1. Introduction

Self-assembly of colloidal spheres was proposed some 20 years ago as a promising approach for making optical materials with photonic band gaps [1–3]. The approach gained traction when it was realized that an inverse face-centered cubic (FCC) crystalline arrangement of touching or slightly overlapping dielectric spheres had an omnidirectional band gap between the 8^th and 9^th bands [4]. Since spherical colloids were well-known to self-assemble into an FCC lattice, self-assembly seemed like a promising approach [5]. However, the band gap opens up only when the refractive index of the matrix is about 2.8 times greater than that of the spheres [6], which is very difficult to achieve at optical frequencies. Moreover, because the band gap appears between the 8^th and 9^th bands, it is very sensitive to disorder.

In contrast to FCC, both the cubic diamond and the pyrochlore lattices exhibit large robust photonic band gaps [4, 7, 8]. The band gaps open up between the 2^nd and 3^rd bands in both lattices and are relatively insensitive to disorder [9]. Direct and inverse structures exhibit band gaps in both lattices for refractive-index contrasts close to 2.0, which is readily achievable in the visible with a variety of optical materials such as TiO₂. The challenge is making these structures, as colloidal spheres do not self-assemble into either lattice. The reason is simple: both lattices are made up of tetrahedrally-coordinated spheres and are very open, with maximum packing fractions of $\sqrt{3}\pi /16\simeq 0.34$ and $\pi /\sqrt{72}\simeq 0.37$, respectively, for cubic diamond and pyrochlore lattices of touching but non-overlapping spheres. By contrast, the maximum packing fraction of the FCC lattice is $\pi /\sqrt{18}\simeq 0.74$.

Hynninen et al. [10] pointed out that the MgCu₂ lattice, one of three related Frank Kasper phases known as Laves phases, consists of two interpenetrating sublattices, a cubic diamond sublattice formed by the Mg atoms and a pyrochlore sublattice formed by the Cu atoms. In the geometrically ideal structure made from hard spheres, the ratio of the diameters of spheres making up the diamond and pyrochlore lattices is $\sqrt{3/2}\simeq 1.225$. Initial attempts to self-assemble this structure from spheres with this size ratio, both in simulation and in experiment, met with very limited success because the two other Laves phases, composed of different sublattices, compete with the MgCu₂ structure such that highly mixed polycrystalline or disordered structures form [10,11].

Recently, we proposed a new approach that involved assembling the MgCu₂ structure from a mixture of spheres and pre-assembled tetrahedral clusters of spheres [12]. The spheres and tetrahedral clusters were coated with a short polymer brush with terminal DNA “sticky ends” programmed so that clusters would bind to spheres, but spheres would not bind to spheres and clusters would not bind to clusters. This approach proved successful: the spheres and tetrahedral clusters self-assembled into interpenetrating sublattices of cubic diamond and pyrochlore, respectively, with the spheres forming the cubic diamond sublattice and the tetrahedral clusters forming the pyrochlore sublattice. The idea would be to self-assemble the MgCu₂ structure with sublattices made from two different materials and then remove one of the sublattices sacrificially leaving behind either the cubic diamond or the pyrochlore lattice alone. Of course, this only works if the remaining sublattice is continuously connected, which for diamond means $\varphi >\sqrt{3}\pi /16\simeq 0.34$ and for pyrocholore $\varphi >\pi /\sqrt{72}\simeq 0.37$.

The success of this new approach to assemble the MgCu₂ structure raises several questions. First, can we optimize the conditions under which the MgCu₂ structure forms? Here, we explore the effect of two critical parameters on the phase diagram: the range of the attractive DNA brush interaction between clusters and spheres and the compression ratio (described below), a parameter that characterizes the different shapes of the tetrahedral clusters we synthesize. We find that the domain where the MgCu₂ structure assemble is greatly expanded by the use of longer range interactions.

A second question raised by the non-geometrically-ideal clusters is how their altered shape affects the photonic band structure of the resultant pyrochlore lattices. Surprisingly, we find that in some cases it dramatically improves the band structure by opening up a much wider photonic band gap. We explore the conditions under which these changes occur, which are very different in direct vs. inverted pyrochlore lattices.

2. Characterization of clusters and spheres

As noted above, the tetrahedral clusters used are not geometrically ideal clusters of non-overlapping spheres. Owing to the synthetic method, each cluster consists of four spheres that partially overlap each other, as depicted in Fig. 1. We characterize the degree of overlap by the compression ratio r_cc/2r₀, defined as the ratio of the distance between the sphere centers r_cc and the diameter 2r₀ of the spheres. The compression ratio can be varied over quite a large range from unity down to 0.5 [13].

 

Fig. 1 Left: Tetrahedral clusters formed by four overlapping polystyrene spheres. The compression ratio r_cc/2r₀ defines the degree of inter-penetration of the constituent spheres. Here, polystyrene spheres are used with r_cc/2r₀ = 0.75. Scale bar: 500 nm. Right: Renderings of clusters with r_cc/2r₀ = 1, 0.75, and 0.5, from top to bottom.

