under vibration should be linked to the elimination of these loose structures [12,54]. Similarly, we traced the variation of a cluster in C3 to investigate its densification evolution. Five snapshots are shown in Fig. 16, which describe the evolution of a cluster from loose to dense state in C3. Since the vibration time used in these cases is 3.0 s with the vibration starts at 1.0 s and ends at 4.0 s, Fig. 16(a) is in the RLP stage of poured packing (before vibration); Fig. 16(b) to (d) are the three stages during vibration; and Fig. 16(e) presents the final stable structure after vibration. As can be seen, the cluster is initially a typical loose structure with particles of few contacts. And there is much space in the cluster with particles A, B, C and D, which are part of an arching structure. After vibration is implemented, the cluster changes greatly. At t = 2.0 s, it has become more uniform with arching structure being eliminated. Especially the CN of the inner particle (particle C) increases greatly to 9, illustrating the positions among the
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Fig. 15. Distribution of particle centers for C1 ((a) front view and (c) top view) and C3 ((b) front view and (d) top view).
Fig. 16. Evolution of a cluster from loose to dense state in C3.
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Fig. 17. Distribution of particle centers for C3 ((a) front view and (c) top view) and C5 ((b) front view and (d) top view).
particles are closer. Although the movement of particles from 2.0 s to 4.0 s is not very obvious, CN of particles changes greatly. From 2.0 s to 3.0 s, CN of some of the particles like B and D increases, whereas CN of the other particles like A and C decreases. From 3.0 s to 4.0 s, CN of some particles like B, C and D increases further with the formation of a denser structure. From 4.0 s to 5.0 s, the cluster almost keeps the same, indicating the structure has become pretty dense and stable. The variation of the cluster explicitly shows the rearrangement among particles to produce a dense structure and more contacts under the vibration condition. And as can be seen, CN of the filling particle C increases steadily to its maximum with small fluctuation, thus is comparable to the “pushing filling” mechanism as identified by An et al. for the vibrated packing of equal spheres [12]. The existence of pushing filling illustrates the vibration intensity for the packing of C3 is not very large, and the densification process is thus steady. This explains why the final structure of C3 can be both dense and uniform with the features of RCP. 3.3.2. Crystallization along the container wall Unlike the densification from C1 to C3, the densification from C3 to C5 witnesses the crystallization process. As can be seen in Fig. 17, crystallization occurs in the bottom layers from the front view of C3 and C5 with particles aligned with ordering. From the top view, crystallization can be more clearly observed from the outer rings with propagation to the inner part. Since the radial crystallization is obvious, the distribution of radial packing density is plotted in Fig. 18 to quantify the effects during densification. It can be seen the peaks near the wall are sharper and decrease with the radial distance from the wall, indicating higher packing densities are formed near the wall. With the increase of packing density, the peaks become sharper. For the packing of C5, the packing
density near the wall indicated from the left two peaks is almost 1, illustrating the two outer layers are almost perfectly ordering. Therefore, it suggests the further densification or crystallization of cubical particles are strongly affected by container walls. Actually, this pattern is consistent with previous studies about the near-wall crystallization phenomena of cylindrical particle packings [55,56]. But it seems the
Fig. 18. Distributions of radial packing density for packings with different packing densities.
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Fig. 19. Snapshots of particle velocity vectors in C5, where: (a) t = 2.89 s; (b) t = 2.91 s; (c) t = 2.94 s.
