Coordination number of binary mixtures of spheres

J. Phys. D: Appl. Phys. 31 (1998) 47 462. Printed in the UK PII: S22-3727(98)8726-3 Coordination number of binary mixtures of spheres D Pinson, RPZou,ABYu, P Zulli and M J McCarthy School of Materials Science and Engineering, The University of New South Wales, Sydney 22, Australia BHP Research - Port Kembla Laboratories, PO Box 22, Port Kembla, NSW 2, Australia BHP Research - Newcastle Laboratories, PO Box 188, Wallsend, NSW 2287, Australia Received 2 September 1997 Abstract. This paper presents an experimental study of the coordination number of binary packings by the use of the liquid bridge technique and provides detailed information about the distributed coordination numbers corresponding to different types of contacts between small and large components and their dependence on particle size distribution. The results indicate that increasing the volume fraction of small component increases the small-to-small and large-to-small contacts and decreases the small-to-large and large-to-large contacts; and this trend is more apparent for a packing with a large size difference. For the packings under gravity, the overall mean coordination number is essentially a constant and independent of particle size distribution. 1. Introduction The coordination number, or number of contact points per particle, is an important parameter in describing the geometrical arrangement of particles in a packing, and is widely used in the evaluation of structural properties related to the connectivity between particles, including the force transmission and tensile strength [1, 2], heat transfer [3], solid solid reaction [4], and others such as phase formation []. A recent study suggests that this parameter is also related to the liquid and powder hold-ups in a packed bed of multiphase flow [6]. In the past, various efforts have been made to understand the coordination number of particle packings mainly by means of two techniques: experiment [7 12] and computer simulation [13 18]. For the packing of monosized spheres, both experimental and computersimulated results indicate: (i) the coordination number distributes within a certain range, although there is an overall mean coordination number that can be used for comparison with that of a regular packing; (ii) the mean coordination number varies with packing method; and (iii) in general, a denser packing gives a higher coordination number. Various correlations have been formulated to relate the mean coordination number to porosity as summarized by Suzuki et al [19]. In particular, for standard packings such as the so-called loose and dense random packings, useful information about the coordination number has been well established in the literature [9, 2]. Author to whom correspondence should be addressed. However, such information is not available for the packing of multisized particles. In fact, contradictory results can be found in the literature. For example, the experimental measurements of Arakawa and Nishino [] and Oda [11] suggest that the mean coordination number varies with particle size distribution while an essentially constant mean coordination number is observed in the computer simulations [1 18], resulting in an uncertainty in assessing the models proposed for predicting the coordination number [21]. Moreover, previous experimental studies do not provide complete information about coordination number. For example, a binary mixture gives three types of contacts: large-to-large, large-tosmall or small-to-large, and small-to-small [22], and each type of contact should have its own distribution; but this information is not available in the literature. While the difficulty in experimentation is recognized, it is useful to generate some concrete results that can be used to verify a simulation technique and/or predictive method that can lead to a comprehensive understanding of coordination number. For this purpose, an experimental study of the coordination number of binary mixtures has been carried out. This paper reports the major findings from that study. 2. Experimental work The liquid bridge technique, initially developed by Smith et al [7], is employed in the present work. In this technique, a packing is first prepared, then a liquid is introduced and drained, resulting in a liquid meniscus held at each contact 22-3727/98/447+6$19. c 1998 IOP Publishing Ltd 47

D Pinson et al point that can be identified as either a real or near contact. Finally the contact points of individual particles are counted for analysis. The technique has been widely used by a number of investigators, although experimental details such as method for preparing a packing and/or liquid for draining may differ [7 11]. The liquid used in this work is a dilute solution of shellac in methylated spirit, since it dries rapidly and produces a marking which is stable to handling whilst easily redissolved in methylated spirit. For each experiment, steel ball bearings of different sizes are mixed and poured into a cylindrical container (diameter = 12 mm, packing height = 13 mm) to form a packing. Then, the shellac solution is slowly introduced from the bottom of the