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Journal ArticleDOI

Periodic layers of a dodecagonal quasicrystal and a floating hexagonal crystal in sedimentation-diffusion equilibria of colloids

14 Sep 2017-Journal of Chemical Physics (AIP Publishing LLC)-Vol. 147, Iss: 10, pp 104902
TL;DR: It is surprising that the formation of layers with dodecagonal, square, and hexagonal symmetries at the relevant pressures in the three-dimensional sedimentation column is observed, and a floating crystal is formed between the colloidal fluid regions.
Abstract: We investigate the behaviour of a system of colloidal particles interacting with a hard-core and a repulsive square shoulder potential under the influence of a gravitational field using event-driven Brownian dynamics simulations. We use a fixed square shoulder diameter equal to 1.4 times the hard-core diameter of the colloids, for which we have previously calculated the equilibrium phase diagram considering two-dimensional disks [H. Pattabhiraman et al., J. Chem. Phys. 143, 164905 (2015) and H. Pattabhiraman and M. Dijkstra, J. Phys.: Condens. Matter 20, 094003 (2017)]. The parameters in the simulations are chosen such that the pressure at the bottom of the sediment facilitates the formation of phases in accordance with the calculated phase diagram of the two-dimensional system. It is surprising that we observe the formation of layers with dodecagonal, square, and hexagonal symmetries at the relevant pressures in the three-dimensional sedimentation column. In addition, we also observe a re-entrant behaviour exhibited by the colloidal fluid phase, engulfing a hexagonal crystal phase, in the sedimentation column. In other words, a floating crystal is formed between the colloidal fluid regions.

Summary (2 min read)

I. INTRODUCTION

  • In the case of colloidal suspensions consisting of particles with sizes on the order of micro-meters, the effect of the gravitational force is not negligible.
  • As a result of this inhomogeneous density distribution in colloidal suspensions, the particles at the bottom of the sediment can crystallise when they reach a certain density.
  • Experimentally, sedimentation is regarded as a prevalent tool to extract information regarding the equilibrium phase behaviour of the system.
  • In their previous studies, 19, 20 the authors have theoretically calculated the phase diagram of a two-dimensional system of particles interacting with a hard-core and repulsive square shoulder (HCSS) potential.

A. Computational methodology

  • The authors perform Event-Driven Brownian Dynamics (EDBD) simulations of N spherical particles of diameter σ HS and buoyant mass m interacting with the HCSS potential in the NVT ensemble.
  • In the EDBD method, a sequence of collision events involving only two particles at any given instant is computed.
  • Previous experiments and simulations show that the structure of the crystalline sediment depends strongly on the Brownian time, the sedimentation time, and the initial volume fraction.
  • The authors note that the direct comparison of pressures between two-and three-dimensional systems may not always hold true.
  • Using systems with different Peclet numbers allows us to study the effect of the settling rate on the formation of the quasicrystal.

A. Formation of layers with dodecagonal symmetry

  • The authors start with the formation of layers with dodecagonal symmetry.
  • First, there are large portions of connected square tilings in the sediment obtained for the lower Peclet number, while the square tilings are more uniformly distributed in the case of the high Peclet number sediment.
  • Here, the particles in the first layer are represented as filled blue circles, while those of the second layer are represented as open black circles.
  • This happens to be the case in their simulations, which leads to the formation of these periodic layers.
  • Additionally, given that this layered structure of the quasicrystal is not a bulk equilibrium structure, it is interesting to know how many layers of the quasicrystal can be obtained using this sedimentation method.

B. Formation of layers with square symmetry

  • As can be noticed in the figure, two layers at the bottom of the column have a large concentration of purple coloured particles, denoting the formation of layers with square symmetry.
  • Further, the authors plot the pressure and density profiles along the sedimentation column, respectively, in Figs. 9(c) and 9(d).
  • Obviously, the settling rate plays an important role in the annealing process.

C. Formation of (suspended) layers with hexagonal symmetry

  • One of the peculiar features exhibited by the HCSS system is the formation of a low-density hexagonal phase, where the particles are separated by a distance equal to the square shoulder diameter δ.
  • The authors objective in this section is two-fold.
  • Combining this time evolution of BOO and the pressure [Fig. 11(c) ] and density [Fig. 11(d) ] profiles along the sedimentation column, the authors find the formation of about twenty layers of hexagonal symmetry.
  • Once it reaches the density where the hexagonal phase is found to be stable, structures with hexagonal symmetry start to form at the bottom layers.
  • This melting process of the first layer of the sediment can also be seen from the particle configurations taken at different times, which is shown in Fig. 13 .

