Powder Technology 317 (2017) 171–180
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DEM dynamic simulation of tetrahedral particle packing under 3D mechanical vibration Bo Zhao a, Xizhong An a,⁎, Yang Wang a, Quan Qian a, Xiaohong Yang a, Xudong Sun b a b
School of Metallurgy, Northeastern University, Shenyang 110004, PR China School of Materials Science and Engineering, Northeastern University, Shenyang 110004, PR China
a r t i c l e
i n f o
Available online 24 April 2017 Keywords: Particle packing Regular tetrahedron 3D vibration Densification mechanism DEM simulation
a b s t r a c t The transition from random loose packing (RLP) to random close packing (RCP) of mono-sized regular tetrahedral particles under 3D mechanical vibration was modeled by using discrete element method. The effects of vibration conditions and container size on the packing densification were systematically studied, and the macro and micro properties such as packing density, coordination number (CN), particle contact type, radial distribution function (RDF), particle orientation correlation, and forces between particles were characterized and analyzed. The randomness of the obtained dense packings and corresponding densification mechanisms were also investigated. The results show that RCP of the tetrahedral particles can be realized by properly controlling the vibration conditions. And the obtained maximum random packing density can reach about 0.7402 by extrapolating the packing densities in different sized containers. The average CN for RLP and RCP are 7 and 6.3, and the CN distribution for RCP is higher and narrower than that for RLP. From RLP to RCP, the frequency of face to face (F-F) contact between two particles increases, while that of vertex to face (V-F) contact and edge to face (E-F) contact decreases, and edge to edge (E-E) contact does not change much. RDF characterization shows four obvious peaks on the curves of RLP and RCP, where the height of the first peak (F-F contact) increases while that of other peaks decreases from RLP to RCP. The densification mechanism can be ascribed to the formation of wagon wheel local dense structures through rearrangement by ‘pushing filling’ and ‘jumping filling’. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Particle packing is an important issue in both the scientific research and industrial applications [1]. In scientific research, it can be used as an effective starting point for studying the structure of complex systems (directly studying these systems is very hard due to the complexity of their dynamics and thermal kinetics) such as liquid [2,3], amorphous state [4,5], crystalline state [6] and their mutual transitions. In industry, it can be used in different areas such as metallurgy, materials, and chemical engineering, etc. Therefore, in the past decades large amount of work was carried out in this regard. The research on particle packing is first started from the packing of equal discs in 2D and mono-sized spheres in 3D. For equal spheres, three packing states have been reproduced, i.e. random loose packing (RLP, with the packing density ρ ≈ 0.60 [7]), random close packing (RCP, with ρ ≈ 0.64 [8,9]), and ordered packing (with the maximum ρ ≈ 0.74 [10–12], corresponding to face centered cubic or hexagonal close packed structure). Compared with the well-identified packing ⁎ Corresponding author. E-mail address:
[email protected] (X. An).
http://dx.doi.org/10.1016/j.powtec.2017.04.048 0032-5910/© 2017 Elsevier B.V. All rights reserved.
structure of mono-sized spheres, the packing of mono-sized nonspherical particles is less studied due to the complexity of their particle shape. For example, for the packings of tetrahedral particles which have been used by scientists to study the structure of different states such as liquid state [13] and glass state [14], there are still some open questions which need to be further studied, including the random dense (or close) packing density and corresponding packing structure, as well as the densification mechanisms. Actually, in 1900, Hilbert proposed the famous 23 problems, where the 18th was about the dense packing of regular tetrahedra [15], which created an open platform for researchers to pursue in the subsequent years. Earlier work on the packing of tetrahedral particles was mainly focused on ordered packing state. For example, in 1970, Hoylman obtained the optimal Bravais lattice packing with ρ = 0.3673 for ordered packing of equal regular tetrahedra [16], one can see that this packing density is extremely low. In 2006, Conway et al. got a higher packing density of 0.7083 by constructing a non-Bravais lattice packing of tetrahedra, and they noted that the densest ordered structure must not be a Bravais lattice packing because the regular tetrahedron is not with the central symmetry [17]. Two years later, Chen et al.'s calculations showed a higher packing density of 0.7786 [18], which illustrated that tetrahedra could
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pack much denser than spheres. In 2009, Torquato et al. proposed an ASC algorithm and generated a tetrahedral particle packing with ρ = 0.8226 [19,20], but this structure is not in long range order, which is also not the so-called Bravais lattice packing. Later, Haji-Akbari pointed out that the packing density of tetrahedral particles can reach 0.8503 [21]. In 2010, Chen et al. constructed a structure with the packing density of 0.8563 [22], which is now known as the densest ordered packing structure for regular tetrahedra. In comparison, some researchers moved their research target to the random packing of regular tetrahedra. Physically, Baker and Kudrolli used regular tetrahedral dices and sequential addition with mechanical vibration to obtain the packing density of 0.64 [23]. Jaoshvili et al. also used the regular tetrahedral