Carina
Karner
*ab,
Christoph
Dellago
a and
Emanuela
Bianchi
bc
aFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090, Vienna, Austria. E-mail: carina.karner@univie.ac.at
bInstitut für Theoretische Physik, TU Wien, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria. E-mail: emanuela.bianchi@tuwien.ac.at
cCNR-ISC, Uos Sapienza, Piazzale A. Moro 2, 00185 Roma, Italy
First published on 18th February 2020
Anisotropy at the level of the inter-particle interaction provides the particles with specific instructions for the self-assembly of target structures. The ability to synthesize non-spherical colloids, together with the possibility of controlling the particle bonding pattern via suitably placed interaction sites, is nowadays enlarging the playing field for materials design. We consider a model of anisotropic colloidal platelets with regular rhombic shape and two attractive sites placed along adjacent edges and we run Monte Carlo simulations in two-dimensions to investigate the two-stage assembly of these units into clusters with well-defined symmetries and, subsequently, into extended lattices. Our focus is on how the site positioning and site-site attraction strength can be tuned to obtain micellar aggregates that are robust enough to successively undergo to a second-stage assembly from sparse clusters into a stable hexagonal lattice.
Finite clusters can serve, for instance, as prototypes of micro-robots to perform tasks at the micro-scale such as delivery and targeting. Recent and successful examples of anisotropic patchy particles assembling into finite clusters are colloidal asymmetric dumbbells that show a rotational propulsion under electric field,15 dielectric cubes with one metallic facet that reconfigure on demand,16 or Janus spheroids that act as encapsulating agents.17
Self-assembled finite clusters can be also used as non-spherical building blocks for further – hierarchical – assembly into extended structures and thereby expand the versatility of the original units.18–22 Colloidal molecules composed of different types of particles, for instance, can be designed to support the assembly of superstructures with target photonic or phononic properties.23
In general, the combination of shape and bond anisotropy leads to an extraordinary control over the crystal structure: polyhedral nanoparticles covered with DNA surface ligands have been shown, for instance, to assemble into crystals fully determined by the size and the symmetry of the particles and by the length of the DNA-ligands.24,25 DNA-based functionalization can also be achieved by creating suitably shaped frames for spherical nanoparticles: two-dimensional, square-like DNA frames with functionalized edges are able to tune the assembly of the resulting complex units from finite clusters (micelles as well as chain-like aggregates) to planar architectures;26 while tetravalent DNA-cages are able to induce the assembly of isotropic nanoparticles into diamond crystals.27
Within this vast realm, colloidal platelets, i.e. colloids with lateral size at least one order of magnitude smaller than the other two dimensions,1 may show, e.g., interesting electronic properties as isolated fluorescent emitters28 or even form tilings with photonic properties.29 In general, colloidal platelets of different shapes can be realized both at the nanometer and the micron scale by, e.g., folding long, single-stranded DNA molecules to create two-dimensional shapes,30,31 by soft lithography10 or by making use of preferred growth mechanisms thus obtaining, e.g., silica polygonal truncated pyramids, lanthanide fluoride nanocrystals or polymer-based platelets.6,14,32,33 Directional bonding can then be added by, e.g., covering the particle edges with ligands14 or immersing the platelets in a liquid crystal.6 While the directional bonding constraints favor particle configurations that maximize the number of bonds, the particle shape favors edge-to-edge contacts; the possible competition between these two driving-agents can lead to self-assembled tilings with tunable properties.34–36
Here we consider hard colloidal platelets with a regular rhombic shape decorated with two mutually attractive patches arranged – in different geometries – along two adjacent particle edges. We define two classes of rhombi characterized by having the two patches enclosing either the big or the small angle and, for each of them, we move the patches either in a symmetric or asymmetric fashion along the particle edges. Thus, the resulting building blocks present a wide range of bonding patterns, the combination of which gives rise to different assembly products: chains as well as micelles with three-, five- or six-fold symmetry. We first investigate the emerging assembly scenarios according to the patch positioning at different interaction energies. For most of the studied systems, we are able to trace a region in the parameter space where micelles are the prevalent assembly product. Hence the second part of our paper investigates if and how systems of micelles with different geometries form a well-defined lattice via hierarchical assembly.