IV. CONCLUSIONS AND OUTLOOK

  • To summarise, the authors studied the sedimentation behaviour of a system of particles interacting with a hard-core and a repulsive square shoulder potential with a fixed shoulder width equal to 1.4 times the hard-core diameter.
  • The authors find that the system forms a two-dimensional layered structure because of energetic arguments.
  • This enables us to validate the formation of the thermodynamically stable phases as predicted by the two-dimensional phase diagram, which is highly surprising.
  • Further ascertaining the validity of the calculated phase diagram, the authors find that the fluid phase, in the case of a low-density hexagonal phase, exhibits a re-entrant phase behaviour along the height of the sedimentation column.

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Periodic layers of a dodecagonal quasicrystal and a floating hexagonal crystal in
sedimentation-diffusion equilibria of colloids
Harini Pattabhiraman, and Marjolein Dijkstra
Citation: The Journal of Chemical Physics 147, 104902 (2017);
View online: https://doi.org/10.1063/1.4993521
View Table of Contents: http://aip.scitation.org/toc/jcp/147/10
Published by the American Institute of Physics
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THE JOURNAL OF CHEMICAL PHYSICS 147, 104902 (2017)
Periodic layers of a dodecagonal quasicrystal and a floating hexagonal
crystal in sedimentation-diffusion equilibria of colloids
Harini Pattabhiraman
a)
and Marjolein Dijkstra
b)
Department of Physics, Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University,
Princetonplein 5, 3584 CC Utrecht, The Netherlands
(Received 29 June 2017; accepted 30 August 2017; published online 14 September 2017)
We investigate the behaviour of a system of colloidal particles interacting with a hard-core and a
repulsive square shoulder potential under the influence of a gravitational field using event-driven
Brownian dynamics simulations. We use a fixed square shoulder diameter equal to 1.4 times the hard-
core diameter of the colloids, for which we have previously calculated the equilibrium phase diagram
considering two-dimensional disks [H. Pattabhiraman et al., J. Chem. Phys. 143, 164905 (2015) and
H. Pattabhiraman and M. Dijkstra, J. Phys.: Condens. Matter 20, 094003 (2017)]. The parameters
in the simulations are chosen such that the pressure at the bottom of the sediment facilitates the
formation of phases in accordance with the calculated phase diagram of the two-dimensional system.
It is surprising that we observe the formation of layers with dodecagonal, square, and hexagonal
symmetries at the relevant pressures in the three-dimensional sedimentation column. In addition,
we also observe a re-entrant behaviour exhibited by the colloidal fluid phase, engulfing a hexagonal
crystal phase, in the sedimentation column. In other words, a floating crystal is formed between the
colloidal fluid regions. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4993521]
I. INTRODUCTION
In the case of colloidal suspensions consisting of particles
with sizes on the order of micro-meters, the effect of the grav-
itational force is not negligible. Under these conditions, the
gravitational energy and the thermal energy of the colloids are
comparable. This leads to the formation of a spatially inho-
mogeneous suspension in which the density of the particles
varies along the height of the suspension. The inhomogeneous
distribution of the colloidal particles along the height under
the influence of gravity is termed as sedimentation.
As a result of this inhomogeneous density distribution in
colloidal suspensions, the particles at the bottom of the sedi-
ment can crystallise when they reach a certain density. Exper-
imentally, sedimentation is regarded as a prevalent tool to
extract information regarding the equilibrium phase behaviour
of the system. For example, the measured concentration pro-
files can be inverted to obtain the osmotic equation of state.
Sedimentation processes can also be used the other way
around, i.e., they can be used to validate the theoretically calcu-
lated equilibrium phase behaviour of a system and thereby its
bulk phase behaviour. For example, a system of hard spheres
has been a model system for sedimentation studies. This sys-
tem exhibits a phase behaviour characterised by a fluid phase at
low densities and a face-centered cubic phase at high densities.
This behaviour, which was earlier theoretically predicted,
1
has
later been corroborated by sedimentation studies.
2
Sedimentation behaviour of various charged particles,
3
mixtures of hard particles,
4
and particles of different shapes
5
which result in the formation of various colloidal crystals has
a)
h.pattabhiraman@uu.nl
b)
m.dijkstra@uu.nl
been extensively studied. However, similar studies involving
the formation of quasicrystals (QCs) by sedimentation has
received less attention. Quasicrystals are solids with long-
range orientational order and no periodicity and may exhibit
intriguing properties including the formation of photonic band
gaps.
611
Hence, the experimental realisation of quasicrystals
is a topic of intense research.
1218
In our previous studies,
19,20
we have theoretically cal-
culated the phase diagram of a two-dimensional system of
particles interacting with a hard-core and repulsive square
shoulder (HCSS) potential. At a shoulder width of 1.4 times
the hard core diameter of the particle, we find a stable random-
tiling dodecagonal quasicrystal. In this work, we investigate
how the phase behaviour in a two-dimensional HCSS sys-
tem extends to three-dimensional systems and whether or not
multiple layers of this quasicrystal can be self-assembled by
sedimentation in three dimensions.
This paper is organised as follows. In Sec. II, we present
the simulation and analysis methods used. In Sec. III, we indi-
vidually discuss the formation of layers with various symme-
tries, and finally we present the conclusions and the direction
of future studies in Sec. IV.
II. METHODS
We first explain the simulation model and computational
methods used for this study in Sec. II A and then give an
account of the analysis methods in Sec. II B.
A. Computational methodology
We perform Event-Driven Brownian Dynamics (EDBD)
simulations of N spherical particles of diameter σ
HS
and buoy-
ant mass m interacting with the HCSS potential in the NVT
0021-9606/2017/147(10)/104902/11/$30.00 147, 104902-1 Published by AIP Publishing.