dices in the cylindrical container and spherical container for packing experiments, in which the effects of the bottom and the side walls were eliminated by extrapolation, and the final packing density can reach 0.76 ± 0.02 [24]. Neudecker et al. studied the variation of coordination number and contact type with the packing density through the packing experiment of frictional tetrahedra [25]. In addition to the above work in physical experiments, numerical simulations were also carried out to study the packing of tetrahedral particles. Jiao et al. used the ASC algorithm to generate the maximum random jammed state (MRJ) of regular tetrahedra, and the packing density is 0.763 ± 0.005 [26], which is consistent with the experimental results of Jaoshvili et al.'s work [24]. Li et al. used the multi-sphere model and the relaxation algorithm to study the random close packing structure of different non-spherical particles, where the packing density of mono-sized regular tetrahedra is 0.6817 [27–29]. In addition, they proposed a quasi-random packing of tetrahedra, which was formed by the random packing of regular tetrahedral clusters. The relaxation algorithm used above can to some extent generate dense packing of tetrahedra, while it is geometrically based and no forces which are very important in real packing are considered therein. i.e. the packings formed by relaxation algorithms in numerical simulations were mainly due to the space filling based on the geometry, they ignored the real description of the packing structure of the mono-sized regular tetrahedral particles under the action of gravity and other external forces. This deficiency can be overcome by the molecular dynamics based discrete element method (DEM) dynamic simulation. Zhao et al. used the polyhedral discrete element method to study the effects of friction, height ratio and eccentricity on packing density and coordination of tetrahedral particles [30]. In comparison, the work by using this method to study the packing densification of regular tetrahedral particles is sparse. Especially the utilization of 3D mechanical vibration to improve the packing densification of mono-sized regular tetrahedral particles in DEM numerical modeling as well as the macro/micro property characterization and densification mechanisms identification are lacking and need to be systematically and thoroughly investigated. In this study, the packing densification of mono-sized regular tetrahedral particles under 3D mechanical vibration is modeled by DEM. The effects of vibration conditions (including vibration time, vibration amplitude, frequency, and vibration intensity) and container size (or wall effects) on the packing densification are systematically studied. The macro and various micro properties such as packing density (ρ), coordination number (CN), radial distribution function (RDF), orientational correlation function and face-face correlation function are characterized and analyzed. Meanwhile, the densification mechanism during vibrated packing is identified by tracking the structure evolution of local clusters through particle translation and rotation as governed by inter-particle forces.
second law of motion: dvp ¼ F p þ mp g dt dωp Ip ¼ Tp dt mp
ð1Þ ð2Þ
Where mp, vp, ωp and Ip are respectively the mass, translational velocity, angular velocity and inertia of particle p; Fp and Tp are the total force and torque acted on the particle p. The description of the geometry of non-spherical particles is directly related to the determination of the particle contact, which will affect the accuracy of the simulation results. Generally, there are two kinds of description methods for non-spherical particles: analytical model [32,33] and multi-sphere model [34–37]. The analytical model adopts the analytic expression to describe particle geometry. Although this method can accurately describe the actual shape of the non-spherical particles, the contact detection for non-spherical particles during packing is much more complicated than that for spherical particles, and different models should be built for different shapes [28]. Furthermore, the application of analytical algorithm is limited by the calculation complexity and efficiency. The limitation of the analytical model can be conquered by the so called multi-sphere model, because in the latter the contact determination of the non-spherical particles is transformed into the determination of their element-spheres. And multi-sphere model is simple and flexible, which can approximate particles of any shape by the combination of spheres. So we use multi-sphere model in this work to describe tetrahedral particles. In multi-sphere model, the total mass mp of a particle p is given by [38]: ns
ni
s¼1
i¼1
mp ¼ ∑ mps − ∑ moverlap;i
ð3Þ
Where ns is the total number of spheres in particle p; ni is the caps pairs in particle p; mps is the mass of sphere s in particle p; moverlap, i is the mass of caps pairs in particle p. The multi-sphere model for representing each tetrahedral particle is composed of 95 uniform element-spheres with the radius of 5 mm as shown in Fig. 1. Also, other element-sphere models (e.g. with 10, 35, 65, and 145 element-spheres) were considered in the numerical simulation. The results indicated that the multi-sphere model composed of 95 element-spheres can meet the requirement of both the accuracy and computation efficiency. The overlap of the element-spheres is 5 mm, which means that distance of the centers between two neighboring element-spheres is 5 mm. Therefore, the volume of a tetrahedral particle is the real volume with the value of 14,868.87 mm3, which is calculated by the total volume of all the element-spheres in that particle minus their overlapped volume. And the edge length (as schematically indicated in Fig. 1) of each tetrahedron in the multi-sphere model is 53 mm.
