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Excluded volumes of clusters in tetrahedral particle packing
Physics Letters A 378 (2014) 835–838
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Excluded volumes of clusters in tetrahedral particle packing Lufeng Liu, Peng Lu, Lingyi Meng, Weiwei Jin, Shuixiang Li ∗ Department of Mechanics & Engineering Science, College of Engineering, Peking University, Beijing, 100871, China
a r t i c l e
i n f o
Article history: Received 30 October 2013 Received in revised form 21 December 2013 Accepted 14 January 2014 Available online 21 January 2014 Communicated by A.R. Bishop Keywords: Excluded volume Cluster Packing Tetrahedra
a b s t r a c t We investigate the excluded volumes of clusters in tetrahedral particle packing using an ideal tetrahedron model and Monte Carlo simulation. Both the influences of the size and topology of clusters on the excluded volume are studied. We find that the excluded volumes of the dimer composed of two tetrahedra and the wagon wheel composed of five tetrahedra are relatively lower than other cluster forms. For large clusters, the excluded volume decreases when the topology of a cluster approaches the wagon-wheel geometry. The results give an explanation to the cluster distribution which demonstrates that the dimer and wagon wheel are the dominative cluster forms in the packing structure of tetrahedra. © 2014 Elsevier B.V. All rights reserved.
1. Introduction As an example of the 18th problem proposed by Hilbert in 1900 about the densest packing of a given shape [1], the packing of regular tetrahedra has achieved significant advances in the last few years, and becomes a prosperous research area in particle packing, granular material and discrete geometry. Chen et al. [2] presented the densest packing of regular tetrahedra so far with a crystalline structure of dimers which has a density about 0.8563. For the disordered packing of tetrahedra, macroscopic and microscopic properties have been well studied via simulations [3–10] and experiments [11–13]. However, the theoretical packing density limit is still unknown, and further investigations should be carried out to explore the dense packing structure of tetrahedra. It has been well observed that tetrahedra assemble into some special cluster forms in various packings both from simulations and experiments. Common cluster forms are the dimer (bipyramid), wagon wheel (pentagonal dipyramid), nonamer, icosahedron and tetrahelix [3]. Our recent investigation [10] showed that the relative amount of particles in clusters increases linearly with the growth of the packing density, and the relation is irrelevant to the packing generation method. We termed the hierarchical random packing of clusters as a “quasi-random packing” [10]. We also found that the dimer and wagon wheel are the two dominative cluster forms in the packing structure of tetrahedra [14]. For instance, in a quasicrystal packing structure of tetrahedra [3], the volume fractions of dimers and wagon wheels are about 23.58% and 13.58%, respectively, which are significantly larger than other
*
Corresponding author. E-mail address: [email protected] (S. Li).
0375-9601/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2014.01.023
cluster forms. However, the reason for the phenomenon is still out of reach. In this work, we try to explain the phenomenon via the concept of excluded volume. The concept of excluded volume was introduced by Werner Kuhn in 1934. In granular and polymer systems, the excluded volume refers to the volume that is inaccessible to other particles as a result of the presence of the first one [15,16]. It was believed that the excluded volume is closely related to packing density, although their relationship is not quite clear yet. Philipse [15] proposed a random contact model and demonstrated that the random packing density has an inverse relation with the excluded volume for long spherocylinders. Gravish et al. [17] and Meng et al. [16] tried to explain the variation of packing density by the excluded volumes of non-convex U particles and curved spherocylinders, respectively. Recently, Baule et al. [18] presented a theoretical framework for packing density prediction based on the concept of the Voronoi excluded volume. Many works have been done to calculate the excluded volume of a single particle, however, the excluded volumes of clusters are rarely referred to in previous works. The excluded volume of a complex shape particle is usually difficult to calculate [19]. Recently, Torquato and Jiao [20] derived an explicit formula for the excluded volume of a convex hyperparticle of arbitrary shape. As for a tetrahedron and derivative clusters, only the tetrahedron and dimer are convex bodies, and all other cluster forms are non-convex. However, it is rather difficult to give an explicit formula for the excluded volumes of non-convex clusters. Fortunately, the Monte Carlo simulation [16,17,21] provides an optional means to estimate the excluded volume. The purpose of this work is to compute and analyse the excluded volumes of clusters in tetrahedral particle packings, which are important to the understanding of the packing density and
836
L. Liu et al. / Physics Letters A 378 (2014) 835–838
packing structure. We will compare the excluded volumes in different cluster sizes as well as in the same cluster size but different cluster types to investigate the effects of the size and topology on the excluded volume. We also compare the excluded volumes of clusters with face–face joints to those of clusters with other face–face contacts. Thereafter, we try to explain the distribution of cluster types in tetrahedral particle packings using their excluded volumes. Finally, we explore how the excluded volume is related to the packing density. 