Shape and size effects on the packing density of binary spherocylinders

Powder Technology 228 (2012) 284–294

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Shape and size effects on the packing density of binary spherocylinders Lingyi Meng, Peng Lu, Shuixiang Li ⁎, Jian Zhao, Teng Li Department of Mechanics & Aerospace Engineering, College of Engineering, Peking University, Beijing, 100871, China

a r t i c l e

i n f o

Article history: Received 19 January 2012 Received in revised form 4 May 2012 Accepted 13 May 2012 Available online 18 May 2012 Keywords: Binary mixture Packing density Spherocylinder Aspect ratio Volume fraction Non-spherical particles

a b s t r a c t The shape and size of particles are the main factors which influence the packing density of binary mixtures. However, investigations of the single effect of size and shape on the binary mixtures of non-spherical particles are much absent. A systematic computational investigation on the binary packing of spherocylinders considering the single effect of shape and size respectively is carried out with the sphere assembly models and relaxation algorithm. Random packings of binary spherocylinders with the same volume, aspect ratio and diameter are simulated numerically to investigate the single effect of shape, size and their combination respectively. The shape effect is observed to present a linear relationship between the packing density and the volume fraction, while the size effect shows a trend of convex. The combined effect, as the case of the same diameter, can be considered as a linear superposition of the shape and size effect, since we find the particle size and shape determine the mixture density independently with no coupling terms. Accordingly, we propose an explicit empirical formula for evaluating the mixture density of binary spherocylinders, and the predicted densities agree well with the simulation results and previous empirical formula in all the cases of the same volume, aspect ratio and diameter, as well as some general cases. For verification, the monophasic packings of spherocylinders are simulated and the results coincide with previous works. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Packing is not only a fundamental physical and mathematical problem, but also widely involved in industry and daily life. Since the Kepler conjecture in 1611 [1], packing problems have been extensively studied to understand the structure of material, and arisen in numerous applications. Packing density is a primary parameter to characterize the properties of particle packings, and the densest packing is always a major concern in packing investigations. According to the diversity of particle shape, packings can be sorted into spherical and non-spherical particle packings. Numerous investigations have been carried out on sphere packings. Weitz [2] summarized several general conclusions of sphere packings: the packing density of random close packing is around 0.64, the value of random loose packing is about 0.56, and the highest packing density with order arrangement is 0.74. Furthermore, the binary [3,4] and polydisperse [5,6] packing of spheres have also attracted a lot of interest and the results with a large disparity of size distributions such as normal [7] and lognormal [8] distributions have been presented by a number of investigations. However, real particles in nature and industry are often nonspherical. In contrast, the packing investigations of non-spherical particles are more complicated due to the additional rotational degrees and limited only in some simple and basic 3D objects, such as

⁎ Corresponding author. E-mail address: [email protected] (S. Li). 0032-5910/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2012.05.033

ellipsoid [9], spherocylinder [10–17], cylinder [18], Platonic solids [19] and superball [20]. The investigations on the mixtures of nonspherical particles with different sizes or shapes are even rare. Along with the great progress in computer technology and packing algorithm, numerical simulation has become the main means in packing investigations. The simulation algorithms of random packings can be generally classified into sequential addition and collective rearrangement. Collective rearrangement algorithms have been widely adopted. The L-S algorithm [21], relaxation algorithm [22], discrete element method [23], molecular dynamics [24], mechanical contraction method [17], and Monte Carlo algorithm [11] have been employed in recent investigations. The relaxation algorithm has been applied in recent studies of non-spherical particle packings for its advantages of simple algorithm, high efficiency and good consistency with previous works [16,22,25–29]. A number of geometrical models have been used to simulate the packings of non-spherical particles. The analytic models [16] are built for only some simple particle shapes. They can accurately detect the contacts between the particles but cost considerable CPU time because of the complexity of contact detection. Moreover, different shapes need different models, and no general applicable analytic model has been found. An alternate method which assembles simple objects to approximate the real particles has been widely employed as well. These approaches have the ability to simulate particles of any shape. Thus, the contact detection is transferred among these simple objects and saves much time. Spheres [22] and pixels [13] are typical elements of assembles in the investigations. The sphere

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assembly model converts the contacts between non-spherical particles to the ones between spheres, therefore, achieves higher efficiency on numerical simulations especially when complex shapes are involved. Furthermore, the model makes it possible to investigate the packing world of non-spherical particles among various shapes using a uniform particle model. Another kind of approximation model for the non-spherical particle packing is to represent the real particle with equivalent simple 3D objects. The ellipsoid [30] and superball [31] are commonly used ones. As a simple particle shape, spherocylinder is very common in packing investigations. A spherocylinder consists of a cylinder and two hemispheres on both sides, which is like a capsule. The aspect ratio w is usually used to describe the shape of a spherocylinder, defined as the ratio of the height of the cylinder H to the diameter D. A spherocylinder has a smooth surface and can be easily described by mathematical or sphere assembly models. It has attracted much attention in recent investigations and sometimes can be used in place of cylinders. Considerable results on the random packing of spherocylinders have already been obtained. Williams and Philipse [10] used mechanical contraction to simulate the random packings of spherocylinders and presented that the packing density reaches a peak value of 0.695 when w equals to 0.4. Abreu et al. [11] carried out a Monte Carlo simulation and obtained a peak value of 0.655 when w = 0.5. Li et al. [22] simulated the packings of spherocylinders with sphere assembly models and relaxation algorithm. When w = 0.35 the packing density reached the top of 0.690. Kyrylyuk et al. [12] simulated the packings of spherocylinders whose aspect ratio varied from 0 to 3.0. Their results indicated that spherocylinders pack to the maximal density of 0.701 when w is about 0.45. Zhao et al. [16] adopted an improved relaxation algorithm and the maximum packing density reached 0.722 when w = 0.5. Although the specific values are different, the tendency of the packing density varied with the aspect ratio is consistent with a maximum around 0.69–0.72 when the aspect ratio is about 0.40. However, for the studies on the influence of particle shape and size, the trend and peak locus of packing density are much more significant than the specific value. Besides the monophasic packings, random packings of mixtures with various particle sizes and shapes are also extensively involved in industry and daily life. Considering a geometric packing, the packing density of mixtures is usually relevant to the size, shape, volume fraction as well as the monophasic packing density of each component. Among these factors, size distribution is well studied [3–8]. Since the contact detection of non-spherical particles is technically complicated, and the numerical simulation is usually computational