2. Simulation method and conditions 2.1. DEM model In DEM model [31], each particle has two types of movement: translational motion and rotational motion, which are governed by Newton's
Fig. 1. Multi-sphere model used in the simulation, where (a) and (b) are respectively 3D morphology and corresponding scenograph of an individual tetrahedron.
B. Zhao et al. / Powder Technology 317 (2017) 171–180
Where Trps is the torque generated by rolling friction.
2.2. Governing equations In the multi-sphere model, the force Fp of each tetrahedral particle p is the sum of forces on its element-spheres, given by: ns
F p ¼ ∑ F ps
ð4Þ
s¼1
Where Fps is the force of sphere s in particle p, and the force at each sphere is the sum of normal force Fnps and tangential force Ftps acting on its contact points: cs F ps ¼ F nps þ F tps ¼ ∑ F npsc þ F tpsc c¼1
ð5Þ
Where cs is the total number of the contact points on each sphere, the n t and tangential force Fpsc are calculated based on Hertz normal force Fpsc model [39] and the work of Mindlin and Deresiewicz [40]. For the energy dissipation in contact, in addition to the coulomb friction, the nonlinear viscous damping which was proposed by Tsuji [41] is used and given by: nd F npsc ¼ F ne psc þ F psc
ð6Þ
4 pffiffiffiffiffi 3=2 F ne R δn n psc ¼ − E 3
ð7Þ
rffiffiffi 5 pffiffiffiffiffiffiffiffiffiffiffi rel β sn m v n 6
F nd psc ¼ −2
ð8Þ
td F tpsc ¼ F te psc þ F psc
ð9Þ
F te psc ¼ −St δt t rffiffiffi 5 pffiffiffiffiffiffiffiffiffiffi rel β st m v t 6
F td psc ¼ −2
ð10Þ ð11Þ
The damping parameter β is defined as: ln e β ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð ln eÞ þ π2
c¼1
ð12Þ
ð13Þ
Where rpsc is the vector between the contact point c and the center of element-sphere s of particle p; and Tnps, which is created by normal forces when the normal force of the element-sphere does not pass through the center of the particle, is given by: T nps ¼ dps F nps
ð14Þ
where dps represents the relative position vector between the centroid of particle p and the center of the element-sphere s, and the torque Tp is the sum of the following three parts: ns T p ¼ ∑ T tps þ T nps þT rps s¼1
2.3. The orientation representation and rotational calculation Compared with the movement of spherical particles, one of the biggest differences in the movement of non-spherical particles lies in the representation of the orientation and the calculation of the rotation. In order to describe the motion of each tetrahedral particle, two coordinate systems are introduced. One is the global coordinate system, which is fixed, and the tetrahedral particles are moving in this coordinate system. The other one is the local coordinate system, which takes the center of the non-spherical particle as the origin; the coordinate axis will rotate with the rotation of the particles. The description of the orientation and rotation of the tetrahedral particles is similar to that in the method of Favier [35], and the details about the rotational calculation can be found in literature [42]. 2.4. Simulation conditions After all models are determined, the packing densification of tetrahedral particles will be simulated by using our self-developed DEM code. At the beginning of the simulation, the particles without overlap between each other are generated at the top of the cylindrical container with random positions and orientations and then move downward under gravity. In this duration, the particles will firstly collide with the bottom of the container and then bounce back and forth. At the same time, the interaction between particles occurs, which leads to the change of particle positions and motion states in the process of packing. Finally the stable static packing is formed, which is called poured packing [2,43]. After the initial packing is generated, the container begins to vibrate, and the periodic sinusoidal vibration is used in three mutually perpendicular directions to realize the densification of the packing structure. The governing equation for the sinusoidal vibration in each direction is given by: Ds ðt Þ ¼ A sin½ωðt−t 0 Þ
ne nd and Fpsc are the normal elastic force and damping force; Where Fpsc te td and Fpsc are the tangential elastic force and damping force; δn and δt Fpsc are the normal and tangential overlap; n and t are the unit vector in the normal and tangential directions; vnrel and vtrel are the relative normal velocity and tangential velocity; R* and m* are the equivalent radius and mass; E* is the equivalent Young's modulus, and e is the restitution coefficient. The torque Ttps which is created by the tangential force of each element-sphere is given by:
cs T tps ¼ ∑ rpsc F tpsc
173
ð15Þ
ð16Þ
Where Ds is the displacement in each direction. A is the amplitude and ω is the vibration frequency. t0 is the vibration starting time. From Eq. (16), one can easily obtain the vibration acceleration equation in each direction as: as ðt Þ ¼ −Aω2 sin½ωðt−t 0 Þ
ð17Þ