104902-2 H. Pattabhiraman and M. Dijkstra J. Chem. Phys. 147, 104902 (2017)
ensemble. The HCSS potential can be written as a sum of a
hard-sphere potential V
HS
(r) and a square-shoulder potential
V
SS
(r), i.e.,
V
HCSS
(r) = V
HS
(r) + V
SS
(r), (1)
where
V
HS
(r) =
, r σ
HS
0, r > σ
HS
(2)
and
V
SS
(r) =
, r δ
0, r > δ
, (3)
where r is the interparticle center-of-mass distance and > 0
is the height of the square shoulder.
In the EDBD method, a sequence of collision events
involving only two particles at any given instant is com-
puted. During the simulation, the velocities of the particles
are randomly adjusted at regular intervals t as
v(t + t) = α
t
v(t) + β
t
v
R
(t), (4)
where v(t) and v(t + t) are, respectively, the velocities of the
particles before and after the stochastic velocity adjustment,
v
R
(t) is a 3-D Gaussian variable with a mean of 0 and vari-
ance of k
B
T/m, with k
B
as the Boltzmann constant and T as
the temperature. Further, α
t
has a value 1/
2 with a proba-
bility νt and 1 otherwise. The temperature is kept constant
by setting β
t
=
q
1 α
2
t
. In accordance to previous EDBD
simulations,
21,22
we set ν to 10τ
1
MD
and t to 0.01τ
MD
, where
τ
MD
is the unit of time of an event-driven molecular dynamics
simulation given as τ
MD
=
m/k
B
T σ
HS
.
The present simulation method is an event driven molecu-
lar dynamics (EDMD) method which is extended with Brow-
nian motion. Previous experiments and simulations show that
the structure of the crystalline sediment depends strongly on
the Brownian time, the sedimentation time, and the initial vol-
ume fraction.
21,23
The faster the sedimentation, the higher the
initial volume fraction, and the less time the system has avail-
able for equilibration during sedimentation and the crystalline
sediment will become more defective. Brownian motion is,
thus, important for equilibration during sedimentation. In this
case, the Brownian time is set by α
t
and ν. It is convenient to
define a dimensionless Peclet number g
*
that is equal to the
inverse gravitational length and a dimensionless particle flux
as defined by the ratio of the Brownian time and sedimentation
time. In our simulations, the Peclet number ranges from 1 to
5, whereas the particle flux is in the range of 0.01-0.05, where
previous simulations and experiments show that there is ample
time for equilibration and least defective crystalline sediments
are obtained.
21,23
The simulation box of volume V has a square cross section
of area A and is elongated in the z-direction. Periodic boundary
conditions are applied along the cross section. In the elongated
direction, the particles are confined between two smooth hard
walls at z = 0 and z = H with H as the height of the sedimen-
tation column. The height H is chosen such that the density
at z =
(
H σ
HS
/2
)
is sufficiently small, which allows us to
consider the system to be infinitely long in the z-direction. We
perform simulations starting with a non-overlapping isotropic
fluid state filling the entire sedimentation column with packing
fraction η = 0.01. To mimic sedimentation experiments, these
particles are further subjected to a gravitational field, which is
expressed as an external potential φ(z) written as
φ(z) =
mgz, σ
HS
/2 z (H σ
HS
/2)
, otherwise
, (5)
where g is the acceleration due to gravity and z is the vertical
coordinate of the particle. The effect of the gravitational field
on the particles is quantified in terms of the gravitational Peclet
number defined as g
= mgσ
HS
/k
B
T.
In this work, we scrutinise the kinetic formation of
the thermodynamic stable phases described for the two-
dimensional HCSS system with δ = 1.40σ
HD
, where σ
HD
is the hard-disk diameter, as used in our previous works.
19,24
To do so, we perform simulations such that the pressure mea-
sured at the bottom of the sedimentation column, i.e., at z = 0,
corresponds to the region of stability of a particular phase in the
2-D phase diagram. This pressure is calculated as P
= βP(z
= 0)σ
3
HS
= g
.ρ
A
, where ρ
A
is the mean areal density defined
as the number of particles at the bottom of the sample per
area ρ
A
= N σ
2
HS
/A, and we compare P
*
with P
2D
= βPσ
2
HD
directly. However, we note that the direct comparison of pres-
sures between two- and three-dimensional systems may not
always hold true. For example, the fluid (hexatic)-solid transi-
tion in the case of hard disks is at P
2D
= 9.17,
25
whereas that
for hard spheres is at βPσ
3
HS
= 11.57.
26
It is good to remind
the readers that we use a three-dimensional system of spheres
here to validate the phase behaviour of a two-dimensional sys-
tem of disks. The underlying reason for this will be explained
in Secs. II B and III AIII C.
The phases considered in this study are dodecagonal qua-
sicrystal, square, low-density hexagonal, and fluid phases. We
especially focus on the possibility of the formation of the qua-
sicrystal and, thus, consider the cross section to be squares with
sides of length 58σ
HS
, which can accommodate a random-
tiling dodecagonal quasicrystal at a density ρ
= 1.07, similar
to that used in Ref. 19. The list of parameters used to simu-
late the different phases is given in Table I. The corresponding
values of pressures at the bottom P
2D
= βP(z = 0)σ
3
HS
are
marked in the phase diagram given in Fig. 1. As a supple-
mentary study, we use two parameter sets having different
Peclet numbers to simulate the quasicrystal. As mentioned
above, a higher value of the Peclet number results in a condi-
tion of high settling rate of the particles, i.e., the particles do
not have enough time to rearrange and equilibrate, and vice
versa. Using systems with different Peclet numbers allows us
TABLE I. System parameters used in the EDBD simulations of a HCSS
system with δ = 1.40σ
HS
under gravity.
N k
B
T/ g
*
βP(z = 0)σ
3
HS
Stable phase at bottom
5 × 10
4
0.25 2.00 30.0 Quasicrystal
2 × 10
4
0.25 5.00 30.0 Quasicrystal
2 × 10
4
0.25 4.00 23.8 Square
5 × 10
4
0.15 0.67 10.0 Low-density hexagonal
1 × 10
5
0.15 0.67 20.0 Fluid