The patch–patch interaction is a square-well attraction given by
A patchy rhombi model of this kind was first detailed in ref. 39, with four attractive patches placed in the center of the edges. In this work, we focus on two-patch rhombi where the patches either enclose the big angle (manta rhombi, referred to as “ma”) or the small angle (mouse rhombi, referred to as “mo”); see the particle sketches reported at the top of Fig. 1.
In either type of system, patches can be placed anywhere on the respective edges, resulting in an – in principle – infinite number of possible two-patch rhombi systems. To methodically characterize the patch positioning, we introduce two patch topologies. Patch topologies prescribe how to move the patches with respect to each other. In the symmetric/asymmetric (s/as) topology, patches are placed symmetrically/asymmetrically with respect to their enclosing vertex. Note that, within a specific topology, the relative distance Δ of one patch with respect to the enclosing vertex also determines the position of the other patch: in the s-topology, both patches are placed at distance Δ from their common vertex, in the as-topology, one patch is at distance Δ from the reference vertex, while the other is at (1 − Δ). With these definitions a two-patch system is fully defined through its patch configuration (ma or mo), its topology (s or as) and its relative position on the edge (Δ). It is important to note that when patches are placed in the edge-center, i.e., at Δ = 0.5, the s- and as-topology collapse into the same topology, referred to as center topology, so that the respective rhombi systems are referred to as ma- and mo-center systems. A summary of the used particle parameters can be found in Table 1.
Parameter | Symbol | Value |
---|---|---|
Angle | α | 60° |
Side length | l | 1.0 |
Patch radius | r p | 0.05 |
Interaction strength | ε | [5.2, 10.2] |
Patch position | Δ | [0.2, 0.8] |
We find that micelles are mostly in competition with chain assemblies, as loops rarely form. While our previous publication detailed the properties of chain assemblies,42 this work focuses on micelles. First, within the framework of a small cluster analysis, we establish for which topologies and patch positions micelles can form (see Fig. 1). Subsequently, we determine for which (Δ,ε)-values micelles are the dominant assembly product in simulations (see Fig. 2–4). Finally, we compress the micelle assemblies with the highest yield and analyze the quality of the second-stage assembly products (see Fig. 5–7).
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Fig. 2 Self-assembly products of symmetric manta (ma-s) and mouse (mo-s) systems. (a) Histogram of yields of different cluster types in the ma-s system at Δ = 0.7 and ε = [5.2, 10.2]kBT. Histograms for the other Δ-values can be found in Fig. S1 of the ESI.† Cluster types of interest are: clusters with size N < 3 (liquid, blue), three np-bonded particle loops (boxes, pink) and clusters with N ≥ 3 (chains/loops, yellow). The barycentric color triangle in (b) maps the distribution of cluster types for all (Δ,ε)-values to the heatmap in (c) (see the ESI† for the detailed description of the barycentric coordinates). (d) Simulation snapshots of ma-s. From top to bottom: chains, loops and boxes at ε = 10.2kBT, Δ = 0.7, boxes at ε = 7.2kBT, Δ = 0.7, rhombi liquid at ε = 5.2kBT, Δ = 0.7. (e) Histogram of yields of different cluster types in the mo-s system at Δ = 0.2 and ε = [5.2, 10.2]kBT. Histograms for the other Δ-values can be found in Fig. S2 of the ESI.† Cluster types of interest are: clusters with N < 5 (liquid, blue), five np-bonded particle loops (5-stars, purple), 6 np-bonded particle loops (6-stars, pink) and clusters with N ≥ 5 (chains/loops, yellow). The barycentric color triangles in (f) and (g) map the distribution of cluster types to the heatmap column/s for Δ = 0.2/[0.3, 0.8] in (h). (i) Simulation snapshots of mo-s. From top to bottom: chains, loops and stars at ε = 10.2kBT, Δ = 0.7, 5-stars at ε = 10.2kBT, Δ = 0.2, 6-stars at ε = 9.2kBT, Δ = 0.3, rhombi liquid at ε = 5.2kBT, Δ = 0.8. |
