Self-Assembly of a Colloidal Interstitial Solid with Tunable Sublattice Doping
L. Filion, M. Hermes, R. Ni, E.C.M. Vermolen, A. Kuijk, C.G. Christova, J.C.P. Stiefelhagen, T. Vissers,
A. van Blaaderen, and M. Dijkstra
Soft Condensed Matter, Debye Institute for NanoMaterials Science, Utrecht University,
Princetonplein 1, NL-3584 CC Utrecht, the Netherlands
(Received 16 May 2011; published 11 October 2011)
We determine the phase diagram of a binary mixture of small and large hard spheres with a size ratio of
0.3 using free-energy calculations in Monte Carlo simulations. We find a stable binary fluid phase, a pure
face-centered-cubic (fcc) crystal phase of the small spheres, and binary crystal structures with LS and LS6
stoichiometries. Surprisingly, we demonstrate theoretically and experimentally the stability of a novel
interstitial solid solution in binary hard-sphere mixtures, which is constructed by filling the octahedral
holes of an fcc crystal of large spheres with small spheres. We find that the fraction of octahedral holes
filled with a small sphere can be completely tuned from 0 to 1. Additionally, we study the hopping of the
small spheres between neighboring octahedral holes, and interestingly, we find that the diffusion increases
upon increasing the density of small spheres.
DOI: 10.1103/PhysRevLett.107.168302PACS numbers: 82.70.Dd, 61.50.Ah, 81.30.Dz
Our understanding of the structure and phase behavior in
colloidal and nanoparticle systems relies greatly on our
knowledge of the hard-sphere system, which can be con-
sidered as a reference system for more realistic systems.
While interactions between such colloids and nanopar-
ticles are complicated, due to, e.g., van der Waals inter-
actions, steric stabilization, electrostatics, and depletion
effects, systems can often be constructed where these
interactions are minimized and the particles behave like
hard particles, e.g., Refs. [1–3]. One of the first observa-
tions of binary crystal structures in colloidal systems was
made by Murray and Sanders in naturally occuring
Brazilian gem opals . The structures they identified
have since been shown to be stable in binary hard-sphere
mixtures . More generally, theoretical and experimental
studies of binary mixtures of hard spheres have found
regions of stability for binary liquids, monodisperse face-
centered-cubic (fcc) crystals, binary crystal phases NaCl,
NaZn13, AlB2, and MgZn2, and substitional solids [5–8].
In this Letter, we demonstrate theoretically and experi-
mentally the self-assembly of a novel thermodynamically
stable interstitial solid solution (ISS) in a binary mixture of
hard spheres, which is constructed by filling the octahedral
holes of an fcc crystal of the large particles with small
particles. In an ISS phase, instead of both species ordering
on a binary crystal lattice, only one of the species is
properly ordered on a lattice, while the other species is
placed irregularily on a sublattice. The resulting ISS phase
can be seen as a compromise between a binary crystal
where both species have crystalline order and a crystal
phase of only large particles in coexistence with a disor-
dered fluid phase, and is hence a truly ‘‘interstitial’’ solu-
tion to the problem of maximizing entropy. The ISS phase
is found in atomic systems; e.g., crystalline iron is
strengthened by the addition of interstitial carbon atoms
yielding steel. However, to the best of our knowledge, a
thermodynamically stable ISS phase has not been reported
in colloidal and nanoparticle mixtures. Surprisingly, we
find that the fraction of octahedral holes filled with a small
particle can be completely tuned from 0 to 1. We are
unaware of another (atomic) system which shows a com-
pletely tunable ISS phase. That entropy alone can stabilize
an ISS phase is not trivial, as the presence of the small
particles may affect the stability of the underlying crystal
lattice of large particles.
To demonstrate the stability of the ISS phase, we
consider a binary mixture of large and small spheres
with diameters ?Land ?Srespectively, and size ratio
q ¼ ?S=?L¼ 0:3. Following Ref. , we first predict
the candidate binary crystal structures for this mixture
where both species exhibit long-range crystalline order.