crystallization of cubical particles can be more obvious, giving insights to the design of fixed beds with different permeability. To further investigate the crystallization mechanism, especially about the reason why cubical particles under vibration can crystallize firstly near the wall easily, we traced the particle velocity variation in the vibration process of C5. Since a single vibration period of C5 is around 0.06 s (ω = 100 Rad/s), three snapshots of particle velocity vectors in a single period are plotted in Fig. 19. As can be seen, instead of the regular flow pattern of particles under 1D vibration [57], the movement of particles under 3D vibration can be much more complicated. Due to the dynamical collisions, particles move towards various directions at t = 2.91 s and t = 2.94 s. It is worth noting that the main vibration energy at t = 2.91 s is actually provided by the right container wall, whereas the main flow stream is directed to the bottom. Similar trend is shown in Fig. 19(c) where many particles finally flow to the left under the vertical vibration. Therefore, the movement of particles is actually affected by the combined effects of 3D vibration which causes the collisions of flow streams of particles. Due to the dynamic collisions, the loose structures such as large pores or arching structures in the packings are thus likely to be eliminated. In addition, after the collision of flow streams, there are always a large number of particles moving towards the container walls at each time as the red arrows marked. The crystallization along the container walls is thus reasonable. 4. Conclusions The DEM simulation is used to study the cubical particle packing under mechanical vibration. Effects of vibration conditions are investigated with the obtained packing structures being discussed. Densification mechanisms are also analyzed. The main findings of the present work are summarized as follows: 1) Mechanical vibration is effective for the packing densification of cubical particles. The degree of densification is significantly affected by vibration amplitude A and frequency ω. Packing density ρ generally increases with A or ω to a maximum value and then decreases. Vibration intensity Γ of 5 with A = 0.15 dv and ω = 100 Rad/s is efficient for achieving a dense packing. 2) Packing structures of different packing densities display different microscopic properties, based on which RLP and RCP of cubical particle packing are identified with ρ = 0.591 and ρ = 0.683, respectively. For cubical particles, a high packing density does not require a high averaged CN, which actually tends to be stable around 6. The CN deviation indicates a turning point of the packing with ρ = 0.683 (RCP). The first splitting peak of RDF with the first component at 1.0 L (edge length of the cube) and the second component at pffiffiffi 5L=2 can indicate the RCP feature of uniform cubes. And the peak pffiffiffi in RDF at ð1 þ 2ÞL=2 indicates the RLP feature. The orientation of cubical particles is easy to exhibit ordering under gravity. But the
orientational ordering decreases with the decrease of packing density, and the ordering is not obvious for the random structures. 3) Two densification mechanisms are identified: the particle rearrangement, which is dominant for realizing the densification from RLP to RCP; the crystallization along the container walls, which is dominant for the densification beyond RCP to ordering state. Pushing filling mechanism exists in the rearrangement of cubical particles during the densification process. In addition to the proximity of cubical particles to the container walls, the crystallization is also due to the vibration effects which causes the movement of particles towards the walls.
Acknowledgements We are grateful to the financial support of the National Natural Science Foundation of China (No. 51374070) and Fundamental Research Funds for the Central Universities of China (No. N130102001). References [1] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Granular solids, liquids, and gases, Rev. Mod. Phys. 68 (1996) 1259–1273. [2] R.M. German, Particle Packing Characteristics, Metal Powder Industries Federation, Princeton, N.J, 1989. [3] J.D. Bernal, A geometrical approach to the structure of liquids, Nature 183 (1959) 141–147. [4] J.D. Bernal, Geometry of the structure of monatomic liquids, Nature 185 (1960) 68–70. [5] J.L. Finney, Random packings and the structure of simple liquids. I. the geometry of random close packing, Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 319 (1970) 479–493. [6] J.L. Finney, Modelling the structures of amorphous metals and alloys, Nature 266 (1977) 309–314. [7] T.G.O. Berg, R.L. McDonald, R.J. Trainor, The packing of spheres, Powder Technol. 3 (1969) 183–188. [8] O. Pouliquen, M. Nicolas, P.D. Weidman, Crystallization of non-Brownian spheres under horizontal shaking, Phys. Rev. Lett. 79 (1997) 3640–3643. [9] Y. Nahmad-Molinari, J.C. Ruiz-Suárez, Epitaxial growth of granular single crystals, Phys. Rev. Lett. 89 (2002) 264302. [10] F.H. Stillinger, T.A. Weber, Packing structures and transitions in liquids and solids, Science 225 (1984) 983–989. [11] M.D. Rintoul, S. Torquato, Computer simulations of dense hard-sphere systems, J. Chem. Phys. 105 (1996) 9258–9265. [12] X.Z. An, R.Y. Yang, K.J. Dong, R.P. Zou, A.B. Yu, Micromechanical simulation and analysis of one-dimensional vibratory sphere packing, Phys. Rev. Lett. 95 (2005) 205502. [13] A.B. Yu, X.Z. An, R.P. Zou, R.Y. Yang, K. Kendall, Self-assembly of particles for densest packing by mechanical vibration, Phys. Rev. Lett. 97 (2006) 265501. [14] E. Arzt, The influence of an increasing particle coordination on the densification of spherical powders, Acta Metall. 30 (1982) 1883–1890. [15] R.M. German, Coordination number changes during powder densification, Powder Technol. 253 (2014) 368–376. [16] S. Bryant, P. King, D. Mellor, Network model evaluation of permeability and spatial correlation in a real random sphere packing, Transp. Porous Media 11 (1993) 53–70. [17] G.J. Cheng, A.B. Yu, P. Zulli, Evaluation of effective thermal conductivity from the structure of a packed bed, Chem. Eng. Sci. 54 (1999) 4199–4209. [18] A. Mehta, G.C. Barker, Vibrated powders: a microscopic approach, Phys. Rev. Lett. 67 (1991) 394–397.