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Fig. 3 Self-assembly products of asymmetric manta (ma-as) and mouse (mo-as) systems. (a) Histogram of yields of different cluster types in the ma-as system at Δ = 0.7 and ε = [5.2, 10.2]kBT. Histograms for remaining Δ values can be found in Fig. S3 of the ESI.† Cluster types of interest are: clusters with size N < 3 (liquid, blue), three np-bonded particle loops (open boxes, pink) and clusters with N ≥ 3 (chains/loops, yellow). The barycentric color triangle in (b) maps the distribution of cluster types for all (Δ,ε)-values to the heatmap in (c) (see the ESI† for the detailed description of the barycentric coordinates). (d) Simulation snapshots of ma-as. From top to bottom: chains, loops and open boxes at ε = 10.2kBT, Δ = 0.7, open boxes at ε = 8.2kBT, Δ = 0.7, rhombi liquid at ε = 5.2kBT, Δ = 0.7. (e) Histogram of yields of different cluster types in the mo-as system at Δ = 0.2 and ε = [5.2, 10.2]kBT. Histograms for the other Δ-values can be found in Fig. S4 of the ESI.† Cluster types of interest for Δ = 0.2 are: clusters with size N < 5 (liquid, blue), five np-bonded particle loops (open 5-stars, purple), six np-bonded particle loops (open 6-stars, pink), and clusters with N ≥ 5 (chains/loops,yellow). The barycentric color triangle in (f) and (g) maps the distribution of cluster types to the heatmap column/s for Δ = 0.2/[0.3, 0.8] in (h). (i) Simulation snapshots of mo-as. From top to bottom: chains, loops and open stars at ε = 10.2kBT, Δ = 0.2, chains, loops and open stars at ε = 8.2kBT, Δ = 0.2, chains, loops and open stars at ε = 8.2kBT, Δ = 0.3, rhombi liquid at ε = 5.2kBT, Δ = 0.2. |
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Fig. 5 Second-stage assembly of boxes at ε = 8.2kBT for symmetric manta systems (ma-s). (a) Left: Simulation snapshots of the system with Δ = 0.3 at P = [10, 30, 100] (from top to bottom). Right: Centers of mass of boxes of the respective snapshots colored according to the hexatic order parameter, Ψ, defined in eqn (3) (see the color wheel in panel c). (b) Eos for box systems at Δ = [0.3, 0.5, 0.7], where ϕ denotes the packing fraction defined in eqn (2). (c) Color wheel denoting the values of Ψ for the center of mass snapshots in panel a. (d) Average fraction of the largest hexagonal domain 〈SL〉 – defined in the text – as function of P; note that the average is taken over all Δ-values; note that the background of the eos-plot in panel a is colored according to 〈SL〉. Error bars denote the standard deviation and their wide extent is due to relatively small system sizes. (e) The radial distribution function g(r) at Δ = 0.3 and P = 100. |
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Fig. 6 Second-stage assembly of open boxes at ε = 8.2kBT for asymmetric manta systems (ma-as). (a) Left: Simulation snapshots of the system with Δ = 0.3 at P = [10, 30, 60, 100] (from top to bottom); clusters are colored to facilitate visual distinction. Right: Centers of mass of open boxes of the respective snapshots colored according to the hexatic order parameter, Ψ, defined in eqn (3) (see the color wheel in panel e). (b) Eos for open box systems at Δ = [0.2, 0.3, 0.4], where ϕ is the packing fraction defined in eqn (2). (c) Left: simulation snapshot of the system at Δ = 0.4 and P = 100. Right: Centers of mass of open boxes of the respective snapshot colored according to Ψ. (d) Left: Simulation snapshot of the system at Δ = 0.2 and P = 100. Right: Centers of mass of open boxes of the respective snapshot colored according to Ψ. (e) Color wheel for the values of Ψ for the center of mass snapshots in panels a, c and d. (f) Average fraction of largest hexagonal domain 〈SL〉 – defined in the text – as function of P for Δ = [0.2, 0.3, 0.4, 0.5], as labeled. The error bars denote the standard deviation and are wide due to relatively small system sizes. (g) The radial distribution function g(r) at Δ = [0.2, 0.3, 0.4] and P = 100, as labeled. (h) Magnified view of the blue rectangle of snapshot of ma-asΔ = 0.3, P = 100. Black lines connect boxes at hexagonal lattice positions; red lines connect boxes that are shifted due to defects. (i) Magnified view of the red rectangle of ma-asΔ = 0.2. Black lines connect boxes at hexagonal lattice positions; red lines connect boxes that are shifted due to defects. (j) Sketch of the perfect hexagonal open box tiling with triangular pores (ot-tiling) as observed in local regions in simulation snapshots. |