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expensive, the studies of the shape effect on the packing density of mixtures are restricted in a few common objects. Binary mixture is the simplest case among the mixture packings. Numerous experiments on binary mixtures have been carried out, but most of them were concerned with spheres [32–39]. Investigations on the mixture of non-spherical particles are rare, especially when the component particles have similar volumes. Previous empirical formulas [33,40] showed that a linear relationship between the packing density and volume fraction was obtained when the two components have similar volumes, so the case of similar volumes is a representative one of special interest. However, the shapes of non-spherical particles used in previous studies were limited and had a significant volume disparity [40–44]. Kyrylyuk et al. [12] simulated the packings of rod-sphere mixtures with the aspect ratio w of the rod varying from 0 to 3.0 and gave a simple relationship between the mixture packing density ϕ and the volume fraction X. The rods in the simulations have an equal diameter D as the sphere and thus the packings combine the size and shape effects together. Similar simulation has been applied to the random packings of binary spherocylinders with the same diameter as well [45]. Bargiel [15] simulated the random packings of binary spherocylinders with the same volume but the results are much limited in special compositions. Besides the numerical simulations, experiments of binary cylinders were carried out and the cylinders could be considered as spherocylinders when they are long enough. Milewski [44] carried out experiments with various mixtures of binary cylinders and gave the relationship between relative bulk volume (defined as the inverse of the packing density) and volume fraction, but the cylinders used were very long and had a significant volume difference. The packing density of binary spherocylinders is affected by both the shape and size of the particles. Investigations in the literature are rarely focused on the single effect of the two aspects. Particles used in most investigations combined the size and shape effects together and usually had different shapes and volumes at the same time. Furthermore, a number of empirical formulas have been developed to predict the packing density of binary mixtures of spherical and some non-spherical particles. A review can be found in part 2 for this issue. However, whether these formulas can be extended to the universal binary mixtures of non-spherical particles is interesting and needs to be verified. Mixtures of binary spherocylinders have three representative cases as illustrated in Fig. 1. The shape and size effects on packing density can be studied through various compositions of the mixture. For instance, spherocylinders with the same volume V but different aspect ratio w reflect the single effects of particle shape on the packing density. By varying the volume of spherocylinders with the same

Fig. 1. Three representative cases of the shape and size effects on the packing density of binary spherocylinders, V1, V2, w1, w2, D1, D2 are the volumes, aspect ratios, and diameters of the two component spherocylinders respectively, X is volume fraction, ϕ is packing density.

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aspect ratio, we study the size effect on the packing density. On the premise of a constant size ratio, different compositions of binary spherocylinders show the variation trend of packing density due to the particle shape. The case with the same diameter D reflects the joint effects of the particles size and shape, which is carried out to compare with the existing results and validate the general rules summarized from the previous cases. The purpose of the paper is to investigate the behavior of the packing density of binary spherocylinders on the effects of particle shape and size, further to validate the prediction results from the previous empirical formula and build a new explicit formula reflecting the shape and size effects independently. To explore the single effects of shape and size, variables such as the aspect ratio and diameter must be elaborately chosen and controlled. We note that only geometric packings are considered in this work. In Section 2, we review some empirical prediction models for binary packings of spherical and non-spherical particles, mostly on the Westman equation and the concept of “equivalent packing diameter”. The sphere assembly models and relaxation algorithm are applied in this work, and they are introduced in Section 3. For the validation of the method, we simulate the monophasic packing of identical spherocylinders in Section 4. Simulation results are compared with the existing simulation results. As for the mixtures of binary spherocylinders in Section 5, the numerical simulations with the same volume are carried out to investigate the single effect of particle shape. The influences of the volume fraction and aspect ratio on the packing density of binary spherocylinders are investigated. In addition, we compare the results with the existing experimental and simulation ones as well as the prediction results from the Westman equation. We also simulate the mixtures of binary spherocylinders with the same shape, which means the two components have an equal aspect ratio. We chose several groups of spherocylinders with different aspect ratios and a constant volume ratio of 1:8. The simulation results are compared with the prediction ones. Furthermore, with the aspect ratio of the two components constant and the size ratio of the mixture changed, both the shape and size effect are incorporated in the binary mixtures. We conclude how the size affects the packing density of the mixture in terms of the volume fraction. The case with the same diameter is also simulated in this work, which combines the size effect and shape effect together. The results are compared with the prediction ones and previous simulations. Based on the simulation results under the same volume and aspect ratio conditions, we propose a new empirical formula described as a linear superposition of two independent functions of the size ratio and aspect ratio. Unlike the implicit expression from the Westman equation, the formula we propose is an explicit one and makes it convenient to compute the mixture packing density. The formula is also compared with the simulation ones under the all three conditions of the same volume, aspect ratio and diameter, as well as some general cases. 2. Previous prediction models on the packing density of binary mixtures Based on the experimental and numerical simulation results, researchers have made great efforts on the prediction of the packing density of binary mixtures. There are a number of prediction models for binary spheres, including the simple packing model [46,47], mixture packing model [48], linear packing model [49,50], linearmixture packing model [51] and the Westman equation [33,40]. Among these models, the Westman equation agrees well to experimental results and has been extended to non-spherical particle mixtures. The Westman equation [33] presents a method concerned with the volume fraction of the mixture and the density of the mono-