Here, the dimensionless vibration intensity Γ is defined by the absolute peak value (Aω2) of the acceleration divided by the gravitational acceleration (g), i.e. Γ = Aω2/g. It needs to note that the vibration in each direction starts from 5.0 s. For simplicity, same A and ω in each direction are used in the 3D vibrated packing densification. The simulation parameters applied in this study are shown in Table 1. Here, it needs to clarify that the properties of the particles in Table 1 are corresponding to resin materials which have been used in our physical experiments. 3. Results and discussion 3.1. Packing densification under 3D vibration Packing density ρ (sometimes called fractional volume fraction) is defined as ρ = Vp/Vc, where Vp and Vc are the volume of the particles and the container, respectively. Here, Vc is determined by the height of particle packing structure. As known, the packing density is probably the simplest and most fundamental parameter in particle packing characterization [42]. It is also an important macroscopic property of particle packing and can be affected by many factors [8,10,11,23,24,44,45]. In this section, the effects of vibration parameters on the macro property
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Table 1 Parameters applied in simulation. Parameters
Basic values
Varying range
Element-sphere diameter, d Particle density, ρt Particle size, dv Number of particles, N Poisson ratio, ν Restitution coefficient, e Shear modulus, G Sliding friction coefficient, μ s Rolling friction coefficient, μ r Vibration time, t Vibration amplitude, A Vibration frequency, ω Container size, D Time step, Δt
1.0 cm 1.2 × 103 kg/m3 30.52 mm 2000 0.35 0.6 3.0 × 108 Pa 0.2 0.002 15 s 2.0 mm 160 Rad/s 0.35 m 3.5 × 10−5 s
– – – 500–20,000 – – – – – 10–20 s 0.5–3.5 mm 80–280 Rad/s 0.2–0.8 m –
(i.e. packing density) were systematically studied and discussed. It needs to clarify that in order to reduce the error to minimum, in our work we use average packing height of the top packing layer for each case to calculate the packing density as indicated in our physical experiments [46]. 3.1.1. Effects of vibration conditions The first vibration parameter considered is the vibration time t because proper choice of t is the precondition for the accuracy and efficiency of the experiments. This has been proved in the numerical and physical experiments on the packing of spheres and non-spherical particles [9,10,47]. The effects of vibration time on the packing density are shown in Fig. 2(a). It can be seen that the packing density in each case initially increases with the vibration time to a high value and then keeps constant when t is more than 10 s. Therefore, unless otherwise specified, the vibration time in the numerical simulations is chosen to be t = 10 s. Fig. 2(b) shows the dynamic evolution of packing density with time when A = 2.0 mm and ω = 160 Rad/s. It is shown that the packing density increases significantly with the time from 5 s to about 8 s when the vibration starts, and then the ρ - t curve fluctuates within a certain range due to the periodical compaction and relaxation during vibration. When the vibration stopped, the particles settled down and formed a stable final packing structure. Here, to identify intuitively the effective role of 3D vibration, two packing morphologies before and after vibration are shown in the inset figures of Fig. 2(b). In addition to the vibration time, the effects of other vibration conditions on the packing density of mono-sized regular tetrahedral particles are shown in Fig. 3, here the container size D = 350 mm. The effects of either amplitude A (with ω being fixed) in Fig. 3(a) or frequency ω (with A being fixed) in Fig. 3(b) on packing density ρ indicate that ρ
initially increases with A or ω to a maximum value and then decreases. And the higher the vibration frequency or amplitude, the lower the vibration amplitude or frequency to realize the packing densification, indicating the left shift of the peak of each ρ - A or ρ - ω curve. The effects of vibration amplitude and frequency can be ascribed to the vibration intensity Γ [9,10,46,48,49]. As shown in Fig. 3(c) that for each case, ρ increases rapidly with Γ to a maximum and then decreases. Therefore, in order to improve the packing densification of tetrahedral particles, appropriate vibration intensity should be chosen. 3.1.2. Container size or wall effects Above work is carried out in a cylindrical container with fixed size. It is known that the container size can affect the packing density due to the existence of the container wall [9,10,24]. In order to study the container size or wall effect, we also conducted numerical simulations in other sized containers. To eliminate the container wall effect, packing densities of mono-sized regular tetrahedral particles in different sized containers under optimal vibration conditions (A = 2.0 mm and ω = 160 Rad/s) were extrapolated and fitted as shown in Fig. 4, where dv is the equivalent diameter of a tetrahedral particle defined by the diameter of a sphere with the same volume of a tetrahedron. It can be seen that an appropriate linear fit for random close packing densities with the inverse of container diameter is identified. Clearly, the larger the container size, the less its inverse, the higher the packing density, and then the less the container wall effect. After extrapolation, the intercept value at vertical axis in Fig. 4 without container wall effect, which corresponds the maximum random packing density of mono-sized regular tetrahedral particles in infinite sized container, can reach about 0.7402. This value is consistent with the result of ρ = 0.749 ± 0.04 [50] and comparable with the densest packing density of about 0.7405 for equal spheres [12], implying that tetrahedra are more efficient objects in obtaining dense packing structure than spheres. 