Y. Wu et al. / Powder Technology 314 (2017) 89–101 [19] P. Philippe, D. Bideau, Numerical model for granular compaction under vertical tapping, Phys. Rev. E 63 (2001) 051304. [20] X.Z. An, R.Y. Yang, R.P. Zou, A.B. Yu, Effect of vibration condition and inter-particle frictions on the packing of uniform spheres, Powder Technol. 188 (2008) 102–109. [21] G.D. Scott, Packing of spheres: packing of equal spheres, Nature 188 (1960) 908–909. [22] C.S. O'Hern, L.E. Silbert, A.J. Liu, S.R. Nagel, Jamming at zero temperature and zero applied stress: the epitome of disorder, Phys. Rev. E 68 (2003) 011306. [23] C. Song, P. Wang, H.A. Makse, A phase diagram for jammed matter, Nature 453 (2008) 629–632. [24] K.J. Dong, R.Y. Yang, R.P. Zou, X.Z. An, A.B. Yu, Critical states and phase diagram in the packing of uniform spheres, Europhys. Lett. 86 (2009) 46003. [25] H.K. Pak, P.R. Behringer, Bubbling in vertically vibrated granular materials, Nature 371 (1994) 231–233. [26] L. Vanel, A.D. Rosato, R.N. Dave, Rise-time regimes of a large sphere in vibrated bulk solids, Phys. Rev. Lett. 78 (1997) 1255–1258. [27] G. Chen, E. Gordo, D. Fray, Direct electrolytic preparation of chromium powder, Metall. Mater. Trans. B Process Metall. Mater. Process. Sci. 35 (2004) 223–233. [28] V. Kozlovskaya, J. Chen, C. Tedjo, X. Liang, J. Campos-Gomez, J. Oh, M. Saeed, C.T. Lungu, E. Kharlampieva, pH-responsive hydrogel cubes for release of doxorubicin in cancer cells, J. Mater. Chem. B 2 (2014) 2494. [29] B.S. John, A. Stroock, F.A. Escobedo, Cubatic liquid-crystalline behavior in a system of hard cuboids, J. Chem. Phys. 120 (2004) 9383–9389. [30] R.P. Zou, A.B. Yu, Evaluation of the packing characteristics of mono-sized non-spherical particles, Powder Technol. 88 (1996) 71–79. [31] J. Baker, A. Kudrolli, Maximum and minimum stable random packings of platonic solids, Phys. Rev. E 82 (2010) 061304. [32] C.R.A. Abreu, F.W. Tavares, M. Castier, Influence of particle shape on the packing and on the segregation of spherocylinders via Monte Carlo simulations, Powder Technol. 134 (2003) 167–180. [33] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Géotechnique 29 (1979) 47–65. [34] J. Zhao, S.X. Li, Numerical simulation of random close packings in particle deformation from spheres to cubes, Chin. Phys. Lett. 25 (2008) 4034–4037. [35] Z.P. Zhang, L.F. Liu, Y.D. Yuan, A.B. Yu, A simulation study of the effects of dynamic variables on the packing of spheres, Powder Technol. 116 (2001) 23–32. [36] G. Lu, J.R. Third, C.R. Mueller, Discrete element models for non-spherical particle systems: from theoretical developments to applications, Chem. Eng. Sci. 127 (2015) 425–465. [37] K.J. Dong, C.C. Wang, A.B. Yu, A novel method based on orientation discretization for discrete element modeling of non-spherical particles, Chem. Eng. Sci. 126 (2015) 500–516. [38] J.F. Favier, M.H. Abbaspour-Fard, M. Kremmer, A.O. Raji, Shape representation of axisymmetrical, non-spherical particles in discrete element simulation using multi-element model particles, Eng. Comput. 16 (1999) 467–480.