container, flooding the packing. After draining and being allowed to dry, this produces a packing with good adhesive strength which can be disassembled one particle at a time without disturbing the remainder of the packing. When removing a particle, its contacts with other particles are marked using different colours depending on the sizes of particles. This method provides detailed information about the types of contacts which was missing in the previous studies of multisized packing [, 11]. The method also affords the ability to distinguish near from real contacts by observing the form of the shellac ring produced at each contact. True contacts form a ring with a clear centre of steel visible while near contacts form a continuous filled circle of shellac, as observed in the previous studies [7 11]. Table 1 details the particle mixtures (involving the use of three sizes of ball bearings) studied in this work. Coordination number, like porosity, may vary with the distance from the wall, because of the so-called wall effect [, 12, 23]. However, it has been reported that the coordination number is almost constant throughout the bulk packing when particles adjacent to a wall are excluded [12]. This has been confirmed in the present work as shown in figure 1. Therefore, in this work, the spheres lying within one or two particle diameters of the wall of the packing, in addition to those close to the top and bottom regions, are excluded from counting, so as to minimize the wall effect. Each experiment involves the marking of around particles, and the contact information is then derived by counting these marked particles. Since the selection of the marked particles for counting is random, it is observed that the statistical outcomes are not sensitive to the number of particles counted, provided that the number is not too small, say, for example, about for monosized packing. 3. Results and discussion The analysis will be focused on the real contact results. To verify the present experimental technique, the coordination number results are first compared with those well established for the packing of monosized spheres. Particular attention is given to the data of Bernal and Mason [9]. These authors measured the coordination number under two packing conditions, giving dense and loose random packings, with their packing densities equal to.38 and.4 respectively. As listed in table 1, the measured porosity Coodination Number 7. 6.8 6.6 6.4 6.2 6..8.6.4.2. 1 2 3 4 Distance from Wall in Sphere Diameters Figure 1. Coordination number averaged at different distances from the wall (sphere diameter = 12.7 mm; container diameter = 12 mm). under the present experimental conditions is.46. This high porosity is mainly due to the wall effect. In fact, the porosity measured by the use of a larger container-toparticle diameter ratio is.38. Note that the wall effects on porosity and coordination number differ significantly. The wall effect on porosity can be observed when the distance from the wall is as high as five sphere diameters [, 23], in contrast to one or two sphere diameters for coordination numbers (figure 1). Therefore, the present results are actually comparable with the dense packing results of Bernal and Mason [9]; and as shown in figure 2, the good agreement between them confirms the validity of the present experimental technique. For the packing of multisized mixtures, a particle can be in contact with particles of different sizes. For a binary packing, this gives several types of contacts, namely largeto-large (C ll ), large-to-small (C ls ), small-to-large (C sl ), and small-to-small (C ss ) contacts. These contacts vary with particle size distribution represented by particle sizes and volume fractions. Figures 3 and 4 show the coordination number distributions with respect to the contacts between small and other particles. The distributions are skew and strongly depend on particle size distribution. For each type of contact, there is a mean coordination which is referred to as the so-called partial mean coordination number [22]. Figure shows the variation of the partial mean coordination number with particle sizes and volume fractions. The results suggest that increasing the volume fraction of small components increases the small-to-small and large-to-small contacts and decreases the small-to-large and large-to-large contacts; and this trend is more apparent for a packing with a large size difference. For each component, there is a coordination number distribution, which is obtained disregarding the types of contacts. Figures 6 and 7 show the measured coordination 48