sized sphere packing to predict the density of binary sphere mixtures by using a quadratic curve, which is



 V−V L X L 2 V−V L X L V−X L −V S X S V−X L −V S X S 2 þ 2G ¼1 þ VS VS V L −1 V L −1

ð1Þ

Where V is the specific volume defined as 1/Φ, Φ is the packing density of the mixture, VL, XL and VS, XS are the specific volumes and volume fractions of the large and small particles respectively. The factor G is determined by the particle size ratio r = ds/dl, where ds and dl are the diameters of the small and large particles respectively. Yu et al. [40] concluded the relationship between G and r from experimental results and proposed an empirical formula, which is 1 ¼ G



1:355r 1:566 1

ðr≤0:824Þ ðr > 0:824Þ

ð2Þ

The existence of the threshold indicates that when r of the two components approaches 1.0, then G = 1 and the previous models give a linear relationship between the mixture specific volume V and volume fraction X, that is V ¼ V L XL þ V SXS

ð3Þ

However, when the two components have a large size disparity, which means the value of r is far away from 1.0, the predicted specific volume of the mixture V is given by a quadratic equation of the general form 2

2

a þ 2Gab þ b ¼ 1

ð4Þ

Where a, b are functions of the volume fraction and specific volume. The relationship behaves as a convex curve. In order to extend the Westman equation to non-spherical particle mixtures, Yu et al. [40] presented a concept named “equivalent packing diameter” to calculate the size ratio r of the mixtures of nonspherical particles. Zou and Yu [52] concluded from the experimental results on particles of different materials, shapes and sizes, and suggested an empirical formula to calculate the equivalent packing diameter dp of a non-spherical particle, which is dp ¼ Ψ

−2:785

exp½2:946ðΨ−1Þdv

ð5Þ

Where dv is the diameter of the sphere with the same volume as the non-spherical particle, and Ψ is the Wadell sphericity, defined as the ratio of the surface area of the sphere having an equal volume as the original particle to the surface area of the particle. Introducing the concept of “equivalent packing diameter” makes it possible to predict the packing density of the mixture of non-spherical particles via the Westman equation, where the mixtures of non-spherical particles are treated as the mixtures of spheres. In particular, since the monophasic packing density of a specific particle is a constant and independent of its size under the conditions of geometric packing and periodic boundary, when we pack the particles of the same shape but different sizes, the results should reflect the single effect of particle volume on the mixture packing density. In this case, Eq. (1) degenerates to a more simple form here,


V −X L V0

2




2 V V−V 0 V−V 0 þ XL þ þ XL þ 2G −X L ¼ 1 ð6Þ V0 V 0 −1 V 0 −1

Where VL = VS = V0 is the specific volume of the monophasic packing and XL is the volume fraction of the larger component. This equation behaves as a convex curve and has some special characters.

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To acquire the value and position of the extreme point on Eq. (6), dV ¼ 0,and we obtain the optimum XLopt and Vopt of the exlet V ′ ¼ dX L

treme point on the curve as a function of V0 and G, which are opt XL

V opt

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 1 V 0− ¼ 2V 0 −1 2ð1 þ GÞ V 20 2V ðV −1Þ þ 0 0 ¼ 2V 0 −1 2V 0 −1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ð1 þ GÞ

ð7Þ

ð8Þ

From Eqs. (7) and (8), when the size ratio of the mixture has a large disparity, the factor G is not a constant of unity, and the XLopt and Vopt are not invariant as well. On the contrary, the G, XLopt and Vopt are all constant when the components of the mixture have similar sizes. Similarly, for a specific value of r (G is a constant), when the packing density of the identical spherocylinders varies in the range of simulated results in this work, though XLopt is not constant, but varies a little. Similar analyses are carried out under the condition of the same volume and diameter. However, the densities in terms of the volume fraction are both monotonic functions and no evident peaks are obtained from the empirical formula. It should be noted that the original Westman equation was developed on the premise of spherical packings. The concept of “equivalent packing diameter” was proposed based on finite samples of experiments, with the spherocylinders excluded. The simulation results in this work are compared with empirical formulas (1), (2) and (5) to verify whether they can be applied to the mixtures of binary spherocylinders. 3. The sphere assembly model and relaxation algorithm The sphere assembly model and relaxation algorithm are used in the simulations and the dual background cell method is adopted for the contact detection in this work. The sphere assembly model transfers the contacts between non-spherical particles to the ones between spheres, which is a universal approach to simulate the nonspherical particles of arbitrary shape. The sphere assembly model of a spherocylinder is constructed by a series of spheres with centers along a straight line. The diameters of all spheres are 1.0 and the distances between the neighbor component

