Experimental observation of structural crossover in binary mixtures of
  colloidal hard spheres
Experimental observation of structural crossover in binary mixtures of colloidal hard
spheres
Jörg Baumgartl1,∗, Roel P.A. Dullens1, Marjolein Dijkstra2, Roland Roth3 and Clemens Bechinger1
12. Physikalisches Institut, Universität Stuttgart, 70550 Stuttgart, Germany
2Soft Condensed Matter Group, Utrecht University, 3584 CC Utrecht, The Netherlands
3Max-Planck-Institut für Metallforschung, 70569 Stuttgart, Germany and
Institut für Theoretische und Angewandte Physik, Universität Stuttgart, 70569 Stuttgart, Germany
Using confocal-microscopy we investigate the structure of binary mixtures of colloidal hard spheres
with size ratio q = 0.61. As a function of the packing fraction of the two particle species, we observe
a marked change of the dominant wavelength in the pair correlation function. This behavior is in
excellent agreement with a recently predicted structural crossover in such mixtures. In addition, the
repercussions of structural crossover on the real-space structure of a binary fluid are analyzed. We
suggest a relation between crossover and the lateral extension of networks containing only equally
sized particles that are connected by nearest neighbor bonds. This is supported by Monte-Carlo
simulations which are performed at different packing fractions and size ratios.
PACS numbers: 82.70.Dd, 61.20.-p
Most systems in nature and technology are mixtures
of differently sized particles. Each distinct particle size
introduces another length scale and its competition gives
rise to an exceedingly rich phenomenology in compari-
son with single-component systems. Already the simplest
conceivable multi-component system, i.e. a binary mix-
ture of hard spheres, exhibits interesting and complex
behavior. Just a few examples include entropy driven
formation of binary crystals [1, 2, 3], frustrated crys-
tal growth [4], the Brazil nut effect [5], glass-formation
[6, 7] and entropic selectivity in external fields [8]. Al-
though interaction potentials in atomic systems are more
complex than those of hard spheres, the principle of vol-
ume exclusion is ubiquitous and thus always dominates
the short-range order in liquids [9]. Accordingly, hard
spheres form one of the most important and successful
model systems in describing fundamental properties of
fluids and solids. It has been demonstrated that many
of their features can be directly transferred to atomic
systems where fundamental mechanisms are often ob-
structed by additional material specific effects [10]. Bi-
nary hard sphere systems are fully characterized by their
size ratio q = σS/σB with σi the diameters of the small
(S) and big (B) spheres and the small and big sphere
packing fractions ηS , ηB , respectively.
The pair-correlation functions, gij(r), are the central
measure of structure in fluids; they describe the probabil-
ity of finding a particle of size i at distance r from another
particle of size j. It is well known that all pair-correlation
functions in any fluid mixture with short-ranged inter-
actions (not just hard spheres) exhibit the same type
of asymptotic decay, which can be either purely (mono-
tonic) exponential or exponentially damped oscillatory
([11] and references therein). This prediction, which is
valid in all dimensions, suggests that all pair-correlation
functions decay with a common wavelength and decay
length in the asymptotic limit. For binary hard-sphere
mixtures where ηB � ηS or ηS � ηB , this is rather obvi-
ous since the system is dominated by either big or small
particles. The pair-correlation functions will asymptot-
ically oscillate with a wavelength determined either by
σB (ηB � ηS) or σS (ηS � ηB). Rather surprising is
that the above statement is also valid for all other rela-
tive packing fractions where the system is not dominated
by particles of a single size ([11, 12]). Accordingly, in the
asymptotic limit the (ηS , ηB) phase diagram is divided
by a sharp crossover line where the decay lengths of the
contributions to gij(r) with the two wavelengths become
identical. Below and above this line, however, the pair-
correlation function is either determined by the diameter
of the small spheres or that of the big spheres [13].
Despite the generic character of structural crossover
and the close relationship between structural and me-
chanical properties, this effect has not been observed in
experiments as the asymptotic limit is difficult to reach
in scattering experiments on atomic and molecular liq-
uids. However, recent calculations suggest that struc-
tural crossover is already detectable at relatively small
distances [12]. Because colloidal particles are directly
accessible in real space, such systems provide an oppor-
tunity to explore the structure of binary fluids and to
investigate structural crossover experimentally.
