Concerning Maximal Packing Arrangements of Binary Disk Mixtures

O. U. Uche^a^,¹, F. H. Stillinger^b^,² and S. Torquato ^,^,^b^,^c^,²

^a Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
^b Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
^c Department of Chemical Engineering, Princeton Materials Institute, Princeton University, Princeton, NJ 08544, USA

Available online 2 July 2004.

1. Introduction

Due to the fact that the structure of a many-particle system is often determined primarily by repulsive interactions, hard-sphere packings serve as useful models for a variety of systems. Examples include simple liquids, colloidal dispersions, fiber-reinforced composites, granular media, and glasses [1, 2, 3, 4 and 5]. Hard spheres only interact when they are in contact with each other, resulting in an infinite repulsion that reflects their impenetrable core [6]. We will deal with hard, circular disks in two dimensions for the purposes of this paper and reserve the three-dimensional extension, which differs from the present two-dimensional problem by virtue of a percolating interstitial space, for later consideration.

Polydispersity in particle size is a fundamental feature of the microstructure of a wide class of many-particle systems. In two dimensions, a monodisperse disk system crystallizes in a triangular close-packed (TCP) lattice with a packing (covering) fraction, ; in general, one is able to exceed this packing fraction by introducing some polydispersity into the system of congruent (equi-sized) disks. Intuitively, the wider the distribution of available disk sizes, the higher the packing efficiency of the system in question as the disk sizes range to the infinitesimally small. However, our studies indicate that this is not necessarily the case.

The effect of polydispersity on microstructure and the effective properties can be dramatic and thus is of great interest. A particular case is one in which particles with conducting properties are prevented from forming a connected network as a result of the relative size and composition of surrounding non-conducting particles. Evidently, the resulting arrangement will have a substantial effect on the binary system properties as compared to the properties of the individual monodisperse systems [6]. Another case in point involves the dissolution of a crystal comprised of polydisperse disks. The large disks will restrict the solubility of the crystal in the solvent [7].

Real-world applications of maximal, polydisperse disk structures include optimal packing of cables in a conduit (viewed in cross section). In addition, maximal ordered arrangements have been studied in the investigation of the phase diagram of two-dimensional binary mixtures of hard disks [8]. That investigation is accomplished by minimizing the area per particle for each given alloy. Alternatively, a particular area of interest for engineers concerns fluid flow in packed bed reactors. An extension of our research to three dimensions should provide information on the optimal packing of spherical catalyst pellets, which can aid in studying the nature of flow patterns and reaction through the reactor.

The understanding of hard-particle packings in confined regions of space [9] is an important subject of applied interest. In particular, the subject of capillary condensation which involves the liquid–gas transition of a fluid in a restricted cavity has been explored for parallel walls [10] and slit-like pores [11]. Freezing in hard-particle systems confined to circular cavities [12] and parallel plates [13] has also been studied. The confining walls play an important role in modifying the thermodynamic fluctuations in the fluid, leading to observable changes in its behavior. Even though the structures formed at freezing are not maximally dense, understanding maximal packing arrangements in confined regions is fundamental to comprehending freezing in such cavities. It is also of interest to note that the Apollonian packing of disks provides another special class of confined disk packings [14].

Several studies have investigated disordered, polydisperse systems in two- and three-dimensional space [15, 16 and 17]. Particularly in two dimensions, results show that for a radius ratio (of large to small) less than five, the density of a binary disk mixture is virtually independent of polydispersity and remains constant at a packing fraction, small phi, Greek ?0.84 [18]. This value for the packing fraction is significantly lower than that associated with ordered packings of comparable polydispersity. Similarly, it is well known that congruent spheres are optimally packed in a face-centered cubic or hexagonal close-packed arrangement [19 and 20]. Clearly, this reveals that a high degree of order must be imparted to a system so as to generate maximally dense structures.

In this paper, we consider binary packings of disks, i.e., two-dimensional disks of two different radii, R_S and R_L, where R_S less-than-or-equals, slant R_L. Ideally, it is desired to obtain the maximal packing fraction, small phi, Greek _max, for given values of small alpha, Greek =R_S/R_L and the mole fraction of disks of size R_S,x_S. Specifically, x_S=N_S/(N_S+N_L), where N_S and N_L are the populations of smaller and larger disks, respectively, in the lattice of choice. The packing efficiency of monodisperse hard-disk arrangements can be improved by introducing polydispersity and order. In particular, Fejes Tóth [21] has reported several two-dimensional binary structures of high packing efficiency for small alpha, Greek gt-or-equal, slanted 0.154701… (see Fig. 1 and Table 1).