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spheres are 0.1. Aspect ratio w can be adjusted by the number of component spheres N, w = (N − 1)/ 20. Fig. 2 shows the sphere assembly models of a spherocylinder with an aspect ratio of 2.0. The volume and surface area of the assembly model can be obtained easily from an AutoCAD system. Because the distances between neighboring assembly spheres are very small, the model is an excellent approximation to the real spherocylinder. As an important parameter to describe the particle shape, sphericity Ψ is defined as the surface area ratio between a sphere and a non-spherical particle of the same volume. The sphericity of a spherocylinder [53] can be written as a function of the aspect ratio, given by

3 2 Ψ¼ ð1 þ wÞ3 =ð1 þ wÞ 2

ð9Þ

For the packings simulated in this work, w varies in the range of [0, 3.5], and Ψranges from 0.80 to 1.0. The original relaxation algorithm was proposed by He et al. [38] to simulate the random packing of spheres. The algorithm was improved to simulate the random packing of non-spherical particles by introducing the torque and rotation of the particles [22]. The relaxation algorithm begins with a randomly placed large overlaps configuration of particles in a cubic region. Afterwards, iterations of the relaxation procedure are carried out to gradually reduce the overlaps of the particles. The boundary of the packing region is enlarged at the end of each iteration step. The final packing is achieved when the maximum overlap rate of all particles is below a predefined value. Details of the algorithm can be found in our previous works [16,29]. The efficiency of neighbor spheres searching heavily affects the computing time of the algorithm. As there are totally two kinds of spheres in this work related to the two kinds of spherocylinders in the mixtures, we adopt the dual background cell method [28], which divides the packing region into two independent background cell systems corresponding to the diameters of the large and small spheres. The dual background cell method obtains the same results as the single background cell method, but improves the efficiency of the algorithm. However, the advantage is significant only when the size ratio of the spheres or the number of the small spheres is large. If the two kinds of spheres have a similar size, this method may increase the computing time because of the complexity of data structures. 4. Random packing of identical spherocylinders

Fig. 2. The sphere assembly model of a spherocylinder with an aspect ratio of 2.0 in 3D solid (above) and 2D wireframe (below) visual styles.

The random packing of identical spherocylinders has been widely investigated in the literature. In this work, we simulate the random packing of 1000 spherocylinders with the aspect ratio varied from 0 to 4.0 using the sphere assembly models and relaxation algorithm. The packing density first increases and then decreases as a function of the aspect ratio and the maximum is 0.691 when the aspect ratio is around 0.35. This result corresponds to that of Williams and Philipse [10], whose peak position is w = 0.4 and the peak value is 0.695. Fig. 3 shows the packing of 1000 spherocylinders with w = 0.35. Note when w = 0, the spherocylinder degenerates to a sphere, and a packing density of 0.643 is obtained which agrees with the commonly accepted value. Fig. 4 gives the comparisons between the simulation results in the literature [10–16] and this work in terms of packing density. Compared with the existing simulations, the tendency of packing densities is the same and the maximum value is among the ones in the literature. The comparison shows that the model and algorithm are applicable to the packing of spherocylinders. Furthermore, it also demonstrates that different simulation methods produce qualitatively comparable but quantitatively different results.

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Fig. 3. A monophasic packing of 1000 spherocylinders (w = 0.35).

5. Mixtures of binary spherocylinders Mixtures of binary spherocylinders with the same volume, aspect ratio and diameter are simulated respectively with the sphere assembly models and relaxation algorithm. Periodic boundary condition is applied in the simulations. The numbers of the particles packed in the simulations are larger than 1000, but varied due to the mixture compositions. The effects of particle shape and size are investigated through the relationship between the mixture packing density, aspect ratio and the volume fraction of two components under three special conditions, the same volume, aspect ratio and diameter. The packing density as a function of aspect ratio w and volume fraction of the mixture X are illustrated. We also compute the nematic order parameter [17] of all the packings to examine the randomness of the mixtures. The parameters obtained are all very small and less than 0.1, indicating the simulations in this work are random packings.

Fig. 5. An example mixture of binary spherocylinders with the same volume, w1 = 0.35, w2 = 2.0, X1 = X2 = 0.5, N1 = N2 = 500, D1 = 1.38, D2 = 1.0, w1, w2, X1, X2 , N1, N2, D1, D2 is the aspect ratios, volume fractions, particle numbers and diameters of the two component spherocylinders, respectively.

The aspect ratios are chosen since w1 = 0.35 corresponds to the maximum packing density of identical spherocylinders, and a relatively longer component of w2 = 2.0. Fig. 6 gives the simulated density curves in terms of volume fraction with various w2 (w1 = 0.35). The simulation results in Fig. 6(a) indicates that all the packing density curves exhibit as linear

5.1. The same volume Mixtures of binary spherocylinders (0 ≤ w ≤ 3.5) with the same volume, which means the individual spherocylinders have the same particle volume, are simulated. Fig. 5 gives an example mixture of binary spherocylinders with the same volume, w1 = 0.35, w2 = 2.0, X1 = X2 = 0.5, N1 = N2 = 500, D1 = 1.38, D2 = 1.0, w1, w2, X1, X2 , N1, N2, D1, D2 are the aspect ratios, volume fractions, particle numbers and diameters of the two component spherocylinders respectively.

