Abstract. In this work, we investigate the phase diagram of a single-component system of
colloidal particles and a binary mixture of hard-core droplets and colloids using means of
molecular dynamics. For the former case, we found a variety of different crystal phases, in
a good agreement with these observed in the simulation work (Engel et al, Nature Materials
14, 109–116 (2015)). In the latter case, it is found that the addition of hard-sphere droplets
in the one-component system extends significantly the isosahedral quasicrystal region, in
contrast to the shrinkage of the hP2 phase region, or even vanish of the BC8 phase region.
Therefore, our findings could provide a promising route for obtaining the icosahedral
quasicystals in the nanoscale systems.

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HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0030
Natural Science, 2018, Volume 62, Issue 8, pp. 48-57
This paper is available online at
SELF-ASSEMBLY OF HARD SPHERE DROPLET-COLLOID MIXTURE INTO
QUASICRYSTAL
Abstract. In this work, we investigate the phase diagram of a single-component system of
colloidal particles and a binary mixture of hard-core droplets and colloids using means of
molecular dynamics. For the former case, we found a variety of different crystal phases, in
a good agreement with these observed in the simulation work (Engel et al, Nature Materials
14, 109–116 (2015)). In the latter case, it is found that the addition of hard-sphere droplets
in the one-component system extends significantly the isosahedral quasicrystal region, in
contrast to the shrinkage of the hP2 phase region, or even vanish of the BC8 phase region.
Therefore, our findings could provide a promising route for obtaining the icosahedral
quasicystals in the nanoscale systems.
Keywords: oscillating pair potential, Pickering effect, molecular dynamics, phase diagram.
1. Introduction
Until the 1980s, scientists believed there are only two types of solid: Crystals composed
of order arrangement atoms and amorphous materials which lacks long-range orders. In 1984,
in a Physical Review Letter entitled metallic phase with long-range orientational order and no
translational symmetry”, Shechtman et al [1] discovered a crystal-like material that contains
atomic arrangement with regular dodecahedrons. Surprisingly, such a crystal is not allowed
in crystallography because it does not possess translational symmetry. Most crystallographers
initially convinced that it was not a new solid form. Later, An-Pang Tsai successfully prepared
a particular alloy belonging to this structure, now called a quasicrystal ladder [2]. Since then,
several types of quasicrystal have been observed in experiments, most of them are metallic alloys
and polymers, for example, AlPdMn, AlLiCu, AlCuFe, AlNiCo, AlCuV, AlMnSi, in addition to
numerous other compositions such as PbUSi, CdYb, InAg, TiZrNi, ZnMgHo, ZnMgSc, Yb [3].
Amongst known quasicrystals, icosahedral quasicrystals (IQCs) are important and popular
in nature. For example, it can be found in structures of virus capsids, fullerenes, or metallic glasses
[4]. Yet, despite a very large quantity of IQCs in metallic compounds at atomic scales, IQCs have
been not been observed in experiments, even in numerical simulations for non-atomic systems, e.g.
mesoporous, colloidal suspensions. Very recently, Glotzer et al [4]. have shown that self-assembly
of colloidal particles into IQCs is possible in a one-component system of particles interacting via
an isotropic potential. Their findings suggest routes to design and preparation IQCs in soft matter
and nanoscale systems.
Received .... Accepted ...
48
Self-assembly of hard sphere droplet-colloid mixture into quasicrystal
Our aim in this paper is to find out any variations in the abundant phase diagram that is
reported by Glotzer and coworkers [4], in the presence of hard-sphere particles. To do this, we
employ means of computer simulations to reproduce the results in Ref [4]. We next investigate
in detail the formation of all crystal structures, especially in IQC region, when the second type of
particles is introduced into the system.
2. Model and simulation method
2.1. Model
We investigate a binary mixture of Nc colloidal particles with diameter σc and Nd droplets
of hard-sphere diameter σd. The total interaction energy U is written as the sum of colloid-colloid,
droplet-droplet, and colloid-droplet interactions,
U =
Nc∑
i<j
φcc (|ri − rj |) +
Nd∑
i<j
φdd (|Ri −Rj |) +
Nc∑
i
Nd∑
j
φcd (|ri −Rj |) , (2.1)
where ri is the center-of-mass position of colloid i, Ri is the center-of-mass position of
droplet j, φcc is the colloid-colloid pair interaction, φcc is the colloid-droplet pair interaction,
andφcc is the droplet-droplet pair interaction.