We find NaCl, NiAs, and a superlattice structure with
LS6stoichiometry where L (S) represents the large (small)
particles (Fig. 1 and Supp. Fig. S1 ). We also examined
larger systems using kinetic Monte Carlo simulations 
and found an ISS nucleus formed spontaneously indicating
that a fcc-based ISS phase should also be considered a
candidate phase. We point out that a NaCl structure is
constructed by filling every octahedral hole in an fcc lattice
of large particles with a single small particle (Fig. 1). Thus,
a random, incomplete filling trivially results in an ISS. The
stoichiometry of the ISS phase is defined as LSnwhere n is
a fractional number in the range [0, 1]. We note that n ¼ 0
corresponds to the fcc phase and n ¼ 1 to a perfect NaCl
An ISS phase can also be constructed by filling some
octahedral holes in the monodisperse HCP lattice; a com-
plete filling yields a NiAs structure. However, the pure fcc
and binary NaCl structures are more stable forhard spheres
than the competing HCP and NiAs phases with free-energy
PRL 107, 168302 (2011)
14 OCTOBER 2011
? 2011 American Physical Society
differences on the order of 0.001 kBT  and 0.002 kBT
 per particle, respectively.
Boltzmann’s constant and T the absolute temperature.
We stress that experimentally, the presence of interstitials
in the holes of the fcc lattice can stem from two main
causes: (i) a lowering of the free energy due to the addition
of the interstitials resulting in a thermodynamically stable
ISS phase, or (ii) a kinetic trapping of the interstitials
leading to nonequilibrium crystal structures. While these
two effects are often difficult to distinguish experimentally,
free-energy calculations using computer simulations can
demonstrate conclusively whether or not ISS phases are
stable compared to fcc crystals of the large and small
spheres and the binary liquid.
To determine the phase behavior of this system, we
calculate the free-energy of the binary NaCl phase, the
and large spheres and the ISS phase. The Gibbs free
energies for the binary fluid and fcc phases were taken
frompreviousstudies ,while the Gibbsfree energies of
the NaCl crystal and LS6crystal structures were calculated
using Einstein integration as in Ref. . For the interstitial
solid we employ the identity 
FðNL;zS;VÞ ¼ FðNL;zS¼ 0;VÞ ?
where FðNL;zS;VÞ denotes the thermodynamic potential
corresponding to the (NL, zS, V) ensemble, the first term at
the right-hand side is the free-energy of the pure large
sphere fcc phase, and the integrand in the second term
can be determined from measuring the average number of
absorbed small particles from a reservoir at fugacity zS
onto a system of NL large spheres in a volume V.
Monte Carlo (MC) simulations in the NPT ensemble
were used to determine the equation of state. In the case
of the ISS we included large displacement moves of the
small particles in addition to typical MC particle moves in
order to allow them to move from one octahedral hole to
another and thus average over the disorder induced by the
small particles. All calculations were performed on 256 to
4000 particles. Using common tangent constructions
of the Gibbs free energies at constant pressure we deter-
mined the coexistence regions and constructed the phase
diagram. The phase diagram in the pressure-composition
(p-xS) representation is shown in Fig. 1, where p ¼ ?P?3
is the reduced pressure, xS¼ NS=ðNSþ NLÞ, NSðLÞ is
the number of small (large) hard spheres, ? ¼ 1=kBT
and V is the volume. From Fig. 1 we see that there is a
large region of parameter space where the ISS phase is
stable. Additionally, the inset in Fig. 1 demonstrates how
the filling of the octahedral holes with small particles in the
coexisting ISS increases as a function of pressure, slowly
approaching the stoichiometry n ¼ 1 of NaCl.
As an illustration of the experimental accessibility of the
ISS phase, we performed sedimentation experiments using
core-shell silica colloids. The structural and thermody-
namic properties of a system under gravity at a certain
height are the same as those of a bulk system at the
corresponding pressure. Sedimentation experiments are
therefore often used in scanning the properties of a system
over a range of pressures in a single experiment .
Typical snapshots obtained by confocal microscopy are
shown in Fig. 2 and Supp. Fig. S3 . The confocal
image in Fig. 2 depicts an experimental realization of an
ISS with stoichiometry n ’ 0:3. We find that the concen-
tration of small particles decreases as a function of height
in the sample (Fig. 2) which is in qualitative agreement
with the phase diagram which predicts an increasing
concentration of small particles with pressure. The con-
focal images and height profile demonstrate not only the
existence of the ISS in this system, but the large range over
which the stoichiometry n of the ISS can be tuned.
