Self-assembly of hard sphere droplet-colloid mixture into quasicrystal

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. 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