Enumerating Finite Sphere Packings. Abstract

Transcription

1 Enumerating Finite Sphere Packings Natalie Arkus, 1 Vinothan N. Manoharan, 1, 2 and Michael P. Brenner 1 1 School of Engineering and Applied Sciences, Harvard University, Cambridge, MA Department of Physics, Harvard University, Cambridge MA Abstract We consider packings of n identical spherical particles that bind to one another without deformation or overlap, and without any long range interactions. Combining graph theoretic enumeration with basic geometry, we analytically solve for packings of n 10 particles satisfying minimal rigidity constraints ( 3 contacts per particle and 3n 6 total contacts). We find that the onset of packings that have different numbers of contacts occurs at n = 10, and that the number of packings that can not be constructed iteratively from < n particle packings proliferates at n 9 particles. 1

2 Colloidal nanoparticles are natural building blocks for self-assembly [1, 2, 3, 4, 5, 6]; however, only a limited number of structures have been assembled from colloidal spheres. The diversity of structures that could in principle be formed by spheres is well illustrated by the construction toy Geomags, consisting of metallic balls that represent centers of spheres and uniformly-sized magnetic rods that represent contacts between the spheres. A complex structure constructed out of Geomags is shown in Fig 1a, while Figs 1b,c show a 6 particle packing constructed out of Geomags and spheres, respectively. The critical problem with using spheres as building blocks is that the number of possible sphere packings increases exponentially with the number of particles, n [7]. Thus the Big Ben structure of Fig 1a would be highly unlikely to assemble spontaneously from interacting colloidal particles unless it were stabilized over all other packings of equivalent n. Directing self-assembly can therefore be broken into two parts: (i) determining the set of all structures that can be self-assembled at a given n, and (ii) devising a method to stabilize any one structure out of the set. The goal of the present study is to tackle the first step: we enumerate a provably complete list of packings of n (colloidal) spheres satisfying minimal rigidity constraints ( 3 contacts per particle, 3n 6 total contacts) for n 9. We also provide a preliminary list of packings for n = 10. Our packings are minimal energy clusters of a infinitesimal short-ranged attractive potential, so that particles adhere upon contact, with an infinite wall on one side preventing particle overlap. We call such packings hard sphere clusters. They include, as a subset, all minimal-second-moment clusters [2, 8, 9]. We expect our results to be relevant to a broad range of physical clustering processes in colloidal and granular systems [10, 11], where, with few exceptions, the particles are hard and the range of any interactions (typically van der Waals forces and electrostatics) is much smaller than the size of the particles. Additionally, comparisons to the jamming of hard spheres [12, 13, 14, 15, 16] will also be of interest. Although minimal-energy clusters have been catalogued for many different potentials (the most studied being the Lennard-Jones potential) [17], hard sphere clusters are conspicuously absent from the catalogue of minimal energy clusters[18]. This is largely due to the nondifferentiable interaction potential, which complicates global optimization methods. We are aware of one previous study enumerating hard sphere clusters [7]; starting from two seeds (a tetrahedron, and an octahedron), n particle packings were constructed iteratively by adding one particle to an n 1 particle packing. It 2

3 was shown that the number of packings grew exponentially with n, though the list was incomplete because the generating set of seeds was incomplete. Here we present a two-step method for enumerating a provably complete set of sphere packings that satisfy minimal rigidity constraints: The first step uses graph theory to construct all possible n particle conformations (a subset of which corresponds to all minimally rigid packings). The second step uses basic geometry to determine which conformations correspond to the minimally rigid packings. Because the number of possible conformations grows exponentially with n, this analytical process must be implemented computationally. Graph Theory Produces the Set of Possible Packings A conformation of n particles can be described by an n n adjacency matrix, A (Fig 1d). A ij = 1 if the i th and j th particles touch, and A ij = 0 if they do not. Because there are n(n 1)/2 possible contacts between n particles, there are 2 n(n 1)/2 possible adjacency matrices [21]. However, most of these are isomorphic due to particle labeling degeneracy, and thus represent the same conformation. Algorithms exist for enumerating non-isomorphic adjacency matrices as a function of n[19]. The number of such matrices is much smaller but still grows exponentially with n (Table 1). Rigidity constraints further restrict the set of possible packings. Rigidity requires (i) there be at least 3 contacts per particle, and (ii) there be at least as many contacts as internal degrees of freedom - thus there must be at least 3n 6 contacts. Imposing these constraints eliminates all but 1 adjacency matrix for n 5 particles, thus identifying a doublet, triangle, tetrahedron and triangular dipyramid as the unique packings for n = 2, 3, 4, 5, respectively. Above n = 5, rigidity constraints alone are insufficient to identify those adjacency matrices that correspond to minimally rigid packings; more information is needed. Algebraic Formulation Each element, A ij, is associated with an interparticle distance, r ij = z i z j, which is the distance between particles whose centers are located at z i, z j, respectively. If A ij = 1, then r ij = 2r, where r is the sphere radius; If A ij = 0, then r ij 2r. For adjacency matrices with 3n 6 contacts, there are precisely as many equations as unknowns[22]. The particle configuration encoded by each A is specified by the distance matrix D, whose elements D ij = r ij (Fig 1e). If any r ij < 2r, the particles overlap and the structure is unphysical. If a continuous set of D corresponds to a given A, the structure is not rigid. The fundamental question is to find an efficient method for mapping A D. Geometric Solutions Numerical approaches for solving these equations can not be guaranteed 3