2. Model and method All the clusters in tetrahedron packings can be treated as a variety of topological structures of tetrahedra. The particle model used in this work is the ideal tetrahedral model with sharp corners, and its roundness ratio [7] γ = 0. Clusters defined here are composed of tetrahedra with continuous face–face joints. A face–face joint can be regarded as a strict face–face contact, which satisfies that the two faces are parallel, contacted and coplanar [10]. The size of a regular tetrahedron is measured by its side length. In the three dimensional Euclidean space, the vertex coordinates can be obtained by using a coefficient matrix which describes the relationship between the vertexes and the centroid of the tetrahedron in a local coordinate system. Other geometric properties of the tetrahedron can be easily expressed in terms of the vertex coordinates. A contact detection algorithm is applied to detect whether two tetrahedra are contacted with each other. In our simulations, we consider two tetrahedra to be contacted if their minimum distance is less than 2.0 × 10−6 (the side length of the tetrahedron is within the spectrum from 1 to 5). First we detect whether their circumscribed spheres are overlapped. If they are not overlapped, the two tetrahedra are surely separated. Then whether the vertexes of a tetrahedron are inside or on the face of the other one is checked. When such vertex exists, the two tetrahedra are overlapped. Afterwards, we detect if one of the edges of a tetrahedron has an intersection point with one of the faces of another tetrahedron. If such edge exists, the two tetrahedra are overlapped. Otherwise, they are separated. The ideal tetrahedron model is employed to construct cluster models. Each cluster consists of a number of tetrahedra with continuous face–face joints. A tetrahedron has four faces, and each of them can respectively generate a new tetrahedron which has a common face with the original one. Therefore, the cluster model can be built by generating tetrahedra in a certain topological order on a tetrahedron. Meanwhile, by detecting whether two tetrahedra belonging to different clusters are contacted, we can judge if the two clusters are overlapped accordingly. Twenty typical cluster types in tetrahedron packings are shown in Fig. 1. We note that only clusters with face–face joints, which are very common in simulation and experimental results [10], are considered in this work, except the last one (2-a) with a face– face contact. The cluster type is unique if the cluster consists of less than four tetrahedra, namely the 1-tetrahedron, 2-dimer and 3-trimer for the cluster size of 1, 2 and 3, respectively. The three types of the clusters consist of four tetrahedra, namely the 4-cluster, are termed as 4-a, 4-b and 4-c shown in Fig. 1. There are seven topological structures for 5-cluster, while only six typical types of 6-cluster are considered in this work. These cluster types are also illustrated in Fig. 1. Additionally, a different type of 2-cluster (2-a) is analysed as well to compare the excluded volume of face–face joint clusters with that of other face–face contact clusters. For a particle with a convex or non-convex shape, the excluded volume can be calculated by the generalised expression as follows
Fig. 1. Typical cluster types in tetrahedron packings with face–face joints. The last one (2-a) is a different two-particle cluster with a face–face contact.
V ex =
p (r , ω)Θ s(r , ω) dr dω
(1)
Ω
where Ω is the whole three dimensional Euclidean space, p (r , ω) is the positional and orientational probability density function, Θ(x) is the unit step function, and s(r , ω) is the minimum distance between the central particle and the particle with the centroid position r and orientation ω . If the two particles are overlapped, the minimum distance is determined as −1. For the particles in a random packing, the probability density is uniform, that is p (r , ω) = 1. We thus obtain the following formula
V ex =
Θ s(r , ω) dr dω
(2)
Ω
In this work, excluded volumes are mostly obtained through the Monte Carlo simulation [16,17,21]. In a Monte Carlo procedure, a cubic box with a side length L is applied as a sampling space. Then we put one particle at the centre of the box with an arbitrary orientation. Another identical particle is generated in the sampling space with an arbitrary position and orientation, and whether it is contacted with the central one is detected. The procedure is carried out iteratively and N c particles are detected to be contacted with the central one. Finally, the excluded volume of the particle can be estimated as
V ex = lim
N →∞
Nc N
Vs
(3)
where V s = L 3 is the volume of the sampling space, N is the total number of tested particles. In fact, Eq. (3) is an equivalent expression of Eq. (2) using a computer simulation scheme. We also introduce the concept of the relative excluded volume which is defined as
V rex =
V ex V
(4)
where V is the volume of the particle. Therefore, the relative excluded volume V rex is dimensionless, and is a constant for a given shaped particle. For example, the V rex is 8 for a sphere. After a number of attempts, we find that the influence of the dimension of the sampling space on the excluded volume can be eliminated when L 50 while the side length of the tetrahedron is within the spectrum from 1 to 5. The effect of the number of tests N on the excluded volume is also explored. The excluded volume changes little when N reaches 109 . The excluded volume of each cluster is computed four times and averaged to reduce errors. With L = 50 and N = 109 , the relative error of the excluded
L. Liu et al. / Physics Letters A 378 (2014) 835–838
837
volume of both single tetrahedron and tetrahedral clusters can be limited to 0.7%. 3. Results and discussion The excluded volumes of 1-tetrahedron and 2-dimer are obtained both through simulation and mathematical deduction to verify the model and method. Torquato and Jiao [20] derived an explicit formula for the excluded volume of a convex particle of arbitrary shape in terms of its d-dimensional volume V , surface area S and radius of mean curvature R. The excluded volume of a three-dimensional convex body is given by
V ex = 2V + 2S R
(5)
So the relative excluded volume can be described as
V ex
V rex =
V
=2+2
SR
(6)
V
Kihara [22] proved that Eq. (5) is equivalent to Eq. (2) when the particle shape is convex. As mentioned above, Eq. (2) is equivalent to Eq. (3). Therefore, Eq. (5) is equivalent to Eq. (3) under the convex condition. The relative excluded volume of a tetrahedron was given directly [19] as tetrahedron V rex
=2+
18
π
3 2
cos−1 −
1
Fig. 2. The relative excluded volumes V rex of 4-clusters.