Fig. 4. Comparison of packing density versus aspect ratio for the monophasic packing of spherocylinders.

Fig. 6. Mixture packing density versus volume fraction of binary spherocylinders with the same volume (w1 = 0.35) (a) simulation results in this work (b) predicted results from the Westman equation.

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functions of the volume fraction when we fix the shape of one component spherocylinder. The slope of these lines first decreases with the increase of w2 ranging from 0 to 0.35, and then increases in the range of 0.35–3.5. Note when w1 = w2 = 0.35, the packing turns to the monophasic packing of spherocylinders whose density is irrelevant to the volume fraction, and the line becomes horizontal. When X2 = 0, the case degenerates to the monophasic packing of identical spherocylinders (w1 = 0.35), and all curves with different w2 converge to one point (ϕ=0.689). According to the empirical formula proposed in Eqs. (5) and (9), the “equivalent packing diameter” is exclusively related to the sphericity. Since the sphericity variation leads to small changes on dp (0.99 b dp b1.08) in the range of 0 ≤ w ≤ 3.5, the predicted results from the Westman equation in Fig. 6(b) behave as a series of lines with different slopes as well. Strictly speaking, we note the packing density curves are approximate lines but the specific volume curves are absolute lines according to Eq. (3). The empirical formulas give very similar results as the simulation in this work, not only the linearity but also the slopes. The linearity is a favorite property and it enhances the significance of the same volume case. Note the packing density (or specific volume) of the monophasic packing involved in the Westman equation is obtained from the simulation results. With the short component spherocylinder fixed, we obtain a series of packing density curves varied with the aspect ratio of the other component with different volume fractions, as shown in Fig. 7. The densities have the same variational tendency, all firstly increasing and then decreasing as a function of the aspect ratio of the other component, which is similar to the monophasic packing of spherocylinder. The peak loci of the curves are constant at a special aspect ratio of 0.35, the same as that of the monophasic packing. The volume fraction does not affect the packing density trend of binary spherocylinders, but determines the absolutely value of packing density. Note that the case of X2 = 0 degenerates to the monophasic spherocylinder packing of w1 = 0.35, and the density curve behaves as a horizontal line, while the case of X2 = 1.0 turns to the monophasic packing of identical spherocylinder with a varied w2, and the density curve is the same as in Fig. 4. When w2 = 0, the case comes to the binary mixture of spheres and spherocylinders, the packing density ranges from the monophasic sphere packing of ϕ=0.643 to the monophasic spherocylinders packing of ϕ=0.689 (w1 = 0.35). Similar numerical simulation on the mixture of binary spherocylinders with the same volume was carried out by Bargiel [15], although only the data of two aspect ratios of 0.4 and 2.0 were presented. Fig. 8 gives the comparison between the simulation results in this work and that of Bargiel under X1 = X2 = 0.5. The simulation results correspond well with those of Bargiel in both the trend and peak locus except the specific values of packing density due to the different simulation methods.

Fig. 7. Packing density versus aspect ratio of binary spherocylinders with the same volume (w1 = 0.35).

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Fig. 8. Comparisons of packing density versus aspect ratio of binary spherocylinders with the same volume in this work and the results of Bargiel [15] (X1 = X2 = 0.5).

If the two component spherocylinders have an equal volume fraction (X1 = X2 = 0.5), we investigate the packing density in terms of the aspect ratio. Fig. 9 gives the curves in the case that w1 is 0.2, 0.35, 0.6 and 1.0 respectively while w2 varies from 0 to 3.5. The densities always have stable maxima in terms of the aspect ratio, no matter how long the other component spherocylinder is, indicating that with one component constant the maximum value of mixture packing density is only determined by the one component whose monophasic packing density is larger. Mixtures of binary spherocylinders with the same volume eliminate the effects of particle size and reflect the shape effects on the packing density. The density curves in terms of volume fraction show linearity which is an evidence of the shape effect. The tendencies of the density curves in terms of aspect ratio are much alike, similar to the packing of identical spherocylinders, and irrespective of the shape of the constant component. Besides, the packing density shows no significant difference in variational tendency with different mixture compositions, but the exact density value is determined by the specific aspect ratios of the two components, indicating that the particle shape is an essential factor to the binary mixtures. 5.2. The same aspect ratio Mixtures of binary spherocylinders with the same aspect ratio are simulated to investigate the size effect on the mixture packing density. We firstly mix up to 8000 small and 1000 large spherocylinders together to obtain the mixtures of binary spherocylinders. The exact numbers of the component spherocylinders are determined by the specific volume fractions. The volume of the large spherocylinders

Fig. 9. Packing density versus aspect ratio of binary spherocylinders with the same volume in the case that w1 is 0.2, 0.35, 0.6 and 1.0 respectively while w2 varies from 0 to 3.5 (X1 = X2 = 0.5).

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Fig. 11. Comparison of packing density versus volume fraction for the binary mixture of spheres with r = 0.5.