3.2. Micro property characterization 3.2.1. Coordination number Compared with the previous analysis of macro property (i.e. packing density) of tetrahedral particle packing, the micro properties can indicate the packing characteristics more effectively. The first micro property to be analyzed is the coordination number CN, which is defined as the number of particles in contact with a considered particle. Here, the contact between two tetrahedral particles is determined by the contact between corresponding element-spheres in each particle. It varies with the definition of ‘contact’, i.e., the minimal or cut-off distance between two spheres from two particles less than which they are regarded to be in contact. In this work, the cut-off distance was set to be 1.005 d (d is the diameter of the element-sphere). In order to identify the CN
Fig. 2. (a) Effect of vibration time on packing density at different amplitudes when D = 350 mm and ω = 160 Rad/s; (b) Dynamic evolution of packing density with vibration time when D = 350 mm, A = 2.0 mm, and ω = 160 Rad/s, where the inset figures represent the packing morphologies at t = 5 s and t = 15.5 s (before and after vibrations), respectively.
B. Zhao et al. / Powder Technology 317 (2017) 171–180
0.690
ω ω ω ω
0.684 0.678
175
= 80 Rad/s = 120 Rad/s = 160 Rad/s = 200 Rad/s
(a)
0.672
0.666 0
1
2
3
4
Amplitude (mm)
Packing density
0.688
A = 1.0 mm A = 1.5 mm A = 2.0 mm A = 2.5 mm
0.684 0.680
(b)
0.676 0.672 0.668
50
100
150
200
250
300
Frequency (Rad/s) 0.684
A = 1.0 mm A = 1.5 mm A = 2.0 mm A = 2.5 mm
0.680
(c)
0.676 0.672 0.668 0
2
4
6
8
10
Vibration intensity Fig. 3. Effects of vibration amplitude (a), frequency (b), and vibration intensity (c) on the packing density when other parameters are fixed, where D = 350 mm.
distribution in different packing structures, we selected four packings with different packing densities as the research target; the packing densities as well as the corresponding vibration parameters are shown in Table 2, where C1 and C4 represent poured initial packing and vibrated dense final packing, respectively. Fig. 5 shows the CN distributions of these packing structures. From the figure one can find that the mean CN of poured packing is 7, corresponding to a loose packing structure. With the increase of packing density, the CN distribution curve shifts to the left, indicating the decrease of average CN, which is quite different from the packing of spheres [8,11,47]. Meanwhile, the curve of CN distribution with higher packing density under vibration becomes narrower and higher, which implies that the contact number of each particle in the vibrated dense packing is more concentrated than that in poured loose packing. This phenomenon can be ascribed to the change of contact type which will be discussed in the next section. The average CN for C4 is 6.3, which is consistent with the
value of 6.3 ± 0.5 obtained from physical experiment [24]. While the CN variation is different from that in literature [25], the reason for this phenomenon may be that the packing density corresponding to the densest packing structure obtained in the literature is 0.622 which is much lower than the packing density of 0.684 in this work, large voids may still exist in their packing structure. 3.2.2. Contact analysis In order to reflect the CN variation of global and local structure in initial loose packing (poured packing, C1) and final dense packing (vibrated packing, C4), we constructed the contact network of the whole structure and corresponding CN distribution of particles in the chosen three cross-sections of the two packings as shown in Fig. 6(a) and (b) respectively, where the ‘ball’ represents the particle center and the ‘stick’ represents the contact between two particles. Different color of each ‘ball’ indicates its CN variation. It can be seen that the particles are randomly distributed in both poured loose initial packing and vibrated dense final packing. And the particles in poured packing are in much darker color than those in vibrated packing, which indicates that the contact number between particles decreases during the densification process. Different from the single contact type in the packing of spherical particles, the contact types of non-spherical particles are more complicated. For tetrahedral particles, there are four types of contacts: face to face contact (F-F), edge to face contact (E-F), vertex to face contact (V-F)
Table 2 Vibration parameters of four packing cases.