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Fig. 7 Second-stage assembly of 5-stars and 6-stars at ε = 9.2kBT for symmetric mouse systems (mo-s). (a) Simulation snapshots of the system at Δ = 0.2 at P = [10, 25, 100] (from top to bottom), where 5-stars are predominant. (b) Eos for star systems at Δ = [0.2, 0.3, 0.4], where ϕ is the packing fraction defined in eqn (2). (c) Simulation snapshots of the system at Δ = 0.3 at P = [10, 25, 100] (from top to bottom), where 6-stars are predominant. (d) The radial distribution function g(r) for the 5-star system at Δ = 0.2 and P = 100. (e) The yield of 5-stars and 6-stars at Δ = 0.2 and Δ = 0.3 as function of P, as labeled. (f) The radial distribution function g(r) for the 6-star system at Δ = 0.3 and P = 100. (g) The red inset is a magnified view of 5-star neighbourhoods at Δ = 0.2 and P = 100. (h) The dark blue inset is magnified view of a 6-star neighborhoods at Δ = 0.3 and P = 100. The black lines connect the star centers and highlight the hexagonal order of this neighbourhood. |
In ma-s, micelles consist of three np-on-bonded particles (also referred to as boxes) and they can form at all Δ-values (see clusters labeled as f in the corresponding panels of Fig. 1). For Δ < 0.5 micelles are the only possible self-assembly product as bonding incompatibilities prevent dimers (see cluster e in the corresponding panel of Fig. 1) from growing into chains and loops (see clusters n and m, respectively). In contrast, for Δ ≥ 0.5, chains and loops can form since the patch positioning does not anymore disfavor the formation of p- and mixed-bonded clusters of sizes bigger than two (see clusters h, i, j and k in the corresponding panels of Fig. 1).
In mo-s, micelles consist either of five or six np-on-bonded particles (referred to as 5- and 6-stars, respectively) and they can form at all Δ-values (see clusters o and q in the corresponding panels of Fig. 1). We note that, micelles of three particles (boxes) would have non-satisfied bonds, thus they are energetically disfavored with respect to the stars. For Δ < 0.5, three-particle clusters with p- and np-bonds are still allowed (see cluster k in the corresponding panel of Fig. 1), thus rendering chains a possible self-assembly product, albeit with a restricted configuration space. For Δ ≥ 0.5, also both p- and mixed-bonded clusters can form (see clusters h, i, j and k in the corresponding panel of Fig. 1).
In both ma-as and mo-as, there are no bonding restrictions for p-bonds, and hence chains, loops and micelles are allowed for all values of Δ. Furthermore, micelles have pores in the center as particles can form np-bonds only off-edge. Therefore, the resulting clusters are referred to as open boxes in the case of ma-as (see clusters g in the corresponding panels of Fig. 1) and open stars in the case of mo-as (see clusters p and r in the corresponding panels of Fig. 1).
It is important to note that while in s-topologies the geometrical form of the assembled micelles is the same at all Δ-values, in as-topologies the cluster geometry does change because of the Δ-dependent opening and closing of the pores. While at Δ = 0.5 (corresponding to ma- and mo-center), the pores are completely closed yielding closed boxes and stars, the pores continuously open as Δ is moved off-center and – as a result – the rhombi corners protrude more and more. As in ref. 34, in the sticky limit, the side length of the pores is given by
lpore = |l − 2Δ|. | (1) |
The yield of a cluster type is defined as the percentage of particles in clusters belonging to the selected cluster type. We define cluster types of interest according to the specific systems. In ma-systems – for both the s- and the as-topology – we distinguish clusters of size N < 3 (classified as liquid), three np-bonded particle loops (classified as boxes/open boxes) and non-box clusters with size N ≥ 3 (corresponding to chains/loops). In mo-systems – both s and as – we distinguish clusters with size N < 5 (liquid), five np-bonded particle loops (5-stars/open 5-stars), six np-bonded particle loops (6-stars/open 6-stars) and non-star clusters N ≥ 5 (chains/loops).
