A precise packing sequence for self-assembled convex structures

Abstract

Molecular simulations of the self-assembly of cone-shaped particles with specific, attractive interactions are performed. Upon cooling from random initial conditions, we find that the cones self-assemble into clusters and that clusters comprised of particular numbers of cones (e.g., 4 -17, 20, 27, 32, and 42) have a unique and precisely packed structure that is robust over a range of cone angles. These precise clusters form a sequence of structures at specific cluster sizes (a ''precise packing sequence'') that for small sizes is identical to that observed in evaporation-driven assembly of colloidal spheres. We further show that this sequence is reproduced and extended in simulations of two simple models of spheres self-assembling from random initial conditions subject to convexity constraints, including an initial spherical convexity constraint for moderate-and large-sized clusters. This sequence contains six of the most common virus capsid structures obtained in vivo, including large chiral clusters and a cluster that may correspond to several nonicosahedral, spherical virus capsids obtained in vivo. Our findings suggest that this precise packing sequence results from free energy minimization subject to convexity constraints and is applicable to a broad range of assembly processes.colloids ͉ self-assembly ͉ simulation ͉ virus assembly T he recent fabrication by means of evaporation-driven selfassembly of colloidal clusters containing micrometer-sized plastic spheres arranged in perfect polyhedra (1) has generated much excitement in the materials community. Precise structures such as these are rare in materials self-assembly problems (2, 3), with some notable, recent exceptions (e.g., refs. 4-6). In biology, however, precise structures are commonplace. A classic example is that of viruses, in which protein subunits, or small groups of protein subunits called capsomers, self-organize into precisely structured shells with icosahedral and other symmetries (7). Although obtained via wholly different processes, the precise packing of colloidal clusters [often referred to as colloidal ''molecules'' (8)] and virus capsids motivates us to investigate, in this article, the minimum set of organizing principles required for the self-assembly of precise structures such as these.One interesting building block shape that self-organizes into precise structures is the cone. Certain amphiphilic nanoparticles (9), molecules (4, 10, 11), and some virus capsomers (12, 13) that self-assemble into precise structures can, to first approximation, be modeled as cone-shaped particles associating via weak attractive interactions. What are the rules that govern the selfassembly of finite numbers of cones into precise clusters? Tsonchev et al. (14) presented a geometric packing analysis to explain the spherical shape resulting from the self-assembly of N cone-shaped amphiphilic nanoparticles in the limit of very large N, in which local hexagonal packing of hard cones is assumed. For clusters of smaller numbers of particles, however, the f...

Surface roughness directed self-assembly of patchy particles into colloidal micelles

“…This flexibility in the cluster shape is due to the relatively small cone angle of the particles, which is ≈17°. For larger cone angles, simulations on cone-shaped particles found clusters with precise convex structures for n ≤ 17 (10,11). Additionally, in analogy with experiments on depletion driven clusters of spheres (32), for n > 6 the smooth sides within the core are found at various iso-energetic configurations in MC simulations and Free Energy calculations.…”

“…The size and geometry of the patches together with the shape of the interparticle potential are expected to determine the formed structures and phases, which may range from empty liquids (4) and crystals (5-7) to finite-sized clusters (1,2,(8)(9)(10)(11), and lead to novel collective behavior (12).…”

Clusters of polyhedra in spherical confinement

“…We note that in a spherical container, in contrast to the situation in infinite space described by Ulam's conjecture, spheres are not the worst packers of all convex bodies at small N. We additionally find that certain values of N, so-called magic numbers (78)(79)(80)(81), correspond to especially high cluster densities of a given particle shape. These magic numbers, however, do not correspond to any particular cluster symmetry, indicating that especially dense clusters exist with a variety of symmetries and structures.…”

“…4; we term them "magic numbers" in deference to the wealth of literature exploring magic numbers in other cluster systems. Typically, magic numbers in these systems correspond to clusters of minimal energy (78,79,81,(99)(100)(101). We deem a cluster at N to be a magic-number cluster if its density ϕ N circ meets three criteria:…”

Self-assembly of amphiphilic peanut-shaped nanoparticles

“…(We note that colloidal silica dumbbells can be synthesized by other routes [19], and that micrometer-scale peanuts show potential as Pickering stabilizers of * swhitelam@lbl.gov arXiv:0907.3227v2 [cond-mat.soft] 19 Feb 2010 2 oil-in-water emulsions [20].) In an attempt to answer our posed question we have constructed a model of interacting peanuts whose minimal character is motivated by the insight into self-assembly afforded by similarly simple model systems [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Our model can be evolved with computational efficiency sufficient to allow observation of collective, thermally-driven dynamics on timescales of seconds.…”