≈ 15.407
3
(7)
As for a dimer with the side length a, the volume V , surface area S and radius of mean curvature R can be respectively written as
√ V = S= R=
2
6
a3
(8)
√
3 3 2 3a 2π
a2
(9)
cos−1 −
1 3
3
− a
(10)
8
Fig. 3. The relative excluded volumes V rex of 5-clusters.
Consequently, the relative excluded volume of the dimer is
√
dimer V rex =2+9 6
3 2π
cos−1 −
1 3
−
3 8
≈ 13.844
(11)
Meanwhile, the excluded volumes of 1-tetrahedron and 2-dimer obtained through the Monte Carlo simulation are 15.413 ± 0.107 and 13.851 ± 0.092, respectively. The simulation results agree well with the analytical expressions in Eqs. (7) and (11). Therefore, the model and algorithm used in this work are applicable for estimating the excluded volume of tetrahedra and tetrahedral clusters. For comparison, the excluded volume of the 2-a type cluster obtained via simulation is 17.241 ± 0.032, which is much larger than that of a dimer, and is even larger than those of all the clusters with face–face joints considered in this work. The fact indicates that the excluded volumes of clusters with face–face joints are smaller than those of clusters with other face–face contacts in the same cluster size, thus the former occupies less space in a random packing than the latter. The face–face joint has been found as the major contact form between two tetrahedra in dense tetrahedral particle packings [3,10]. The low excluded volumes of face–face joint clusters should explain this phenomenon. Since the explicit formula for the excluded volume of a nonconvex body is rather difficult to deduce, we employ the Monte Carlo simulation [16,17,21] to estimate the value. The excluded volumes of the 4-clusters, 5-clusters and 6-clusters in various topological types are respectively illustrated in Figs. 2, 3 and 4. All the values are ranked in an incremental order. The error bars in the
Fig. 4. The relative excluded volumes V rex of 6-clusters.
figures represent the standard deviation and are averaged over four independent simulation runs. The clusters in Fig. 2 all consist of four tetrahedra, namely the 4-clusters. As shown in Fig. 2, the cluster of 4-a type has a smaller excluded volume than the other two 4-cluster types. Note that the 4-a type cluster is mostly approximate to the wagon wheel (5-a type) in 4-clusters. As Fig. 3 shows,
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L. Liu et al. / Physics Letters A 378 (2014) 835–838
Nevertheless, we note that the excluded volume is not the only factor affecting the cluster distribution in tetrahedral particle packings. The geometric compatibility of the clusters and the configurational entropy of the packing structure are also effective factors, and are expected to be investigated in future works. 4. Conclusions
Fig. 5. The relative excluded volume V rex vs. the size of clusters in the packing structure of tetrahedra.