Fig. 4. Extrapolation of packing densities in different sized containers when A = 2.0 mm and ω = 160 Rad/s, where the equation inside is the linear fit.
Cases
Packing density (D = 0.35 m)
Vibration amplitude, mm
Vibration frequency, Rad/s
C1 C2 C3 C4
0.660 0.672 0.676 0.684
0 2.0 1.5 2.0
0 80 160 160
176
B. Zhao et al. / Powder Technology 317 (2017) 171–180
randomness of non-spherical particle packing structure, besides the RDF analysis based on particle center distribution, the correlation of the particle orientation should also be taken into account. Therefore, based on RDF, two correlation functions, i.e. orientational correlation function C(r) and face-face correlation function F(r), were successfully utilized to characterize the angular correlation and face correlation between particles at different distance r given by [24–26,50]:
C ðr Þ ¼
F ðr Þ ¼
Fig. 5. Coordination number distributions of packing structures under different packing conditions.
and edge to edge contact (E-E) as shown in Fig. 7, which respectively provide three, two, one and one constraints [24–26,51]. Fig. 8 shows the evolution of average contact number per particle with the packing density for each contact type. It can be seen that more particles are in V-F contact and less particles are in F-F contact within each packing. With the increase of the packing density, the contact number of both V-F and E-F all decreases, while F-F contact number increases and E-E contact number does not change much. It also shows that the point and line contacts transfer to face contact with the increase of packing density, which is beneficial to the packing densification. Table 3 gives the average contact and constraint number. In our work, the DOF (Degree Of Freedom) of 9.868 is much greater than the isostatic limit of 6 [25], but slightly less than about 12 in others' work [24,26]. The reason is that unlike the packing of frictionless particles, the friction between the particles can also provide part of constraint in real packing condition. And it can also be seen from Table 3 that the F-F contact number is much smaller than that in literatures [24] and [26], but consistent with the results in literature [51]. This is because taking into account of the real packing situation, if small face to face angle is permitted, such an F-F contact is not a stable surface contact structure which cannot provide a translational and two rotational constraints [24]. And it will lead to the existence of errors in computing constraints. So that's why we use the cut-off distance of 1.005 d in determining the contact types. 3.2.3. Radial distribution function In comparison, radial distribution function (RDF) is also an effective micro property characterization method of a packing. RDF is the probability of finding one particle center (here, the particle refers to a tetrahedral particle) at a given distance from the center of a given particle, defined by: g(r) = [ΔN(r)]/(4πr2Δrρ0), where ΔN(r) is the number of particle centers situated at a distance between r and r + Δr from the centre of a given particle and ρ0 is the average number of particle per unit volume in the packing. Fig. 9 shows the RDF of the poured packing and vibrated packing, where R is the inradius of the tetrahedral particle (here is the minimum distance from tetrahedral particle center to surface). It can be seen that some peaks were appeared in the two packings in short distance. Compared with the poured packing, the first peak of RDF curve in vibrated packing is higher, then the curve decreases significantly and no obvious peaks were identified. It indicates that there is no long-range translational order in the final packing structure, which is in good agreement with the RDF of sphere packing [47]. 3.2.4. Particle orientation correlation The difference between the packings of non-spherical and spherical particles lies in the orientation of the former. In the evaluation of the
1 4 q l ∑i¼1 ni ni 4 D
E min nqi nlj
ð18Þ
ð19Þ