4 to converge to all D, and existing analytical approaches are effective at low n, but do not scale practically with n [20]. Instead, we use basic geometry to construct rules associating patterns of 1 s and 0 s in A s with either a given relative distance, D ij, or an unphysical conformation (in which case no D 2r exists). Let us associate with each sphere a neighbor sphere of radius 2r, whose surface defines where a touching particle must lie. When two spheres are in contact, their neighbor spheres intersect, and form an intersection circle of radius 3r (Fig 2a). Thus any sphere that touches these two spheres must lie on that intersection circle. Additionally, any sphere touching multiple doublets must lie at the intersection of the doublet s associated intersection circles. Because two circles can intersect in at most two points, any A with more than 2 intersections of intersection circles is unphysical. This eliminates all A s for which any 2 of the set {A jk, A jp, A kp } equal 1, and > 2 i s for which A ij = A ik = A ip = 1. All unphysical matrices are eliminated by this rule at n = 6, whereas 17/21; 298/425; 9095/13780; and / are eliminated at n = 7, 8, 9, 10, respectively. Figures 2b,c show intersection circle constructions that lead to 6 additional rules (3 that eliminate A s, and 3 that map A D). Fig 2b illustrates that the distance between 2 particles mutually touching a triplet is given by calculating the distance between the 2 intersection points of the triplet s 3 mutually intersecting intersection circles. Basic trigonometry yields this distance is (4 2/3)r. Thus, any A implying these particles touch is unphysical. Fig 2c shows that the maximum number of spheres that can mutually touch a doublet is 5, because a maximum of 5 points a distance 2r apart can lie along an intersection circle s circumference. The distance between the nearly touching particles on the ring is = (8 6/9)r 2.16r, and the distance between the 1st and 4th particles (and also the 2nd and 5th) = (10/3)r; any A implying any of these particles contact is unphysical. As n increases, new types of structures appear, and hence the number of geometrical rules that are needed to eliminate and solve for adjacency matrices increases [23]. Thus, deriving individual rules to eliminate A s does not scale effectively with n. To decrease the number of A s for which rules must be derived, we note that packings can be separated into two types: iterative packings and new seeds, distinguished as those n particle packings that can be (cannot be) constructed solely by combining known lower order packings; the most common of which is adding a single particle to a known n 1 particle packing. Put 4

