# Dense Packing and Symmetry in Small Clusters of Microspheres

Original entry: Lidiya Mishchenko, APPHY 226, Spring 2009

Second entry: Xu Zhang, APPHY 225, Fall 2009

## Contents

## Reference

V.N. Manoharan *et al*. *Science* **301**, 483 (2003);

## Keywords

Colloidal packing, colloidal stability, interfaces, emulsions, capillarity

## Abstract

"When small numbers of colloidal microspheres are attached to the surfaces of liquid emulsion droplets, removing fluid from the droplets leads to packings of spheres that minimize the second moment of the mass distribution. The structures of the packings range from sphere doublets, triangles, and tetrahedra to exotic polyhedra not found in infinite lattice packings, molecules, or minimum–potential energy clusters. The emulsion system presents a route to produce new colloidal structures and a means to study how different physical constraints affect symmetry in small parcels of matter."

## Soft Matter Example

The way these sphere clusters are created applies many soft matter concepts, such as emulsions, capillarity, and surface charge.

The sphere packing is produced as follows: Polymer (PS) colloidal spheres are dispersed in toluene, and then water is added to create an oil-in-water emulsion consisting of small droplets of toluene with colloidal spheres at the interface. The colloids are strongly bound to the interface because of surface tension. As the toluene is preferentially evaporated, the spheres form spherical packing at the surface, followed by deformation of the interface as the oil is removed further. At the final stages of evaporation, strong capillary forces rapidly rearrange the spheres to form clusters of various numbers of colloids (See Figure 1).

The interesting part of the packing is the surface interaction of the sphere interfaces. Sulfate terminated polystyrene spheres have a negative surface charge in water, but are almost neutral in toluene. Also, van der Waals forces are much stronger between particles in water than toluene. Thus, in toluene, the particles act as hard spheres, and interact mostly through short-range steric repulsion. Since the particles interact like hard spheres when they pack, they only stick together through van der Waals forces as all the toluene is evaporated. Also, as the toluene is evaporated, more and more of the particle surface area is exposed to water, leading to surface charges, preventing aggregation with other clusters and allowing them to be stable in the suspension.

The forces constraining this system are complicated. Capillary forces provide a spherically symmetric compressive strain on the system until packing constraints break the symmetry (again, this all happens as the toluene is evaporated). In the end, it was found that up to a certain number of particles, the configuration that minimizes the moment of inertia of the cluster is the one that forms (See Figure 2).

Though the paper does not propose a unifying minimization criterion that predicts packing for smaller and larger clusters alike, these structures are reproducible and can be easily separated by centrifuging. Thus, suspensions of any one of these clusters can be used for assembly of entirely new crystal/glass structures.

The difference between finite and bulk packing is mentioned here as well. Some finite packing arrangements cannot be repeated for bulk, crytalline phases or are inconsistent with the normal fcc packing for particles subject to Lennard-Jones potential (as many molecules and colloidal suspensions are). These packing arrangements are however observed in glasses and liquids. This issue of packing and order of matter at different length scales is very important in soft matter.

In general this paper is important in a fundamental way because it allows for the study of packing phenomena under different constraints (not necessarily interparticle attraction) and leads to entirely new sphere-packing motifs.

## Relevant Article

Interestingly, ellipsoids pack more closely than spheres during random orienting. See Packing in the Spheres for more information.

# Dense Packing and Symmetry in Small Clusters of Microspheres(Summarized by Xu Zhang)

## Keywords

cluster packing, spherical packing, capillary force, emulsion

## Summary

This paper reports a method for making large quantities of identical colloidal particles with complex shapes consisting of equal-sized colloidal spheres. It is shown that under a compressive force, small numbers ( n=2 to 15) of hard spheres pack into distinct and identical polyhedra for each value of n. The clusters are formed in a three-phase colloidal system consisting of evaporating oil droplets suspended in water, with n micrometer-sized polymer spheres attached to the droplet surfaces. In this system, capillary forces provide a compressive force that is spherically symmetric until packing constraints break the symmetry. Then the oil is dried out forcing the hard-sphere-like particles in each droplet to pack together.

Figure 1 shows how the entire packing process works.Equal-sized, crosslinked polystyrene microspheres (844nm in diameter, with sulfate groups covalently bonded to the surface) are chosen as the packing components and toluene is chosen as the solvent. The spheres are dispersed in toluene(Fig.1A), water is added and mixed to create an oil-in-water emulsion consisting of small droplets of toluene ranging from 1 to 10 <math>\mu m</math> in diameter(Fig.1B). The particles are strongly bound to the droplet interfaces by surface tension. Then the toluene is evaporated from the system, forcing the hard-sphere-like particles in each droplet to pack together (Fig.1C). Removing more oil causes the droplet to deform, generating capillary forces that ultimately lead to a rapid (<33ms) rearrangement of the particles(Fig.1D).

After all the toluene has evaporated, a suspension of clusters of different sizes is left. The clusters are separated using centrifugation in a density gradient on the basis of differences in sedimentation velocity. The separation yields a set of sharp, well-separated bands(Fig.2), showing that only specific configurations emerge from the packing process.

Clusters of a given n are all identical and the structures are shown in Fig.3. The structures are considered to be optimal arrangements in the context of minimization of the second moment of the mass distribution <math>M_2=\sum_{i=1}^{n} {\mid r_i-r_0 \mid}^2 </math>, where <math>r_i</math> is the center coordinate of the ith sphere and <math>r_0</math> is the center of mass of the cluster.

## Soft Matter Connection

What defines an optimal packing of a set of n identical spheres has been a pertinent and pervasive question in mathematics and science. For packings of an infinite number of spheres, the obvious measure of optimality is the bulk density. But for a finite group of spheres there is no compelling definition of density, and optimality in finitesphere packings can be defined by the minimization of a certain physically reasonable variable. The minimization principle that governs the shapes of finite packing is figured out in this paper.

Because the method introduced in this paper yields large quantities of colloidal molecules in pure form, their phase behavior and glass transition can be investigated. This will shed light on fundamental condensed-matter issues and will lead to new colloidal crystal structrues. For instance, making 3D photonic crystals with a band gap in the visible spectrum would be much easier if colloids could form crystals with a diamond lattice. Tetrahedral colloids might do just that.

This research has brought the investigation of the collective behavior of more complex colloidal molecules within reach. Because the method requires stability of the initial colloidal spheres in both the oil and the water phase, it will not be easy to extend the method to colloids with different compositions. But the exciting prospects for model studies and new materials will provide a strong impetus for extending research.

## Reference

1.V.N. Manoharan, M.T. Elsesser, and D.J. Pine, Dense Packing and Symmetry in Small Clusters of Microspheres, Science 301: 483–487 (2003)

2.Alfons van Blaaderen , Colloidal Molecules and Beyond,Science 301 (5632), 470 (2003).