Fig. 10. The packing density versus volume fraction of binary spherocylinders with the same aspect ratio (a) simulated results (b) predicted results from the Westman equation (r = 0.5, XL is the volume fraction of the larger spherocylinders).

simulated in this case are 8 times as the small components (r = 0.5) and their aspect ratios are varied from 0 to 2.0. Fig. 10 gives the simulated and predicted packing density curves in terms of various aspect ratios (0 ≤ w ≤ 2.0). XL is the volume fraction of the large spherocylinders, and w is the aspect ratio of both the large and small spherocylinders. Fig. 10(a) shows the packing density behaves as a series of convex curves as a function of the volume fraction. The existence of filling and occupation effect [51] in binary mixtures proposed by Yu and Standish may explain the tendency of the curves. For a constant size ratio of 0.5 but different aspect ratios, the packing density behaves as a series of equidistant curves and has a similar trend with a stable peak locus around 0.60 with only a small fluctuation. The aspect ratio has no effects on the trend of the curves, but determines the exact density value. These phenomena indicate that the shape and size of spherocylinders affect the mixture density independently. Compared to the simulated results with the predicted ones from the Westman equation, as in Fig. 10(a) and (b) respectively, for all cases as w varies from 0 to 2.0, the curves of the simulated results agree well with the predicted results from the Westman equation, in both the trend and peak locus. The peak loci of the predicted curves also have a small fluctuation as the discussion of Eq. (7) in Section 2. Nevertheless, the simulated density curves are always lower than the predicted ones due to the different packing conditions between the simulations and experiments. Note that when w=0, the packing degenerates to the mixture of binary spheres. Fig. 11 shows the comparison of the packing density results of simulated binary mixture of spheres with those in the literature [6,37–39]. The simulation results in this work of r=0.5 generally agree with the

ones in the literature, with an intervenient density value and a similar peak locus, indicating our method is applicable to binary mixtures. As an example of the mixture of binary spherocylinders with the same aspect ratio, Fig. 12 shows the mixture of 3200 small and 600 large spherocylinders with an identical aspect ratio of w = 0.35. The volume fraction of the larger component is 60%, which is the peak locus of the density curve. We also use up to 8000 small spherocylinders with w = 0.35 to obtain the binary mixtures under different size ratios. The numbers of the larger components are correspondingly changed with the variations of the size ratios. Beside the fixed shape diversity between the components, the only difference in the group of simulations lies in the size ratio, which should reflect the single size effect on the packing density. The packing density curves in terms of the volume fraction are all convex, as Fig. 13(a) shows. The curves become more flat with the increase of the size ratio r, which indicates the filling or occupation effect is positively related to the size ratio of the particles. The tendency is reasonable since when r = 1, a monophasic packing is obtained, and the density curve becomes a horizontal line with no relevance to the volume fraction. Note that all density curves converge to points of monophasic packing, as XL = 0.0 and XL = 1.0 in Fig. 13, and the densities at the two points are the same under the condition of geometric packing. We also compare the simulation results to the prediction ones from the Westman equation. As Fig. 13 shows, the density

Fig. 12. An example packing of binary spherocylinders with the same aspect ratio, NS = 3200 NL = 600, w = 0.35, r = 1/2, XL = 0.6, NS and NL are the numbers of the small and large component spherocylinders in the binary mixture, respectively.

L. Meng et al. / Powder Technology 228 (2012) 284–294

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Fig. 14. An example packing of binary spherocylinders with the same diameter, w1 = 0.35, w2 = 2.0, X1 = X2 = 0.5, N1 = 1311, N2 = 500.

we have a constant component spherocylinder (w1 = 0.35), when the other component is relatively short, the volume difference is consequently small, and the density curves behave as a series of lines. However, as w2 increases and when the spherocylinder is long enough, the size diversity gradually makes a difference on the packing density, and the density curves become a series of convex curves. Furthermore, the curves get more convex with the increase of w2.

Fig. 13. Mixture packing density versus volume fraction of binary spherocylinders with the same aspect ratio (w = 0.35) but different size ratios (a) simulation results in this work (b) predicted results from the Westman equation.

curves are much similar to the predicted ones, in both the trend and peak locus. However, the packing density of the simulation is lower than the prediction due to the different packing conditions. The volume fraction of the larger component in the peak locus gradually increases from 0.60 to 0.70 while the size ratio decreases from 1/2.0 to 1/3.5, while the one from the predicted results generally keeps stable at XL = 0.60. Mixtures of binary spherocylinders with the same aspect ratio eliminate the effect of the particle shape and reflect the single effect of the particle size on the packing density. The density behaves as a series of convex curves. The curves become more flat with the increase of the size ratio r, indicating the size ratio or the volume disparity also has significant effects on the packing density of binary spherocylinders. 5.3. The same diameter Mixtures of binary spherocylinders with the same diameter are also simulated to investigate the combined effects of particle size and shape, as well as to compare with previous prediction models and simulations. Under the same diameter condition, both the shape and size difference contribute to the packing density. The number of spherocylinders in the simulations are larger than 1000, and the specific value is determined by the exact volume fraction. Fig. 14 gives an example of the binary mixture with the same diameter (w1 = 0.35, w2 = 2.0, X1 = X2 = 0.5, N1 = 1311, N2 = 500). Fig. 15 gives the simulated and predicted density curves in terms of volume fraction with various w2 (w1 = 0.35). In the case of the same diameter, the volume difference between a single sphere and spherocylinder is more distinct with increasing aspect ratio. Since

Fig. 15. Mixture packing density versus volume fraction of binary spherocylinders with the same diameter (w = 0.35) (a) simulation results in this work (b) predicted results from the Westman equation.