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Fig. 1. Six two-dimensional binary structures for which >0.2. These structures were originally reported by Fejes Tóth [21].

Table 1. Relevant parameters for the binary structures displayed in Fig. 1

Our research was motivated by a desire to discover more dense binary disk arrangements than those presented by Fejes Tóth [21]. This amounts to finding the maximal packing arrangements of the small disks within the cavities of the TCP lattice of the large disks. Thus, the problem becomes one of determining the optimal arrangement of equi-sized disks within a tricusp: the non-convex cavity between three close-packed large disks (see Fig. 2). The binary structures are generated by making use of our particle-growth Monte Carlo algorithm. These optimal structures have packing fractions between that of the single-species TCP lattice value and the maximal allowable value of small phi, Greek _tl + _tl(1?_tl)?0.991332… . Ultimately, we seek the best estimate to the function _max( small alpha, Greek ,x_S) over the range 0< less-than-or-equals, slant 0.154701… and 0<x_S1.

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Fig. 2. Three identical disks in a triangular close-packed arrangement. The cavity within this close-packed triad of disks is termed a tricusp.

In Section 2, we describe the particle-growth Monte Carlo algorithm for generation of maximal packing arrangements. In Section 3, we tabulate and illustrate our findings. Upper bounds for the packing fraction associated with varying values of small alpha, Greek are also discussed in that section. We compare packing structures and geometric properties for the tricusp and flat-sided equilateral triangle in Section 4. In the concluding section, we discuss jamming categories for the arrangements. In addition, some remarks have been directed to the occurrence of ordered arrangements in quasicrystals and our conclusions have been summarized. A derivation of the asymptotic upper bound on the packing fraction is presented in Appendix A.

2. Numerical procedure

Monte Carlo simulations are proven methods for obtaining representative configurations of molecules in equilibrium [22]. However, we note that for binary systems, there is an inherent inefficiency associated with these methods. In particular, selection of move sizes for the particles is important. Reasonable displacements for the small particles may not be so reasonable for the large particles. As large displacements will provide higher probability for overlap of the large disks which in turn lead to rejections of these trial moves, we are constrained to use small displacements. However, the choice of small displacements to allow for rearrangements of all disks will force the system to evolve slowly [6 and 22].

Focusing on displacing only the small particles circumvents the above issue. In effect, this modified approach for the generation of geometric packing structures involves performing Monte Carlo (MC) simulations of disks within the container of choice, a tricusp. The disks are grown in size during the simulation. The algorithm for this MC simulation is as follows:

• Initial setup: The container perimeter is specified. The disks are initiated as very small particles. Position coordinates are assigned to all disks using a random sequential addition (RSA) method [6]. The disks are randomly distributed subject to the condition that: • There is no overlap between any of the particles in the system. • All particles must occupy the interior of simulation box/container.
• Proceed with MC simulation. During each MC cycle, potential displacements of two types occur: • First, attempt moving each disk (in sequence) such that its displacement is less than or equal to the specified maximum step size. The direction of displacement is entirely random. If a particular disk cannot be moved over the random displacement, skip the disk and attempt moving the next disk. The simulation is performed in three stages during which different step sizes are implemented. All three step sizes are proportional to the initial diameter of the disks, D_S,0. During the initial stage, the step size is assigned a value of 10^?2D_S,0. The step size is reduced to a tenth of its value in the previous stage for the latter stages. In other words, the intermediate stage step size is 10^?3D_S,0 and the step size in the final stage is 10^?4D_S,0. • Second, attempt growth of disks. The growth increment is specified as a fixed fraction of the present disk diameter. In our simulation, we have used a growth increment of 0.025%. To keep sizes uniform, either grow all or none of the particles during each cycle. Ensure all particles have room for growth for the specified growth increment by displacing nearest neighbors, if necessary: If two particles are sufficiently close that an incremental change in size will result in overlap, randomly displace both particles in opposite directions along the line of centers until there is enough room for growth. If a particle is sufficiently close to the container boundary that an incremental size change will result in the particle crossing the boundary, randomly displace only that particle towards the interior of the container.
• Repeat previous step until the hard-particle system approaches its jammed state. The jammed state is reached when the average gap between neighboring disks is less than 10^?5R_S where R_S is the final disk radius.