2.1.1. Colloid-colloid pair interaction
Mihalkovi and Henley [5] defined the family of oscillating pair potentials (OPP) as follows,
Vcc(r) = C1r
−η1 + C2r−η2 cos(kr + Φ) (2.2)
where C1, C2 are constants, k is the wave vector and Φ is the phase shift and r is the interparticle
distance. For the sake of simplicity, we set C1 = C2 = 1 for the energy scale and the length
scale. The first exponent η1 = 15 responsible for the repulsive core that is similar to a value used
by Mihalkovi and Henley [5] and by the Zetterling potential [6]. The second exponent, η2 = 3 is
chosen to resemble Friedel oscillations [7] due to
VFriedel(r) ≈ cos(2kF r + θ)/r3, (2.3)
The oscillating pair potential (OPP) with three-wells and six parameters mimics the atomic
interactions of many metallic systems [5]. However, here we use the OPP as a generic potential
for colloidal suspensions.
2.1.2. Droplet-droplet pair interaction
The droplet-droplet interaction is hard-sphere like [8]
Vdd(r) =
{ −1 + (σdr )36, r < σd
0 otherwise,
(2.4)
with σd being the diameter of droplet spheres.
49
.Table 1: Dimensionless physical quantities.
physical quantity definition
wave vector k∗ = kσ
phase shift Φ
interparticle distance r∗ = r/σ
temperature T ∗ = kBT/
volume V ∗ = V/σ3
particicle density ρ∗ = ρσ3
Hard-sphere model are widely used as model particles in the statistical mechanical theory
of fluids and solids. They are defined simply as impenetrable spheres that cannot overlap in space.
They mimic the extremely strong at very close distances. Hard spheres systems are studied by
molecular dynamics simulations, and by the experimental study of the certain colloidal model
system.
2.1.3. Colloid-droplet pair interaction
The colloid-droplet interaction is aimed at the Pickering emulsions. The adsorption energy
when a colloid absorbs at the droplet-solvent interface is modeled by a simple parabolic well [8]
Vcd(r) =
{
a(r −B)2 + c r < σc+σd2
0 othewise,
(2.5)
where a = (−+
√
2 + r20)/2r
2
0, c = (−
√
2 + r20)/2 and B = (σc + σd)/2− r0. The value
r0 = 0.1 determine the contact angle between droplets and colloids to 150◦. Different from a
dynamic evaporation process of the oil droplets simulated in our previous works [9, 10, 11], here
we restrict ourselves to the case of static oil droplets. Therefore, the droplet diameter does not
change during the simulation time.
2.2. Simulation methods
The simulations are performed with the HOOMD molecular dynamics package [12, 13, 14]
that allows us to perform a broad parameter sweep and equilibration of medium size systems over
long simulation trajectories. We use the following dimensionless quantities: the frequency k, the
phase shift Φ, and the interparticle distance r. Table 1 shows how the units are defined. For
simplicity, from now “asterisks” are omitted in physics quantities.
The minimum system size to yield useful information is a thousand particles which are
sufficient to form a few hundred unit cells for most crystals and a few icosahedral clusters for
quasicrystal. The molecular dynamics are carried out using 108 steps with step size of ∆t = 0.01.
We carry out the simulations in the range of OPP parameters (k,Φ) to explore the phase space. The
reproduction is determined by using a matrix of 13× 8 simulations in the (k,Φ) parameter plane
ranging from (k,Φ) = (5.75, 0.38) to (k,Φ) = (8.75, 0.38) at regular intervals of ∆k = 0.25
and ∆Φ = 0.06. For a given value of (k,Φ), a thousand particles are initialized randomly in the
cubic box with the periodic boundary condition. The temperature decreases linearly from T = 0.4
to T = 0.1 in time over 108 molecular dynamics steps. Hard sphere droplets are added to the
50
Self-assembly of hard sphere droplet-colloid mixture into quasicrystal
colloidal system in such a way that the ratio of the number of colloid particles over the number of
hard spheres is 5 : 1. The diameter ratio of colloids and droplets, σd/σc is set to 1.5.