5 another way, iterative packings correspond to A s that contain only other packings as subgraphs (i.e. having no new graph structure). All iterative packings can be solved using a simple geometrical principle: 2 points are fixed in 3 dimensional space if they can be related to a common triangular base. Let there exist two particles i, j whose interparticle distance, r ij, is unknown. If there also exist 3 particles, k, p, q, with known interparticle distances (r kp, r kq, r pq ), and if the distances between i, j and the 3 particle base (r ip, r ik, r iq, r jp, r jk, r jq ) are also known; then there exists an analytical relationship for the resulting r ij (see supplementary information for derivation). These 5 points, i, j, k, p, q, make up a triangular dipyramid (Fig 3) - thus we call this the triangular dipyramid rule. By applying this rule sequentially to all unknown distances in an A, we can solve for the relative distances for which r = (r ik, r ip, r iq, r jk, r jp, r jq, r kp, r kq, r pq ) is known. The set of {r ij } thus calculated must satisfy particle overlap and consistency constraints; consistency requires that the triangle inequality is satisfied for each i, j, k (r ij r ik + r kj ), and that all bases lead to the same {r ij }. The violation of any constraint implies the adjacency matrix is unphysical. By definition, new seeds contain relative distances that do not exist in lower order packings; thus at least one distance of r will be unknown. Therefore, the triangular dipyramid rule can not be directly applied, and individual rules must still be derived for new seeds. The following steps thus solve any A: (i) Identify whether any part of A corresponds to a seed. If it does, the corresponding interparticle distances are known from individual geometrical rules and substituted in. (For example, the octahedron, which is the n = 6 new seed, is identified by the existence of a closed 4 ring, A ij = A jk = A kp = A pi = 1, where the cross particles don t touch, A ik = A jp = 0, and 2 particles touch all points in the 4 ring, A mj = A mk = A mp = A mi = 1 for two m s). (ii) Once all seeds are identified within an A, the triangular dipyramid rule is applied to solve the remainder of A. (iii) If A can not be completely solved by these 2 steps, then A is non-iterative, and a new rule (of the type depicted in Fig. 2 must be derived. This is a reasonable proposition if there are only a few non-iterative A s. For n 9, this is the case (Table 1). However, at n = 10, non-iterative A s grow significantly, and this is no longer feasible. Figures 4 and 5 summarize the central result of our study. Figure 4 shows all packings and new seeds up to n = 8 (n = 9, 10 are omitted due to their large number of packings). Fig. 5 shows the growth of new seeds for n = 9, 10. The supplementary information contains analytically derived coordinates for all packings up to n = 10, as well as their corresponding A s and D s. 5

6 The results presented are provably complete except for the new seed list at n = 10. Here, the new seeds were constructed by brute force, by parsing through the 126 non-iterative A s with Geomags. For n < 10, individual geometrical rules were derived to rigorously eliminate or solve every non-iterative A (see supplementary information). For 6 n 8 there is a single new seed for each n and it is a shell. For n 9, the proliferation of new seeds increases significantly. Here, in addition to complete shells, new seeds that are combinations of shells and lower order packings arise. Additionally, non-convex shells arise. Several intriguing packings occur at n = 9, 10 that will have important implications for the thermodynamics of finite packings of colloidal particles. At n = 9, all packings have the same number of contacts, and hence the same contact energy. However, one of the new seeds at n = 9 is not rigid, presumably implying higher entropy. The constraints we applied ( 3 contacts per particle, 3n 6 total contacts) were necessary but not sufficient conditions for rigidity. This non-rigid 9 particle seed gives rise to four 10 particle packings, only one of which is rigid. There also exists a non-rigid new seed of 10 particles. Additionally, at n = 10, packings with larger than 3n 6 = 24 contacts arise. Out of the 208 packings, 206 have 24 contacts and 2 have 25 contacts. Thus while for n 9 particles, the contact energies of packings are identical per n, for n = 10 particles, two packings have lower contact energies than the rest. In conclusion, we have combined graph theory with basic geometry to solve for all packings of n 9 particles, and to produce a putative list of packings at n = 10. Having a complete list of packings provides the essential ingredient for addressing many questions. For example, using this list, we have proved that the minimal second moment clusters identified by Sloane and Conway [8] are correct for n 9 (and have preliminarily confirmed the second moment cluster of n = 10). Our method could be generalized to enumerate packings in a closed container, which allow for explicit tests of the Edwards conjecture [11] for the entropy of a granular material. Ultimately we regard this work as showing the set of possible self-assembled structures of spherical colloidal nanoparticles. In addition to finding methods for extending these results to higher n, future work will focus on determining the equilibrium probabilities for each of these packings with and without binding specificity. This will begin to shed light on how to direct the self-assembly of sphere packings. We thank John Lee for consultations and help in writing some of the computer code and 6