The particle-growth Monte Carlo algorithm can be extended to any simulation space of choice. In Table 2, we compare results generated by the above algorithm to reported results from Melissen [23]. In both cases, the maximum separation distance d_{N_S} of optimal packings of two-dimensional disks within a flat-sided equilateral triangle are presented. d_{N_S} is defined as the ratio of the disk diameter to the side length of the smallest equilateral triangle that contains all particle centers. The choice of measure, d_{N_S}, is consistent with previously published conventions for disk packings within a flat-sided equilateral triangle. The reader should note that the bounding triangle we use in our MC routine contains the entirety of the disks, not just their centers as in the Melissen calculations. Conversion between these conventions is not a trivial matter and can amplify discrepancy between pairs of entries in Table 2. However, it is obvious that results generated from our algorithm have good agreement with the published data. Small differences in d_{N_S} between both sets of data can be reduced by executing our simulation at even lower growth rates and with a tighter jamming criterion than specified above.

Table 2. Comparison of maximum separation distances for optimal packings of disks within a flat-sided equilateral triangle

N_S is the number of disks arranged in the equilateral triangle, d_{N_S} is the maximum separation distance of packings obtained by the particle-growth MC algorithm, and d_{N_S}^* is the maximum separation distance reported by Melissen [23].

3. Results

3.1. Maximal packings in the tricusp

The basic rationale in constructing the maximal structures lies in the knowledge that these packings can be formed within an original arrangement of the equi-sized larger disks in the TCP lattice. Increasing numbers of smaller disks are then inserted in optimal arrangements within the tricusps formed by the larger disks. We make use of the algorithm presented in Section 2 for the generation of these maximal structures. We perform Monte Carlo simulations of N_S small disks within the tricusp of large disks. It should be noted that the effective value of N_L is 1/2 when considering the N_S small disks arranged in a single tricusp because each of the three large disks is part of six tricusps in the TCP lattice. We present the first 19 members of an infinite sequence, with larger and larger numbers of smaller and smaller disks filling the interstices of the triangular large-disk lattice. Structures generated via this method are displayed in Fig. 3, Fig. 4, Fig. 5 and Fig. 6. Relevant parameters for these structures are reported in Table 3.

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Fig. 3. Structures for N_S=1–6 of an infinite series of packing arrangements in which small disks are optimally located within the cavity of three large disks arranged in the TCP lattice. These maximal structures were generated via the particle-growth MC algorithm.

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Fig. 4. As in Fig. 3, except N_S=7–12.

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Fig. 5. As in Fig. 3, except N_S=13–17.

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Fig. 6. As in Fig. 3, except N_S=18 and 19.

Table 3. Relevant geometric parameters for the structures displayed in Fig. 3, Fig. 4, Fig. 5 and Fig. 6

Reported data includes number of small disks within the tricusp N_S, radius ratio , mole fraction of small disks x_S=2N_S/(2N_S+1), and the unit cell packing fraction .

The packing fraction for this infinite sequence approaches the limiting value [6]

It should be noted that the packing fraction does not monotonically increase with decreasing radius ratio (increasing N_S) for this infinite sequence. In fact, local maxima in the packing fraction exist at N_S=1,3,6,10,18,… for structures of this type. Initially, this observation may seem counter-intuitive, but results suggest that the local maxima are associated with a relatively high degree of symmetry (see Table 4).

Table 4. Classification of the symmetry of jammed disk subsets displayed in Fig. 3, Fig. 4, Fig. 5 and Fig. 6

Note that in assigning these symmetries, rattler disks have been placed at their average positions over their respective displacement domains.

Several of the arrangements, e.g. those corresponding to N_S=2,5,6, and 7 can have varying orientation. This can introduce a lack of periodicity in an extended binary crystal. Specifically, the five small disks in the bottom center panel of Fig. 3 can be oriented in three different ways within the large-disk tricusp. In view of the fact that the small disks can be independently arranged in each of the tricusps, these structures may be viewed as local arrangements in a degenerate family of packings that possess local orientational disorder. Also, in two-dimensional particle packings, disks that are not necessarily in contact with three or more other disks are referred to as rattlers. In particular, the number of rattlers that occur in the maximally dense packings for N_S=7,10,15, and 17 are 1,3,3, and 1, respectively. Table 4 shows a classification of the jammed disk subsets displayed in Fig. 3, Fig. 4, Fig. 5 and Fig. 6 in various symmetry categories.

The effect of the choice of step size on the numerical accuracy of the packing fraction of the maximally dense structures should not be underestimated. In Table 5, we report the percent error in the packing fraction for different values of the step size for the maximally dense structure of N_S=6. For this case, an exact value of the packing fraction ( small phi, Greek =0.964607…) can be obtained by algebraic techniques. Clearly, the sequentially reduced step size discussed in Section 2 yields the most accurate packing fraction.

