ARTICLE pubs.acs.org/JPCC
New High-Density Packings of Similarly Sized Binary Spheres Patrick I. O’Toole and Toby S. Hudson* School of Chemistry, Building F11, The University of Sydney, NSW 2006, Australia
bS Supporting Information ABSTRACT: Binary sphere packings with a density greater than that of phase-separated face-centered cubic (fcc) packings are important structural candidates for the self-assembly of mixtures of stabilized, hard, spherical particles. Details are presented of an A3B binary hard sphere packing obtained by simulated annealing in an isopointal set 62cd/c. The structure has a higher density than all known others when the sphere radius ratio rA/rB is between 0.619 and 0.659786. This extends the known range over which compound binary sphere packings can exceed the density of phase-separated fcc packing. Previously, no packings were known to exceed this mark between a radius ratio of 0.62 and 1.0. The structure of interest has a peak density of 0.747857, which occurs at a radius ratio of 0.647989. A related dense packing is also discussed, which has a peak density of 0.7573 at a size ratio of 0.5147. This latter structure is the densest known noninterstitial compound packing when the mixture has three small spheres for every large sphere, at any size ratio. Both structures are likely candidates for stability in binary hard sphere phase diagrams at this composition, but neither has so far been observed experimentally.
’ INTRODUCTION Packing objects densely in space is a widespread constraint and ordering principle observed throughout the natural sciences, but a sufficiently difficult problem that has occupied thinkers since at least the time of Aristotle.1 Even the simplest of threedimensional objects, the sphere, required four centuries between Kepler’s identification of the close-packed structures and the proof that they were optimal.2 When a second size of sphere is introduced to the system, we have an easily enunciated, but rich, geometrical problem, because, in principle, it can have a different solution at every different size ratio. Moreover, each of these solutions is of potential relevance to the physical chemistry community, because many self-assembly problems involve particles with a distribution of sizes. Binary hard sphere packing has received longstanding and increasing attention because of the relationship between packing and crystallization, at the atomic,3 nanoparticle,4 and colloidal crystal5 size scales. A rich variety of self-assembled binary sphere structures have been produced in solvent-evaporation experiments. These include structures analogous to the atomic crystals NaCl, CuAu, NiAs, AlB2, MgZn2, MgNi2, Cu3Au, Fe4C, CaCu5, CaB6, and NaZn13,6 most of which are well-known for achieving reasonably high packing densities at the relevant size ratios.7 The counterintuitive rationale for this, the space-filling principle, is that, as the available volume diminishes, entropy is maximized by ordering into the crystal structure able to form with the greatest density. Although configurational entropy is sacrificed during this process, it is the best way to maintain the maximal free volume entropy. At any particular high solution concentration, the space around the particle sites is at its greatest when they are arranged in the structure that could be compressed the most.8,9 Some of r 2011 American Chemical Society
these structures have a suboptimal density, which suggests that other interparticle interactions are involved in their stability. However, all are subject to packing constraints, and thus any theoretical approach to structure prediction must start by identifying dense candidate structures. At many size ratios, there is no clear conjecture as to which structures are densest, but recent results have improved the bounds. At extreme ratios between the sphere sizes, the densest packings known involve simply filling the interstices of a closepacked array of large spheres as densely as possible with smaller ones.10 This is possible until a radius ratio of 0.4142, at which point the binary packing of a single small sphere inside the octahedral sites of the face-centered cubic (fcc) lattice of large spheres corresponds to a NaCl lattice. When the spheres are more similar-sized than this, the structures must rearrange into a variety of crystal types in order to maximize their density. Structures theoretically thought to be important to the hard sphere binary phase diagram include NaCl,9,11AlB2,12 NaZn13,13,14 MgZn2,15,16 and some computationally generated structures isopointal to the chemical structures FeB (set 62c/c),17,18 and Ag2Se (set 62c2/c).19,20 One or other of these structures are denser than phase-separated fcc packing at all radius ratios up to 0.62.18 When spherical particle sizes are more similar than this, prior to this work, no binary structure was known with a density above that of phase-separated fcc structures. It is important to know where this limit is, because, at that point, the expected binary phase diagram changes considerably. Beyond this point, as Received: June 28, 2011 Revised: August 10, 2011 Published: August 12, 2011 19037
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Figure 2. Crystal fragment of structure 1 showing the optimal packing at rA/rB = 0.5147. Figure 1. Optimal known packing fraction for a range of stoichiometries as a function of the sphere size ratio rA/rB. Heavy black curve: A3B structures found during simulated annealing of the 62cd/c and 82g3/g isopointal sets. Dashed green curve: AB structures.17 Solid red curve: A2B structures.20 Thin indigo curve: NaZn13 structures.14 Dotted black line: phase-separated fcc packing. Circles: the maximum A3B packings found by Filion and Dijkstra using a genetic algorithm.19 Structures marked 1 and 2 are discussed in detail in this paper.