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Note when w2 = 0, the packing turns to the case of binary spherespherocylinder under the same diameter condition, and since the sphere and spherocylinders (w1 = 0.35) have a small difference in volume, the density curve behaves as a straight line with the endpoints of densities of monophasic spherocylinder packing. When w1 = w2 = 0.35, the packing degenerates to the case of monophasic spherocylinder, and the line becomes horizontal. When X2 = 0, the case degenerates to the packing of identical spherocylinders (w1 = 0.35), and all curves with different w2 converge to one point (ϕ = 0.689). According to Eqs. (5) and (9), since the function of sphericity varies little (from 0.99 to 1.08), dp depends mostly on dv. As a result, the density of spherocylinders with similar volumes nearly has a linear function with the volume fraction. However, when the other spherocylinder is long enough (w2 ≥ 1.6), the size ratio drops below the critical value (0.824) and the Westman equation approaches a series of convex curves. Comparing the simulated density curves to the predicted ones from the Westman equation, though the specific values have a small difference, it can be found that the two groups of curves are consistent well both in shape and trend. With the aspect ratio of one spherocylinder constant (w1 = 0.35), we obtain a series of packing density curves in terms of the aspect ratio of the other component. The density curves are much similar to the case of the same volume in Fig. (7) and vary as the ones of the monophasic spherocylinder packing. Fig. 16(a) shows the packing density in terms of aspect ratios of the two components in the system. In this work, the aspect ratio of the spherocylinders simulated is ranged from 0 to 3.0. When the mixture is of an equal volume fraction, which is X1 = X2 = 0.5, the packing density in terms of the aspect ratios of both components varies similarly to the case of monophasic packing. For all cases with different

constant components, the density increases as the aspect ratio and then decreases with a steady maximum position of 0.35. The density curves may have a little interaction in different cases, but have no effect on the tendency of the curves. Note that when w2 = 0, the packing degenerates to the binary mixture of spheres and spherocylinders. Kyrylyuk et al. [12] and Lu et al. [28] both simulated the rod-sphere mixture packings, which are composed of spheres and spherocylinders with the same diameter. The trends of the density curves in this work and that by Kyrylyuk et al. as well as Lu et al. are generally consistent. The results are comparable to the predictions as well, indicating the model and algorithm in this work are applicable to the binary mixtures of spherocylinders. We also compare the simulation results of mixtures of binary spherocylinders with those of Kyrylyuk and Philipse [45] in Fig. 16(b) under the same diameter condition. The exact packing densities obtained in this work are lower probably because of the sphere assembly models. However, the tendency of the density curves coincide with each other, revealing a maxima with a peak locus of a short rod (w = 0.35 in this work, while w = 0.5 in [45]). Mixtures of binary spherocylinders with the same diameter combine the effects of particle shape and size together. The trends of density curves of binary spherocylinders are different as the increase of the size and shape disparity, changing from a series of lines to convex curves as a function of the volume fraction, and they agree well with the prediction results from the Westman equation. Besides, the general trend of the packing density with the increase of both the two aspect ratios is similar to that of the monophasic spherocylinder packings, and irrespective of the volume fraction. The simulation results indicate that the particle shape and size are both important factors to the packing density of the mixture. 5.4. An explicit empirical formula of the mixture packing density The packing density of binary mixtures is affected by several factors, including the size, shape and the volume fraction of the components. The relationships between the packing density, volume fraction and aspect ratio under the conditions of the same volume, aspect ratio and diameter show the single effect of particle shape, size and the their combinations respectively. From the simulation results in Fig. 10(a), the densities behave as a series of equidistance curves of a similar varying trend and peak locus. Therefore, we note that the particle size and shape influence the mixture packing density of binary spherocylinders independently. The size ratio determines the convex shape of the density curve, while the aspect ratio determines the exact density value. Hence, the mixture packing density can be described as a linear superposition of two functions of the size ratio and the aspect ratio with no coupling terms. Based on the observation, we propose a new explicit empirical formula as follows, ϕðr; w; X Þ ¼ f ðr; X L Þ þ g ðw1 ; w2 ; X 1 Þ

ð10Þ

Where f ðr; X L Þ ¼ C 1 ðX L Þr 2 þ C 2 ðX L Þr þ C 3 ðX L Þ g ðw1 ; w2 ; X 1 Þ ¼ X 1 hðw1 Þ þ ð1  X 1 Þhðw2 Þ

ð11Þ

Unlike previous works using dp to obtain the size ratio in Eq. (5), here r is the ratio of the diameter of the smaller sphere with the same volume as the non-spherical particle to the diameter of the larger one. Hence the term of f(r, XL)reflects the single size effect while the one of g(w1, w2, X1)reflects the single shape effect. C1, C2, C3 are three functions of XL, and the fitting results are 3

Fig. 16. Packing density versus aspect ratio of binary spherocylinders with the same diameter (X1 = X2 = 0.5) (a) simulation results in this work (b) simulation results by Kyrylyuk and Philipse [45].

