DEM simulation of binary sphere packing

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DEM simulation of binary sphere packing densification under vertical vibration Xizhong An, Fei Huang, Kejun Dong & Xiaohong Yang To cite this article: Xizhong An, Fei Huang, Kejun Dong & Xiaohong Yang (2018) DEM simulation of binary sphere packing densification under vertical vibration, Particulate Science and Technology, 36:6, 672-680, DOI: 10.1080/02726351.2017.1292335 To link to this article: https://doi.org/10.1080/02726351.2017.1292335

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PARTICULATE SCIENCE AND TECHNOLOGY 2018, VOL. 36, NO. 6, 672–680 http://dx.doi.org/10.1080/02726351.2017.1292335

DEM simulation of binary sphere packing densification under vertical vibration Xizhong Ana

, Fei Huanga, Kejun Dongb, and Xiaohong Yanga

a

School of Metallurgy, Northeastern University, Shenyang, P R China; bInstitute for Infrastructure Engineering, Western Sydney University, Penrith, NSW, Australia ABSTRACT

KEYWORDS

The densification of random binary sphere packings subjected to vertical vibration was modeled by using the discrete element method (DEM). The influences of operating parameters such as the vibration conditions, sphere size ratio (diameter ratio of larger versus small spheres), and composition (volume fraction of large spheres) of the binary mixture on the fractional packing density � (defined as the volume of spheres divided by the volume of container) were studied. Two packing states, i.e., random loose packing (RLP) and random close packing (RCP), were reproduced and their micro properties such as the coordination number (CN), radial distribution function (RDF), and force structure were characterized and compared. The results indicate that properly controlling vibration conditions can realize the transition of binary packing structure from the RLP to RCP state when the sphere size ratio and composition are fixed, and the fractional packing density for RCP after vibration can reach �RCP ≈ 0.86. Different packing characteristics from RLP and RCP identify that RCP shows much denser and more uniform structure than RLP. The current modeling results are in good agreement with those obtained from both the physical experiments and the proposed empirical models in the literature.

Binary sphere mixture; DEM simulation; densification; particle packing; vibration

1. Introduction Particle packing is an important subject in scientific research and industrial applications (German 1989; Bideau and Hansen 1993). Among the large amount of work on particle packing in the past decades, the study on random particle packing densification has attracted many researchers’ interests because these random packing structures have become effective starting points to study the structures of the amorphous state or glass state in phase diagram (Rintoul and Torquato 1996). According to the fractional packing density / (defined as the volume of spheres divided by the volume of container) of equal spheres, two random packing states have been identified and well accepted. That is, random loose packing (RLP, which corresponds to natural packing) with /RLP ≈ 0.60 and random close packing (RCP, which refers to a random packing where the particles have been agitated to attain the closest packing possible without introducing long-range order or deformation) with /RCP ≈ 0.64 (Scott 1960). Previously, both computer simulation and physical experiments indicate that by using one-dimensional (1D) vertical vibration, the transition of RLP to RCP for the packing of equal spheres can be reproduced (Scott 1960; Finney 1970; Nolan and Kavanagh 1992; Knight et al. 1995; Philippe and Bideau 2002; An et al. 2005, 2009). Parallel to the packing densification of equal spheres, people also targeted to the random packing densification of binary sphere mixtures, since the latter can create much denser packing structures and richer characteristics than the former by properly controlling the operation parameters. Actually, the researches on the random packing densification

of binary spheres have been conducted for years. For example, physical experiments regarding random packing densification of binary spheres have been performed since 1930s (Furnas 1931; Westman 1936; McGeary 1961; Yerazunis, Bartlett, and Nissan 1962; Ayer and Soppet 1965; Visscher and Bolsterli 1972; Pinson et al. 1998; Rassouly 1999; An et al. 2016); based on the experimental results, a number of analytical and geometrical models have been proposed for the prediction of fractional packing density from known characteristics of binary sphere mixtures (Furnas 1931; Westman 1936; Yerazunis, Cornell, and Wintner 1965; Aim and Goff 1968; Fedors and Landel 1979; Goff, Leclerc, and Dodds 1985; Yu and Standish 1987, 1991; Clarke and Wiley 1987; Zheng, Carlson, and Reed 1995; Rassouly 1999; Liu and Ha 2002; Kristiansen, Wouterse, and Philipse 2005; Lochmann, Oger, and Stoyan 2006; Desmond and Weeks 2009; Hopkins et al. 2011; Zou, Gan, and Yu 2011; Brouwers 2013; Desmond and Weeks 2014; Meng, Lu, and Li 2014). Even though the above numerical and physical studies can to some extent realize the random binary sphere packing structures and predict the corresponding fractional packing densities, however, each study has at least one of the following limitations: (1) no sufficient high fractional packing densities can be obtained; (2) the numerical and physical conditions are rather limited (e.g., most of them are focusing on a small size ratio of large and small spheres); (3) prediction equations or formulas are empirical; (4) the results are mainly focusing on the macro-properties (e.g., fractional packing density), much less were on micro-properties of the packing structures. Even some numerical models can to some extent characterize