As colloidal suspension we used an aqueous binary
mixture of small melamin particles (σS = 2.9µm) and
big polystyrene spheres (σB = 4.8µm). Addition of
salt screens residual electrostatic interactions thus lead-
ing to an effective hard sphere system. Since melamin
has a higher density (ρM = 1.51g/cm3) than polystyrene
(ρP = 1.05g/cm3) the sedimentation velocities are sim-
ilar and, hence, we obtain a homogeneous system after
mixing. The suspension was contained in a cylindrical
sample cell with a silica bottom plate to allow optical
imaging with an inverted confocal microscope in reflec-
tion mode (Leica TCS SP2). From the images, particle
positions were obtained with digital video microscopy
[14]. Strong layering at the bottom wall allowed us to
image only the first two-dimensional bottom layer of the
three-dimensional system. We define the packing fraction
Figure 1: Different paths with constant total packing fraction
η = ηS + ηB in the (ηS , ηB)- plane. Experimental data (open
symbols: η = 0.72, q = 0.61) are sorted into ten bins. The
bin size is indicated by the ’error bars’. Closed symbols cor-
respond to the MC-simulations (N: η = 0.62, q = 0.4) and
(•: η = 0.57, q = 0.5). For convenience all samples are la-
beled with numbers increasing in the direction indicated by
the arrows.
as ηi = πσ2i /4, with ρi the number density of component
i. Variation of the relative packing fractions of the par-
ticles was achieved by addition of small particles to a
suspension of big spheres (Fig.1). Thus, the total pack-
ing fraction in the two-dimensional bottom layer remains
constant for all samples: η = 0.72. In the following we
will refer to the different samples by the sample numbers
(No.) as given in Fig.1.
Typical snapshots of the system for different packing
fractions of big and small particles are shown in Figs.2A-
C. The images demonstrate how the structure of the bot-
tom layer changes from being rich in small particles (No.
1, Fig.2A) to being rich in big particles (No. 10, Fig.2C).
Fig.2B (No. 5) corresponds to about the same number
density of small and big spheres. In order to analyze the
samples for a possible structural crossover, we calculated
the pair correlation function from the determined particle
positions. To minimize statistical noise we did not distin-
guish between big and small spheres. This is justified be-
cause the crossover has been predicted to be visible in all
pair-correlation functions and thus also in any linear com-
bination [11, 12]. The dominating wavelength in the os-
cillations is identified by computing the total correlation
function htot(r) =
i,j xixjhij(r) =
ij xixj [gij(r)−1],
with the mole fraction xi = ρi/
i ρi of component i [12].
Fig.2D exemplarily shows ln |htot(r)| for samples No. 1,5,
and 9. Note that in this representation the oscillation
wavelength is halved. The correlation functions of sam-
ples No.1 and 9 clearly oscillate with a single wavelength,
respectively, given by ≈ σB/2 and ≈ σS/2. In contrast,
sample 5 does not show a dominating wavelength but
an interference of different length scales which is typical
near the structural crossover. It is important to mention,
Figure 2: A-C) Typical snapshots of the bottom layer of a
binary mixture observed with a confocal microscope used in
reflection mode. The mixtures correspond to sample 10 (A), 5
(B) and 1 (C). The field of view is 40×40µm2. D) Logarithmic
plot of the total correlation functions htot(r) for the experi-
mental binary mixtures with η = 0.72 ± 0.04. Correlation
functions are plotted for sample numbers 1,5 and 9 (compare
Fig.1) and are shifted in vertical direction for clarity. The hor-
izontal bars correspond to σB/2 and σS/2, respectively. E)
Fourier-transforms of htot(r) for the experimental data points
(compare Fig. 1). Vertical lines indicate the wave vectors k
corresponding to the diameters of the small (S) and big par-
ticles (B), respectively. (color online).
that this intermediate behavior is only observed for sam-
ples No.5 and 6, i.e. only for about 10% of the entire
range over which ηB and ηS was varied. The experimen-
tally identified crossover-region is in excellent agreement
with the theoretically calculated value of ηS ≈ 0.3 at
those size ratios, which were determined from the decay
of the pair correlation functions calculated within density
functional theory in the test particle limit [15]. Fig.2E
Figure 3: Visualization of the different bond-types as determined by a Delaunay triangulation: big-big (black), big-small
(yellow) and small-small (red). Different plots correspond to the sample numbers as indicated in Fig.1. The field of view is
180× 180µm2.
shows the Fourier transforms of htot(r) for all samples
where the rather sudden change of the dominating wave-
length is seen more clearly [16]. At small and high pack-
ing fractions, the correlations are clearly dominated by
frequencies corresponding to either small or large parti-
cles (vertical lines) while around sample No.5 hardly any
dominating frequency is observed. This experimentally
confirms structural crossover as well as its occurrence at
finite particle distances.