the excluded volume of a wagon wheel (5-a type) is smaller than any other 5-cluster types. Clusters of 5-b and 5-c types also have smaller excluded volumes than those of 5-d, 5-e, 5-f and 5-g types. Meanwhile, both the 5-b and 5-c type clusters contain the 4-a type cluster which has the smallest excluded volume in 4-clusters. Clusters of 5-b and 5-c types are geometrically closer to a wagon wheel than other types. Only some typical topological types in 6-clusters are selected to contrast their excluded volumes which are shown in Fig. 4. The 6-a type cluster which contains a wagon wheel has the smallest excluded volume in 6-clusters. Comparing with the 6-e and 6-f types, clusters of 6-b, 6-c and 6-d types have smaller excluded volumes. Similar to the 5-b and 5-c types, the 6-b, 6-c and 6-d type clusters include a 4-a type as well, and they are geometrically closer to a wagon wheel than the clusters of 6-e and 6-f types. From Figs. 2, 3 and 4, we find that with the same cluster size, topological types which approach to the wagon-wheel geometry have a smaller excluded volume than other types. Meanwhile, a wagon wheel has the smallest relative excluded volume among all these clusters. Hence, wagon wheel is a special topological structure among clusters in tetrahedral particle packings. It was found that wagon wheel is a dominative cluster form in dense tetrahedron packings [14], and the fairly low excluded volume should be the reason. Fig. 5 illustrates the excluded volumes of different clusters in terms of cluster size. Note that the type with the minimum excluded volume is chosen in each cluster size. The values of tetrahedron (size 1) and dimer (size 2) are analytical results from Eqs. (7) and (11), respectively. The excluded volume varies nonmonotonically with the change of cluster size. Especially, dimer and wagon wheel are the local minima of the excluded volume. The excluded volume of a wagon wheel is obviously a valley on the curve shown in Fig. 5, and the value of dimer is slightly smaller than those of the 3-trimer and 4-a type clusters. These results well agree with our recent investigation that dimer and wagon wheel are the two dominative forms in the packing structure of tetrahedra [14]. We believe that the smaller excluded volume of clusters leads to a higher packing density in tetrahedral particle packing, and a dense packing structure should have dominative clusters with low excluded volumes.
We employ the ideal tetrahedron model and the Monte Carlo simulation to compute the excluded volumes of the clusters in tetrahedral particle packing. Both the influences of the topological type and cluster size on the excluded volume are investigated. Excluded volumes of clusters with face–face joints are smaller than those of clusters with other face–face contacts. This is coincident with the phenomenon that face–face joints can be found as the major contact form between two tetrahedra in dense tetrahedral particle packings. Dimer and wagon wheel are the local minima of relative excluded volume in terms of cluster size. The results correspond to the cluster distribution in tetrahedral particle packings which demonstrates that dimer and wagon wheel are the dominative cluster forms in the packing structure. They also give an explanation to the cluster distribution. Wagon wheel has the smallest relative excluded volume among all these clusters, and the clusters have smaller relative excluded volumes when their topologies approach the wagon-wheel geometry. Wagon wheel is a special and important cluster type in tetrahedral particle packings and should be further studied in the future. The findings in this work are important to the understanding of the packing density and packing structure of tetrahedral particle packings. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 11272010) and the National Basic Research Program of China (Grant No. 2010CB832701). References [1] T. Aste, D. Weaire, The Pursuit of Perfect Packing, 2nd ed., Taylor & Francis, Boca Raton, FL, 2008. [2] E.R. Chen, M. Engel, S.C. Glotzer, Discrete Comput. Geom. 44 (2010) 253. [3] A. Haji-Akbari, M. Engel, A.S. Keys, X. Zheng, R.G. Petschek, P. Palffy-Muhoray, S.C. Glotzer, Nature 462 (2009) 773. [4] Y. Jiao, S. Torquato, Phys. Rev. E 84 (2011) 041309. [5] K.C. Smith, M. Alam, T.S. Fisher, Phys. Rev. E 82 (2010) 051304. [6] K.C. Smith, T.S. Fisher, M. Alam, Phys. Rev. E 84 (2011) 030301. [7] J. Zhao, S. Li, W. Jin, X. Zhou, Phys. Rev. E 86 (2012) 031307. [8] J.P. Latham, Y. Lu, A. Munjiza, Geotechnique 51 (2001) 871. [9] S. Li, J. Zhao, X. Zhou, Chin. Phys. Lett. 25 (2008) 1724. [10] S. Li, P. Lu, W. Jin, L. Meng, Soft Matter 9 (2013) 9298. [11] J. Baker, A. Kudrolli, Phys. Rev. E 82 (2010) 061304. [12] A. Jaoshvili, A. Esakia, M. Porrati, P.M. Chaikin, Phys. Rev. Lett. 104 (2010) 185501. [13] M. Neudecker, S. Ulrich, S. Herminghaus, M. Schröter, Phys. Rev. Lett. 111 (2013) 028001. [14] W. Jin, P. Lu, S. Li, unpublished results, 2014. [15] A.P. Philipse, Langmuir 12 (1996) 1127. [16] L. Meng, S. Li, P. Lu, W. Jin, Phys. Rev. E 86 (2012) 061309. [17] N. Gravish, S.V. Franklin, D.L. Hu, D.I. Goldman, Phys. Rev. Lett. 108 (2012) 208001. [18] A. Baule, R. Mari, L. Bo, L. Portal, H.A. Makse, Nat. Commun. 4 (2013) 2194. [19] L. Onsager, Ann. N.Y. Acad. Sci. 51 (1949) 627. [20] S. Torquato, Y. Jiao, Phys. Rev. E 87 (2013) 022111. [21] N. Ibarra-Avalos, A. Gil-Villegas, A.M. Richa, Mol. Simul. 33 (2007) 505. [22] T. Kihara, Rev. Mod. Phys. 25 (1953) 831.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Excluded volumes of clusters in tetrahedral particle packing Lufeng Liu, Peng Lu, Lingyi Meng, Weiwei Jin, Shuixiang Li ∗ Department of Mechanics & Engineering Science, College of Engineering, Peking University, Beijing, 100871, China
a r t i c l e
i n f o
Article history: Received 30 October 2013 Received in revised form 21 December 2013 Accepted 14 January 2014 Available online 21 January 2014 Communicated by A.R. Bishop Keywords: Excluded volume Cluster Packing Tetrahedra
a b s t r a c t We investigate the excluded volumes of clusters in tetrahedral particle packing using an ideal tetrahedron model and Monte Carlo simulation. Both the influences of the size and topology of clusters on the excluded volume are studied. We find that the excluded volumes of the dimer composed of two tetrahedra and the wagon wheel composed of five tetrahedra are relatively lower than other cluster forms. For large clusters, the excluded volume decreases when the topology of a cluster approaches the wagon-wheel geometry. The results give an explanation to the cluster distribution which demonstrates that the dimer and wagon wheel are the dominative cluster forms in the packing structure of tetrahedra. © 2014 Elsevier B.V. All rights reserved.
