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Building Materials by
Packing Spheres
Vinothan N. Manoharan and David J. Pine
Abstract
An effective way to build ordered materials with micrometer- or submicrometer-sized
features is to pack together monodisperse (equal-sized) colloidal particles. But most
monodisperse particles in this size range are spheres, and thus one problem in building
new micrometer-scale ordered materials is controlling how spheres pack. In this article,
we discuss how this problem can be approached by constructing and studying packings
in the few-sphere limit. Confinement of particles within containers such as micropatterned
holes or spherical droplets can lead to some unexpected and diverse types of polyhedra
that may become building blocks for more complex materials. The packing processes
that form these polyhedra may also be a source of disorder in dense bulk suspensions.
Keywords: colloids, microspheres, optical materials, packing, structure.
Introduction
Monodisperse colloidal particles are
natural building blocks for composite materials with micrometer-scale features, but
these particles have one inherent limitation:
they are nearly always spheres. Because a
colloidal particle is much larger than its
constituent atoms, its shape is determined
by surface tension, not by its interior
covalent bonds. The equilibrium and nonequilibrium behavior of colloidal suspensions is controlled primarily by the hard
repulsive interaction between particles,
and dense suspensions of identical hard
spheres are known to form only fcc colloidal crystals1,2 or disordered, glassy
packings. Despite this limitation, several
new kinds of materials based on colloids
have emerged in the past decade.
Materials such as photonic crystals and
macroporous media are made by drying
or flocculating a dense suspension so that
the particles touch and form a rigid, colloidal sphere packing. Since the colloidal
length scale is commensurate with optical
wavelengths, fcc colloidal crystallization
is an inexpensive and simple way to prepare ordered materials that diffract light.
Dried colloidal crystals can be used as
templates3–5 to make photonic crystals
that can function as optical semiconductors
in a number of devices. The same templating procedures can be applied to either ordered or disordered particle packings to
make macroporous materials, which can
serve as flow-through catalyst supports,
filters, and lightweight structural materials.6
MRS BULLETIN/FEBRUARY 2004
The properties of these materials depend
sensitively on the microstructure of the colloidal sphere packing. In photonic crystals,
grain boundaries formed during crystal
growth and cracks formed during drying
can incoherently scatter light, reducing the
diffraction efficiency, whereas other types
of defects such as vacancies can act as
resonant cavities for specific frequencies,
allowing us to create localized states in the
optical spectrum. Unfortunately, there is
no simple way to control the number and
type of defects when making photonic
crystals by packing spheres, nor is there a
way to control the most important characteristic of the microstructure, the crystal
structure itself. The fcc structure diffracts
light much less efficiently than structures
with a lower degree of symmetry, such as
the diamond lattice.7 However, the spherical symmetry of colloidal particles leaves
us with few possibilities for achieving the
uniformly low coordination of diamond,
in which each particle touches only four
others. It is a challenge to control even the
average coordination, which determines
the connectivity of the pore network and
hence the permeability in macroporous
materials.
Thus, there is great incentive to control
the packing in colloidal materials. Several
recent approaches have shown that we
may at least be able to control the bulk
crystal structure. We briefly mention a few
methods here (and some are discussed in
other articles in this issue). One approach
is to change how the particles interact,
either through synthesis or external fields.
For example, spheres with a soft repulsive
potential can form crystal structures such
as bcc8 or A15 lattices.9 Other equilibrium
phases can form if the spherical symmetry
of the interactions is broken by inducing a
dipole moment in the particles.10 Another
approach is to perturb the natural crystal
spacing of the particles, either by the addition of smaller spheres11 or by the use of
a patterned substrate that defines a different lattice spacing.12 At sufficiently large
perturbations, alternative crystal structures
nucleate.
Here, we discuss a more general approach to controlling not just the crystal
structure, but also the defect structure and
connectivity of colloidal materials. Instead
of building materials by packing individual spheres together, we propose building
them in two steps: first, packing the spheres
into small clusters, or “finite packings,”
then packing the packings. The advantage
of this approach is that finite packings, unlike bulk sphere packings, are infinitely
malleable: their structures vary with the
definition of the boundaries that confine
and the forces that consolidate the particles.
As we show in the next section, finite packings can form a wider variety of structures
than are found inside the fcc arrangement
or its stacking variants. In fact, recent experiments on finite packing in colloidal
systems have shown that it is possible to
create large numbers of colloidal clusters
with specific shapes and symmetries. These
clusters, which we describe in the third
section of this article, could become the
building blocks of more complex materials.