The obtained yields are reported as a function of ε for each Δ-value (see panels a and e of Fig. 2 and 3 for an example Δ-value, while for the full overview of Δ-values see Fig. S1–S4 in the ESI†). Through a mapping to a barycentric coordinate system (see ESI† for the mapping and calculation details), we obtain heatmaps for the four investigated systems (ma-s, mo-s, ma-as and mo-as) where at each (Δ,ε)-value we identify the predominant self-assembly scenario. The heatmaps reflect the relative dominance of the different cluster types: whenever one cluster type has a yield higher than 2/3, the heatmap obtains the color of the respective triangle edge (blue for liquid, yellow for chains/loops, pink and purple for micelles) and we call this cluster type 2/3-dominant; if no cluster type dominates by more than 2/3, one of the mixed colors in the center of the triangle is adopted (see panels c and h of Fig. 2 and 3 for the heatmaps, see panels b, f and g in Fig. 2 and 3 for the respective barycentric triangles). Simulation snapshots are displayed in Fig. 2d and i for ma-s/mo-s systems and in Fig. 3d and i for ma-as/mo-as systems. Micelle yields for all systems are given in Fig. 4.
We start our discussion with ma-s systems (Fig. 2, panels from a to d). When ε ≤ 6.2kBT such systems remain liquid over the whole Δ-range. On increasing Δ, the self-assembly process takes place: at ε = 7.2kBT and ε = 8.2kBT, boxes are the 2/3-dominant self-assembly product at all Δ-values; while for ε ≥ 9.2kBT there exists a range of Δ-values where chains/loops prevail over the micelles (see Fig. 2c). The yield of boxes increases monotonically with ε for all Δ-values until ε = 9.2kBT (see Fig. 4a). For ε ≥ 9.2kBT and Δ < 0.5, the yield of the boxes reaches values above 0.95, while at Δ = 0.5, we observe a sudden drop: for Δ ≥ 0.5 bonding restrictions allow chains/loops to form and compete with boxes and as soon as the interaction strength is ε ≥ 9.2kBT, chains/loops prevail over the boxes. The dip in the box-yield at ma-center becomes deeper on increasing ε and while for ε = 9.2kBT the yield recovers to 0.8 at Δ = 0.8, for ε = 10.2kBT the yield remains below 0.45 at Δ = 0.8. Summarizing, in ma-s, boxes prevail over chains/loops over a wide range of (Δ,ε)-values.
In mo-s systems (Fig. 2, panels from e to i), we observe characteristic self-assembly products only for ε ≥ 8.2kBT. At such high interaction strengths, micelles emerge with a non-negligible yield at Δ = 0.2 in the case of 5-stars and at Δ = 0.3 and 0.4 in the case of 6-stars. For these Δ-values, the star-yields rise monotonically with ε (see Fig. 4b for the 5-star-yield and Fig. 4c for the 6-star-yield), but only reach values above 2/3 for ε ≥ 9.2kBT (see Fig. 2h). The highest 5-star-yield is reached at ε = 10.2kBT and Δ = 0.2 with 0.838 ± 0.04. On increasing Δ, the 5-star-yield falls to 0.084 ± 0.018 at Δ = 0.3 and becomes smaller than 0.025 at Δ = 0.5. In contrast, 6-stars reach their highest yields at ε = 9.2kBT, with the highest value 0.754 ± 0.028 at Δ = 0.4. On increasing Δ, the 6-star-yield drops to ≈0.2 at mo-center and proceeds to drop gradually below 0.05 at Δ = 0.8. With star-yields below 0.2 for Δ ≥ 0.5, chains/loops prevail as dominant clusters at high ε-values (see Fig. 2h for phase boundaries).