1. Introduction As an example of the 18th problem proposed by Hilbert in 1900 about the densest packing of a given shape [1], the packing of regular tetrahedra has achieved significant advances in the last few years, and becomes a prosperous research area in particle packing, granular material and discrete geometry. Chen et al. [2] presented the densest packing of regular tetrahedra so far with a crystalline structure of dimers which has a density about 0.8563. For the disordered packing of tetrahedra, macroscopic and microscopic properties have been well studied via simulations [3–10] and experiments [11–13]. However, the theoretical packing density limit is still unknown, and further investigations should be carried out to explore the dense packing structure of tetrahedra. It has been well observed that tetrahedra assemble into some special cluster forms in various packings both from simulations and experiments. Common cluster forms are the dimer (bipyramid), wagon wheel (pentagonal dipyramid), nonamer, icosahedron and tetrahelix [3]. Our recent investigation [10] showed that the relative amount of particles in clusters increases linearly with the growth of the packing density, and the relation is irrelevant to the packing generation method. We termed the hierarchical random packing of clusters as a “quasi-random packing” [10]. We also found that the dimer and wagon wheel are the two dominative cluster forms in the packing structure of tetrahedra [14]. For instance, in a quasicrystal packing structure of tetrahedra [3], the volume fractions of dimers and wagon wheels are about 23.58% and 13.58%, respectively, which are significantly larger than other
*
Corresponding author. E-mail address: [email protected] (S. Li).
0375-9601/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2014.01.023
cluster forms. However, the reason for the phenomenon is still out of reach. In this work, we try to explain the phenomenon via the concept of excluded volume. The concept of excluded volume was introduced by Werner Kuhn in 1934. In granular and polymer systems, the excluded volume refers to the volume that is inaccessible to other particles as a result of the presence of the first one [15,16]. It was believed that the excluded volume is closely related to packing density, although their relationship is not quite clear yet. Philipse [15] proposed a random contact model and demonstrated that the random packing density has an inverse relation with the excluded volume for long spherocylinders. Gravish et al. [17] and Meng et al. [16] tried to explain the variation of packing density by the excluded volumes of non-convex U particles and curved spherocylinders, respectively. Recently, Baule et al. [18] presented a theoretical framework for packing density prediction based on the concept of the Voronoi excluded volume. Many works have been done to calculate the excluded volume of a single particle, however, the excluded volumes of clusters are rarely referred to in previous works. The excluded volume of a complex shape particle is usually difficult to calculate [19]. Recently, Torquato and Jiao [20] derived an explicit formula for the excluded volume of a convex hyperparticle of arbitrary shape. As for a tetrahedron and derivative clusters, only the tetrahedron and dimer are convex bodies, and all other cluster forms are non-convex. However, it is rather difficult to give an explicit formula for the excluded volumes of non-convex clusters. Fortunately, the Monte Carlo simulation [16,17,21] provides an optional means to estimate the excluded volume. The purpose of this work is to compute and analyse the excluded volumes of clusters in tetrahedral particle packings, which are important to the understanding of the packing density and
836
L. Liu et al. / Physics Letters A 378 (2014) 835–838
packing structure. We will compare the excluded volumes in different cluster sizes as well as in the same cluster size but different cluster types to investigate the effects of the size and topology on the excluded volume. We also compare the excluded volumes of clusters with face–face joints to those of clusters with other face–face contacts. Thereafter, we try to explain the distribution of cluster types in tetrahedral particle packings using their excluded volumes. Finally, we explore how the excluded volume is related to the packing density. 2. Model and method All the clusters in tetrahedron packings can be treated as a variety of topological structures of tetrahedra. The particle model used in this work is the ideal tetrahedral model with sharp corners, and its roundness ratio [7] γ = 0. Clusters defined here are composed of tetrahedra with continuous face–face joints. A face–face joint can be regarded as a strict face–face contact, which satisfies that the two faces are parallel, contacted and coplanar [10]. The size of a regular tetrahedron is measured by its side length. In the three dimensional Euclidean space, the vertex coordinates can be obtained by using a coefficient matrix which describes the relationship between the vertexes and the centroid of the tetrahedron in a local coordinate system. Other geometric properties of the tetrahedron can be easily expressed in terms of the vertex coordinates. A contact detection algorithm is applied to detect whether two tetrahedra are contacted with each other. In our simulations, we consider two tetrahedra to be contacted if their minimum distance is less than 2.0 × 10−6 (the side length of the tetrahedron is within the spectrum from 1 to 5). First