In the final stage of the computer simulations, each configuration is examined by visual
inspection, radial distribution function, and diffraction pattern to identify all possible structures.
The detailed information for these analysis can be found in Ref. [4].
3. Results and Discussion
3.1. Single-component system (OPP spheres)
We first consider a single component system of colloidal particles, i.e. the system only
consists of colloids interacting via the OPP potential. Figure 1 shows the phase diagram in the
(k,Φ)-plane representation. In agreement with the results reported by Ref. [4], we observe a
variety of different phases, including the hP2-, BC8-, cP8- structure types, clathrates, icosahedral
quasicrystal, and disordered phase. For each typical structure, we described in detail a structural
motif by a representative snapshot, diffraction pattern, together with its radial distribution function.
Fig. 1. Phase diagrams of a single system of colloids with the OOP potential in the (k,Φ)
representation
3.1.1. hP2-structure type
The hP2 crystal, sometimes called A3 type structure, is Pearson symbol of hexagonal close
packing of spheres (Fig. 2). The space group is P 63/mmc. Elements belonging to this structure
are Cd, Mg, and Zn. In this structure, the ideal ratio of c/a is 1.63. The snapshot obtained from a
visual tool (VMD) [15] shows an arrangement of particles (Fig. 2(a)), together with its diffraction
pattern shown in Fig. 2(c). From these figures, every sphere has six nearest neighbors. Hexagonal
close-packing corresponds to an ABAB stacking. This structure is confirmed by a presence of
unique characteristic peaks in the radial distribution function (Fig. 3.2(d)).
3.1.2. cl16-structure type
The cl16, sometimes called BC8 type structure (Fig. 3), is Pearson symbol of tetragonally
bonded structure which packs more efficiently than diamond. It is found experimentally and in
first-principles calculations of silicon. The BC8 structure can be understood as a periodic tiling of
51
.(a)
(b)
(c)
(d)
Fig. 2: The crystal structure hP2: (a) simulation snapshot, (b) diffraction pattern, (c)
representative model, (d) radial distribution function, at (k,Φ) = (6.75, 0.74).
interpenetrating prolate and oblate Penrose rhombohedra [4]. The space group of BC8 is Ia-3 (No.
206). Elements which have this structure are Si and Ge.
(a)
(b)
(c)
(d)
Fig. 3: The crystal structure BC8: (a) simulation snapshot, (b) diffraction pattern, (c)
representative model, (d) radial distribution function, at (k,Φ) = (6.5, 0.56).
52
Self-assembly of hard sphere droplet-colloid mixture into quasicrystal
3.1.3. cP8-structure type
cP8, also called A15 type structure (Fig. 4), is Pearson symbol of a body-centered packing
of edge-sharing icosahedra. cP8 is a Frank-Kasper phase, its space group is Pm-3n. V3Si and
Nb3Ge have this type of structure. Some of this structure type are superconducting materials.
(a)
(b)
(c)
(d)
Fig. 4: The crystal structure cP8: (a) simulation snapshot, (b) diffraction pattern, (c)
representative model, (d) radial distribution function, at (k,Φ) = (6.25, 0.56).
3.1.4. cP54 Cla I structure
Clathrates are obtained at intermediate k and low Φ (Fig. 5). We distinguish two variants:
Cla I (clathrate type I), Cla IV (clathrate type IV). Our observations suggest a dominance of Cla
I towards lower k and a dominance of Cla IV towards higher k. All large cages of the clathrates
are filled with a particle at their centre. Clathrate I is a cubic structure with its space group being
Pm3n.
3.1.5. hP47 Cla IV structure
The hP47 crystal structure (Fig. 6), atomic analogue to A4B3 alloys (Zr4Al3), can be
regarded as a modification of the CaZn5 structure where the Ca atoms lie along the hexagonal
axes of the simple kagome net at z=0, and is replaced by two smaller atomic spheres at z=1/4 and
z=3/4.