Summarizing, in mo-s, stars have a significantly narrower (Δ,ε)-range of prevalence compared to boxes in ma-s (see Fig. 2c for ma-s and Fig. 2h for mo-s). While boxes are 2/3-dominant for all Δ at intermediate ε-values, stars reach yields above 2/3 only at high ε-values and only for Δ < 0.5. At high ε and for Δ ≥ 0.5 extended chains become available to mo-s and are formed at a higher rate.
In ma-as systems (Fig. 3, panels from a to d), the yield of micelles (open boxes, in this case) rises monotonously with ε: open boxes become 2/3-prevalent for intermediate interaction strengths (i.e., at ε = 7.2kBT and 8.2kBT) over the whole Δ-range, with the highest yields – above 0.9 – achieved at ε = 8.2kBT (see Fig. 3c for the heatmap and Fig. 4d for numerical values of yields). At high interaction strengths, i.e., for ε ≥ 9.2kBT, the open box yield drops for all Δ, first to 0.4 at ε = 9.2kBT and then to 0.2 at ε = 10.2kBT. The drop in the micelle-yield is due to the emergence of chains, that can form at all Δ-values and prevail over the open boxes when ε ≥ 9.2kBT.
In mo-as systems (Fig. 3, panels from e to h), micelles (open stars, in this case) do not become dominant at any (Δ,ε)-value. In particular, the yield of open 5-stars stays below 0.03 for all Δ-values and does not increase with ε, meaning that these micelles are always negligible compared to chains/loops (see Fig. 4e). In contrast, the yield of open 6-stars does increase with ε for all but the most extreme Δ-values (i.e., Δ = 0.2 and 0.8); nonetheless, as the maximum yield stays below 0.25 (see Fig. 4f), also these micelles are negligible over the whole Δ-range. While for ε ≤ 7.2kBT, the systems are mostly liquid, for ε ≥ 7.2kBT, chains become the most dominant cluster type at all Δ-values.
Concluding, micelles in as-topologies have a smaller window of prevalence compared to micelles in s-topologies because in as-topologies extended p- and mixed bonded chains are allowed at all Δ-values and they dominate over micelles for high interaction strengths. In the following, we describe the properties of the observed micelles, mostly focusing on their packing properties. For the detailed characterization of the chain/loops assemblies we refer the reader to ref. 42.
Upon increasing pressure, micelles pack and tend to form crystalline domains with local hexagonal order. We define the packing fraction as
![]() | (2) |
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Upon increasing pressure, boxes assemble into a lattice with hexatic/hexagonal order at all selected Δ-values. On the left hand side of Fig. 5a, we display simulation snapshots for boxes with Δ = 0.3 at different pressures, while on the right hand side we show the same snapshots with the box centers colored according to their respective Ψ (see Fig. 5c for Ψ-color wheel). We note that, while at P = 10, the system of boxes is liquid, at P ≈ 20 the system becomes hexatic, i.e., with local hexagonal order but no long range order. As P increases further, long range order becomes stronger. At P = 100, remaining single rhombi induce grain boundaries into the otherwise perfectly hexagonally ordered domains. We confirm the hexatic/hexagonal order at P = 100 with the radial distribution function (see Fig. 5e). Further, we calculate 〈SL〉 as function of the pressure (see Fig. 5d) to visualize the growth of ordered domains we colored the background of the eos-curves (see Fig. 5b). The same scenario is observed at all selected Δ-values: from Δ = 0.3 to Δ = 0.7 the equations of state collapse onto one curve (see Fig. 5b). The maximum packing at the highest pressure P = 100 is ϕ = 0.793 ± 0.006 for all Δ-values. Therefore we conclude that for these boxes, the specific Δ-value does not influence the second-stage assembly.
It is important to note that although significant parts of the systems are ordered hexagonally at high pressures, the small system sizes render it impossible to determine whether the system is hexagonal on a long range scale. See ref. 43 for detailed studies on the liquid-hexatic-hexagonal phase transitions.
As with the boxes, we also measure the positional order with g(r), 〈SL〉 and Ψ for the open boxes. The peak pattern of the g(r) shows that hexagonal order is still pronounced at Δ = 0.4, significantly deteriorates already at Δ = 0.3 and ceases completely at Δ = 0.2 (see Fig. 6g). Similarly, 〈SL〉 indicates smaller ordered domains for Δ = 0.3 and Δ = 0.2 (note that due to large error-bars – that are a result of the relatively small system sizes – we can only report trends for 〈SL〉). The progressive reduction of the hexatic/hexagonal order with increasing Δ can also be directly observed in the simulation snapshots reported in Fig. 6a–d as well as in the insets 6h and i. In conclusion, in ma-as the hexatic/hexagonal order gets destroyed as Δ shifts towards off-center values.