we detect whether their circumscribed spheres are overlapped. If they are not overlapped, the two tetrahedra are surely separated. Then whether the vertexes of a tetrahedron are inside or on the face of the other one is checked. When such vertex exists, the two tetrahedra are overlapped. Afterwards, we detect if one of the edges of a tetrahedron has an intersection point with one of the faces of another tetrahedron. If such edge exists, the two tetrahedra are overlapped. Otherwise, they are separated. The ideal tetrahedron model is employed to construct cluster models. Each cluster consists of a number of tetrahedra with continuous face–face joints. A tetrahedron has four faces, and each of them can respectively generate a new tetrahedron which has a common face with the original one. Therefore, the cluster model can be built by generating tetrahedra in a certain topological order on a tetrahedron. Meanwhile, by detecting whether two tetrahedra belonging to different clusters are contacted, we can judge if the two clusters are overlapped accordingly. Twenty typical cluster types in tetrahedron packings are shown in Fig. 1. We note that only clusters with face–face joints, which are very common in simulation and experimental results [10], are considered in this work, except the last one (2-a) with a face– face contact. The cluster type is unique if the cluster consists of less than four tetrahedra, namely the 1-tetrahedron, 2-dimer and 3-trimer for the cluster size of 1, 2 and 3, respectively. The three types of the clusters consist of four tetrahedra, namely the 4-cluster, are termed as 4-a, 4-b and 4-c shown in Fig. 1. There are seven topological structures for 5-cluster, while only six typical types of 6-cluster are considered in this work. These cluster types are also illustrated in Fig. 1. Additionally, a different type of 2-cluster (2-a) is analysed as well to compare the excluded volume of face–face joint clusters with that of other face–face contact clusters. For a particle with a convex or non-convex shape, the excluded volume can be calculated by the generalised expression as follows
Fig. 1. Typical cluster types in tetrahedron packings with face–face joints. The last one (2-a) is a different two-particle cluster with a face–face contact.
V ex =
p (r , ω)Θ s(r , ω) dr dω
(1)
Ω
where Ω is the whole three dimensional Euclidean space, p (r , ω) is the positional and orientational probability density function, Θ(x) is the unit step function, and s(r , ω) is the minimum distance between the central particle and the particle with the centroid position r and orientation ω . If the two particles are overlapped, the minimum distance is determined as −1. For the particles in a random packing, the probability density is uniform, that is p (r , ω) = 1. We thus obtain the following formula
V ex =
Θ s(r , ω) dr dω
(2)
Ω
In this work, excluded volumes are mostly obtained through the Monte Carlo simulation [16,17,21]. In a Monte Carlo procedure, a cubic box with a side length L is applied as a sampling space. Then we put one particle at the centre of the box with an arbitrary orientation. Another identical particle is generated in the sampling space with an arbitrary position and orientation, and whether it is contacted with the central one is detected. The procedure is carried out iteratively and N c particles are detected to be contacted with the central one. Finally, the excluded volume of the particle can be estimated as
V ex = lim
N →∞
Nc N
Vs
(3)
where V s = L 3 is the volume of the sampling space, N is the total number of tested particles. In fact, Eq. (3) is an equivalent expression of Eq. (2) using a computer simulation scheme. We also introduce the concept of the relative excluded volume which is defined as
V rex =
V ex V
(4)
where V is the volume of the particle. Therefore, the relative excluded volume V rex is dimensionless, and is a constant for a given shaped particle. For example, the V rex is 8 for a sphere. After a number of attempts, we find that the influence of the dimension of the sampling space on the excluded volume can be eliminated when L 50 while the side length of the tetrahedron is within the spectrum from 1 to 5. The effect of the number of tests N on the excluded volume is also explored. The excluded volume changes little when N reaches 109 . The excluded volume of each cluster is computed four times and averaged to reduce errors. With L = 50 and N = 109 , the relative error of the excluded
L. Liu et al. / Physics Letters A 378 (2014) 835–838
837
volume of both single tetrahedron and tetrahedral clusters can be limited to 0.7%. 3. Results and discussion The excluded volumes of 1-tetrahedron and 2-dimer are obtained both through simulation and mathematical deduction to verify the model and method. Torquato and Jiao [20] derived an explicit formula for the excluded volume of a convex particle of arbitrary shape in terms of its d-dimensional volume V , surface area S and radius of mean curvature R. The excluded volume of a three-dimensional convex body is given by
V ex = 2V + 2S R
(5)
So the relative excluded volume can be described as
V ex
V rex =
V
=2+2
SR
(6)
V
Kihara [22] proved that Eq. (5) is equivalent to Eq. (2) when the particle shape is convex. As mentioned above, Eq. (2) is equivalent to Eq. (3). Therefore, Eq. (5) is equivalent to Eq. (3) under the convex condition. The relative excluded volume of a tetrahedron was given directly [19] as tetrahedron V rex
=2+
18
π
3 2
cos−1 −
1
Fig. 2. The relative excluded volumes V rex of 4-clusters.