104902-3 H. Pattabhiraman and M. Dijkstra J. Chem. Phys. 147, 104902 (2017)
FIG. 1. Phase diagram in the (reduced) pressure-temperature plane for a two-
dimensional HCSS system with shoulder width δ = 1.40σ
HD
. The reduced
quantities are defined as P
2D
= βPσ
2
HD
and T
= k
B
T/. The phases marked
are fluid (FL), low-density hexagonal (LDH), square (SQ), dodecagonal qua-
sicrystal (QC), and high-density hexagonal (HDH). The crosses denote the
state points corresponding to the pressures at the bottom of the sediment.
to study the effect of the settling rate on the formation of the
quasicrystal.
B. Structural analysis
In order to characterise the different phases, we employ
an analysis method that is two-dimensional in nature since the
phases observed in the sedimentation column have a layered
structure and resemble the phases observed for the HCSS sys-
tem in 2-D. Specifically, we first identify different layers of the
sediment and then carry out the following analysis procedure
in these layers. We construct the polygonal tiling of the layer
and calculate the two-dimensional m-fold bond orientational
order parameter (BOO) of each particle j in layer l, χ
l
m
(j), and
the average BOO of each layer χ
l
m
as explained in Ref. 20.
The polygonal tiling of each layer is constructed by draw-
ing bonds between the neighbouring particles of each particle j,
which are at a center-of-mass distance smaller than the square
shoulder diameter δ from particle j.
We, then, calculate the m-fold BOO of each particle j in
layer l as
χ
l
m
(j) =
1
N
B
(j)
N
B
(j)
X
k=1
exp(imθ
r
jk
)
2
, (6)
where m is the symmetry of interest, r
jk
is the center-of-mass
distance vector between two neighbours j and k, θ
r
jk
is the angle
TABLE II. Method of classification of particle j belonging to layer l
according to its bond orientational order (BOO) χ
l
m
(j).
Symmetry BOO conditions Colour
Fluid/other (OT) χ
l
4
(j), χ
l
6
(j), χ
l
12
(j) < 0.5 Orange
Crystal χ
l
4
(j), χ
l
6
(j), χ
l
12
(j) > 0.5
Square (SQ) χ
l
4
(j) > χ
l
6
(j), χ
l
12
(j) Purple
Hexagonal (HX) χ
l
6
(j) > χ
l
4
(j), χ
l
12
(j) Green
Dodecagonal (QC) χ
l
12
(j) > χ
l
4
(j), χ
l
6
(j) Red
FIG. 2. Colour scheme for classes of particles based on the BOO classification
described in Table II.
between r
jk
and an arbitrary axis, and N
B
(j) is the number of
neighbours of particle j in the same layer. For each particle j, we
calculate χ
l
4
(j), χ
l
6
(j), and χ
l
12
(j), respectively, representing
square, hexagonal, and dodecagonal symmetries.
The particles are classified based on their BOO according
to the method given in Table II. We consider a particle to be
fluid-like if each of the three χ
l
m
(j) is less than 0.5. On the other
hand, if each of χ
l
m
(j) is greater than 0.5, then a particle is said
to have symmetry m1 if χ
l
m1
(j) is greater than the other two,
namely, χ
l
m2
(j) and χ
l
m3
(j). Further, we identify and colour the
particles according to the following scheme: particles of square
symmetry in purple, those of hexagonal in green, dodecagonal
in red, and fluid-like in orange as shown in Fig. 2.
After calculating the BOO of each particle, the average
BOO of each layer is then evaluated as
27
χ
l
m
=
1
N
l
N
l
X
j=1
χ
l
m
(j), (7)
where N
l
is the number of particles in each layer.
III. RESULTS AND DISCUSSION
In this section, we consider individually the different sed-
imentation simulations carried out to obtain the various stable
phases calculated for the two-dimensional HCSS system.
A. Formation of layers with dodecagonal symmetry
We start with the formation of layers with dodecagonal
symmetry. In this section, we present the sedimentation sim-
ulations using two different Peclet numbers in order to assess
the effect of the settling rate on the formation of the quasicrys-
tal. We first compare the formation of the quasicrystal formed
in these simulations and then analyse the driving force behind
the formation of these layers. Finally, we review the valid-
ity of the phase diagram given in Fig. 1 by comparing the
phases formed in the sedimentation column. The dodecago-
nal quasicrystal (QC) which, as seen in the phase diagram, is
sandwiched between two periodic crystal phases with square
and hexagonal symmetries. Thus, it is interesting to note if and
how the interfaces between the quasicrystal and the periodic
crystals are formed in the sedimentation column.
We first present a typical configuration of the sediment
forming quasicrystalline layers in Fig. 3. The panels on the
left correspond to simulations with a Peclet number g
*
= 5.0
and those on the right are obtained for g
*
= 2.0. The particles
here are coloured according to the convention explained in
Fig. 2. We notice the formation of about two quasicrystalline
layers for g
*
= 5.0 and about four quasicrystalline layers for
g
*
= 2.0. This difference in the number of layers is due to a