Finite Sphere Packings
To illustrate the variability of finite
packings, we consider a finite packing
problem related to the “kissing problem”13
in mathematics: how many identical
spheres can touch a central sphere of the
same size, and what configurations do
they form? A simple calculation shows that
each particle touching the central sphere
subtends a solid angle of 2 3 steradians. Even though the surface of the
central sphere can accommodate nearly 15
[i.e., 42 3 objects of this size, only
12 spheres can fit.14 Much of the available
coordination area is therefore unused.
The most symmetrical arrangement of
the 12 outer spheres is the icosahedral packing shown in Figure 1a. This structure has
12 axes of fivefold rotational symmetry.
Figure 1b shows a very different arrangement, the asymmetrical “weary icosahedron,”15 made by rolling the spheres of the
icosahedron toward one of its vertices. The
empty region on the surface shows just how
91
Building Materials by Packing Spheres
this packing is about 8% lower than that of
a close-packed cluster when the spheres
interact through a Lennard–Jones 6–12
potential, a mathematically simple potential
used to model pairwise-additive attractions
in atomic systems.16 If the particles are hard
spheres, all packings have the same potential energy, so optimal packing must be
defined by a geometrical criterion. As we
discuss in the next section, a restriction on
the mass distribution of spheres leads to
the weary icosahedron.15 On the other hand,
minimizing the volume of the packing, as
defined by its convex hull, leads to an
arrangement in which the particles form
linear chains called sausages (see Figure 1g).17 If we change the number n of
spheres for a given set of constraints, the
optimal structures may differ even more
dramatically. For hard spheres, it is not always obvious what physical constraints
correspond to the geometrical conditions.
Colloidal Clusters
Figure 1. (a) An icosahedral arrangement of 13 spheres. (b) The “weary icosahedron.” (c) A
13-sphere subelement of the fcc lattice. (d) The close-packed cluster of (c) shown as a
stacking of dense layers. (e) A 13-sphere subelement of the hcp arrangement, generated by
rotating the upper layer of (c) by 180 . (f) The close-packed structure of (e), shown as a
stacking of dense layers. (g) A 13-sphere “sausage,” the finite packing that occupies the
least amount of volume.
much coordination area is unused in any
three-dimensional (3D) sphere packing.
Because of the excess area, the 12 kissing
spheres can be rearranged into infinitely
many configurations.
Two of the possible configurations correspond to local packings found in closepacked bulk structures. Figure 1c shows
the local packing in the fcc lattice. Here,
the centers of the outer spheres are located
at the vertices of a cuboctahedron, which
can also be viewed as a stacking of dense
layers (see Figure 1d). The alternate stacking, in which the top layer of Figure 1d is
92
rotated by 180 degrees, is a local packing
of the hcp arrangement (see Figures 1e
and 1f), which has the same density as the
fcc lattice. Any finite packing that corresponds to a local packing of an fcc or hcp
crystal is called a close-packed cluster.
If the only constraint on our 13-sphere
finite packings is that they be kissing
arrangements, then each of the configurations is equally valid. But other conditions
or constraints can favor a single optimal
structure. For example, if the spheres attract one another, the icosahedron is the
most stable configuration: the energy of
Creating clusters of hard spherical colloids requires first confining or isolating
particles from a bulk suspension, consolidating them, and then “freezing” the structure by bonding the particles together. In
two dimensions, the particles can be confined by functionalized patches18,19 or
holes20,21 on a patterned substrate. As a
suspension of spheres flows over the substrate, the confining wells isolate small
groups of particles. Drying the suspension
then consolidates each n-sphere group.
The structures that result from this twodimensional (2D) finite packing process
are optimal circle packings22 whose configurations depend on the shape and size
of the confining well. If the well is circular,
the particles form sphere doublets, triangles, squares, pentagons, and hexagons.
All of these structures except the pentagon
are elements of bulk 2D and 3D close
packings. It is also possible to change the
depth and cross section of micropatterned
holes to allow the 2D structures to stack.
In the resulting polyhedra, known as antiprisms, the upper layer is rotated with respect to the lower by one-half the vertex
angle, yielding the tetrahedron (n 4),
octahedron (n 6), and square antiprism
(n 8), which is also a non-close-packed
cluster. More complex arrangements are
possible in either larger wells or in wells
with different shapes.