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DNA coated particles are composed of an organic or inorganic spherical hard core, decorated with polymers terminated with DNA sticky ends [14], as described above. For micron sized particles, a typical DNA brush is about 15 nm thick. The length of the brush defines the range and softness of the effective interaction between spheres with complementary DNA sticky ends. The strength of the interaction can be adjusted by changing various parameters: the temperature, the number of nucleotides and sequence of bases in the sticky ends, and various other parameters [15–17].

3. Phase diagram

In previous work [12], we used simulations to investigate the nucleation and growth of the MgCu₂ structure using tetrahedral clusters and spheres interacting through a very short range attractive potential. The simulations showed that there are two key geometrical parameters that influence the formation of the MgCu₂ structure: the compression ratio r_cc/2r₀, introduced above and in Fig. 1, and the ratio R/r₀, where R is the radius of the singlet spheres and r₀ is the radius of the individual spheres that make up the clusters. Using a 96-48 Lennard-Jones potential to model the short range DNA interaction, we found that the MgCu₂ structure does not form unless the compression ratio is significantly less than the geometrically ideal value of 1, which came as something of a surprise. This result was confirmed in experiments using clusters and spheres coated with complementary DNA with r_cc/2r₀ = 0.75 and R/r₀ = 1.1. The simulation result was later reproduced by Zanjani et al. [18] using a similar short-range interaction potential.

Using DNA to construct an attractive effective potential between spheres and clusters affords one a great deal of flexibility in setting the range and softness of the interaction. Such changes can be realized, for example, by varying the density [19,20], thickness [21], or stiffness [22] of the polymer brush. Here we use simulations to explore the effect of the range and softness of the potential on the phase diagram.

To explore the effect of the interaction range on the self-assembly of tetrahedral clusters and spheres, we performed Brownian dynamics simulations using HOOMD-blue [23,24] in a three-dimensional square box with periodic boundary conditions. As in the experiments, we use an attractive potential U_A(r) between clusters and spheres and a repulsive potential U_R(r) between clusters and clusters and between spheres and spheres. We also add a weak attractive interaction between spheres to model the depletion interaction observed in the experiments. The potentials we use are given by

Figure 2(a) plots U_A(r) for different values of n over this range. The repulsive potential U_R(r) is the standard WCA potential [25] but with the same value of n used in U_A(r), since the repulsive part of the potential in both cases is governed by the softness of the polymer brushes.

 

Fig. 2 Phase diagrams for different potentials. (a) Attractive potential U_A(r) between spheres and clusters for various values of n in Eqs. (1) and (2). (b) Phase diagrams showing domains of compression ratios r_cc/2r₀ sphere ratios R/r₀ where the MgCu₂ lattice forms for different values of the exponent n, with the correspondence between color and n for both panels indicated in panel (b). For each value of n, the structure is amorphous outside its corresponding colored boundary.

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For a typical simulation run, a box containing spheres and clusters with a number ratio 1:2 is slowly cooled down, taking the system through the gas/solid phase transition. The total volume fraction of particles is 0.05. The final structure is characterized by direct visualization and by calculating the sphere-sphere, cluster-cluster and cluster-sphere partial radial correlation functions. Depending on the compression ratio r_cc/2r₀, size ratio R/r₀, and range of the potential (set by the exponent n), the system condenses into either an amorphous or MgCu₂ crystalline structure. Figure 2(b) summarizes the simulation results.

The darkest red region in Fig. 2(b) shows the range of compression and size ratios over which the MgCu₂ structure forms for the case n = 48. For this region, r_min in the potential occurs at a distance that is about 1.44% larger than the hard sphere diameter σ (for n = 48, (r_min − σ)/σ ≃ 0.0144). The domain over which the MgCu₂ structure forms is limited to compressed clusters with r_cc/2r₀ below 0.8 and R/r₀ between 0.8 and 1.1. These results are consistent to those we reported earlier.

For the next region in Fig. 2(b), colored dark orange, n = 24 and the minimum in the potential occurs at a distance that is about 2.93% larger than the hard sphere radius. The MgCu₂ structure forms for all values of compression and size ratios within this region, including the dark red region. As n further decreases to 18 and ultimately down to 6, the region in the phase diagram over which the MgCu₂ structure forms grows dramatically, both in r_cc/2r₀ and in R/r₀. With a thicker interaction shell, spheres and clusters can be a bit further away from each other and still participate in the stabilization of the crystal. For n ≤ 12, uncompressed clusters can be used to assemble the crystal. We also observe that the quality of the crystal formed in the simulation box is generally improved—there are fewer defects—for the softer potentials.

We have shown previously that using n = 24, which corresponds to a fairly short-range stiff potential, this model provides a good description of the formation MgCu₂ crystals for 800-nm particles with a 15-nm DNA brush. However, these new results show that an MgCu₂ structure is more readily realized using softer long-range interactions. This can be accomplished by increasing the thickness of the DNA brush relative to the particle size. This could be done either by using smaller particles and/or long polymer brushes. One can also use a DNA tether that is largely double-stranded DNA to make the DNA sticky extend further into the solvent (water).