CONTACT X. Z. An [email protected] School of Metallurgy, Northeastern University, Shenyang 110004, P R China. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/upst. © 2017 Taylor & Francis

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" the micro-properties, they are mostly geometrically based, lacking of the analysis of particulate dynamics which are of key importance for real process. Therefore, how to overcome above limitations and realize not only the dense binary random packing structures but also the macro- and microproperty characterizations need yet to be further studied. Under these circumstances, a molecular dynamics based discrete element method (DEM) aided by 1D vertical vibration can solve above problems, which can model the densification process of binary sphere mixtures both dynamically and realistically. In this paper, the packing densification of binary spheres subjected to 1D (vertical) continuous mechanical vibration is numerically simulated using discrete element method. The effects of operating parameters such as vibration amplitude A, frequency s, volume fraction of large spheres XL (composition) and sphere size ratio d (dL/dS, dL, and dS represent the diameter of large and small spheres, respectively) on the fractional packing density / are systematically analyzed, and the micro-properties such as coordination number (CN), radial distribution function (RDF), and forces among packed spheres for different random packing states are characterized and compared. The objectives of the current work can be ascribed to: (a) realize the transition of binary sphere packing from RLP to RCP by properly controlling the operation parameters; (b) obtain the binary RCP structure and corresponding optimal operation parameters; (c) characterize the macro- and micro-properties of the obtained RLP and RCP binary sphere packings; (d) validate the proposed empirical models.

2. Numerical method and simulation conditions 2.1. DEM model DEM simulation is effective in the microscopic study of the packing of particles, as it treats particle packing as a dynamic process where interparticle forces are explicitly considered (Liu, Zhang, and Yu 1999; Yang, Zou, and Yu 2000; An et al. 2005). In a DEM simulation, a particle can possess two types of motion, namely, translational motion and rotational motion, which can be described by Newton’s second law of motion: dvi X n ðFij þ Fsij Þ þ mi g ¼ dt j � � dxi X � � ^ iÞ Ii ¼ ðRi � Fsij lr Ri �Fnij �x dt j mi

ð1Þ ð2Þ

where vi, ωi, and Ii are, respectively, the translational and angular velocities and moment of inertial of particle i. Ri represents the vector running from the center of particle i to the contact point, and Ri, particle radius, is the scalar value of vector Ri. mi is the mass of particle i. lr is the rolling friction coefficient. The normal and tangential contact forces, Fnij and Fsij , between two contacting particles i and j are given by (Langston, Tüzün, and Heyes 1995; Liu, Zhang, and Yu 1999; Yang, Zou, and Yu 2000; An et al. 2005): � � pffiffiffipffiffiffiffi 2 pffiffiffi ^ ij Þ n ^ij ð3Þ Fnij ¼ E Rn3n=2 cn E R nn ðvij � n 3

Fsij

¼

sgnðns Þls jFijn j

1

� 1

minðns ; ns;max Þ ns;max

673

�32 # ð4Þ

where parameter E ¼ Y =ð1 r~2 Þ, Y and r~ are the Young’s ^ ij is a unit vector running from modulus and Poisson’s ratio, n the center of particle j to the center of particle i, and R ¼ Ri Rj =ðRi þ Rj Þ. vij is the velocity vector between particles i and j. ls is the sliding friction coefficient. nn and ns are the total normal and tangential displacement of particles during contact, and ns;max ¼ l½ð2 r~Þ=2ð1 r~Þ�nn (Langston, Tüzün, and Heyes 1995). The normal damping constant γn can be treated as a material property directly linked to the normal coefficient of restitution (Schwager and Pöschel 1998; Martin, Bouvard, and Shima 2003). 2.2. Simulation conditions The container is an open-topped box with square base (with the edge length of 20d, d is the diameter of small spheres) which can move in different direction according to the vibration mode as governed by the equation Dis(t) ¼ A sin [s(t t0)], where D is represents the displacement of the base, A and s are the vibration amplitude and frequency, respectively. t0 is the starting time of vibration. From the governing equation, the vibration acceleration can be derived as α(t) ¼ A s2 sin[s(t t0)]. In this work, only 1D continuous vibration was used, i.e., the container can vibrate along vertical direction. Periodical boundary conditions are applied in horizontal directions to avoid container wall effects. Two different sized spheres with different size ratio and weight are chosen in our numerical experiments, and the parameters used in DEM simulation are listed in Table 1. The container is assumed to have the same properties as the particles. Unless otherwise specified, the effect of each variable was examined while others were fixed at their respective base values that are also listed in Table 1. All simulations started from randomly generating large and small spheres with no overlap in a rectangular container. The spheres were allowed to fall down under gravity. After 1 sec when all spheres were settled to form a stable random loose packing, the container base was then vibrated according to different vibration conditions. In order to avoid boundary effects from the top and bottom of the packing, we take the zone of 1/4Zmax 3/4Zmax to calculate /, CN, and RDF in all simulations, where Zmax is the maximum packing height at each moment. The measurement for fractional packing density starts after the generation of all the spheres and ends when the final stable packing structure is formed. While the measurement for CN and RDF is mainly focusing on the final static packing state.

