Superlattice formation in mixtures of hard-sphere colloids
Neil Hunt, Roger Jardine, and Paul Bartlett*
Department of Chemistry, University of Bath, Bath BA2 7AY, United Kingdom
Received 4 February 2000
We report a detailed experimental study of the superlattice structures formed in dense binary mixtures of
hard-sphere colloids. The phase diagrams observed depend sensitively on the ratio =R
S
/ R
L
of the radii of
the small S and large L components. Mixtures of size ratio =0.72, 0.52, 0.42, and 0.39 are studied. The
structures of the colloidal phases formed were identified using a combination of light-scattering techniques and
confocal fluorescent microscopy. At =0.39, ordered binary crystals are formed in suspensions containing an
equal number of large and small spheres which microscopy shows have a three-dimensional structure similar
to either NaCl or NiAs. At the larger size ratio, =0.52, we observe LS
2
and LS
13
superlattices, isostructural
to the molecular compounds AlB
2
and NaZn
13
, while at =0.72 the two components are immiscible in the
solid state and no superlattice structures are found. These experimental observations are compared with the
predictions of Monte Carlo simulations and cell model theories.
PACS numbers: 82.70.Dd, 64.70.Dv, 64.75.+g
I. INTRODUCTION
Hard spheres constitute probably the simplest yet also one
of the most important models of condensed-matter physics
1. The phase behavior of hard spheres is determined by
minimizing the free energy F =U -TS or, since hard spheres
are forbidden to interpenetrate and the internal energy U is a
constant, by maximizing the entropy S. It was highly surpris-
ing, therefore, given the simplicity of the interactions in hard
spheres, when experiments on colloidal particles 2 demon-
strated that hard spheres of different sizes form equilibrium
crystalline superlattices with large and highly complicated
unit cells. Subsequent computer simulations 3 confirmed
that a mixture with a radius ratio =R
S
/ R
L
of 0.58 formed
two binary crystals, with stiochiometries LS
2
and LS
13
where L denotes the large particle. Clearly, given the
subtlety of entropic effects at this size ratio, it is interesting
to ask what structures might be stable at other size ratios. In
general, the stability of a binary crystal, L
m
S
n
, depends on
three variables, namely the size ratio , the total packing
fraction =
L
+
S
, and the relative numbers of small and
large spheres n
S
/ n
L
. Clearly a determination of the phase
behavior as a function of these three variables, , , and
n
S
/ n
L
, represents a formidable task.
The present paper presents a first step in this direction
where we report a comprehensive experimental study of the
phase behavior of hard-sphere mixtures at size ratios
=0.72, 0.52, 0.42, and 0.39. Our motivation is threefold.
First, while the stability of superlattice phases at size ratios
other than =0.58 has attracted considerable theoretical in-
terest in recent years 4,5, there has been little experimental
work that critically tests these theories. In the current paper
we use a sterically stabilized colloidal system which other
studies have demonstrated 6 provides a close experimental
realization of classical hard spheres. Our experiments there-
fore model accurately a mixture of hard spheres. Second, the
statistical properties of mescoscopic systems of particles are
frequently dominated by entropic as opposed to enthalpic
interactions. Consequently, the phase behavior of many col-
loidal systems can be appropriately mapped onto an effective
hard-sphere system. Thus, for instance, LS
2
and LS
13
struc-
tures have been observed in a range of mesoscopic systems
including mixtures of charge-stabilized polystyrene spheres
7, two-dimensional arrays of gold clusters 8, and gem
opals 9,10 in which the interactions are far from hard-
sphere-like. Experiments on colloidal hard-sphere mixtures
are important therefore as a simple reference case. Third,
colloidal superlattices provide examples of three-
dimensional photonic crystals with lattice parameters of the
order of the wavelength of visible light and a range of dif-
ferent lattice symmetries. Such crystals could be used to cre-
ate photonic band gaps frequency ranges where light will
not propagate because of multiple Bragg reflections which
are predicted to have unique and highly useful optical prop-
erties 11,12.
The paper is organized as follows. In Sec. II we summa-
rize the current theoretical predictions for freezing in binary
hard-sphere mixtures. The colloidal mixtures used are out-
lined in Sec. III and our detailed observations of the phase
behavior are presented in Sec. IV. We then describe the scat-
tering from a random-stacked LS superlattice in Sec. V and
its microscopy in Sec. VI. Finally, we compare our results
with theoretical predictions in Sec. VII and we collect our
conclusions in Sec. VIII.
II. BINARY MIXTURES
At first sight the idea that entropy, which is normally
taken to be a force-favoring disorder, should generate com-
plex superlattice structures seems counterintuitive. It is fre-
quently assumed, for instance, incorrectly, that complex in-
teractions between particles are needed to generate complex
phases. However, the physical origin of superlattice forma-
tion in hard-particle suspensions is both extremely simple
and general 13. The resolution of this paradox relies on
distinguishing two components to the total entropy. The en-
tropy of a system of spheres is composed of contributions *Corresponding author.
PHYSICAL REVIEW E JULY 2000 VOLUME 62, NUMBER 1
PRE 62 1063-651X/2000/621/90014/$15.00 900 ©2000 The American Physical Society