104902-4 H. Pattabhiraman and M. Dijkstra J. Chem. Phys. 147, 104902 (2017)
FIG. 3. Comparison of the quasicrys-
tal (QC) sediment formed for Peclet
numbers g
*
= 5.0 (left) and 2.0 (right).
Side view of a configuration of the sed-
imentation column obtained at t
MD
= 800 (top). The particles are coloured
according to their individual BOO: qua-
sicrystal (red), square (purple), hexag-
onal (green), and fluid (orange). The
m-fold BOO of each layer with time cal-
culated for symmetries m = 4 (middle)
and 12 (bottom) showing, respectively,
the formation of layers with square and
dodecagonal symmetries.
difference in the height range that corresponds to the pres-
sure range of the stable quasicrystal phase. This height range
decreases with increasing g
*
. Additionally, we observe that
most of the particles seen in these layers are coloured either
in purple or red which, respectively, represent square or
dodecagonal symmetries. Therefore, we follow the dynam-
ics of the formation of these layers by calculating the BOO
χ
l
4
and χ
l
12
of each layer as a function of time. The time
evolution of χ
l
4
is given in the middle panel in Fig. 3 and
that of χ
l
12
is given at the bottom. In these time evolution
heat maps, the time scale t
MD
is plotted on the horizon-
tal axis and the layer number is plotted along the vertical
axis.
We make the following observations from these plots: (1)
In both cases, the value of χ
l
12
is higher than that of χ
l
4
, which
confirms the dodecagonal symmetry of these layers. (2) With
increasing time, we find that the fraction of fluid in the sed-
imentation column decreases, as seen by the receding blue
region in these plots. Alternatively, this means that more crys-
talline layers are formed with time. (3) The value of χ
l
12
at a
given time decreases as we go up in the sediment indicating that
the layers on the top are more fluid-like than the bottom layers.
(4) Finally, we see that χ
l
12
obtained for the sediment at higher
Peclet numbers is larger than that at lower Peclet numbers. This
is counter-intuitive as this suggests that faster settling of the
particles result in the formation of a less-defective quasicrystal.
We investigate this further by plotting the polygonal tiling
constructed for the bottom two layers as a function of time for
both the sediments. The top view of these tilings is given in
Fig. 4 for g
*
= 5.0 and in Fig. 5 for g
*
= 2.0. Two striking fea-
tures are conspicuous from these polygonal tilings. First, there
are large portions of connected square tilings in the sediment
obtained for the lower Peclet number, while the square tilings
are more uniformly distributed in the case of the high Peclet
number sediment. Second, the position of the tiles in the first
and second layers seems to be on top of each other. Let us now
evaluate each of these observations separately.
Firstly, we analyze the tilings and quantify the square tiles
by calculating the ratio of areas occupied by the square tiles
to those of the triangle tiles. This also relates to the square-
triangle tiling description of a dodecagonal quasicrystal, where
the maximum entropy of the tiling corresponds to equal areas
of squares and triangles.
2830
In the current sediments, we find
that the ratio of the areas of squares to triangles for g
*
= 2.0
is 1.40 ± 0.05 and for g
*
= 5.0 is 1.15 ± 0.03. In other words,
in both cases, there are more squares formed than in an ideal
dodecagonal tiling. This excess of squares is larger for the low
Peclet number sediment. This can be explained as follows.
A lower Peclet number refers to a lower rate of sedimentation
and, thus, a longer relaxation time for the particles to rearrange.
The density at the bottom of the sample increases slowly since
the beginning of the sedimentation simulation, as more and

