Minimal-energy clusters of hard spheres

Sloane, N. J. A.; Hardin, R. H.; Duff, Timothy; Conway, John H.

doi:10.1007/bf02570704

Cited by 113 publications

(86 citation statements)

References 30 publications

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“…If the nanoparticle size is increased to N np ϭ 55 (nearly spherical, d np ϭ 2.1R g ) and N np ϭ 75 (slightly oblong largest available Lennard-Jones cluster, 21 d np ϭ Our results show that functionalized copolymers can be successfully used to assemble ordered structures of nanoparticles. Although the order will be mainly dictated by the polymer subsystem, the nanoparticles are not silent observers but direct the process of structure formation, either by stabilizing the polymer order (in the case of the gyroid phase) or by significantly rearranging micelles (in the case of square columnar phase).…”

Section: Resultsmentioning

confidence: 72%

Nanoparticle Ordering via Functionalized Block Copolymers in Solution

Sknepnek¹,

Anderson²,

Lamm

et al. 2008

ACS Nano

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We consider nanoparticles and functionalized copolymers, block copolymers with attached end groups possessing a specific affinity for nanoparticles, in solution. Using molecular dynamics, we show that nanoparticles are able to direct the self-assembly of the polymer/ nanoparticle composite. We perform a detailed study for a wide range of nanoparticle sizes and concentrations. We show that the nanoparticles order in a number of distinct phases: simple cubic, layered hexagonal, hexagonal columnar, gyroid, and a novel square columnar. Our results show that nanoparticles ordered with functionalized block copolymers can provide a simple and efficient tool for assembling novel materials with nanometer scale resolution. B lock copolymers in solutions or melts are known to self-assemble into mesophases with one-, two-, or threedimensional periodic order with typical repeating distances in the 10Ϫ200 nm range. 1,2 In solutions, the self-assembly of the different phases can be controlled by tuning external conditions such as pH or temperature. The wide availability and relatively inexpensive synthesis of block copolymers as well as the ability to exquisitely control so many diverse phases with periodic order make them compelling candidates for the building blocks of novel materials. Thin films of block copolymers can provide "bottom-up" templates for fabrication of devices on sub-30-nm length scales that are inaccessible to standard lithography techniques, opening the door for molecular size electronic components and ultrahigh density magnetic storage media. Applications include growth of nanowires 3,4 and nanocrystals used in devices such as flash memory, 5 metal-oxidesemiconductor capacitors, 6,7 arrays of quantum dots, 8,9 and photonic crystals. 10 For most applications, the proposed fabrication process consists of forming a thin copolymer film on a substrate, which is then chemically treated to remove one of the copolymer components. The remaining component is used as a mask for growing the active component. It would be even more desirable to include the active component (e.g., metallic or silica nanoparticles) from the outset and form the desired structure in a single pass. Direct application of this method is limited mainly due to nanoparticle aggregation and incompatibility of nanoparticle surface and ionic solutions with the polymers. 11 These problems can be circumvented by the use of self-assembling functionalized block copolymers, that is, polymers with covalently attached end groups that show specific affinity for nanoparticles, as agents for assembling ordered nanoparticle structures. 12,13 The theoretical understanding of the factors that lead to a successful self-assembly of nanocomposites is relatively limited. Recent studies include investigations of the influence of nanoparticles on the selfassembly and nanostructure formation of diblock copolymer melts based on solving cell dynamical system equations, 14 Monte Carlo simulations, 15 and self-consistent mean field/density functional theory. 16,17 ...

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Section: Resultsmentioning

confidence: 72%

Nanoparticle Ordering via Functionalized Block Copolymers in Solution

Sknepnek¹,

Anderson²,

Lamm

et al. 2008

ACS Nano

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“…26, the potential energy of the colloidal system can be estimated from a sum over all particles of the product of the net force on the particle and the distance of the particle to the droplet center, which is a quadratic function and has the form of the second moment of the mass distribution, M ϭ ͚ iϭ1 N ͉r i Ϫ r 0 ͉ 2 , with a prefactor. Mathematical solutions for ''minimal second moment of mass distribution'' structures have been reported in the literature (27) but not subject to a convexity constraint.…”

Section: Resultsmentioning

confidence: 99%

A precise packing sequence for self-assembled convex structures

Chen¹,

Zhang²,

Glotzer

2007

Proc. Natl. Acad. Sci. U.S.A.

115

111

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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...

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“…Physically, they represent all possible colloidal molecules that can be formed from spherical particles with no barriers to bond angle rotation. Our packings include as subsets many different structures previously observed and described in the literature, such as minimal-second-moment clusters [17], colloidal clusters observed through capillary-driven assembly [13,18], and Janus particle clusters [19].…”

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confidence: 99%

Minimal Energy Clusters of Hard Spheres with Short Range Attractions

2009

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The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. We calculate the ground states of hard-sphere clusters, in which n identical hard spherical particles bind by isotropic short-ranged attraction. Combining graph theoretic enumeration with basic geometry, we analytically solve for clusters of n 10 particles satisfying minimal rigidity constraints. For n 9 the ground state degeneracy increases exponentially with n, but for n > 9 the degeneracy decreases due to the formation of structures with >3n À 6 contacts. Interestingly, for n ¼ 10 and possibly at n ¼ 11 and n ¼ 12, the ground states of this system are subsets of hexagonal close-packed crystals. The ground states are not icosahedra at n ¼ 12 and n ¼ 13. We relate our results to the structure and thermodynamics of suspensions of colloidal particles with short-ranged attractions.

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Minimal-energy clusters of hard spheres

Abstract: What is the tightest packing of N equal nonoverlapping spheres, in the sense of having minimal energy, i.e., smallest second moment about the centroid? The putatively optimal arrangements are described for N < 32. A number of new and interesting polyhedra arise.

Cited by 113 publications

References 30 publications

Nanoparticle Ordering via Functionalized Block Copolymers in Solution

Nanoparticle Ordering via Functionalized Block Copolymers in Solution

A precise packing sequence for self-assembled convex structures

Minimal Energy Clusters of Hard Spheres with Short Range Attractions

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