≈ 15.407
3
(7)
As for a dimer with the side length a, the volume V , surface area S and radius of mean curvature R can be respectively written as
√ V = S= R=
2
6
a3
(8)
√
3 3 2 3a 2π
a2
(9)
cos−1 −
1 3
3
− a
(10)
8
Fig. 3. The relative excluded volumes V rex of 5-clusters.
Consequently, the relative excluded volume of the dimer is
√
dimer V rex =2+9 6
3 2π
cos−1 −
1 3
−
3 8
≈ 13.844
(11)
Meanwhile, the excluded volumes of 1-tetrahedron and 2-dimer obtained through the Monte Carlo simulation are 15.413 ± 0.107 and 13.851 ± 0.092, respectively. The simulation results agree well with the analytical expressions in Eqs. (7) and (11). Therefore, the model and algorithm used in this work are applicable for estimating the excluded volume of tetrahedra and tetrahedral clusters. For comparison, the excluded volume of the 2-a type cluster obtained via simulation is 17.241 ± 0.032, which is much larger than that of a dimer, and is even larger than those of all the clusters with face–face joints considered in this work. The fact indicates that the excluded volumes of clusters with face–face joints are smaller than those of clusters with other face–face contacts in the same cluster size, thus the former occupies less space in a random packing than the latter. The face–face joint has been found as the major contact form between two tetrahedra in dense tetrahedral particle packings [3,10]. The low excluded volumes of face–face joint clusters should explain this phenomenon. Since the explicit formula for the excluded volume of a nonconvex body is rather difficult to deduce, we employ the Monte Carlo simulation [16,17,21] to estimate the value. The excluded volumes of the 4-clusters, 5-clusters and 6-clusters in various topological types are respectively illustrated in Figs. 2, 3 and 4. All the values are ranked in an incremental order. The error bars in the
Fig. 4. The relative excluded volumes V rex of 6-clusters.
figures represent the standard deviation and are averaged over four independent simulation runs. The clusters in Fig. 2 all consist of four tetrahedra, namely the 4-clusters. As shown in Fig. 2, the cluster of 4-a type has a smaller excluded volume than the other two 4-cluster types. Note that the 4-a type cluster is mostly approximate to the wagon wheel (5-a type) in 4-clusters. As Fig. 3 shows,
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Nevertheless, we note that the excluded volume is not the only factor affecting the cluster distribution in tetrahedral particle packings. The geometric compatibility of the clusters and the configurational entropy of the packing structure are also effective factors, and are expected to be investigated in future works. 4. Conclusions
Fig. 5. The relative excluded volume V rex vs. the size of clusters in the packing structure of tetrahedra.