So far, structural crossover has been discussed in terms
Figure 4: Averaged radii of gyration 〈Rig〉 (normalized to L/2
with L2 the size of the field of view) of networks formed by
large (solid symbols) and small particles (open symbols) as a
function of the sample number for A) the experimental data,
B) the MC-simulations at η = 0.57 and q = 0.5 and, C) the
MC-simulations at η = 0.62 and q = 0.4. The correspond-
ing packing fraction of small particles ηS is indicated as well.
The grey area and the dashed line respectively indicate the
crossover as inferred from the correlation functions and from
density functional theory. (color online).
of pair correlation functions, i.e. spatially averaged quan-
tities. Since our experiments naturally provide detailed
structural information, we investigate what the reper-
cussions are of the structural crossover on the real-space
structure. We first subjected a Delaunay triangulation to
the set of particle centers and identified nearest-neighbor
bonds between big-big (black), big-small (yellow), and
small-small (red) particles, respectively (see Fig.3). As
observed in Fig.3, sample 1 predominantly consists of big-
big bonds which form a large network spreading across
the entire field of view. With increasing sample No., i.e.
increasing ηS , the number of small-small bonds increases,
which leads to fragmentation of the big-big network into
smaller, randomly distributed patches. At large sample
numbers, the role of big and small particles is inverted
and small-small bonds form a network spanning the en-
tire area (No.10). Having distinguished between differ-
ent bond-types, a natural and well-known measure of the
spatial extend of a network formed by ni particles of size
i at positions ~xik (k = 1 . . . n
i) is given by the radius of
gyration Rig =
k=1(~x
k − ~R
2, with ~Ri0 the cen-
troid position of the network. Computing this quantity
for all, say N iC , networks formed by connected particles
of size finally yields a weighted averaged radius of gyra-
tion 〈Rig〉 =
m=1 ni(m)R
g(m) where N
i denote the
total number of particles i. We calculated 〈Rig〉 for net-
works consisting of connected big or small particles and
plotted these values for our experimental data in Fig.4A
as a function of the sample number. At small and high
sample numbers the quantities saturate while a relatively
sharp transition with an intersection point occurs around
sample 6. This location is indeed in very good agreement
with the crossover transition as determined from the cor-
relation functions in Fig.2 and density functional theory
(also indicated in Fig. 4A). This suggests that the struc-
tural crossover corresponds to a competition between the
sizes of networks consisting of connected big or small par-
ticles, respectively.
As structural crossover is also predicted for other size
ratios and packing fractions, we use Monte-Carlo (MC)
simulations to test our findings for more dilute systems
with size ratios q = 0.5 and q = 0.4. The corresponding
paths through the phase diagram (see closed symbols in
Fig.1) were obtained from 2-dimensional simulations with
a fixed number of particles of about 0 < N < 3000 for
both species and box areas of about 1500σ2B employing
periodic boundary conditions. From the configurational
snapshots we first determined the region of crossover by
analyzing htot(r) (the correlation functions are sampled
using 104 MC cycles per particle). Then, we performed
the above described Delaunay triangulation to calculate
〈Rig〉 for networks of connected big or small particles, re-
spectively. The corresponding radii of gyration are plot-
ted in Fig.4B and C and show a similar behavior as in the
experiment. Again, the intersection points are consistent
with the crossover region as inferred from the correlation
functions and DFT calculations. Note that the crossover
region sensitively depends on the size ratio and packing
fractions. Both the experiment and Monte-Carlo sim-
ulations show that structural crossover is accompanied
by a pronounced change in the typical size of networks
consisting of connected big and small particles. By in-
troducing small particles into a system of big spheres,
connections between big particles are broken and, at the
same time, connections between small particles are made.
This sensitively affects the typical size of networks con-
taining connected, equally-sized particles and thereby the
chance of finding another particle with the same size at a
relatively large distance. Consequently, the change from
〈RBg 〉 > 〈RSg 〉 to 〈RSg 〉 > 〈RBg 〉 (and vice versa) provides
a simple real-space argument why the oscillation wave-
length of the gij(r) in the asymptotic limit is either set
by σB or σS .
We have experimentally demonstrated the structural
crossover in a binary colloidal hard sphere system. Fur-
thermore, we show that structural crossover is strongly
coupled to the size of networks containing connected
equally-sized particles only. Going across the structural
crossover, the size ratio of such networks comprised by
either connected big or small particles is reversed. We be-
lieve this real-space configurational picture of structural
crossover is not just applicable to binary hard spheres,
as structural crossover is a generic feature of mixtures
with competing length scales. Moreover, it shows inter-
esting similarities with force chains in granular matter
[17] and glassy systems [6, 7, 18] of dissimilar sized parti-
cles. Therefore, our finding may help to gain more insight
into structure-related properties in binary systems at an
universal level.
∗Electronic address: j.baumgartl@physik.uni-
stuttgart.de
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