An even simpler way to confine the particles is to adsorb them onto a fluid interface. Solid particles attach to an oil–water
interface to minimize the total surface
energy of the three-phase system.23 The
binding potential U per particle is given by
U r 21 cos 2,
(1)
MRS BULLETIN/FEBRUARY 2004
Building Materials by Packing Spheres
where r is the particle radius, is the oil–
water surface tension, and is the bulk
solid/oil/water contact angle. For particles 1 m in diameter, the binding
potential is thousands of times the thermal
energy. This effect can be exploited to convert the surface of an emulsion droplet into
a 3D particle trap.24–26
Such droplets can consolidate as well as
confine the particles.27 Removing the liquid
from a droplet containing bound particles
generates compressive forces that draw the
particles together. Once the spheres touch
on the surface of the droplet, the receding
interface forms menisci between them, and
the resulting capillary forces collapse the
particles into a cluster. After all the oil has
evaporated, van der Waals attractions
freeze the structure. Surprisingly, this type
of compression leads to structures that are
unique and consistent for each value of n,
as shown in Figure 2. The preferred structures range from familiar polyhedra such
as tetrahedra, triangular dipyramids, and
octahedra to more unusual polyhedra that
have not been observed in either Lennard–
Jones clusters or other systems with attractive potentials. The symmetry varies erratically with n, with two-, three-, four-, and
fivefold rotational symmetry all occurring
as a result of packing constraints alone.
Two general properties of the packing
sequence stand out. First, configurations
of up to n 11 can be reproduced by a
simple mathematical criterion: minimization of the second moment,
M
r r ,
enough. Thus, all eight deltahedra occur
under certain packing conditions.
Assembling Materials from
Finite Packings
n
i
0
2
(2)
i1
of a set of hard spheres, where ri is the center coordinate of the ith sphere and r0 is the
center of mass of the cluster.15 This type of
packing has not been observed before. For
n 12, the observed clusters are not
minimal-moment structures, but they
lower their second moment during the
packing process. Although we do not yet
understand why the second moment defines the density in these systems, the correlation may provide some clues about the
physics of the packing constraints.
The second general property of the sequence is the occurrence of deltahedra, or
polyhedra constructed from equilateral triangles. There are eight possible regular
convex deltahedra,28 the first seven of
which correspond to the observed colloidal
clusters and the theoretical minimalmoment packings for n 4 through
n 10. The eighth deltahedron is the
icosahedron, but neither 12- nor 13-sphere
minimal-moment clusters are icosahedra.
In fact, the 13-sphere structure is the
weary icosahedron of Figure 1b, and the
MRS BULLETIN/FEBRUARY 2004
Figure 2. Scanning electron micrographs
of clusters formed by particles on the
surface of an evaporating droplet.
Schematic illustrations show the sphere
packings and polyhedra that minimize
the second moment of the mass
distribution.
13-sphere colloidal cluster we observe has
one sphere displaced from this arrangement. The 12-sphere icosahedron appears
to be a metastable arrangement; it rearranges to the minimal-moment structure when the drying forces are large
Any of the colloidal clusters can be frozen
in their optimal configurations and redispersed. In the surface templating method,
this is done by sintering the particles and
dissolving the substrate, a process that
yields about 105 clusters of any type, of
which 5% contain defects. In the emulsion trapping method, minimal-moment
clusters of different n can be separated by
centrifugation to yield 108–1010 identical,
colloidally stable clusters.
These clusters could let us tune the
microstructure of colloidal sphere packings. In the simplest scenario, we may dope
a bulk colloidal crystal with clusters made
from the same type of spheres. The surface
templating method is most appropriate for
such experiments, since it yields a small
amount of highly configurable clusters.
When a colloidal crystal dries, the particles
touch and form a close-packed fcc arrangement, but the addition of non-close-packed
clusters would introduce specific defects.
For example, hexagonal clusters of six particles can be made by packing spheres into
circular micropatterned holes with posts
at the centers. Such packings are essentially
close-packed clusters that are missing a
central sphere; they would be an excellent
tool for introducing vacancies in photonic
crystals. Other types of defects are also
possible. The interstices in an fcc crystal
are coordinated by either tetrahedral or
octahedral arrangements of particles, while
minimal-moment clusters can contain interior voids that are coordinated in many
other ways. Doping a dried crystal with
such clusters must therefore reduce both
the density and the average coordination
number. When the cluster has fivefold
symmetry, as does the seven-sphere
minimal-moment cluster or the 12-sphere
icosahedron, it may introduce a significant
amount of disorder into the bulk packing.
We could use these methods to control
the permeability in a macroporous material.
As shown in the 13-sphere finite packing
problem, 12-fold local coordination alone
does not guarantee maximum density; thus
it should be possible to control the two variables independently by adding the appropriate type and quantity of clusters. We may
even be able to create disorder by design—
using spheres and clusters to make bulk aggregates with specific structure factors.29,30
The clusters may also allow us to control nucleation in colloidal crystals, and
thereby to improve the crystal quality for
optical materials. Yin and Xia showed that
adding a small amount of square (n 4)
93
Building Materials by Packing Spheres
clusters to a dense suspension influences
the epitaxial growth direction of colloidal
crystals on flat substrates;20 however, little
is known about how doping affects bulk
crystal formation. There may be specific
structures that enhance nucleation of
metastable phases, for example, or that
can lead to controlled stacking sequences
in crystalline phases.