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Journal ArticleDOI
TL;DR: It is argued that identifying and following a crossover line in the phase diagram towards higher densities where the solid phase(s) occur is a good strategy for finding quasicrystals in a wide variety of systems.
Abstract: We investigate the liquid state structure of the two-dimensional model introduced by Barkan et al. [Phys. Rev. Lett. 113, 098304 (2014)], which exhibits quasicrystalline and other unusual solid phases, focusing on the radial distribution function $g(r)$ and its asymptotic decay $r\ensuremath{\rightarrow}\ensuremath{\infty}$. For this particular model system, we find that as the density is increased there is a structural crossover from damped oscillatory asymptotic decay with one wavelength to damped oscillatory asymptotic decay with another distinct wavelength. The ratio of these wavelengths is $\ensuremath{\approx}1.932$. Following the locus in the phase diagram of this structural crossover leads directly to the region where quasicrystals are found. We argue that identifying and following such a crossover line in the phase diagram towards higher densities where the solid phase(s) occur is a good strategy for finding quasicrystals in a wide variety of systems. We also show how the pole analysis of the asymptotic decay of equilibrium fluid correlations is intimately connected with the nonequilibrium growth or decay of small-amplitude density fluctuations in a bulk fluid.

23 citations

Journal ArticleDOI
TL;DR: The stable phases for binary hard-sphere systems with varying diameter ratios are determined using Monte Carlo simulations and analytical equations of state available in literature and calculate the corresponding stacking diagrams.
Abstract: Colloidal photonic crystals, which show a complete band gap in the visible region, have numerous applications in fibre optics, energy storage and conversion, and optical wave guides. Intriguingly, two of the best examples of photonic crystals, the diamond and pyrochlore structure, can be self-assembled into the colloidal MgCu2 Laves phase crystal from a simple binary hard-sphere mixture. For these colloidal length scales thermal and gravitational energies are often comparable and therefore it is worthwhile to study the sedimentation phase behavior of these systems. For a multicomponent system this is possible through a theoretical construct known as a stacking diagram, which constitutes a set of all possible stacking sequences of phases in a sedimentation column, and uses as input the bulk phase diagram of the system in the chemical potential plane. We determine the stable phases for binary hard-sphere systems with varying diameter ratios using Monte Carlo simulations and analytical equations of state available in literature and calculate the corresponding stacking diagrams. We also discuss observations from event-driven Brownian dynamics simulations in relation to our theoretical stacking diagrams.