3. Results and discussion 3.1. Characterization on macro properties 3.1.1. Influences of operating parameters on packing density The effect of vibration frequency s on fractional packing density / of the binary sphere packing with volume fraction of large spheres XL ¼ 0.6 and XL ¼ 0.7 is shown in Figure 1 when

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Table 1. Parameters used in DEM simulation. Parameter Base value Particle size, d Particle density, r0 Numbers of particles, N Young’s modulus, Y Poisson’s ratio, r~ Sliding friction coefficient, ms Rolling friction coefficient, mr Damping coefficient, gn Vibration frequency, s Vibration amplitude, A Size ratio, d (dL/dS) Volume fraction of large particles, XL

1 cm 2.5 × 103 kg•m 5000 1.0 × 107 N•m 0.29 0.3 0.002 2 × 10 5 s 1 300 Rad/s 0.1 d 7 0.7

Varying range 3 2

0.875–7 cm 3000–8000

100–500 Rad/s 0.06–0.25 d 1–8 0–1

the amplitude is fixed to be A ¼ 0.1d and the size ratio d ¼ 7:1. As indicated, at each composition, / increases with s to a maximum value and then decreases, which means that there exists an optimal s to realize very dense packing, too large/ small s will not be helpful to the densification of the binary mixture. The fractional packing density for RCP can reach /RCP ≈ 0.86, which is much denser than the packing of equal hard spheres. And this value is between the optimal fractional packing density 0.84 and 0.88 estimated by others (Yerazunis, Bartlett, and Nissan 1962; Ayer and Soppet 1965; Ridgway and Tarbuck 1968), which is in good agreement with the fractional packing density of 0.86 of homogeneously mixed binary sphere mixtures (German 1989). Compared with the effect of s, similar trend can be identified from the influence of vibration amplitude A on the fractional packing density of the binary sphere mixture as shown in Figure 2. That is, for each vibration frequency, there is an optimal amplitude to generate the maximum packing density. When other parameters are fixed, the role of A or s in the binary packing densification can be ascribed to the vibration intensity Γ (peak value of the vibration acceleration) given by Γ ¼ As2/g (g is the gravitational acceleration) to show their combined effects (Knight et al. 1995; Philippe and Bideau 2001). Increasing A or s can lead to the increase of Γ, which

can increase the external energy or force that used to transform a packing from a loose to dense state. Γ must be high enough to break down the inter-particle locking or bridges among spheres formed during packing. However, too large Γ may over excite the packing, and the final packing state is produced toward that of loose packing by size segregation. From Figures 1 and 2, one can find that the high fractional random close packing density /RCP ≈ 0.86 occurs when A ¼ 0.1d, s ¼ 300 Rad/s, XL ¼ 0.7, and d ¼ 7:1. The variation trends of fractional packing density with vibration conditions have also been identified in our recent physical experiments of copper powder packing densification (An, Xing, and Jia 2014) and packing densification of binary spheres (An et al. 2016) subjected to vibrations. Figure 3 shows the influence of sphere size ratio d on the final fractional packing density / of totally 5200 particles at different volume fraction of large spheres XL when the amplitude A and frequency s are fixed to be 0.1d and 300 Rad/s,

Figure 1. Effects of vibration frequency s on fractional packing density / when amplitude A ¼ 0.1d and size ratio d ¼ 7:1, where: �, volume fraction of large particles XL ¼ 0.7; and D, XL ¼ 0.6.

Figure 3. Effects of particle size ratio d on fractional packing density / when amplitude A ¼ 0.1d and frequency s ¼ 300 Rad/s, where: �, XL ¼ 0.7; and D, XL ¼ 0.6.

Figure 2. Effects of vibration amplitude A on fractional packing density / when frequency s ¼ 300 Rad/s and sphere size ratio d ¼ 7:1, where: �, XL ¼ 0.7; and D, XL ¼ 0.6.

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Figure 4. Effects of volume fraction of large spheres XL on fractional packing density / when sphere size ratio d ¼ 7:1, amplitude A ¼ 0.1d, and frequency s ¼ 300 Rad/s.

respectively. It can be indicated that with each XL, the fractional packing density / increases monotonously and the trends of / d curves tend to be leveled off with the further increase of d, which implies that increasing d can assist the densification but its role on packing density increment becomes weaker when d turns larger, which can be indicated from the slope of each / d curve. Clearly, when d ¼ 1, the maximum fractional packing density obtained is 0.64, however, when d ¼ 7, the fractional packing density for XL ¼ 0.7 can reach / ≈ 0.86. On other words, the high binary fractional packing density occurs when the large spheres provide a fixed bed for the small spheres to fill, further increasing the sphere size ratio (e.g., with the case of d > 7) will not create significant influence on the fractional packing density increment of binary sphere mixtures (German 1989). Compared with sphere size ratio d, the volume fraction of large spheres XL is also an important factor to create effects on the fractional packing density of the binary mixtures, whose influence can be identified from Figure 4 when particle size ratio d ¼ 7:1, amplitude A ¼ 0.1d, and frequency