The eos-curve for 5-stars (red dots in Fig. 7b) lies slightly above the 6-star curves for all intermediate and high P-values, indicating that 5-stars pack less close at the same P with respect to 6-stars (green squares and blue triangles in Fig. 7b). The packing fraction for 5-stars at P = 100 is 0.696 ± 0.0214, while for 6-stars at Δ = 0.3 it is 0.706 ± 0.018.
The analysis of the g(r) at P = 100 (panels d and f of Fig. 7) suggests that in both star systems there exists a residual hexagonal order. However, the resulting structures can not be described as hexagonal lattices due to the many non-star clusters present already in the initial configuration – as the star-yields are low compared to those observed for the boxes. Simulation snapshots in Fig. 7a and c show that these clusters often have the structure of non-completed or misshaped stars. These “broken stars” induce defects to an extent that destroys local and, subsequently, long range order.
Moreover, the 5-star-yield reduces on increasing pressure (see Fig. 7e): while the yield of 6-stars at Δ = 0.3 only slightly decreases with P, the 5-star yield at Δ = 0.2 decreases from 0.75 ± 0.03 at P = 10 to 0.54 ± 0.04 at P = 100. This is due to the fact that, as the pressure is increased, the void where a sixth star would fit, starts to fill up. The result are conglomerates of destroyed 5-stars (see Fig. 7b, red inset). Together with already present grain boundaries, these conglomerates completely break the hexatic/hexagonal lattice order.
In contrast, for 6-stars, we observe the presence of residual areas with hexagonal ordering and triangular pores independent of Δ (see Fig. 7b, dark blue inset).
We explore dynamically accessible self-assembly products at finite temperatures to understand which self-assembly scenarios would emerge in experiments. Depending on the patch arrangement and interaction strength we find micelles (minimal loops) or chains to be the prevalent self-assembly product. We observe that equivalent simulation runs produce comparable chain and micelle yields over simulation times of about 107 MC sweeps.
While an earlier work studied the properties of chains,42 this work focuses on micelles. In general, we distinguish two types of micelles: boxes, that emerge for systems where the patches enclose the big angle (manta), and stars that are formed by systems where the patches enclose the small angle (mouse).
Boxes and stars can be stabilized for systems where both patches are distributed symmetrically at a distance less than 0.5 (Δ < 0.5) from their enclosing angle. For systems with Δ ≥ 0.5, micelles and chains compete at high interaction strengths.
In systems, where patches are distributed asymmetrically – i.e., the distance of one patch to the enclosing angle is Δ, while the distance of the second patch is (1 − Δ) – stars and boxes have a pore in the center and we call them open boxes and open stars. We note that while for large as well as small Δ-values the pore size is maximal, the pore closes off completely for Δ = 0.5 and symmetric and asymmetric patch distributions are identical. We find that, while open boxes are prevalent at intermediate interaction strength, open stars never reach a yield over 0.25. At high interaction strength chains prevail for all Δ-values in both, manta and mouse systems.
We note that in our investigations the patch size is fixed to 1/10th of the side length. As this feature can be tuned in experimental systems, for instance by changing the length of the sticky-ended DNA strands, it would be interesting to investigate how the self-assembly process is influenced by a change in the patch size. From our results we speculate that, a larger yield of chains would be observed on increasing the patch size, as the larger patch areas would imply a stronger interaction strength.
In the second part of the paper we compress the systems of boxes, open boxes and stars and characterize the emerging lattices. Of course, the yield of the first structures plays an important role in the second-stage assembly process,19 so we compress only those systems where micelle yield is sufficiently high.
While boxes yield a hexagonal lattice independent of Δ, open boxes self-assemble into a hexagonal lattice with a Δ-dependent long range order. As Δ becomes more extreme, gaps and pores within the lattice grow larger, while the long range order decays. In contrast, the self-assembly product of stars displays no long range hexagonal order for any Δ-value.