the size ratio increases, the phase diagram is thought to evolve through azeotropic, to eutectic, to spindle-type behavior, with the two pure fcc phases as the stable phases.21 Because large-scale diffusive demixing is required for crystallization into two pure phases, systems with similar-sized spheres can be exploited to study glassy dynamics.22 For simulations of such systems, binary systems (including those with soft potentials) are often specifically chosen that have radius ratios above 0.62, in order to frustrate crystallization.23 25 This size range is also of most interest to those studying metallic alloys, because the periodic table provides only a limited range of neutral metallic radii, and more often than not, their ratios fall within the range of 0.62 1.0.26 In this paper, we describe two dense A3B compound sphere packings that have a peak density above that of separated fcc crystals. The first has a peak density of 0.7573 at a size ratio of 0.5147. The second has a radius ratio of 0.647989, and is denser than separated fcc crystals until a radius ratio of 0.659786. This latter structure thus extends the range over which compound packings are optimal.
’ SIMULATIONS Monte Carlo simulated annealing was performed in the manner described in detail in our previous work.17,20 The starting point for this search method is to choose an isopointal set, a space group together with a list of occupied Wyckoff positions. This choice limits the structural search to the configuration space defined by a small number of parameters that define individual structures within the set. This concept is similar to, but somewhat broader than, the crystallographic concept of structure types. Multiple chemically distinct structure types are often within the same isopointal set, and a computational search of the isopointal set (a simulated annealing Monte Carlo method) can, in principle, access those, and other structure types that are not seen in natural crystals. It should be noted that, although the isopointal set is used as a generator, the optimal solutions may show additional symmetry if
the variable parameters evolve to particular special values (e.g., an orthorhombic search may result in a tetragonal or cubic structure if the adjustable lengths of cell edges converge). Our previous studies included surveys of the homogeneous isopointal sets (those with only one occupied Wyckoff site for each species) representing the natural AB17 and A2B20 inorganic crystal structures. As the stoichiometric ratio increases, the number of possibilities expands, and we see some nonhomogeneous structures with high densities. In principle, it is possible to systematically enumerate all isopointal sets. However, until that is done, we make do with studying sets related to inorganic crystal structures with high sphere-packing densities. For the A3B stoichiometry, we have run simulations in the isopointal sets corresponding to the structures of crystals with structure types Be3Nb (166 bch/ac), Fe3C (62cd/c), Ni3P (82g3/g), Re3B (63cf/c), ReB3 (194af/c), AuCu3 (221c/a), Cr3Si (223c/a), Fe3N (182g/c), and Ni3Sn (194h/c). Of those tested, the isopointal set 62cd/c (which includes the structure types Fe3C, YF3, and ClF3) achieved the greatest density at most radius ratios, only outdone by set 82g3/g over small ranges. The optimal density was often above the fcc density, including the two notable structures reported below. Figure 1 shows a plot of the best simulated packing fraction for A3B structures accessible from these two isopointal sets, as a function of the sphere size ratio rA/rB.