However, we did not observe the LS6superlattice structure
as this occurs at high pressures where it is difficult
to observe equilibrium behavior experimentally. Further-
more, in examining the motion of the small particles using
confocal microscopy, we noted that some small particles
near the top of the samplewere moving between interstitial
lattice sites. Near the bottom of the sample, this was not
observed. Motivated by this intriguing observation, we
examined the motion of the small particles in an ISS phase
in more detail using event driven MD simulations . In
these simulations the large (small) particles had a mass
FIG. 1 (color online).
sphere mixture in the composition xs¼ NS=ðNSþ NLÞ-reduced
pressure p ¼ ?P?3
‘‘ISS’’ and ‘‘fluid’’ denote the interstitial solid solution and
the binary fluid consisting of large and small particles. The
labels ‘‘ISS þ fluid’’,
‘‘LS6þ FCCðSÞ’’ denote the coexistence regions where the
tie-lines that connect the two coexisting phases are horizontal.
fcc(S) denotes a face-centered cubic crystal of the small spheres.
Inset: The stoichiometry n of the coexisting ISS phase as a
function of the pressure. A typical configuration of an ISS with
localized interstitials is also displayed. Right: Unit cells of
(a) fcc, (b) NaCl and (c) the LS6crystal structure. Note that
NaCl is constructed by filling the octahedral holes in the fcc
lattice of large particles with small particles.
Left: Phase diagram of the binary hard-
Lplane. The stable one-phase regions labeled
‘‘LS6þ fluid’’, ‘‘ISS þ LS6’’,and
PRL 107, 168302 (2011)
14 OCTOBER 2011
mLðmSÞ, where mS¼ mL?3
in units of ?L
drawn randomly from a Maxwell-Boltzmann distribution
after which the center of mass was fixed. A typical trajec-
tory taken by a single small particle is shown in Fig. 3
which demonstrates clearly that the particles spend most of
their time in the octahedral holes of the lattice. However,
the particles do not hop directly between the octahedral
holes but rather hop via a neighboring tetrahedral hole in
the fcc lattice. We point out that the tetrahedral holes are
significantly smaller than the octahedral holes, 0:225?Lin
comparison to 0:414?Lat close packing.
At close packing of the underlying fcc crystal, the
interstitials are prevented from moving between neighbor-
ing holes by the presence of the large particles. However, at
lower pressures due to larger lattice constants in combina-
tion with the motion of large particles around their lattice
sites (phonons), the small particles can travel between the
octahedral holes. We calculated the mean square displace-
ment (MSD) h?r2
time. From this MSD, we determined the long-time self-
number density for various stoichiometries (Fig. 3). As
evident in Fig. 3, D increases with stoichiometry n when
the number density of the large particles is kept fixed. This
is in contrast to most systems where the diffusion constant
decreases with density; in general as the density increases
the available space in the system decreases leaving less
room for particles to move resulting in a lower diffusion
constant. However, in this case, as the number of small
particles in the system is increased, the mean square dis-
placement of the large particles from their ideal lattice sites
increases (Supp. Fig. S5 ), likely arising from an
increase in the depletion interaction. This in turn lowers
. The initial velocities were
L, and time was measured
SðtÞi of the small particles as a function of
the free-energy barrier between octahedral and tetragonal
sites (Fig. 3) leading to an increase of the diffusion coef-
ficient of the small spheres. The free-energy barriers
were calculated using event driven MD simulations with
NL¼ 864 large particles. Each small particle coordinate
was mapped to the closest position on the line connecting
the ideal lattice positions of an octahedral hole with a
tetragonal hole and the probability (PðxÞ) of finding a small
particle at position x along that line was determined.
Number of particles
Height in sample z (µm)
FIG. 2 (color online).
and large particles from a sedimentation experiment. Right:
Confocal image of an interstitial solid solution with size ratio
q ¼ 0:3 and n ’ 0:3 In this image two (111) planes of the
crystal, one consisting of mostly small (red) particles with the
other consisting of mostly large (green) particles have been
overlayed in order to be viewable in the same figure. The larger
and darker red regions correspond to defects, such as vacancies,
in the underlying lattice of green particles which have been filled
by many small (red) particles. Scale bar is 10 ?m.