7 acknowledge support from the NSF Division of Mathematical Sciences, the Harvard MRSEC, and DARPA s programmable matter. [1] G. M. Whitesides and B. Grzybowski, Science 295, 2418 (2002). [2] V. N. Manoharan, M. T. Elsesser, and D. J. Pine, Science 301, 483 (2003). [3] D. Y. Wang and H. Mohwald, J. Materials Chem. 14, 459 (2004). [4] P. L. Biancaniello, A. J. Kim, and J. C. Crocker, Physical Review Letters 94 (2005), [5] A. D. Dinsmore, J. C. Crocker, and A. G. Yodh, Current Opinion in Colloid and Interface Science 3, 5 (1998). [6] D. G. Grier, Mrs Bulletin 23, 21 (1998). [7] M. Hoare and J. McInnes, Faraday Disc. Chem.Soc. 61, 12 (1976). [8] N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, Disc. Comp. Geom. 14, 237 (1995). [9] E. Lauga and M. P. Brenner, Phys. Rev. Lett. 93 (2004). [10] H. M. Jaeger, S. R. Nagel, and R. P. Behringer, Rev. Mod. Phys. 68, 1259 (1996). [11] S. F. Edwards, in Granular Matter: an Interdisciplinary Approach, edited by A. Mehta (Springer, New York, 1994), pp [12] A. R. Kansal, S. Torquato, and F. H. Stillinger, Journal of Chemical Physics 117, 8212 (2002). [13] A. R. Kansal, S. Torquato, and F. H. Stillinger, Physical Review E 66 (2002), [14] A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, Journal of Applied Physics 95, 989 (2004). [15] A. Donev, S. Torquato, and F. H. Stillinger, Physical Review E 71 (2005), [16] A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, Journal of Computational Physics 225, (2007). [17] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, Grundlehren der mathematischen Wissenschaften ; 290. (Springer, New York, 1999), 3rd ed., j.h. Conway, N.J.A. Sloane ; with additional contributions by E. Bannai... [et al.]. ill. ; 25 cm. [18] D. J. Wales, J. P. K. Doye, A. Dullweber, M. P. Hodges, F. Y. Naumkin, F. Calvo, J. Hernndez- Rojas, and T. F. Middleton (2008), URL [19] B. McKay, Congressus Numerantium 30, 45 (1981), URL bdm/nauty/. [20] B. Buchberger, Proceedings of EUROCAST p (2001). 7

8 [21] 2 n(n 1)/2 because there are 2 possibilities, either the particles touch or they don t, over the n(n 1)/2 relative distances (i.e. the n(n 1)/2 elements of A). [22] Without loss of generality, we can set one particle to reside at the origin, another to reside along a single axis (for example the y-axis), and a third to reside in one plane (such as the xy-plane) 6 coordinates are then fixed, leaving 3n 6 instead of 3n coordinates. [23] S. Weinberger (U. Chicago) has proven the following theorem: Suppose there are rules that give all rigid packings up to some n; then there exists an n > n such that this list of rules is insufficient to diagnose rigidity. 8

9 n A s Non-Isomorphic A s Minimally Iterative A s Non-Iterative A s rigid A s , , ,097,152 1, ,435,456 12, ,668 13,828 13, ,005, , , Table 1:The number of adjacency matrices (constructed by [19]) decreases rapidly as isomorphism and rigidity constraints are imposed. Iterative and non-iterative are defined in the text. 9

10 n Packings Total Packings New Seeds Non-Rigid from [7] (Current Study) Packings Table 2: Total number of packings found, compared to those of [7], who used only the tetrahedral and octahedral (n = 6) seed. Our list of packings is provably complete for n 9. 10

11 FIG. 1: (a) Big Ben, created out of Geomags. Picture taken from Palais du Challoit, Paris. (b) 6 particle polytetrahedral packing created out of Geomags. (c) Corresponding 6 particle sphere packing. (d) Corresponding 6 particle adjacency matrix, A. (e) Corresponding relative distance matrix, D. 11

12 FIG. 2: Geometrical rules for solving and eliminating packings are based on (a) intersection circles. Each particle is surrounded by its neighbor sphere (dashed), on whose surface the center of a neighboring particle must lie. Two neighbor spheres intersect in a circle of radius 3r. (b) Only two particles (red points) can touch three touching particles. The red points occur at the intersection of the intersection circles. (c) The maximum number of spheres that can mutually touch a doublet is 5, since only five touching particles can lie on an intersection circle. (Solid lines represent particles that are in contact, and a dashed line or no line signifies particles that are not in contact). 12

13 FIG. 3: Geometrical construction for finding r ij in the triangular dipyramid rule. k, p, q is the potentially irregular triangular base, and i, j are the 2 particles whose interparticle distance is unknown. 13

14 FIG. 4: Tree of packings for 4 n 8. Rows are different n s and columns show the occurrence of different seeds. A single new seed arises at each of n = 6, 7, 8. 14

15 FIG. 5: Special Packings. The top part of the figure denotes new seeds for n = 9 and n = 10. The boxed packing in each case is a nonrigid seed (despite having 3n 6 total contacts and 3 contacts per sphere). At n = 10 there are non-rigid packings that derive from the nonrigid n = 9 seed, and there are also two packings with 25 contacts. 15