’ STRUCTURES Structure 1 is shown in Figure 2 and has a peak density of 0.7573 at a size ratio of 0.5147. Details of the structure at its peak density are given in Table 1. At this size ratio, the only known denser compounds have an A2B stoichiometry (with a structure related to that of AuTe2, and a packing fraction of about 0.770).19,20 For structures with an A3B stoichiometry, the previous optimal packing at a nearby radius ratio was 0.744 at a radius ratio of 0.5 in the isopointal set 19lf/b.19 Although isopointal to Cd3Er (63cg/c), our structure is not closely related to any known inorganic crystal structures. If it can be experimentally realized, the topology of this structure may prove useful for the self-assembly of nanowire arrays (by using metallic small particles and insulating large particles). Structure 2 is shown in Figure 3 and has a packing fraction of 0.747857 at a radius ratio of 0.647989. This structure is of greater theoretical significance. It is the densest known structure at this size ratio, among compounds of all stoichiometries; this density 19038
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Table 1. Details of the Optimal A3B Packing at rA/rB = 0.5147a Wyckoff
x/a
y/b
z/c
A1
8g
0.67179
0.44353
0.25000
A2 B
4c 4c
0.00000 0.00000
0.11488 0.66645
0.25000 0.25000
a
It can be specified by space group 63 with the above occupancies and fractional positions. When the scale is fixed with rB = 1, the values of the cell edge lengths are a = 2.99612, b = 5.202857, and c = 2.0000. Each unit cell contains 16 spheres in total.
Table 2. Details of the Optimal A3B Packing at rA/rB = 0.647989a Wyckoff
x/a
y/b
z/c
A1
4f
0.209945
0
0.608886
A2 B
2b 2a
0 0
0.5 0
0.700326 0.108886
a
It can be specified by space group 59 with the above occupancies and fractional positions. When the scale is fixed with rB = 1, the values of the cell edge lengths are a = 3.08647, b = 2.17521, and c = 3.0305. Each unit cell contains eight spheres in total.
Figure 3. Crystal fragment of structure 2 showing the optimal packing at rA/rB = 0.647989.
exceeds the packing fraction of phase-separated pure fcc crystals. No other structure is known that achieves this at such a high radius ratio. Details of the structure at its peak density are given in Table 2. The structure produced by the simulation was examined analytically by requiring neighboring spheres to be in exact contact. The values reported here are the solutions to the resulting systems of simultaneous equations. Away from the peak radius ratio, slight variations in the free parameters allow a family of closely related structures to continue this dense packing over a range of radius ratios. This structural family includes the best-known binary sphere packing for the range between rA/rB = 0.619 and rA/rB = 0.659786. The upper end of this range extends the known range over which compound binary sphere packings can exceed the density of phase-separated face-centered cubic packing.
’ LATTICE-SWITCH SIMULATIONS Because structure 2 has the highest known density among all structures near its peak radius ratio, it should be thermodynamically stable relative to other known solid phases at high concentrations in hard sphere suspensions. Using lattice-switch Monte Carlo simulations,27 we have calculated free energy differences between our structure and phase-separated fcc at a range of volume fractions (Figure 4). Simulations were performed at system sizes of 216 and 864 particles, and only extremely small differences were seen. We predict that structure 2 is the preferred phase at all densities down to those achievable in colloidal experiments. ’ DISCUSSION The densest known binary sphere packings as a function of radius ratio are summarized in Table 3.
Figure 4. Helmholtz free energy difference per particle between the reference fcc and target A3B crystal (FA3B FfccA+fccB)/NkT at rA/rB = 0.647989, as a function of the relative packing fraction, as measured in the lattice-switch simulations. The final data points at low density indicate the point at which the crystals became unstable against disordering to a liquid.