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s ¼ 300 Rad/s. As indicated, the inherent packing densities (here, inherent packing density refers to the fractional packing density of mono-sized spheres (German 1989). For small spheres (XL ¼ 0) and large spheres (XL ¼ 1), they are the same as 0.64, which corresponding to the RCP structure of monosized sphere packing. The fractional packing density / increases with XL to a maximum value and then decreases, and the optimal packing corresponding to XL ¼ 70% which agrees well with our physical results (An, Li, and Qian 2016) and others’ results obtained by different methods (Meng, Lu, and Li 2014). Ideally, the predicted RCP binary packing for infinite size ratio should be /RCP ¼ /L þ (1 /L)/S, where /RCP, /L, and /S corresponding to the fractional packing density of binary RCP mixture, inherent packing densities for large and small particles, respectively. Therefore, /RCP ¼ 0.64 þ (1 0.64)0.64 ≈ 0.87 is much comparable with our obtained maximum fractional packing density of about 0.86 for finite sphere size ratio, further proving that the obtained binary packing structure is in RCP state. 3.1.2. Two random packing states Similar as the packing of mono-sized spheres, two random packing states for binary packing of spheres can also be reproduced as shown in Figure 5. While other parameters are fixed as d ¼ 7, s ¼ 300 Rad/s, and XL ¼ 0.7, the final fractional packing densities for Figure 5a and b can reach / ≈ 0.78 and / ≈ 0.86 with A ¼ 0.25d and A ¼ 0.1d, respectively. It can be seen that the small spheres randomly distributed between the interstices left by large spheres. 3.2. Characterization on micro properties The micro structures of the obtained two binary sphere packing states were analyzed in terms of commonly used parameters such as coordination number (CN), radial distribution function (RDF), and forces. CN is defined as the number of spheres in contact with a considered sphere. In experiment, it often depends on the accuracy of experimental techniques and varies with the definition of “contact”, i.e., the minimal or cutoff distance between two spheres less than which they are regarded to be in contact. In this study, the cut-off distance

Figure 5. Final packing morphology for (a) random loose packing, A ¼ 0.25d; and (b) random close packing, A ¼ 0.1d. In both cases, d ¼ 7, s ¼ 300 Rad/s, and XL ¼ 0.7.

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Figure 6. CN distribution for obtained random loose packing of (a) A ¼ 0.25d; and random close packing of (b) A ¼ 0.1d, where: d ¼ 7:1, s ¼ 300 Rad/s, and XL ¼ 0.7.

between two spheres for CN is 1.005d, similar as that in the packing of equal spheres (An et al. 2008). Figure 6 shows the CN distribution for random loose packing (a) and random close packing (b). For binary mixtures, four different contacts existed, they are L–L, L–S, S–L, and S–S, where L and S represent large and small spheres, respectively. It can be seen from Figure 6 that the CN distribution of L–L and S–S shift to the right from loose to dense packing, which indicates that more L–L contacts and S–S contacts appear in the calculated zone in random close packing; meanwhile, the packing of small spheres is much denser with the peak value of CN distribution from 5 to 6, which is in good agreement with the packing transition from RLP to RCP for equal spheres (Scott 1960). Even the L–S contact distribution in the inset figure of Figure 6 is discontinuous compared with other CN curves due to the less number of large spheres involved in the binary packing, it can also indicate the tendency that higher probability of large spheres with more neighbors of small particles for RCP than that for RLP. Aim and Goff (1968) and Goff et al. (1985) proposed an estimated equation for coordination number (Nc) of large spheres: Nc ¼ 3.0 þ 3.16 (dL/dS)2. According to this equation, the predicted Nc ¼ 158, which agrees well with our peak value of Nc ¼ 158 for random close binary packing (see the inset figure in Figure 6). Also, we extracted some local packing structures from Figure 5b as shown in Figure 7. As indicated, the large spheres are in good contact with their neighbors, no large pores left on the surface of large spheres. RDF is the probability of finding one sphere center at a given distance from the center of a given sphere, defined by: DNðrÞ gðrÞ ¼ 4pr 2 Drq , where ΔN(r) is the number of sphere centers

situated at a distance between r and r þ Δr from the center of a given sphere and ρ0 is the average number of particle per unit volume in the packing. Figure 8 shows that the peaks of RDF for RCP binary structure are more characteristic compared with RLP binary structure. Since the sphere size ratio is d ¼ 7:1, the RDF curve in inset figure of Figure 8 starts from 4d for large spheres involved. One advantage of the present simulation technique is that not only the structure but also the forces acting on particles can be obtained (Yang, Zou, and Yu 2000; An et al. 2005; Dong et al. 2006). We have analyzed the forces with special reference to the normal forces between particles as done elsewhere (An et al. 2005). Figure 9 indicates the force structure of the two binary sphere packing states, where the thickness of the “stick” is proportional to the magnitude of the force. Three different kinds of “sticks” can be identified, i.e., long, medium, and short, which correspond to the forces between large–large, large–small, and small–small sphere interactions. It is observed that no matter for RLP or for RCP, the whole packing is randomly arranged and its weight is mainly supported by the forces between large spheres (see the long stick network for large forces). In the whole packing, there are some randomly distributed concentrated areas which correspond to the positions of large spheres, and the densely radiated segments represent the forces between large spheres and their neighbors. The number of forces acted on large spheres indicates more contacts with their neighbors, therefore, for most of the large spheres inside the packing, the number of their average neighbors is more than 50. The distribution of large spheres inside the packing and their corresponding forces is much more uniform for RCP than RLP, which implies the higher stability

0

Figure 7. Some close contacts in local packing extracted from the structure of Figure 5b.