The separation process for minimalmoment clusters yields enough material
to pursue some more ambitious goals, including studying the phase behavior of
some fundamental nonspherical shapes.
The clusters themselves might then become building blocks for new ordered
materials. Most of the minimal-moment
structures are incompatible with an fcc
arrangement of spheres, and there are few
theoretical predictions of the types of
phases that might be observed, although
we can speculate based on the symmetry.
It would be interesting to see, for example,
if suspensions of tetrahedra could under
some conditions form a colloidal crystal
with diamond symmetry, the basis of an
advanced photonic crystal. Suspensions of
fivefold symmetrical clusters cannot crystallize, but they might form quasi-crystals.
And mixtures of different clusters could
form all kinds of packings, disordered and
ordered, with a vast array of possibilities
for controlling the microstructure of bulk
packings.
Conclusions
An experimental study of the effect of
colloidal clusters on bulk packing has yet
to be completed, but there are numerous
areas to explore, some of which we have
listed here. Even more possibilities arise
when we consider how other patterns of
finite packing might arise. Some alternate
strategies are suggested by the various
methods of creating non-fcc packings. For
example, we could break the spherical
symmetry of the compressive forces on
the colloidal particles stuck to a droplet
surface by stretching the droplet or by inducing dipoles in the spheres. Also, mixing
different sizes of spheres on the droplet
surface may perturb the cluster polyhedra
just as it perturbs the crystalline unit cell
in a bulk system. More systematic variation of cluster configuration is possible
with micropatterned templates.
Some of these methods may unearth
other mathematical sphere packing motifs,
which may in turn allow us to develop
some general empirical rules about local
structure in a complicated configuration
space. This is the inverse problem of assembling spheres into specific bulk packings: Can we “disassemble” or interpret
the structure of disordered materials in
94
terms of finite packings? Frank16 first
speculated that the local arrangements in
disordered systems of attractive spheres
correspond to finite packings that are more
stable than close-packed clusters.31 Scattering experiments have now verified this
hypothesis for metallic liquids.32,33 In these
systems, the interatomic forces are well
approximated by a Lennard–Jones potential, a type of interaction that favors the
13-sphere icosahedron, as we have seen.
Icosahedra, whose fivefold symmetry is
incompatible with long-range order, are
indeed local packings in the liquid state.
The stability of the icosahedron relative to
a close-packed cluster entails an energetic
barrier to rearrangement, thus hindering
crystallization and allowing the liquid to
persist at temperatures well below the
freezing point.
Relating finite packings to bulk phenomena in hard-sphere systems is more
difficult. In a dense, disordered bulk packing, the forces on a local arrangement
depend on the positions of all the other
particles in the system—we cannot neglect
the effect of particles far away. Near the
glass transition of a hard-sphere colloidal
system, for example, large groups of particles move simultaneously when the
system is compressed.34,35 Researchers
studying “jammed”36 systems in which
particles cannot flow and glassy systems
have only recently made progress in characterizing the forces and the local arrangements that can bear them. So while there is
some evidence that local fivefold symmetry
occurs in dense random packings37 and
supercooled fluids38 of hard spheres, it is
not yet clear if these local structures correspond to some kind of optimal finite packing. On the other hand, we have seen that
specific kinds of compression lead to preferred finite packing structures. The
isotropic compression of the evaporating
emulsion droplet may be similar to the
compression of a local arrangement in a
disordered system. Though icosahedra have
been the focus of most studies of local packing in hard-sphere systems, our clusters
suggest that compressed hard spheres
more generally prefer to form deltahedra—
and as Bernal noted in 1960,39 none of the
deltahedra with n 6 or combinations
thereof can fill space. Perhaps nucleation
in disordered packings is suppressed by
deltahedral local packing rather than by
short-range pentagonal or icosahedral
order alone. Further experiments on colloidal systems should reveal whether
minimal-moment structures are found in
hard-sphere glasses, supercooled fluids,
or jammed bulk packings.
Given all of these possibilities, we believe
that the microstructure of materials derived
from spheres is more understandable and
mutable than previously imagined. Viewing dense colloidal suspensions as collections of individual particles limits the way
we approach organizing them. The physical
reality is that the particles are correlated,
and we might do better to think of dense
suspensions as mixtures of stable and
transient finite packings. Understanding
the conditions that favor certain finite
configurations will let us control defect
placement, coordination, density, and
crystal structure in the next generation of
colloidal materials.
Acknowledgments
We thank the National Science Foundation (grants CTS-0221809 and DGE9987618) and Unilever Corporation for
supporting our work on colloidal clusters.
We also thank Mark Elsesser for contributions to the experiments.
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