Leveraging Hierarchical Self-Assembly Pathways for Realizing Colloidal Photonic Crystals

Colloidal open crystals are attractive materials, especially for their photonic applications. Self-assembly appeals as a bottom-up route for structure fabrication, but self-assembly of colloidal open crystals has proven to be elusive for their mechanical instability due to being low-coordinated. For such a bottom-up route to yield a desired colloidal open crystal, the target structure is required to be thermodynamically favored for designer building blocks and also kinetically accessible via self-assembly pathways in preference to metastable structures. Additionally, the selection of a particular polymorph poses a challenge for certain much sought-after colloidal open crystals for their applications as photonic crystals. Here, we devise hierarchical self-assembly pathways, which, starting from designer triblock patchy particles, yield in a cascade of well-separated associations first tetrahedral clusters and then tetrastack crystals. The designed pathways avoid trapping into an amorphous phase. Our analysis reveals how such a two-stage self-assembly pathway via tetrahedral clusters promotes crystallization by suppressing five- and seven-membered rings that hinder the emergence of the ordered structure. We also find that slow annealing promotes a bias toward the cubic polymorph relative to the hexagonal counterpart. Finally, we calculate the photonic band structures, showing that the cubic polymorph exhibits a complete photonic band gap for the dielectric filling fraction directly realizable from the designer triblock patchy particles. Unexpectedly, we find that the hexagonal polymorph also supports a complete photonic band gap, albeit only for an increased filling fraction, which can be realized via postassembly processing.

with any numerical integrals calculated using Gauss-Legendre quadrature with 20 points.
The scaled difference in free energy between two the polymorphs, ∆A HT-CT /N k B T , where ∆A HT-CT = A HT − A CT and k B is the Boltzmann constant, was calculated as a function of the reduced temperature T at the ideal density of the crystals, ρ * = N σ 3 /V = 0.645, as determined by crystal structure prediction for the AB-triblock patchy particles considered here.

Annealing protocols
In each case of AA-and AB-triblock patchy particles, a system of 500 particles was gradually cooled in a series of five independent virtual-move Monte Carlo (VMMC) simulation runs.
In the case of the AA system, we initially followed the cooling schedule: T = 1.0 → 0.6 with ∆T = −0.2 and then T = 0.5 → 0.2 with ∆T = −0.1, running 1 million steps at each temperature. The formation of tetrahedral clusters from the AA-triblock patchy particles was observed below T = 0.2. We therefore used larger numbers of VMMC steps and gradually smaller cooling step sizes to ensure equilibration. The cooling schedule was continued with 2 million steps at each of the following temperatures T = 0.18, 0.16, 0.15, 0.14, 0.135, 0.13.
As the temperature was reduced to T ≤ 0.125, the number of tetrahedral clusters present in the system increased significantly, requiring considerably more VMMC steps to ensure equilibration: T = 0.125 → 0.05 with ∆T = −0.005, involving 5 million steps at each temperature.
In the case of the AB system, we initially followed the cooling schedule: T = 1.0 → 0.6 with ∆T = −0.2, running 1 million steps at each temperature; T = 0.5 → 0.3 with ∆T = −0.1, running 2 million steps at each temperature. For the system of AB-triblock patchy particles, a near 100% yield of tetrahedral clusters was observed at T = 0.2 in terms of discrete tetrahedra, where each particle is part of a single tetrahedron. For each run, 5 million VMMC steps were carried out at T = 0.2 and four configurations, 1 million steps apart along the trajectory, were chosen to be subject to further cooling. The original objective for taking more steps at T = 0.2 was to have larger sample size for the purpose of investigating the second stage of assembly in detail. We thus have 20 runs below T = 0.2 for the AB-triblock patchy particles, grouped into four batches, each following identical annealing protocols: 2 million steps at T = 0.18, 20 million steps at T = 0.16 and 10 million steps at T = 0.155. Finally, we followed the following cooling schedule: T = 0.15 → 0.1 with ∆T = −0.005, running 5 million steps at each temperature.

Ring statistics at low temperatures
Supporting Figure S4a shows that for the system of AB-triblock patchy particles, the numbers of 6-, 7-, and 8-member rings on average remain practically constant and 4-and 5member rings are essentially non-existent below T = 0.15 upon crystallization. In this case, we note the presence of 7-member rings in small numbers over this temperature range, presumably due to fluctuations around weak bonds that the strongly bound tetrahedra form through the second stage of assembly. Across a narrow temperature window where crystallization takes place, while the numbers of 6-and 8-members rings sharply grow, the number of 7-member rings evolves to a low-level presence. In contrast, in the case of the system of AA-triblock patchy particles shown in Supporting Figure S4b, the numbers of 5-and 7-member rings present grow along with those of 6-and 8-member rings, though the growth of the numbers of 6-and 8-member rings are not as much as observed for the system of AB-triblock patchy particles.

Crystallization pathways
In order to glean deeper physical insight into crystallization pathways, we probed the evolution of the size of the largest crystalline cluster along each of the three Brownian dynamics trajectories at T = 0.15, where crystallization took place. This analysis critically relies upon identifying crystalline bonds between particles. For any two particles i and j, we calculate the following normalized scalar product of the respective complex vectors q 6 : 4,5 d 6 (i, j) = Re q 6 (i) · q * 6 (j) |q 6 (i)||q 6 (j)| Two particles are considered bonded if 0.25 ≤ d 6 (i, j) ≤ 0.6, where these limits were determined by calculating the probability distribution of d 6 (i, j) in the cubic and hexagonal tetrastack structures along a Monte Carlo trajectory at T * = 0.15. Additionally, any particle is labeled to be crystalline if it has six crystalline bonds. Using an iterative depth-first search algorithm we then calculate the size of the largest crystalline cluster, N cluster , where two crystalline particles are considered to be a part of the same cluster if they share a crystalline bond. Supporting Figure S6 shows the evolution of the fraction of particles in the largest crystalline cluster, f cluster = N cluster /N , along each of the three Brownian dynamics trajectories. We find that the size of the largest crystalline cluster along each trajectory correlates well at long times with the sum of f CT + f HT + f IF shown in Figure 5, implying that the largest crystalline cluster then contains nearly all crystalline particles for these three crystallization pathways, which thus involve a single nucleus. We note that the growth of the nucleus along the second trajectory occurs relatively slowly, yielding a higher a proportion of the cubic content relative to the hexagonal content. While this correspondence is of interest, establishing it unequivocally requires a much larger pool of crystallization pathways, which is beyond the scope of the present study.

Associated content
Data are available on request from the corresponding author. Figure S1: Representations of cubic (a) and hexagonal (b) tetrastack structures formed by the designer triblock patchy particles, highlighting the presence of ABCABC and ABABAB sequences of triangular planes, respectively, by color coding.      Here r is the radius of the spheres of the particles and a is the lattice constant of the conventional unit cell of the cubic tetrastack structure as predicted for the set of parameters considered here for the AB-triblock patchy particles. Figure S8: Photonic band gap maps for the inverse cubic and hexagonal tetrastack structures composed of air spheres embedded in a high-dielectric medium. Photonic band gap maps are presented as a function of r/a, where r is the radius of the spheres and a is the lattice constant of the conventional unit cell of the cubic tetrastack structure as predicted for the set of parameters considered here for the AB-triblock patchy particles, and are calculated at a dielectric contrast value of 12. Figure S9: Relative gap width of the photonic bandgap, ∆ω/ω m , in percentage for the cubic tetrastack structure supported by the designer AB-triblock patchy particles as a function of the AA / BB ratio, where we set BB = 1.