Because structure 2 has the highest known density among all structures over a range of radius ratios from 0.619 to 0.659786, it (or some even denser structure) must be the thermodynamically stable phase at high concentrations in hard sphere suspensions. Indeed, our free energy calculations show that, at the ideal radius ratio, this structure is preferred to segregated fcc even for lower concentrations. Some experiments have used size ratios in this range, but none have observed the structure we propose. In 70 °C solvent evaporations of PbSe/CdSe nanoparticle systems corresponding to a radius ratio of approximately 0.65, the predominant phases were phase-separated single-component lattices, with minute fractions of CaCu5 and MgZn2 superlattice structures.28 The same study, using a Au/PbSe system at a radius ratio of ≈0.64 also found phase separation, with a small fraction of the CuAu superlattice structure. Two studies using CdTe and CdSe particles with an effective radius ratio of ≈0.62 and ≈0.63 observed three types of superlattice structures: cub-NaZn13, ico-NaZn13, and CaCu5.29,30 The occurrence of these phases with clearly inferior packing densities suggests the influence of additional interparticle interactions. Indeed, free energy calculations conclude that the densest of these candidates, CaCu5 (with a packing density of 0.700489 at a radius ratio of 0.647853), is not thermodynamically stable for hard binary sphere mixtures in this region.28 Some earlier studies used larger, harder particles made of PMMA with their surface stabilized by PHSA, but the closest size ratio of 0.62 ( 0.01 is borderline for our structure. Therefore, 19039
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densest known structure
(0, ≈0.3)
fcc lattice of B particles with multiple A particles in
(≈0.3, 0.414)
the interstitial sites10 fcc lattice of B particles with a single A particle in every octahedral interstitial site (corresponding to the NaCl structure)3
(0.414, 0.4460)
an AB structural family within the 62c/c isopointal set (with eight spheres in the unit cell), described as structure 1 in Marshall and Hudson18
(0.4460, 0.460)
an A2B structural family within the 19aa/a isopointal
(0.460, 0.4784)
set19,20 an A2B structural family within the 62cm3/c isopointal set19,20
(0.4784, 0.5274)
branch falling away from the left side of the AlB2 double peak; the best packings are distorted from that structure, into the 19aa/a isopointal set19,20
(0.5274, 0.619)
hexagonal AlB2 double peak3
(0.619, 0.659786) A3B, described in this work as structure 2 (0.659786, 1)
segregated fcc-A and fcc-B
the occurrence of only AlB2 and NaZn13 superlattice structures is inconclusive.13 It is possible that polydispersity or kinetic effects also play a role in the absence, to date, of an experimental observation of A3B. Nevertheless, we claim that our A3B structure 2 is the most thermodynamically stable phase among known structures. Thus, its absence lays out a challenge to experiment. It is less clear-cut that our A3B structure 1 will be thermodynamically favored, even around its ideal size ratio 0.5147. At this ratio is in competition with dense structures with both A2B and AB stoichiometry (see Figure 1). Consider a system where the composition was deliberately skewed so that it was A-rich. At the high concentration limit, it is more favorable to phase separate into (A2B + fcc-A) than either (A2B + A3B) or (A3B + fcc-A). At a mole fraction of XA = 0.75 (corresponding to the A3B stoichiometry), the packing fraction possible in a phase-separated (A2B + fcc-A) system is 0.767, greater than the 0.7573 obtained by our structure. Furthermore, moving away from the ideal radius ratio, the packing efficiency of our structure rapidly falls away from its peak. Relevant experimentation would require very accurate size control. To our knowledge, there have not been any studies that closely corresponded with the ideal size ratio. The closest we know of, at radius ratios of ≈0.487 and ≈0.5331,28, assembled into bcc-AB6 and AlB2 superlattice structures respectively. Nevertheless, our structure is a highdensity candidate, and judicious choice of additional interparticle interactions may stabilize this structure.
’ ASSOCIATED CONTENT
bS
Supporting Information. Supporting Information is available, which gives analytic packing curves in the vicinity of structure 2. This material is available free of charge via the Internet at http://pubs.acs.org.
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’ ACKNOWLEDGMENT The authors gratefully acknowledge valuable discussions with Peter Harrowell. T.S.H. wishes to acknowledge funding support from the Australian Research Council. ’ REFERENCES (1) Aristotle On the Heavens; Vol. III, p 8.
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’ AUTHOR INFORMATION Corresponding Author
*E-mail:
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