Left: The density profiles of the small
FIG. 3 (color online).
for a volume fraction of the large particles ?L¼ 0:6. Note that
the trajectories connecting the lattice sites are neither vertical nor
horizontal indicating that the small particle does not take the
direct route between neighboring holes, but rather hops first
to a tetrahedral hole, and then to an octahedral hole. A
Bottom: Dimensionless, long-time self-diffusion coefficient
metries n. Note that the D?increases with stoichiometry n. Inset:
Free-energy barriers in an interstitial solid solution felt by small
particles hopping from an octahedral hole to a tetragonal hole
and back to an octahedral hole for ?L¼ 0:6. Note that the height
of the free-energy barriers decreases as a function of stoichi-
ometry (filling fraction) n.
Top: The trajectory of a single particle
isshown on the left.
=?Las a function of the number density of large
L) for various interstitial solid solution stoichio-
PRL 107, 168302 (2011)
14 OCTOBER 2011
The free-energy barrier was then determined from
?UðxÞ ¼ ?logPðxÞ.
The phase behavior of this system can be compared to
previous experimental and theoretical studies of binary
hard-sphere mixtures. Vermolen et al. examined various
methods for growing binary NaCl hard-sphere crystals in-
and dielectrophoretic compression from a binary mixture
withsizeratioq ¼ 0:3.However,thecrystalstructures
the small colloids ( > 10%). This result is in keeping with
the phase behavior observed in Fig. 1. Hunt et al. also
examined binary hard-sphere mixtures of particles with
q ¼ 0:39 and 0.42 and reported the presence of NaCl
. However, from their presented results it is impossible
to distinguish between an ISS and a NaCl crystal. On the
using free-energy calculations are the phase diagrams pre-
sented for binary hard-sphere mixtures with q ¼ 0:2 
and0.414.Forq ¼ 0:2,thephasebehaviorisexpectedto
be significantly different than that for q ¼ 0:3 as the small
particles can fit in both the tetragonal and octahedral holes.
However, the phase behavior for a binary hard-sphere mix-
turewith q ¼ 0:414 is likely to be qualitatively the same as
that of q ¼ 0:3. In both cases, the smaller particles both fit
and are restricted to reside in the octahedral holes at
close packing. MD snapshots included in Ref.  for the
q ¼ 0:414 mixture show a coexistence between a solid and
fluid phase which the authors identified as a monodisperse
fcc phase and a binary liquidphase. However, the fcc phase
depicted in their snapshots contains some small particles,
and is more likely an ISS phase. To summarize, the NaCl
phases previously identified experimentally  and theo-
that there are a vast number of other colloidal and nano-
we have also found an fcc-based ISS phase in experiments
on binary mixtures of oppositely charged colloids with size
ratio ?S=?L¼ 0:73 (Supp. Figs. S6 and S7 ) thereby
systems. Moreover, we would also expect binary mixtures
of charged colloids to form body-centered-cubic (BCC)
based ISSs since the BCC lattice is stable for monodisperse
particles interacting via a Yukawa potential.
In conclusion, we have demonstrated that the ISS is
thermodynamically stable in binary hard-sphere mixtures
and is likely stable for a wide variety of particle interac-
tions. Our finding on the stability of ISS phases is of vital
importance for a wide range of applications as it provides a
method to grow large, defect-free single colloidal ISS
crystals with unprecedented control of the sublattice dop-
ing. For instance, a transistor based on a regularly doped
semiconductor has been realized with a binary nanocrystal
solid . Similarly, intriguing structural color tuning has
previously been achieved by interstitial doping of an fcc
photonic crystal with light absorbing nanoparticles .
Extending this to the ISS phase present in our model, we
would expect more control over the color tuning. More
generally, the availability of a simple model system which
is both theoretically and experimentally realizable will be
ofgreatvalue forthe understandingofthe manyISSphases
arising in atomic and molecular systems. The nucleation of
molecular ISSs can be very complex due to the different
time scales on which both species equilibrate. Realizing a
stable ISS phase in a colloidal system allows one to study
the nucleation process in real space on reasonable time
scales. Additionally, this ISS is an ideal system for exam-
ining interactions between equilibrium interstitial defects
for any concentration.
We acknowledge J. Hoogenboom for particle synthesis
and F. Smallenburg and M. Marechal for fruitful discus-
sions. We acknowledge financial support from Nanodirect,
NWO-CW, a NWO-VICI grant, NanoNed, and FOM.
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