2

C 1 ðX L Þ ¼ 1:4112X L þ 1:8053X L  0:4041X L 3 2 C 2 ðX L Þ ¼ 2:2556X L −2:7228X L þ 0:4833X L −0:0031 3 C 3 ðX L Þ ¼ −0:845X L þ 0:9179X 2L −0:0792X L þ 0:0027

ð12Þ

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h (w1) and h (w2) can be described as a piecewise function as  hðwÞ ¼

2

−0:39ðw−0:35Þ þ 0:6890 w≤0:5 −0:0423ðw−0:35Þ þ 0:6924 w > 0:5

ð13Þ

w1, w2 are the aspect ratios of the two component spherocylinders respectively. Note that when the aspect ratio is very large, it is better to employ an empirical formula fitted from the specific volume results rather than the ones of packing density. The predicted results from the explicit formula of Eq. (10) are compared with the simulation ones of the same volume in Fig. 6(a), the same aspect ratio in Figs. 10(a) and 13(a), the same diameter in Fig. 15(a), and they agree well with each other. We also examine the explicit formula in a wide range of aspect ratios and some more general cases of binary mixtures beyond the same volume, aspect ratio and diameter, and they generally correspond to the simulation results. Fig. 17 gives a comparison of the simulated density to the ones obtained from the explicit formula in terms of the volume fraction in a general mixture (wS = 1.0, wL = 0.5, DS = 2.0, DL = 4.5, r = 0.5, VS:VL = 1:8). Both results coincide with each other well, indicating the universal applicability of the explicit formula on the mixtures of binary spherocylinders, which means the explicit formula based on the simulation results of the same volume and aspect ratio is applicable to predict the mixture packing density in the cases which combines the size and shape effects together. Unlike previous methods which usually combined the shape and size effects together and the shape effect was transferred into size effect through certain equivalent methods, as the concept of “equivalent packing diameter”, our finding brings a new idea for predicting the mixture density of non-spherical particles, which should lead to a more precise but simple prediction approach for binary mixtures. Besides, previous predictions are usually implicit formulas, as the Westman equation, and one should solve a complicated quadratic equation to obtain the mixture packing density, which may be not convenient for engineering purpose. On the contrary, the new empirical formula we proposed is an explicit one using polynomials, which can be computed easily, and the linear superposition of the size and shape effects is believed to reflect the physical essence of the mixtures packings of non-spherical particles. However, this essence should be verified in other non-spherical particle mixtures before it can be accepted as a general rule. 6. Conclusions A systematic investigation on the binary mixture of spherocylinders considering the single effects of the shape and size respectively is carried out by the sphere assembly models and relaxation algorithm. The

293

random packings of binary spherocylinders with the same volume, aspect ratio and diameter are simulated to investigate the single effects of particle shape, size, and their combinations respectively. Under the condition of the same volume, the density curves in terms of volume fraction show good linearity which is an evidence of the shape effect, while the density behaves as a series of convex curves in terms of the volume fraction under the condition of the same aspect ratio, indicating the single size effect. Under the third condition of the same diameter, which combines the size and shape effects together, the trend of density in terms of the volume fraction are different and changed from a series of straight lines to convex curves, and the curves become more convex with a larger size and shape disparity. The tendencies of the density curves in terms of aspect ratio are much alike, first increase and then decrease with a stable peak locus of w = 0.35, which is similar to the packing of identical spherocylinders, and irrespective of the shape of the constant component. More important, we find the particle size and shape influence the mixture packing density independently, and the density can be described as a linear superposition of two independent functions of the particle size ratio and aspect ratio. Additionally, based on the simulation results of binary mixtures with the same volume and aspect ratio, we propose a new explicit empirical formula composed with two independent functions of particle size and shape respectively, in which the size ratio and aspect ratio of the two components affect the mixture density separately with no coupling terms. It is believed to reflect the physical essence of the binary mixture of non-spherical particles. Unlike the previous formula, as the Westman equation, where a quadratic equation needs to be solved, the formula which uses polynomials is an explicit one, and is convenient in engineering practice. The cases with the same diameter and more general mixtures which combine the size and shape effects together are carried out for validation, and the formula is demonstrated to be applicable to predict the mixture packing density of binary spherocylinders. The simulation results of the monophasic and binary packings of spheres and spherocylinders are carried out and compared with the previous simulations and empirical formula for verification. We believe the conclusions and formulas obtained in this work are applicable to general binary mixtures of spherocylinders and have significant values in practice and industry. However, the simulations of the mixtures carried out in this work are only for binary spherocylinders. Whether the conclusions and formulas are universally applicable for other non-spherical particles needs to be investigated in future works. Additionally, we note that the order factor should be taken into account when various shapes are involved. Besides that, since a ternary or polydisperse mixture can be considered as binary packings of more steps, the behavior and property of these mixtures of spherocylinders with a size and shape distribution is another interesting and valuable issue to be explored. Acknowledgments The authors thank Prof. Aibing Yu for his valuable discussion on the binary mixtures. This work was supported by the National Natural Science Foundation of China (Grant No. 10772005) and the National Basic Research Program of China (Grant No. 2010CB832701). References

Fig. 17. Comparison of the simulated packing density of a general mixture with the ones from the explicit formula versus volume fraction (wS = 1.0, wL = 0.5, DS = 2.0, DL = 4.5, r = 0.5, VS:VL = 1:8).

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