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Figure 8. RDF distribution for obtained random loose packing of (a) A ¼ 0.25d; and random close packing of (b) A ¼ 0.1d, where d ¼ 7:1, s ¼ 300 Rad/s, and XL ¼ 0.7.

and performance for the former when subjected to compaction under pressure in powder metallurgy or ceramics industry. 3.3. Entropy analysis of RLP and RCP states The RCP and RLP states have been widely discussed in uniform sphere packings (Makse et al. 2004; Song, Wang, and Makse 2008; Baule et al. 2016). According to Edwards Ensemble theory of granular materials, they can be linked to the phase transitions in terms of the critical changes of entropy (Oakeshott and Edwards 1992). In this theory, packing fraction can be analogous to temperature in the statistical thermodynamics, while the statistics description of local packing fraction is important in modeling the related partition function. The local packing fraction of a particle is often determined based on the Voronoi cell volume (Vcell). Aste and Di Matteo (2008) proposed the Gamma distribution for Vcell in uniform sphere packings and calculated the entropy of the packings accordingly. Song, Wang, and Makse (2008) used

statistical analysis to coarse-grain Vcell over a mesoscopical length scale based on the coordination number. Such a coarse-grain model has been extended to the packing of polydisperse spheres within a certain size ratio range (d < 2.0) (Danisch, Jin, and Makse 2010). Yet, how to apply Edwards Ensemble theory to the packings of binary spheres with large size ratios is still an open question. Here, we analyze the distribution of the reduced Voronoi cell volume (Vc ¼ Vcell/ Vparticle) in our packings. Firstly, we find that the distribution of Vc always splits into two peaks representing large and small components respectively, similar to what we have found in previous studies on the packings of ternary spheres (Yi et al. 2012) and mixtures of spheres with log-normal distributions (Yi et al. 2015). Considering that the number of large particles in our system is relatively low, which brings observable fluctuations in the distribution, here we mainly focus on the distribution of small particles. Figure 10a shows the distribution of Vc for small particles at RLP and RCP states obtained in case 1 (A ¼ 0.25d) and case 2 (A ¼ 0.1d) respectively (other parameters are the same: d ¼ 7,

Figure 9. Force network of the final binary packings for (a) RLP, A ¼ 0.25d; and (b) RCP, A ¼ 0.1d, where d ¼ 7:1, s ¼ 300 Rad/s, XL ¼ 0.7. The thickness of the ‘stick’ is proportional to the magnitude of the force.

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Figure 10. Entropy analysis of the RLP case (A ¼ 0.25d) and RCP case (A ¼ 0.1d), where d ¼ 7:1, s ¼ 300 Rad/s, XL ¼ 0.7: (a), distribution of reduced Voronoi cell volume at final state: ●, RLP case; D, RCP case; lines are fitted log-normal distributions; and (b), S calculated from the distribution as a function of qs: ●, packings selected in the RLP case; D, packing selected in RCP case; the vertical dashed lines show the critical changes of S.

s ¼ 300 Rad/s, and XL ¼ 0.7). We find that the distribution of Vcell cannot be fitted with Gamma distribution but that of Vc can be fitted with lognormal distribution, as shown in the figure. In fact, we have selected different dense packing states in the vibration processes of case 1 and case 2, and found that in all these packings, the distributions of Vc can always be well described by log-normal distributions, given by: 1 f ðVc Þ ¼ pffiffiffiffiffi r 2pðVc

1Þ

exp½ ðlnðVc

1Þ

gÞ2 =2r2 �

ð5Þ

where g and r are the two shape parameters for the distribution. This is in accordance with the observations in the packings of cohesive spherical particles (Yang, Zou, and Yu 2002) and non-spherical particles in 2D (Wang, Dong, and Yu 2015) and 3D (Dong, Wang, and Yu 2016). We then calculate the entropy of the log-normal distribution by pffiffiffiffiffi S ¼ log½r expðg þ 12Þ 2p�. Figure 10b shows the calculated entropy versus the mean packing fraction (given by ρs ¼ 1/ hVci) for small particles. It can be seen that the data points from different cases collapse in most regions, indicating the relationship between ρs and S is independent of the vibrational conditions. The change of S with ρs is shown to be similar to that of uniform sphere packings. From ρs ¼ 0.54 to ρs ¼ 0.575, S generally decreases with ρs, which is because the more compact the packing becomes, less possibilities are left to the spheres to explore equivalent configurations. However, with the further increase of ρs, S suddenly increases and then enters a scattered region, showing a critical change at this point, which is probably corresponding to the maximally random jammed (MRJ) state for small particles in this packing system. Note here the critical packing fraction is less than that of RCP for uniform sphere packings (0.64), which is probably due to the effect of large particles in the packings. We can also find that when ρs further increases beyond 0.585, S keeps almost as a constant, showing another critical change, which may also result from the effect of large particles and need further studies. Interestingly, for those packings selected from case 1 which generates the RLP packing at the final state, ρs just goes beyond the first critical point but does not reach the second one.