13 citations

Journal ArticleDOI
TL;DR: This work uses Monte Carlo simulations to explore the types of crystal structures that can form in a simple hard-core square shoulder model that explicitly incorporates two favored distances between the particles, and shows the stability of a rich variety of crystal phases, such as body-centered orthogonal (BCO) lattices not previously considered for the square shoulders.
Abstract: In many cases, the stability of complex structures in colloidal systems is enhanced by a competition between different length scales. Inspired by recent experiments on nanoparticles coated with polymers, we use Monte Carlo simulations to explore the types of crystal structures that can form in a simple hard-core square shoulder model that explicitly incorporates two favored distances between the particles. To this end, we combine Monte Carlo-based crystal structure finding algorithms with free energies obtained using a mean-field cell theory approach, and draw phase diagrams for two different values of the square shoulder width as a function of the density and temperature. Moreover, we map out the zero-temperature phase diagram for a broad range of shoulder widths. Our results show the stability of a rich variety of crystal phases, such as body-centered orthogonal (BCO) lattices not previously considered for the square shoulder model.

12 citations

Journal ArticleDOI
TL;DR: In this article , a simplified two-dimensional effective-solvent model of triblock Janus particles, consisting of three interaction sites in a linear configuration, a core particle, and two particles modeling the attractive patches at the poles, is developed to study the mechanism of nucleation and self-assembly in triblockJanus particles.
Abstract: A simplified two-dimensional effective-solvent model of triblock Janus particles, consisting of three interaction sites in a linear configuration, a core particle, and two particles modeling the attractive patches at the poles, is developed to study the mechanism of nucleation and self-assembly in triblock Janus particles. The potential energy parameters are tuned against phase transition temperatures and free energy barriers to the nucleation of crystalline phases, calculated from our previous detailed model of Janus particles. Vapor-liquid equilibria and critical temperatures are calculated by grand-canonical molecular dynamics simulations for particles of different patch widths. With metadynamics, phase equilibria, mechanism of nucleation, and free energy barriers to nucleation are investigated. The minimum free energy path to nucleation indicates two steps. The first step, with a higher free energy increase, consists of the densification of the fluid into a disordered cluster. In the second step, of a lower free energy barrier, the inner particles of the disordered cluster reorient to form a crystalline nucleus. This two-step mechanism of nucleation of a kagome lattice is in complete agreement with the experiment and with our previous simulations using a detailed model of Janus particles. Large systems at a slight supersaturation generate multiple crystalline domains, which are misaligned at the grain boundaries. In complete agreement with the experiment and with previous simulation results, we observe a two-step mechanism for crystal growth: melting of the smaller (less stable) crystallites to a fluid followed by recrystallization at the surface of neighboring bigger (more stable) crystallites. A comparison of the present softer modeling of a Janus particle with harder models in the literature for self-assembly of Janus particles indicates that softer potentials stabilize open lattices (e.g., kagome) more than dense lattices (e.g., hexagonal). Also, experimental locations of phase transition points and barrier heights to nucleation are better reproduced by the present model than by the existing simple models.

9 citations

References
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Journal ArticleDOI
TL;DR: In this paper, large-scale computer simulations of the hard disk system at high densities in the region of the melting transition are presented, where the authors reproduce the equation of state, previously obtained using the event-chain Monte Carlo algorithm, with a massively parallel implementation of the local Monte Carlo method and with event-driven molecular dynamics.
Abstract: We report large-scale computer simulations of the hard-disk system at high densities in the region of the melting transition. Our simulations reproduce the equation of state, previously obtained using the event-chain Monte Carlo algorithm, with a massively parallel implementation of the local Monte Carlo method and with event-driven molecular dynamics. We analyze the relative performance of these simulation methods to sample configuration space and approach equilibrium. Our results confirm the first-order nature of the melting phase transition in hard disks. Phase coexistence is visualized for individual configurations via the orientational order parameter field. The analysis of positional order confirms the existence of the hexatic phase.