Table 5. The effect of step size on the packing fraction for the maximally dense structure of N_S=6

Fifty or more realizations were generated for each value of N_S using random initial conditions. Table 6 presents the detection frequency of the maximally dense structures for N_S=8,9,…,19. The decreasing frequency of occurrence of the maximal structures as N_S increases is apparent. There is a range in small phi, Greek over all jammed structures for a fixed N_S. For the most part, we find that the range in is reduced as N_S is increased. For example, the least dense jammed configuration for N_S=6 yields a packing fraction that is 98.8% of its maximal packing fraction. In contrast, the least dense jammed packing has a packing fraction that is 99.4% of the maximal value for N_S=18.

Table 6. Detection frequency of the maximal structures displayed in Fig. 4, Fig. 5 and Fig. 6

3.2. Upper bound on the packing fraction

In binary systems, numerical computations suggest that the packing fraction is bounded above by the case of one large disk and two small disks, mutually touching one another as shown in Fig. 7 [21]. This simple bound s( small alpha, Greek ) can be expressed in the form:

The above two equations are based on the assumption that the small disks are arranged in the TCP lattice within the tricusp. Details concerning the derivation of (3) and (4) are provided in Appendix A.

Fig. 8 compares s( small alpha, Greek ) to the packing fractions for the structures depicted in Fig. 3, Fig. 4, Fig. 5 and Fig. 6. An important point that can immediately be observed from the plot is that the packing fractions of these maximal structures do not coincide with the upper bound. This suggests the possibility that polydisperse systems (three or more different particle sizes) are likely to approach the upper bound more closely. As N_S? infinity ( small alpha, Greek ?0), it should be noted that Eq. (3) becomes a more accurate estimate for the upper bound on small phi, Greek than the upper bound derived by Florian [24]. Table 7 shows a comparison of the packing fraction for maximal structures for N_S=1,3,6,10,15 and the corresponding upper bound from Eq. (3). We can expect packing structures generated within the tricusp to approach the limiting packing fraction ( small phi, Greek _limit=0.991332…) for N_S? infinity slowly, as indicated by the reciprocal square root term in Eq. (4).

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Fig. 8. Relationship between packing fraction and radius ratio for periodic lattices built from structures in Fig. 3, Fig. 4, Fig. 5 and Fig. 6. Eq. (2) is plotted as a function of for the upper bound curve.

Table 7. Comparison of packing fraction for maximal structures generated within a curve-sided tricusp for N_S=1,3,6,10,15

4. Comparison to packings in the flat-sided equilateral triangle

In this section, we compare the packing structures and geometric properties of disks optimally packed entirely within two similar containers: the flat-sided equilateral triangle and the tricusp of large disks. The tricusp can be viewed as an equilateral triangle with curved sides.

In some cases, similar optimal packings appear for both the flat-sided equilateral triangle and the tricusp with small positional differences imposed by the curved boundaries in the latter case. For an example, see the six-disk arrangements displayed in Fig. 9. Note that upon going from the flat-sided triangle to the tricusp, the symmetry of the arrangement has been reduced. However, for other cases, the optimal structures differ markedly from each other. The five-disk arrangements shown in Fig. 10 display this behavior. Essentially, the curved sides of the tricusp (in the right panel) force the relocation of the shaded disk and convert it to a rattler.

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Fig. 9. Maximal packings of six disks in two different geometries. Left panel: equilateral triangle with flat sides. Right panel: equilateral triangle with curved sides.

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Fig. 10. Maximal packings of five disks in two different geometries. Left panel: equilateral triangle with flat sides. Right panel: equilateral triangle with curved sides. Note the large displacement of the shaded disk.

In addition, we compare packing fractions of maximal structures for both simulation regions. The packing fraction small zeta, Greek is the area occupied by the particles relative to the total area of the encompassing equilateral triangle or tricusp, respectively. Table 8 displays the relevant data. _flat is the packing fraction for the packing arrangements associated with the flat-sided equilateral triangle and small zeta, Greek _curved is the packing fraction for those structures within the tricusp. An inspection of Table 8 reveals that _flat is consistently higher than _curved. One would expect such behavior as the equilateral triangle possesses some unique characteristics. In particular, the flat sides of the equilateral triangle mimic the straight edges of disks in a tiered TCP arrangement. Also, the vertex angle of the equilateral triangle is identical to the angle between disks in a triangular close-packed arrangement. The curved edges of the tricusp do not possess the above properties, and hence lead to less efficient packings.