3.1.6. cF160 Cla II structure
In a few cases, we observe small patches of clathrates (Fig. 7) that resemble axially
symmetric random tiling quasicrystals. Clathrates can form patches that resemble dodecagonal
(twelve-fold) and decagonal (ten-fold) axial quasicrystals. Sometimes we observe such random
tilings, but they are often highly defective and cannot be equilibrated on the time scale of our
simulations. This makes their interpretation as an axial quasicrystal.
53
.(a)
(b)
(c)
(d)
Fig. 5: The crystal structure cP54: (a) simulation snapshot, (b) diffraction pattern, (c)
representative model, (d) radial distribution function, at (k,Φ) = (7.25, 0.44).
(a)
(b)
(c)
(d)
Fig. 6: hP47 Cla IV structure: (a) simulation snapshot, (b) diffraction pattern, (c)
representative model, (d) radial distribution function, at (k,Φ) = (7.75, 0.38).
3.1.7. Icosahedral quasicrystal structure
The snapshot obtained from a visual tool (VMD) shows an arrangement of particles
(Fig. 8a), together with its diffraction pattern shown in Fig. 8b. These figures demonstrate
that nearest-neighbor bonds point exclusively in two-fold, three-fold and five-fold symmetry
54
Self-assembly of hard sphere droplet-colloid mixture into quasicrystal
(a)
(b)
(c)
(d)
Fig. 7: cF160 Cla II structure: (a) simulation snapshot, (b) diffraction pattern, (c)
representative model, (d) radial distribution function, at (k,Φ) = (7.25, 0.38).
directions. Although the radial distribution function in the icosahedral region (Fig. 8d) is
qualitatively most similar to the radial distribution function of BC8 and the clathrate phases, the
observation of the 5- fold and 2-fold axes are clear evidence of the formation of the icosahedral
quasicrystal.
(a)
(b)
(c)
(d)
Fig. 8: Icosahedral quasicrystal structure: (a) simulation snapshot, (b) diffraction pattern, (c)
representative model, (d) radial distribution function, at (k,Φ) = (6.75, 0.5).
55
.3.2. Binary mixture of droplets and colloidal particles
It is well-known that an addition of a second colloidal species in the system of interest has
a significant effect on the resulting phase diagram due to the relevance of control parameters: the
size ratio, the total packing fraction and the relative composition, pair particle-particle interaction.
Here, for simplicity, we only consider a simple interaction of hard-core spheres with simulated
parameters given in Section 2.
Fig. 9. Phase diagrams of a binary mixture system of colloids and dropletsl in the (k,Φ)
representation
Compared to the phase diagram in the one-component system (shown in Fig. 1), the phase
diagram of the binary mixture system has changed significantly (Fig. 9). The disordered region
becomes larger and dominant in the region with intermediate values k and high values, where
the parameter space for hP2 phase shrinks to a small region. In addition, the BC8 structure also
disappears and is replaced by the quasicrystal region. Furthermore, in the limitation of the present
analysis methods we do not distinguish the crystallographic forms of clathrates and quasicrystals
in this assembly map as a result of the more disordered arrangement in the presence of hard-sphere
particles under consideration.
4. Conclusion
The study concerns the structural behavior of self-assembled colloidal suspensions. By
varying the shift phase and wave vector in a simple isotropic three-well potential model, we
can reproduce a rich variety of phases, including structure type of hP2, BC8, cP8, clathrates,
particularly of the icosahedral quasicrystal. Our results show a good agreement with those reported
previously by Glotzer and coworkers. We extend the Glotzers one-component model to a binary
mixture of colloidal particles and hard-core spheres to examine the role of hard spheres on the
resulting phase diagram. We find an elongation of the disordered liquid phase, in contrast to a
shrink of the hP2 phase. In particular, the BC8 structure in many state points in the phase diagram
is replaced by the quasicrystal phase region. We note that while the icosahedral quasicrystal is
found abundantly in atomic scales, it has not yet been observed in colloidal scales. Therefore, our
findings could help in the preparation of icosahedral quasicrystal at colloidal scales.
56
Self-assembly of hard sphere droplet-colloid mixture into quasicrystal
Acknowledgment.
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