216 citations

Journal ArticleDOI
13 Feb 2014-Nature
TL;DR: Two-dimensional hard disks decorated with step-like square-shoulder repulsion that mimics, for example, the soft alkyl shell around the aromatic core in dendritic micelles are studied, finding a family of quasicrystals with 10-, 12-, 18- and 24-fold bond orientational order which originate from mosaics of equilateral and isosceles triangles formed by particles arranged core-to-core and shoulder- to-sh shoulder.
Abstract: Over the past decade, quasicrystalline order has been observed in many soft-matter systems: in dendritic micelles, in star and tetrablock terpolymer melts and in diblock copolymer and surfactant micelles The formation of quasicrystals from such a broad range of 'soft' macromolecular micelles suggests that they assemble by a generic mechanism rather than being dependent on the specific chemistry of each system Indeed, micellar softness has been postulated and shown to lead to quasicrystalline order Here we theoretically explore this link by studying two-dimensional hard disks decorated with step-like square-shoulder repulsion that mimics, for example, the soft alkyl shell around the aromatic core in dendritic micelles We find a family of quasicrystals with 10-, 12-, 18- and 24-fold bond orientational order which originate from mosaics of equilateral and isosceles triangles formed by particles arranged core-to-core and shoulder-to-shoulder The pair interaction responsible for these phases highlights the role of local packing geometry in generating quasicrystallinity in soft matter, complementing the principles that lead to quasicrystal formation in hard tetrahedra Based on simple interparticle potentials, quasicrystalline mosaics may well find use in diverse applications ranging from improved image reproduction to advanced photonic materials

189 citations

Journal ArticleDOI
TL;DR: In this paper, the integral equation for the radial distribution function of a fluid of rigid spherical molecules has been integrated numerically in the Kirkwood approximation and in the Born-Green approximation over a wide range of densities.
Abstract: The integral equation for the radial distribution function of a fluid of rigid spherical molecules has been integrated numerically in the Kirkwood approximation and in the Born‐Green approximation over a wide range of densities. The distribution functions so obtained have been used to calculate the equation of state and excess entropy of the fluid. The results are compared with those obtained by means of the free volume theory of the liquid state.

171 citations

Journal ArticleDOI
TL;DR: It is found that it is very difficult to find an accurate probe of the local particle density over a wide range of volume fractions by using standard colloidal suspensions, and conventional polarized light scattering gives results which depend also on interparticle interactions.
Abstract: Suspensions of model colloidal particles represent, besides their intrinsic interest,a very important system for the study of basic properties of liquids because they show length scales accessible to optical scattering techniques, and interparticle potentials simple enough to allow quantitative tests of liquid state theories [1]. The usual experimental approach is that of deriving the static structure factor from scattering experiments. However, it is known that the measurement of the equation of state would constitute a more stringent test of the properties of the system, because of the greater sensitivity to the details of the interparticle interaction potential [2]. It has been noted [3–5] that a single measurement of the particle density profile of a colloidal suspension at sedimentation equilibrium under the gravitational field can directly yield the osmotic equation of state of the suspension, but sufficiently accurate data are not available, except for measurements of the osmotic pressure at melting in settling suspensions of polystyrene particles by Hachisu and Takano [4] and for the study of order-disorder transitions in suspensions of colloidal silica performed by Davis et al. [6]. It is very difficult to find an accurate probe of the local particle density over a wide range of volume fractions by using standard colloidal suspensions. Indeed, conventional polarized light scattering gives results which depend also on interparticle interactions.

139 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied the transmission properties of the same systems by using the multiple-scattering method and found that the 12fold triangle-square tiling is indeed very good for the realization of photonic gaps.
Abstract: A recent publication [Nature (London) 404, 740 (2000)] claimed that absolute photonic gaps can be realized in 12-fold quasicrystalline arrangement of small airholes in a matrix of silicon nitride or glass. The result is rather surprising since silicon nitride $(n=2.02)$ and in particular, glass $(n=1.45)$ have rather low refractive index. In this work, we have studied the transmission properties of the same systems by using the multiple-scattering method. We found that the 12-fold triangle-square tiling is indeed very good for the realization of photonic gaps and we found absolute gaps in systems with airholes in dielectric, dielectric cylinders in air, and metal cylinders in air. However, for the case of air-holes in a dielectric background, absolute gaps appear only when the dielectric contrast is sufficiently high, and both silicon nitride and glass have refractive indices below the threshold.

132 citations

Frequently Asked Questions (2)
Q1. What contributions have the authors mentioned in the paper "Periodic layers of a dodecagonal quasicrystal and a floating hexagonal crystal in sedimentation-diffusion equilibria of colloids" ?

In this paper, the sedimentation behavior of a system of particles interacting with a hard core and a repulsive square shoulder potential with a fixed shoulder width equal to 1.4 times the hard core diameter was studied. 

For future work, it is interesting to study the formation of the dodecagonal quasicrystal in detail and to determine the optimal pressure and settling rates for its formation.