Where ﹤﹥denotes the average over all particle pairs (q, l), i, j = 1, 2, q 3, 4. ni is the normal vector of particle q and the lth particle is chosen to q maximize the scalar product ni ⋅nli. C(r) is used to measure whether two particles are completely aligned at different radial distances. C(r) = 1 when particles are completely aligned. F(r) is used to measure the face to face correlation of two particles at different radial distances. F(r) = −1 when particles are in face to face alignment. C(r) and F(r) evolutions with r in the vibrated dense packing structure are shown in Fig. 10. It can be seen that both C(r) and F(r) increase with the distance and then maintain constant after 3Rmin (Here, Rmin is the minimum distance between two particle centers with face-face contact). It indicates that orientation correlation and face-face correlation only exist in the short range of particles being in contact with each other, which is consistent with previous results [24]. Combined with RDF, it shows that there is no long range orientational correlation between particles in the vibrated final dense packing structure and the whole packing is in random state. 3.2.5. Densification mechanism analysis During the transition from initial poured loose packing to finial vibrated dense packing, we find that the improvement of the packing density is mainly due to the generation of the so called wagon wheel structure, which has been observed in the dense packings from other researchers' work [18–20,51]. The statistics on the number of wagon wheel structures in poured packing and vibrated packing were made and shown in Table 4. One can see that the number of wagon wheel structure increases significantly after vibrated packing densification. While a question still remains, that is how these wagon wheel structures are formed during vibration? To answer this question, we tracked the evolution of the local structure in the packing densification process. In the simulation, we found that two formation mechanisms for the wagon wheel structure can be identified to aid the packing densification, which are respectively caused due to the constraints between particles. Fig. 11 gives the evolution of a local cluster in the lower part of the packing when the particles in the cluster are close to each other but with different contact type, which corresponds to more constraints. It can be seen that particles A, B, C, D and E are in contact with each other in the form of V-F and E-E contacts in poured static packing before vibration, and there is a large pore formed by particles A, B, and E which makes the initial packing be very loose. When the vibration starts (from 5.0 to 6.0 s), particles B, C and particles D, E which are in V-F contact with small face to face angle gradually form more stable F-F contact. Since the vibration time was too short, the vibration energy input into the packing was not enough to destroy the large pore structure in initial packing formed by particles A, B and E. From 6.0 s to 8.0 s, particle A rotates accordingly with the tendency of forming F-F contact with its neighboring particles B and E. At 9.0 s, F-F contact is formed between particles A, B and particles A, E. With the further increase of vibration time (e.g. t = 12.0 s), the full wagon wheel structure can be formed by the rearrangement of neighboring particles. In this case, the pore structure generated by these particles is effectively eliminated. From 12.0 s to 15.0 s, the wagon wheel structure does not change.
B. Zhao et al. / Powder Technology 317 (2017) 171–180
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Fig. 6. (a) Contact network of initial natural packing (left, poured packing) and final dense packing (right, vibrated packing) (b) CN distribution of particles in different cross-sections of the two packing structures.
Correspondingly, we also tracked the formation of the wagon wheel structure for the particles with less constraints (i.e. no contacts between the selected particles in the initial poured packing before vibration) in the upper part of the packing. Fig. 12 shows the evolution of the selected particles to form a wagon wheel structure in the upper part of the tetrahedral particle packing. One can see that in addition to particles 1 and 2 which are in V-F contact, the positions of other particles are far from
Fig. 7. Four types of contacts in the packing of tetrahedral particles.
Fig. 8. Evolution of average contact number per particle with the packing density for each contact type.