The critical changes of S are similar to the observations in the packings of mono-sized spheres (Aste and Di Matteo 2008; An, Dong, Yang, et al. 2016) and also partly in accordance with the theoretical speculations (Oakeshott and Edwards 1992). This suggests that the Edwards Ensemble theory could be applicable to the packings of binary spheres with large size ratios. However, due to the complexity of this problem here we can only demonstrate a very preliminary attempt. Further studies are well deserved on this topic yet that would be out of the scope of this work. 3.4. Comparison with the proposed empirical models Actually, various empirical equations have been developed in the literature to predict the fractional particle packing density of a binary system (Westman 1936; Yerazunis, Cornell, and Wintner 1965; Fedors and Landel 1979; Yu and Standish 1987, 1991). According to Westman (Westman 1936), � �2 � � the equation can be written as: V VVSL XL þ2G V VVSL XL � � � �2 V X L VS X S V XL VS XS ¼ 1, where: 1/G ¼ 1.355d1.5666 þ VL 1 VL 1

Figure 11. Comparison of the current modeling results with those predicted by others’ proposed models.

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when d � 0.824 or 1/G ¼ 1 when d > 0.824. The specific volume V is defined as the inverse of /. Yerazunis et al. (1965) proposed different equations for large d and small d: / ¼ 0.64/[1 (0.362 0.315(dS/dL)0.7)XL þ 0.955(dS/dL)4 (XL2/(1 XL))] – small d; / ¼ 0.64/(1 0.362XL) – large d. Figure 11 gives the comparisons between the modeling results in this work and those predicted ones. As indicated, our results are much comparable with those predicted by Yerazunis et al. in their work when the size ratio d is large, and they are higher than the predicted values by using the Westman model (Westman 1936), where the latter is effective mainly for the ideal case of the binary sphere packings (German 1989).

4. Conclusions Random packings of spheres with two different sizes under one-dimensional (1D) vertical vibration were modeled by the discrete element method (DEM). The effects of operating parameters such as the vibration amplitude (A), frequency (s), sphere size ratio (d), and volume fraction of large spheres (XL) are studied and analyzed. Two random packing states, i.e., random loose packing (RLP) and random close packing (RCP) are produced and their macro property such as fractional packing density / and micro properties such as coordination number (CN), radial distribution function (RDF), and forces among spheres are characterized and compared. Following conclusions can be drawn: 1. Fractional packing density / of binary sphere packing mixture increases with amplitude A or frequency s to a high value and then decreases with further increase of A or s. Increasing particle size ratio d can densify the packing mixture, however, the fractional packing density increment decreases with the increase of d when it is very large. When other parameters are fixed, the fractional packing density increases linearly with the volume fraction of large particles XL to a maximum and then decreases with the further increase of XL. 2. Macro property characterization indicates that by properly controlling the operating conditions, the fractional packing density for random close packing of binary sphere mixture can reach /RCP ≈ 0.86, and the corresponding optimal parameters are: d ¼ 7, XL ¼ 0.7, A ¼ 0.1d, and s ¼ 300Rad/s. 3. Micro property characterization shows the difference between random loose and random close binary packing structures. 4. The modeling results of this work are comparable with those obtained from proposed empirical equations.

Acknowledgments The authors thank Prof. A.B. Yu from Monash University for the constructive discussions and suggestions about this work.

ORCID Xizhong An

http://orcid.org/0000-0003-4287-6767

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Funding The authors 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 (N130102001).