Table 1. Binary mixtures studied in the present work. Coordination number of binary mixtures of spheres Volume fractions (%) Overall mean Mixture 2.4 mm 12.7 mm 6.4 mm coordination number Porosity 1 6.24.46 2 28 72 6.442.388 3 72 28 6.199.379 4 28 72 6.18.34 6.428.38 6 72 28 6.27.299 1. 6.8 Cumulative Frequency Distribution.6.4.2 Dense Packing Loose Packing 4 3 2 Current Work. 2 4 6 8 Figure 2. Comparison of the present coordination number distribution with those of Bernal and Mason [9] for the packing of monosized spheres. number distributions of individual components for different volume fractions and size ratios. As expected, the distributions vary with particle size distribution, but the results indicate that this trend is much more pronounced for large components. Arakawa and Nishino showed that the coordination number distribution of a component could be approximated by the normal distribution function []. However, this is not so for the present results because, as seen in figures 6 and 7, linear relationships are not observed in the probability plot. It can be shown that even the log-normal distribution function cannot satisfactorily describe the coordination number distribution. This remark is obviously also applicable to the coordination number distribution for individual types of contacts when examining figures 3 and 4. The contact between particles is governed by two factors: statistical and geometrical. Statistically, the contacts involving a given component increase with the increase in the amount of that component; geometrically, a large particle should have a high coordination number. Obviously, the partial mean coordination numbers for individual components should be a collected outcome of the two factors. For example, as shown in figure, 1 2 3 4 3 2 2 1 (a) 2 4 6 8 (b) Figure 3. Coordination number distributions for different volume fractions of small component when small-to-large size ratio is 1:2: (a) small-to-large contacts (C sl ) and (b) small-to-small contacts (C ss ). because of the statistical effect, the large-to-large contacts decrease, while as a result of the geometrical effect the 49

D Pinson et al 4 4 3 3 2 2 1 1 2 3 4 3 (a) % Partial Mean Coordination Number 14 12 8 6 4 2 C ll C sl 2 4 6 8 Volume Fraction of Small Component (a) C ls C ss 2 % 8 7 Cls 6 2 1 2 4 6 8 (b) Figure 4. Coordination number distributions for different volume fractions of small component when small-to-large size ratio is 1:4: (a) small-to-large contacts (C sl ) and (b) small-to-small contacts (C ss ). Partial Mean Coordination Number 6 4 3 2 1 Cll Csl 2 4 6 8 Volume Fraction of Small Component (b) Figure. Partial mean coordination numbers for binary sphere packings when the small-to-large size ratio is (a) 1:2, (b) 1:4. Css 4 3 2 large-to-small contacts increase with the volume fraction of small component. Since the geometrical effect is more significant, the partial mean coordination number of large component increases with the volume fraction of small component, as shown in figure 8. A similar discussion applies to the small component but the statistical effect is more dominant. In fact, mainly based on this understanding, a few attempts have been made to develop mathematical models for predicting the partial mean coordination numbers, but none of them can predict distributed coordination numbers [22, 24, 2]. Further work is therefore necessary in order to properly describe coordination numbers and their distributions. By averaging the number of contacts irrespective of particle sizes, there is an overall mean coordination number for each packing. For the packing of monosized spheres, it has been reported that this mean coordination number varies with porosity. Figure 9 shows the literature data, suggesting that an increase in porosity corresponds to a decrease in the mean coordination number. Probably due to the difference in experimental technique, the correlation based on the measurements of Smith et al [7] and Arakawa and Nishino [] differs from that based on the measurements of Oda [11]. These authors also measured the coordination numbers of a few particle mixtures and suggested that the 46

Coordination number of binary mixtures of spheres 1. Cumulative Frequency Distribution....1 2 4 6 8 12 14 16 18 Figure 6. Coordination number distributions of individual components for different volume fractions of the small components, when the small-to-large size ratio is 1:2 (open symbols for small components and full symbols for large components). Cumulative Frequency Distribution 1.....1.1. % 2 4 6 Figure 7. Coordination number distributions of individual components for different volume fractions of the small component, when small-to-large size ratio is 1:4 (open symbols for small components and full symbols for large components). Coordination Number 1 2 4 6 8 Volume Fraction of Small Component Figure 8. Mean coordination numbers for individual components when small-to-large size ratio is 1:4: overall mean coordination number ( ), partial mean coordination number for large components ( ), and partial mean coordination number for small components ( ). Coordination Number 9 8 7 6 4 3 2 1.2.3.4..6.7 Porosity Figure 9. Overall coordination number versus porosity from Smith et al [7] ( ); Arakawa and Nishino [] (, ); Oda [11] (, ); and present work (, ). Open symbols indicate monosize spheres, full symbols multisize spheres. resulting correlation was valid for the packing of multisized spheres. This issue is examined in the present work in a more systematic way by studying directly the effects of particle size and volume fraction for binary packings. However, the results in table 1 indicate that the mean coordination number is essentially a constant, equal to 6.29 on average, and independent of particle size distribution. This is clearly seen in figure 9, when compared with the previous measurements [7,, 11]. There are two factors which should be carefully controlled in the quantification of porosity and coordination number: one is packing method and the other particle size distribution. By varying the packing method, different porosity and mean coordination numbers can result, even for monosized packings. In this way, correlation between mean coordination number and porosity can be established. However the effect of particle size distribution on these two parameters should be studied with a fixed packing method. It seems that this was not properly achieved in 461