We note that, in box systems defects in the hexagonal order are exclusively due to either vacancies or not fully formed boxes. In open box and star systems, on the other hand, defects inherent to the particle shape arise.
For what concerns open boxes, the best hexagonal order would be achieved by the lattice reported in Fig. 8a. We refer to this lattice as open triangular tiling (ot-tiling) as it has triangular pores, whose size depend on Δ. In the ot-tiling, all rhombi are aligned in a non-parallel fashion. Although such an alignment is able to incorporate the protruding corners while maximizing edge-to-edge contact, other on-edge alignments are possible and can poison the local order. As a consequence, in simulations, we observe orientational point defects due to the fact that rhombi belonging to different boxes align in a parallel way, as shown in Fig. 8b. While for closed boxes this kind of parallel alignment is commensurable with the closed-packed hexagonal lattice, in open box systems, lattice distorting defects are introduced and the more extreme Δ is, the more the lattice is distorted. While at Δ = 0.4 and Δ = 0.3 open boxes still achieve long range order and high packing, at Δ = 0.2 the orientation defects destroy the lattice completely: no long range order is observed and a lower packing is reached.
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Fig. 8 Tilings of open boxes where the clusters are depicted in different colors to facilitate visual distinction. (a) Perfect hexagonal tiling with triangular pores, referred to as open triangular tiling (ot-tiling). The black lines connect the centers of neighboring boxes and highlight the hexagonal order of the tiling. (b) Representation of a lattice defect: the red open box is rotated by 60° with respect to the perfect ot-tiling in panel a. Due to the rotation, the rhombi of the red open box are aligned parallel with their neighboring rhombi belonging to a different box. The hexagonal lattice is thus distorted: while neighboring boxes in correct lattice positions are still connected by black lines, boxes that had to shift due to the orientational defect are connected with red lines. (c) Perfect hexagonal tiling of open boxes with triangular and hexagonal pores, referred to as open triangular-hexagonal tiling (oth-tiling). This tiling – also known as kagome lattice – was found in ref. 34 in systems of patchy rhombi with four patches. |
We note that, the ot-tiling is reminiscent of the perfect kagome lattice observed in ref. 34, that is a hexagonal lattice with two kinds of pores: triangular and hexagonal ones. Both pore sizes are dependent on Δ, with the pores becoming larger at more extreme Δ-values. We refer to such a tiling as open triangular-hexagonal tiling (oth-tiling) and we sketch it in Fig. 8c. To map the ot-tiling to the oth-tiling, the on-edge alignment of the unbonded edges – i.e., those edges that are not decorated with patches – must shift to an off-edge alignment. This shift would open up the hexagonal pores of the oth-tiling. Clearly, the ot-tiling prevails over the oth-tiling because hard particles prefer arrangements that maximise edge-to-edge contacts. To stabilize a defect-free kagome lattice, neighboring boxes must be forced into an off-edge alignment. One strategy to reach this goal might involve the presence of patches in the outer perimeter of the open boxes in positions that stabilize the off-set edge contacts. The specific topology supporting the oth-tiling in ref. 34 can in fact be interpreted as a four patch extension of the ma-as topology that forms the open boxes.
Analogue to open boxes, we speculate that, if the star yield was high enough, an ot-tiling could be the assembly product of the 6-stars systems (see Fig. S5c in the ESI†). In this case, the distortion from such an hexagonal perfect lattice would be less extreme with respect to the open boxes as neighboring stars have no choice but to align parallel, thus leading to line defects (see Fig. S5d in the ESI†). Local hexagonal arrangements and line defects can be observed in simulations; however the hexagonal order does not prevail there as stars are not stable under compression. To enhance the stability of stars – and thus the quality of the second-stage assembly products – parallel bonds between the rhombi should be disfavored as it is this type of bonding that allows broken stars and disfavors chains (see Fig. S6a in the ESI†). A strategy to disfavor parallel bonds might consist in having different kinds of patches, where different types of patches attract each other (see Fig. S6b in the ESI†).
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0sm00044b |
This journal is © The Royal Society of Chemistry 2020 |