References Aim, R. B., and P. L. Goff. 1968. La coordinance des empilements desordonnes de spheres. Application aux melanges binaires de spheres. Powder Technology 2:1–12. doi:10.1016/0032-5910(68)80027-0. An, X. Z., K. J. Dong, R. Y. Yang, R. P. Zou, C. C. Wang, and A. B. Yu. 2016. Quasi-universality in the packing of uniform spheres under gravity. Granular Matter 18:1–7. doi:10.1007/s10035-016-0612-6. An, X. Z., C. Li, and Q. Qian. 2016. Experimental study on the 3D vibrated packing densification of binary sphere mixtures. Particuology 27:110–14. doi:10.1016/j.partic.2015.03.009. An, X. Z., C. X. Li, R. Y. Yang, R. P. Zou, and A. B. Yu. 2009. Experimental study of the packing of mono-sized spheres subjected to one-dimensional vibration. Powder Technology 196:50–55. doi:10.1016/j.powtec.2009.06.016. An, X. Z., Z. T. Xing, and C. C. Jia. 2014. Cold compaction of copper powders under mechanical vibration and uniaxial compression. Metallurgical and Materials Transactions A 45A:2171–79. doi:10.1007/ s11661-013-2160-6. An, X. Z., R. Y. Yang, K. J. Dong, R. P. Zou, and A. B. Yu. 2005. Micromechanical simulation and analysis of one-dimensional vibratory sphere packing. Physical Review Letters 95:205502. doi:10.1103/ physrevlett.95.205502. An, X. Z., R. Y. Yang, R. P. Zou, and A. B. Yu. 2008. Effect of vibration condition and inter-particle frictions on the packing of uniform spheres. Powder Technology 188:102–09. doi:10.1016/ j.powtec.2008.04.001. Aste, T., and T. Di Matteo. 2008. Emergence of gamma distributions in granular materials and packing models. Physical Review E 77:021309. doi:10.1103/physreve.77.021309. Ayer, J., and F. E. Soppet. 1965. Vibratory compaction: I, compaction of spherical shapes. Journal of the American Ceramic Society 48:180–83. doi:10.1111/j.1151-2916.1965.tb14708.x. Baule, A., F. Morone, C. S. O’Hern, and H. A. Makse. 2016. Edwards Statistical Mechanics for Jammed Granular Matter. arXiv:1602.04369. Bideau, D., and A. Hansen. 1993. Disorder and granular media. Amsterdam: Random Materials and Process. Brouwers, H. J. H. 2013. Random packing fraction of bimodal spheres: An analytical expression. Physical Review E 87:032202. doi:10.1103/ physreve.87.032202. Clarke, A. S., and J. D. Wiley. 1987. Numerical simulation of the dense random packing of a binary mixture of hard spheres: Amorphous metals. Physical Review B 35:7350–56. doi:10.1103/physrevb.35.7350. Danisch, M., Y. Jin, and H. A. Makse. 2010. Model of random packings of different size balls. Physical Review E 81:051303. doi:10.1103/ physreve.81.051303. Desmond, K. W., and E. R. Weeks. 2009. Random close packing of disks and spheres in confined geometries. Physical Review E 80:051305. doi:10.1103/physreve.80.051305. Desmond, K. W., and E. R. Weeks. 2014. Influence of particle size distribution on random close packing of spheres. Physical Review E 90:022204. doi:10.1103/physreve.90.022204. Dong, K. J., C. C. Wang, and A. B. Yu. 2016. Voronoi analysis of the packings of non-spherical particles. Chemical Engineering Science 153:330–43. doi:10.1016/j.ces.2016.07.013. Dong, K. J., R. Y. Yang, R. P. Zou, and A. B. Yu. 2006. Role of interparticle forces in the formation of random loose packing. Physical Review E 96:145505. doi:10.1103/physrevlett.96.145505. Fedors, R. F., and R. F. Landel. 1979. An empirical method of estimating the void fraction in mixtures of uniform particles of different size. Powder Technology 23:225–31. doi:10.1016/0032-5910(79)87011-4. Finney, J. L. 1970. Random packings and the structure of simple liquids: I. The geometry of random close packing. Proceedings of the Royal

680

X. AN ET AL.

Society A: Mathematical, Physical and Engineering Sciences 319:479–93. doi:10.1098/rspa.1970.0189. Furnas, C. C. 1931. Grading aggregates. I. Mathematical relations for beds of broken solids of maximum density. Industrial & Engineering Chemistry 23:1052–58. doi:10.1021/ie50261a017. German, R. M. 1989. Particle packing characteristics. Princeton: Metal Powder Industries Federation. Goff, P. L., D. Leclerc, and J. Dodds. 1985. The structure of packed beds: Continuity of research in Nancy and some new results. Powder Technology 42:47–53. doi:10.1016/0032-5910(85)80037-1. Hopkins, A. B., Y. Jiao, F. H. Stillinger, and S. Torquato. 2011. Phase diagram and structure diversity of the densest binary sphere packings. Physical Review Letters 107:125501. doi:10.1103/physrevlett.107.125501. Knight, J. B., C. G. Fandrich, C. N. Lau, H. M. Jaeger, and S. R. Nagel. 1995. Density relaxation in a vibrated granular materials. Physical Review E 51:3957–63. Kristiansen, K. L., A. Wouterse, and A. Philipse. 2005. Simulation of random packing of binary sphere mixtures by mechanical contraction. Physica A 358:249–62. doi:10.1016/j.physa.2005.03.057. Langston, P. A., U. Tüzün, and D. M. Heyes. 1995. Discrete element simulation of granular flow in 2D and 3D hoppers: Dependence of discharge rate and wall stress on particle interactions. Chemical Engineering Science 50:967–87. doi:10.1016/0009-2509(94)00467-6. Liu, L. F., Z. P. Zhang, and A. B. Yu. 1999. Dynamic simulation of centripetal packing of mono-sized spheres. Physica A 268:433–53. doi:10.1016/s0378-4371(99)00106-5. Liu, S. J., and Z. Y. Ha. 2002. Prediction of random packing limit of multimodal particle mixtures. Powder Technology 126:283–96. doi:10.1016/ s0032-5910(02)00075-x. Lochmann, K., L. Oger, and D. Stoyan. 2006. Statistical analysis of random sphere packings with variable radius distribution. Solid State Sciences 8:1397–1413. doi:10.1016/j.solidstatesciences.2006.07.011. Makse, H. A., J. Brujic, and S. F. Edwards. (2004). The Physics of Granular Media. New York: Wiley-VCH. Martin, C. L., D. Bouvard, and S. Shima. 2003. Study of particle rearrangement during powder compaction by the discrete element method. Journal of the Mechanics and Physics of Solids 51:667–93. doi:10.1016/s0022-5096(02)00101-1. McGeary, R. K. 1961. Mechanical packing of spherical particles. Journal of the American Ceramic Society 44:513–22. doi:10.1111/j.11512916.1961.tb13716.x. Meng, L. Y., P. Lu, and S. X. Li. 2014. Packing properties of binary mixtures in disordered sphere systems. Particuology 16:155–66. doi:10.1016/j.partic.2014.02.010. Nolan, G. T., and P. E. Kavanagh. 1992. Computer simulation of random packing of hard spheres. Powder Technology 72:149–55. doi:10.1016/ 0032-5910(92)88021-9. Oakeshott, R. B. S., and S. F. Edwards. 1992. Local theory of the statistics of sphere packings. Physica A: Statistical Mechanics and its Applications 189:188–207. doi:10.1016/0378-4371(92)90134-c. Philippe, P., and D. Bideau. 2001. Numerical model for granular compaction under vertical tapping. Physical Review E 63:051304. doi:10.1103/ physreve.63.051304. Philippe, P., and D. Bideau. 2002. Compaction dynamics of a granular medium under vertical tapping. Europhysics Letters 60:677–83. doi:10.1209/epl/i2002-00362-7.