Table 8. Comparison of packing fractions for optimal structures generated within an equilateral triangle and within a curve-sided tricusp

A point of note is that the flat-sided equilateral triangle shows the same kind of non-monotonic increase in area fraction small zeta, Greek that we observed for the curved-sided structures in Section 3.1. However, the local maxima generally occur at different values of N_S. For the flat-sided equilateral triangle, they occur at N_S=1,3,6,10,12,15…, while they exist at N_S=1,3,6,10,18… for the curve-sided tricusp. The disparity shows the effect of dissimilar maximal structures induced at high N_S by the curved sides of the tricusp. We expect both cases involve infinite sequences of maxima, with mostly unequal N_S values.

5. Discussion and conclusions

It is of interest to classify the optimal binary packings that we have found into their jamming categories. An individual particle is locally jammed if it has at least d+1 contacting neighbors not in the same semicircle or hemisphere, where d is the system dimension [6 and 25]. Clearly, this requirement is satisfied for the majority of configurations displayed in Fig. 3, Fig. 4, Fig. 5 and Fig. 6. Packings can be classified according to the following jamming categories: local, collective, and strict jamming [25 and 26]. A locally jammed packing is defined to be one in which all particles are locally jammed. A collectively jammed packing is a locally jammed packing in which no subset of particles can simultaneously be displaced so that the system unjams. A strictly jammed packing is collectively jammed and remains fixed despite attempted globally uniform area (volume)-maintaining deformations of the system boundary.

By virtue of the TCP lattice being a strictly jammed packing [25], all structures (TCP lattice of large disks and the small disk subset) in Fig. 3, Fig. 4, Fig. 5 and Fig. 6 are strictly jammed. For the subset of small disks within each tricusp, packings without rattlers are invariably locally jammed and are often collectively and strictly jammed. It should be noted that the likelihood of the incidence (though not necessarily the fraction) of rattlers may increase as small alpha, Greek is reduced. The curved edges of the tricusp, as opposed to the straight edges of a triangle, contribute to the asymmetric structures. The presence of these isolated rattlers does not affect the jammed network as they can be removed without affecting the remainder of the network.

It is possible to create a non-periodic structure using disks of the same radius ratio as in the top right panel of Fig. 1 arranged in side-sharing modular units that are the filled squares, supplemented with close-packed triangles of large disks. Fig. 11 and Fig. 12 show the structure and its random tiling of squares and equilateral triangles. Such non-periodic structures are termed quasicrystals [27]; they possess long-range bond orientational order but lack spatial periodicity. Unlike crystals, quasicrystals do not have a simple unit cell that repeats infinitely in all directions but they do have a finite number of local patterns that repeat irregularly. Multi-component systems of metals form quasicrystals, and examples include alloys of Al,Mn,Fe, and Cr [28]. Many well-known two-dimensional quasicrystals resemble Penrose tilings, which use two different rhombi as basic building blocks to cover an infinite plane in complex, interlocking patterns [29 and 30]. Although we have no proof, it seems unlikely that other binary quasicrystals composed of internally jammed squares and triangles could exist for .

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Fig. 11. Depiction of a quasicrystal. The radius ratio for the quasicrystal is .

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Fig. 12. Random square tiling composed of squares and triangles. The tiling is the underlying framework of the structure in Fig. 11.

In conclusion, we have presented a particle-growth MC method for the generation of geometric disk packings in any container of choice. Our algorithm has been tested for packings within the flat-sided equilateral triangle and has yielded excellent agreement with published results [23]. In addition, the particle-growth MC algorithm has been used to generate the first 19 members of an infinite sequence of disk packings within the tricusp. We have compared the geometric properties of hard disks packed within the above two analogous geometric boundaries.

An important finding is the non-monotonic increase in the area fraction of optimal disks arranged within the interior of a flat-sided equilateral triangle and the tricusp of large disks. For the flat-sided triangle, the local maxima occur at N_S=1,3,6,10,12,15…, whereas they appear at N_S=1,3,6,10,18… for the curved-sided tricusp. For the latter case, we conjecture that this observation is due to the greater success of the appropriate number of disks to arrange themselves with a relatively high degree of symmetry within the curved walls. In addition, we derive an asymptotic upper bound on the packing fraction. We observe a slow approach to the upper bound suggesting that one would have to go to very high values of N_S to approach closely the limiting value of the packing fraction for structures formed within the tricusp.

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