D Pinson et al the previous work, in addition to the lack of systematic investigation. For example, in Oda s work [11], the packings of porosities greater than.4 were obtained by mixing 12.3 and 1.2 mm glass balls and then analysed, with the contribution of small balls to both porosity and coordination number completely ignored; the results, when used in correlating coordination number with porosity, are hence quite misleading. The present work should overcome these problems but provides strong experimental evidence to support the argument, obtained from computer simulations [1 18], that for the packings under gravity, the mean coordination number is not sensitive to particle size distribution [2]. It is likely that this constancy argument is valid for the mixtures of spheres with their minimum small-to-large size ratio greater than.14, the critical ratio of entrance [26]. For such packings, the so-called mixing effect should be the dominant packing mechanism, with negligible size segregation and inhomogeneity [18, 27]. 4. Conclusions The coordination number of binary packings of (small-tolarge) size ratios. and.2 has been studied in detail by the use of the liquid bridge technique. The results indicate that there are distributed coordination numbers corresponding to different types of contacts between the small and large components, and the resulting distributions cannot satisfactorily be described by a simple distribution function like the normal or log-normal distribution function. They also confirm that increasing the volume fraction of small component increases the small-to-small and large-tosmall contacts and decreases the small-to-large and large-tolarge contacts; and this trend is more apparent for a packing with a large size difference. However, for the packings under gravity, contrary to the previous experimental observations [, 11], the overall mean coordination number is essentially a constant and independent of particle size distribution. Acknowledgments The project is financially supported by the Australian Research Council (ARC), Energy Research and Development Corporation (ERDC) and BHP Research. References [1] Johnson K L, Kendall K and Roberts A D 1971 Proc. R. Soc. A 324 31 [2] Cundall P A and Strack O D L 1979 Geotechnique 29 47 [3] Wakao N and Kaguei S 1982 Heat and Mass Transfer in Packed Beds (New York: Gordon and Breach) [4] Hao Y-J and Tanaka T 199 Can. J. Chem. Eng. 68 81 [] German R M 1989 Particle Packing Characteristics (Princeton, NJ: Metal Powder Industries Federations) [6] Pinson D, Yu A B, Zulli P and McCarthy M J 1997 Powder entrapment in a multi-phase flow packed bed Processing and Handling of Powders and Dusts ed T P Battle and H Henein (Warrendale, PA: TMS Publication) p 27 [7] Smith W O, Foote P D and Busang P F 1929 Phys. Rev. 34 1271 [8] Wadsworth J 196 Mechanical Engineering Report MT-41 (Canadian National Research Council) [9] Bernal J D and Mason J 196 Nature 188 9 [] Arakawa M and Nishino M 1973 J. Soc. Mater. Sci. Japan 22 68 [11] Oda M 1977 Soils Foundations 17 29 [12] Goodling J S and Khader M S 198 Powder Technol. 44 3 [13] Tory E M, Cochrane N A and Waddell S R 1968 Nature 22 23 [14] Bennett C H 1972 J. Appl. Phys. 43 7 [1] Powell M J 198 Powder Technol. 2 4 [16] Suzuki M and Oshima T 198 Powder Technol. 44 213 [17] Jullien R, Meakin P and Pavlovitch A 1993 Growth of packings Disorder and Granular Media ed D Bideau and A Hansen (Amsterdam: Elsevier) p 3 [18] Oger L, Troadec J P, Bideau D, Dodds J A and Powell M J 1986 Powder Technol. 46 121 [19] Suzuki M, Makino K, Yamada M and Linoya K 1981 Int. Chem. Eng. 21 482 [2] Bideau D and Hansen A (eds) 1993 Disorder and Granular Media (Amsterdam: Elsevier) [21] Pinson D, Yu A B and Zulli P 1994 Calculation of the coordination number of particle mixtures: an assessment of the predictability of some proposed packing models Int. Ceramic Monographs - Proc. Int. Ceramics Conf. Austceram 94 ed C C Sorrell and A J Ruys (Sydney: Australasian Ceramic Society) p 892 [22] Dodds J A 198 J. Colloid Interface Sci. 77 317 [23] Zou R P and Yu A B 1996 Wall effect on the packing of spheres in a cylindrical container Excellence in Chemical Engineering - Proc. Chemeca 96 vol, ed G Weiss (Sydney: The Institute of Engineers) p 13 [24] Ouchiyama N and Tanaka T 198 Ind. Chem. Fundam. 19 34 [2] Suzuki M and Oshima T 1983 Powder Technol. 3 19 [26] Graton L C and Fraser H J 193 J. Geol. 43 78 [27] Yu A B and Standish N 1991 Ind. Eng. Chem. Res. 3 1372 462