the excluded volume of a wagon wheel (5-a type) is smaller than any other 5-cluster types. Clusters of 5-b and 5-c types also have smaller excluded volumes than those of 5-d, 5-e, 5-f and 5-g types. Meanwhile, both the 5-b and 5-c type clusters contain the 4-a type cluster which has the smallest excluded volume in 4-clusters. Clusters of 5-b and 5-c types are geometrically closer to a wagon wheel than other types. Only some typical topological types in 6-clusters are selected to contrast their excluded volumes which are shown in Fig. 4. The 6-a type cluster which contains a wagon wheel has the smallest excluded volume in 6-clusters. Comparing with the 6-e and 6-f types, clusters of 6-b, 6-c and 6-d types have smaller excluded volumes. Similar to the 5-b and 5-c types, the 6-b, 6-c and 6-d type clusters include a 4-a type as well, and they are geometrically closer to a wagon wheel than the clusters of 6-e and 6-f types. From Figs. 2, 3 and 4, we find that with the same cluster size, topological types which approach to the wagon-wheel geometry have a smaller excluded volume than other types. Meanwhile, a wagon wheel has the smallest relative excluded volume among all these clusters. Hence, wagon wheel is a special topological structure among clusters in tetrahedral particle packings. It was found that wagon wheel is a dominative cluster form in dense tetrahedron packings [14], and the fairly low excluded volume should be the reason. Fig. 5 illustrates the excluded volumes of different clusters in terms of cluster size. Note that the type with the minimum excluded volume is chosen in each cluster size. The values of tetrahedron (size 1) and dimer (size 2) are analytical results from Eqs. (7) and (11), respectively. The excluded volume varies nonmonotonically with the change of cluster size. Especially, dimer and wagon wheel are the local minima of the excluded volume. The excluded volume of a wagon wheel is obviously a valley on the curve shown in Fig. 5, and the value of dimer is slightly smaller than those of the 3-trimer and 4-a type clusters. These results well agree with our recent investigation that dimer and wagon wheel are the two dominative forms in the packing structure of tetrahedra [14]. We believe that the smaller excluded volume of clusters leads to a higher packing density in tetrahedral particle packing, and a dense packing structure should have dominative clusters with low excluded volumes.
We employ the ideal tetrahedron model and the Monte Carlo simulation to compute the excluded volumes of the clusters in tetrahedral particle packing. Both the influences of the topological type and cluster size on the excluded volume are investigated. Excluded volumes of clusters with face–face joints are smaller than those of clusters with other face–face contacts. This is coincident with the phenomenon that face–face joints can be found as the major contact form between two tetrahedra in dense tetrahedral particle packings. Dimer and wagon wheel are the local minima of relative excluded volume in terms of cluster size. The results correspond to the cluster distribution in tetrahedral particle packings which demonstrates that dimer and wagon wheel are the dominative cluster forms in the packing structure. They also give an explanation to the cluster distribution. Wagon wheel has the smallest relative excluded volume among all these clusters, and the clusters have smaller relative excluded volumes when their topologies approach the wagon-wheel geometry. Wagon wheel is a special and important cluster type in tetrahedral particle packings and should be further studied in the future. The findings in this work are important to the understanding of the packing density and packing structure of tetrahedral particle packings. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 11272010) and the National Basic Research Program of China (Grant No. 2010CB832701). References [1] T. Aste, D. Weaire, The Pursuit of Perfect Packing, 2nd ed., Taylor & Francis, Boca Raton, FL, 2008. [2] E.R. Chen, M. Engel, S.C. Glotzer, Discrete Comput. Geom. 44 (2010) 253. [3] A. Haji-Akbari, M. Engel, A.S. Keys, X. Zheng, R.G. Petschek, P. Palffy-Muhoray, S.C. Glotzer, Nature 462 (2009) 773. [4] Y. Jiao, S. Torquato, Phys. Rev. E 84 (2011) 041309. [5] K.C. Smith, M. Alam, T.S. Fisher, Phys. Rev. E 82 (2010) 051304. [6] K.C. Smith, T.S. Fisher, M. Alam, Phys. Rev. E 84 (2011) 030301. [7] J. Zhao, S. Li, W. Jin, X. Zhou, Phys. Rev. E 86 (2012) 031307. [8] J.P. Latham, Y. Lu, A. Munjiza, Geotechnique 51 (2001) 871. [9] S. Li, J. Zhao, X. Zhou, Chin. Phys. Lett. 25 (2008) 1724. [10] S. Li, P. Lu, W. Jin, L. Meng, Soft Matter 9 (2013) 9298. [11] J. Baker, A. Kudrolli, Phys. Rev. E 82 (2010) 061304. [12] A. Jaoshvili, A. Esakia, M. Porrati, P.M. Chaikin, Phys. Rev. Lett. 104 (2010) 185501. [13] M. Neudecker, S. Ulrich, S. Herminghaus, M. Schröter, Phys. Rev. Lett. 111 (2013) 028001. [14] W. Jin, P. Lu, S. Li, unpublished results, 2014. [15] A.P. Philipse, Langmuir 12 (1996) 1127. [16] L. Meng, S. Li, P. Lu, W. Jin, Phys. Rev. E 86 (2012) 061309. [17] N. Gravish, S.V. Franklin, D.L. Hu, D.I. Goldman, Phys. Rev. Lett. 108 (2012) 208001. [18] A. Baule, R. Mari, L. Bo, L. Portal, H.A. Makse, Nat. Commun. 4 (2013) 2194. [19] L. Onsager, Ann. N.Y. Acad. Sci. 51 (1949) 627. [20] S. Torquato, Y. Jiao, Phys. Rev. E 87 (2013) 022111. [21] N. Ibarra-Avalos, A. Gil-Villegas, A.M. Richa, Mol. Simul. 33 (2007) 505. [22] T. Kihara, Rev. Mod. Phys. 25 (1953) 831.