101
[39] H. Hertz, Ueber die Berührung fester elastischer Körper, J. Reine Angew. Math. 1882 (1882) 156–171. [40] R.D. Mindlin, H. Deresiewicz, Elastic spheres in contact under varying oblique forces, J. Appl. Mech. Trans. ASME 20 (1953) 327–344. [41] Y. Tsuji, T. Tanaka, T. Ishida, Lagrangian numerical-simulation of plug flow of cohesionless particles in a horizontal pipe, Powder Technol. 71 (1992) 239–250. [42] W. Nan, Y. Wang, Y. Liu, H. Tang, DEM simulation of the packing of rodlike particles, Adv. Powder Technol. 26 (2015) 527–536. [43] Z.Y. Zhou, R.P. Zou, D. Pinson, A.B. Yu, Dynamic simulation of the packing of ellipsoidal particles, Ind. Eng. Chem. Res. 50 (2011) 9787–9798. [44] P.A. Langston, M.A. Al-Awamleh, F.Y. Fraige, B.N. Asmar, Distinct element modelling of non-spherical frictionless particle flow, Chem. Eng. Sci. 59 (2004) 425–435. [45] M. Lu, G.R. McDowell, The importance of modelling ballast particle shape in the discrete element method, Granul. Matter 9 (2006) 69–80. [46] J.C. Macrae, W.A. Gray, Significance of the properties of materials in the packing of real spherical particles, Brit. J. Appl. Phys. 12 (1961) 164–172. [47] C.X. Li, X.Z. An, R.Y. Yang, R.P. Zou, A.B. Yu, Experimental study on the packing of uniform spheres under three-dimensional vibration, Powder Technol. 208 (2011) 617–622. [48] X.Z. An, C.X. Li, R.Y. Yang, R.P. Zou, A.B. Yu, Experimental study of the packing of mono-sized spheres subjected to one-dimensional vibration, Powder Technol. 196 (2009) 50–55. [49] R.P. Zou, A.B. Yu, The packing of spheres in a cylindrical container: the thickness effect, Chem. Eng. Sci. 50 (1995) 1504–1507. [50] R.Y. Yang, R.P. Zou, A.B. Yu, Computer simulation of the packing of fine particles, Phys. Rev. E 62 (2000) 3900–3908. [51] Z.Y. Zhou, D. Pinson, R.P. Zou, A.B. Yu, Discrete particle simulation of gas fluidization of ellipsoidal particles, Phys. Rev. E 66 (2011) 6128–6145. [52] G.W. Delaney, J.E. Hilton, P.W. Cleary, Defining random loose packing for nonspherical grains, Phys. Rev. E 83 (2011) 051305. [53] R.Y. Yang, R.P. Zou, A.B. Yu, S.K. Choi, Characterization of interparticle forces in the packing of cohesive fine particles, Phys. Rev. E 78 (2008). [54] Y. Wu, X. An, F. Huang, DEM simulation on packing densification of equal spheres under compression, Mater. Res. Innov. 18 (S4) (2014) 1082–1086. [55] W. Zhang, K.E. Thompson, A.H. Reed, L. Beenken, Relationship between packing structure and porosity in fixed beds of equilateral cylindrical particles, Chem. Eng. Sci. 61 (2006) 8060–8074. [56] L. Pournin, M. Weber, M. Tsukahara, J.A. Ferrez, M. Ramaioli, T.M. Liebling, Three-dimensional distinct element simulation of spherocylinder crystallization, Granul. Matter 7 (2005) 119–126. [57] M. Bourzutschky, J. Miller, “Granular” convection in a vibrated fluid, Phys. Rev. Lett. 74 (1995) 2216–2219.