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Table 3 Average contact and constraint number. Contact type
CF-F
CE-F
CE-E
CV-F
Ctotal
Average contact number DOF
0.836 9.868
1.926
1.347
2.161
6.270
each other without any contact in poured packing. From 5 s to 10 s, significant translational and rotational motions for each particle in Fig. 12 can be observed. In this duration, the particles move closer to each other. And the contact between particles 1 and 2 transforms from V-F to E-E. At 12 s, the cluster structure is formed with E-E, E-F and V-F contacts between the particles. From 12 s to 14 s, particles 2, 4 and 1, 5 formed F-F contact respectively, and there still exists a large pore structure formed by particles 1, 2 and 3. From 14 s to 14.75 s, particle 3 continuously rotates by relatively a large angle to fill the pore and the final wagon wheel structure is formed. Through analysis we find that the evolution mechanism of local particles to form the wagon wheel structure is similar to the ‘pushing filling’ and ‘jumping filling’ in the vibrated packing densification of equal spheres in our previous study [8]. The only difference lies in that for tetrahedral particles the formation of local dense structure is mainly determined by the contact constraints between particles under the optimal vibration conditions, while for spheres the vibration intensity is in dominant. In addition, we also analyzed the force evolution of the whole packing structure during vibrated packing densification as shown in Fig. 13, where the color and thickness of the ‘stick’ represents the magnitude of the normal contact force between two particles. It can be seen that in the static packing states before and after vibration, the normal contact forces between particles located in the lower part of the packing are significantly greater than those in the upper part due to the gravity of the packing. And after vibration (Fig. 13(e)), the density of the contact forces is larger than that in poured packing (Fig. 13(a)), and greater and more concentrated contact forces appear in the lower part of the packing because of the vibration and the formation of the cluster. During vibration (Fig. 13(b)–(d)), one can find that the concentration of contact forces decreases from the top to bottom of the packing, which agrees well with the contact number evolution in Fig. 6(b), indicating that the particles in the lower part are subjected to a much greater constraint for motion. This phenomenon has been identified in different particle rearrangements in Figs. 11 and 12, which further proves that the particles in the upper part of the packing need large translational and rotational movement to form the local wagon wheel structure. After the wagon wheel structure has ever been formed, it will be maintained in the final packing structure.
Fig. 10. Evolution of orientational correlation function C(r) and face-face correlation function F(r), where Rmin is the minimum distance between two particle centers with face-face contact.
4. Conclusions The packing densification of mono-sized regular tetrahedral particles under 3D mechanical vibration was simulated by discrete element method. The effects of vibration conditions such as vibration time t, vibration amplitude A, frequency ω, vibration intensity Γ, and container size D (wall effects) on the packing densification were studied and optimized. Various macro and micro properties of the obtained packings were characterized and compared, and corresponding densification mechanisms were identified. Following conclusions can be drawn. 1) The transition from random loose packing (RLP) to random close packing (RCP) of equal regular tetrahedral particles can be numerically reproduced under 3D mechanical vibration by properly controlling the vibration conditions. Through analysis, the optimal vibration conditions are ω = 160 Rad/s and A = 2.0 mm. The container size effects are mainly indicated by container wall. After extrapolation to eliminate the wall effects, the maximum packing density obtained in infinite sized container can reach about 0.7402. 2) The average coordination number (CN) for RLP and RCP are 7 and 6.3, respectively. For the latter, more F-F contact can be formed, and corresponding CN distribution becomes higher and narrower. With the increase of packing density, the number of both V-F contact and E-F contact decreases, while F-F contact number increases and E-E contact number does not change much. Except the first peak, other obvious peaks in RDF diminish after vibration, indicating that the obtained final dense packing is in random state, which has also been proved by the orientation correlation function and face correlation function analyses. The variation of RDF is consistent with that of contact type. 3) There are two mechanisms for the formation of local wagon wheel structures which has been regarded as the main factor for improving the packing densification. When the constraints between the particles are high, the rearrangement of particles to form the wagon
Table 4 The variation of the number of wagon wheel structure in different packings.
Fig. 9. Radial distribution function of poured packing and vibrated packing of tetrahedral particles.
Packing types
Number of wagon wheel structure
Poured packing Vibrated packing
9 43
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179
Fig. 11. Formation process of a wagon wheel structure during vibrated packing densification of equal regular tetrahedral particles with more constraints in the lower part of the packing when A = 2.0 mm, ω = 160 Rad/s, and D = 0.35 m.
wheel structure is in small extent just like the ‘pushing filling’ in the vibrated packing densification of equal spheres. While, when the constraints between the particles are low, the wagon wheel structure can be formed by large translational and rotational movement of particles with farther positions between each other, like the ‘jumping filling’ mechanism in the vibrated packing densification of equal spheres. Future work will be focused on the dynamics and mechanisms of other local clusters aiding the packing densification of equal tetrahedral
particles with different materials properties. Meanwhile, systematic physical experiments will be carried out to further identify and validate the numerically obtained results. Acknowledgement The authors are grateful to the National Natural Science Foundation of China (No. 51374070) and Fundamental Research Funds for the Central Universities of China (No. N162505001) for the financial support of current work.
Fig. 12. The formation process of a regular wagon wheel structure during vibrated packing densification of tetrahedral particles with less constraints in the upper part of the packing when A = 2.0 mm, ω = 160 Rad/s, and D = 0.35 m.
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Fig. 13. Force network of the packing during vibration, where (a) t = 5 s; (b) t = 7 s; (c) t = 9 s; (d) t = 11 s; (e) t = 15.5 s.
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