Pinson, D., R. P. Zou, A. B. Yu, P. Zulli, and M. J. McCarthy. 1998. Coordination number of binary mixtures of spheres. Journal of Physics D: Applied Physics 31:457–62. doi:10.1088/0022-3727/31/4/016. Rassouly, S. M. K. 1999. The packing density of ‘perfect” binary mixtures. Powder Technology 103:145–50. Ridgway, K., and K. J. Tarbuck. 1968. Particulate mixture bulk densities. Chemical Engineering and Processing 49:103–05. Rintoul, M. D., and S. Torquato. 1996. Metastability and crystallization in hard-sphere systems. Physical Review Letters 77:4198–4201. Schwager, T., and T. Pöschel. 1998. Coefficient of restitution of viscous particles and cooling rate of granular gases. Physical Review E 57:650–54. doi:10.1103/physreve.57.650. Scott, G. D. 1960. Packing of spheres. Nature 188:908–09. doi:10.1038/ 188908a0. Song, C., P. Wang, and H. A. Makse. 2008. A phase diagram for jammed matter. Nature 453:629–32. doi:10.1038/nature06981. Visscher, W. M., and M. Bolsterli. 1972. Random packing of equal and unequal spheres in two and three dimensions. Nature 239:504–07. doi:10.1038/239504a0. Wang, C., K. J. Dong, and A. B. Yu. 2015. Structural characterization of the packings of granular regular polygons. Physical Review E 92:062203. doi:10.1103/physreve.92.062203. Westman, A. E. R. 1936. The packing of particles: empirical equations for intermediate diameter ratios. Journal of the American Ceramic Society 19:127–29. doi:10.1111/j.1151-2916.1936.tb19809.x. Yang, R. Y., R. P. Zou, and A. B. Yu. 2000. Computer simulation of the packing of fine particles. Physical Review E 62:3900–08. doi:10.1103/ physreve.62.3900. Yang, R. Y., R. P. Zou, and A. B. Yu. 2002. Voronoi tessellation of the packing of fine uniform spheres Physical Review E 65:041302. doi:10.1103/physreve.65.041302. Yerazunis, S., J. W. Bartlett, and A. H. Nissan. 1962. Packing of binary mixtures of spheres and irregular particles. Nature 195:33–35. doi:10.1038/195033a0. Yerazunis, S., S. W. Cornell, and B. Wintner. 1965. Dense random packing of binary mixtures of spheres. Nature 207:835–37. doi:10.1038/207835a0. Yi, L. Y., K. J. Dong, R. P. Zou, and A. B. Yu. 2012. Radical tessellation of the packing of ternary mixtures of spheres. Powder Technology 224:129–37. doi:10.1016/j.powtec.2012.02.042. Yi, L. Y., K. J. Dong, R. P. Zou, and A. B. Yu. 2015. Radical tessellation of the packing of spheres with a log-normal size distribution. Physical Review E 92:032201. doi:10.1103/physreve.92.032201. Yu, A. B., and N. Standish. 1987. Porosity calculation of multi-component mixtures of spherical particles. Powder Technology 52:233–41. doi:10.1016/0032-5910(87)80110-9. Yu, A. B., and N. Standish. 1991. Estimation of the porosity of particle mixtures by a linear-mixture packing model. Industrial & Engineering Chemistry Research 30:1372–85. doi:10.1021/ie00054a045. Zheng, J. M., W. B. Carlson, and J. S. Reed. 1995. The packing density of binary powder mixtures. Journal of the European Ceramic Society 15:479–83. doi:10.1016/0955-2219(95)00001-b. Zou, R. P., M. L. Gan, and A. B. Yu. 2011. Prediction of the porosity of multi-component mixtures of cohesive and non-cohesive particles. Chemical Engineering Science 66:4711–21. doi:10.1016/j.ces.2011. 06.037.