Another important conclusion of these simulations is that it is not necessary start from compressed clusters to assemble the MgCu₂ lattice, provided the attractive potential between the spheres and clusters is sufficiently long-ranged and soft. This may make synthesis of the clusters and assembly easier when working with hard materials like silica or titania that do not so readily interpenetrate like polymer spheres.

4. Photonic band structures

The pyrochlore lattice was identified as having a robust large photonic band gap some 12 years ago [8,26]. However, none of the investigations considered the possibility of making a pyrochlore lattice using compressed tetrahedra, which has recently been realized experimentally. Since experimentalists have a great deal of latitude in constructing pyrochlore lattices with compressed clusters, it is useful to know which compression ratios produce the widest band gaps for both the direct or inverse lattices.

To this end, we have performed a series of photonic band structure calculations using the MIT Photonic Bands (MPB) software [27], estimating the bands along the high symmetry points of the FCC Brillouin zone. Both the direct and inverse pyrochlore lattices have complete photonic band gaps between the second and third bands, so the photonic band calculations were performed for each of the 5 lowest energy bands at 79 k-vectors with the primitive unit cell discretized on a 16 × 16 × 16 grid (spatial resolution of ∼ 1/10^th of a sphere diameter). Performing the simulations on 64 × 64 × 64 and 128 × 128 × 128 grids did not perceptibly change the plots.

Figure 3(a) and (b) show the photonic band structures for the direct and inverse pyrochlore lattices, respectively, for a refractive index contrast of m = 2.6 and without any compression of the clusters (r_cc/2r₀ = 1). For r_cc/2r₀ = 1, the direct pyrochlore lattice exhibits a band gap, indicated by the blue band in Fig. 3(a), while the indirect lattice does not have a band gap. For m = 2.6, a band gap does open up in the inverse lattice for compressed clusters when r_cc/2r₀ < 0.96.

 

Fig. 3 Photonic band diagrams and renderings for four different crystals at an optical contrast m = 2.6, where f is the frequency, a is the lattice constant for the underlying FCC unit cell, and c the speed of light in vacuum. The presence of an omnidirectional photonic band gap is highlighted by the blue band. (a) The geometrically ideal pyrochlore lattice with no compression. (b) The geometrically ideal inverse pyrochlore lattice, for m = 2.6 this crystal does not have a complete photonic band gap. (c) The pyrochlore formed from compressed clusters with r_cc/2r₀ = 0.95. (d) The inverse of a pyrochlore formed from clusters with r_cc/2r₀ = 0.60. INSETS: Renderings of the corresponding lattices

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The question arises, therefore, as to how changing the compression ratio affects the photonic band gap. In general, one prefers structures for which a band gap opens at the lowest possible refractive index contrast, as this puts fewer demands on the material used and generally gives the largest band gap for a given optical contrast m > m_min. In Fig. 4(a), therefore, we plot the minimum refractive index contrast m_min at which a band gap opens up as a function of the compression ratio r_cc/2r₀. For the direct pyrochlore lattice, we see that reducing r_cc/2r₀ from 1 to about 0.9 results in a modest improvement with m_min going from 2.19 to 2.12. Further reduction in r_cc/2r₀ results in a dramatic increase in m_min. By contrast, reducing r_cc/2r₀ below 1 for the inverse pyrochlore results in a precipitous drop in m_min. The reduction in m_min begins to plateau around r_cc/2r₀ ≈ 0.6 where it even dips a bit below 2. Thus we see that the lowest m_min achieved for the direct lattice is 2.12 and occurs at a very modest degree of compression, r_cc/2r₀ ≃ 0.92, while for the inverse lattice m_min is 2.0 and occurs for r_cc/2r₀ ≃ 0.5, which represents a significant degree of compression. Further reduction in r_cc/2r₀ for the indirect lattice does not lower m_min. As noted previously, the volume fraction ϕ of a pyrochlore lattice of uncompressed clusters is r_cc/2r₀ = 0.37. From there, the volume fraction increases linearly to ϕ = 0.65 for r_cc/2r₀ = 0.50.

 

Fig. 4 (a) Minimum refractive index contrast m_min for which a band gap opens (i.e. Δf/f_c > 0.0001) for the direct and inverse pyrochlore lattices built using compressed clusters. (b) Relative band gap δ = Δf/f_c as a function of the compression ratio for various optical contrast m for direct (solid lines) and inverse (dashed lines) pyrochlore lattices.

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We now turn our attention to the actual values of the band gaps and how they depend on m and r_cc/2r₀. We characterize band gaps by the center frequency f_c and the dimensionless width of the band gap δ = Δf/f_c, where Δf is the width of the omnidirectional band gap (see Fig. 3(a)). In Fig. 4(b) we plot the dimensionless band gap δ as a function of the compression ratio r_cc/2r₀ for different values of the refractive index contrast m. Once again, we see that the direct pyrochlore lattice is optimized—it gives the largest dimensionless band gaps δ—for compression ratios just below 1, while the inverse lattice is optimized for the compression ratios near 0.5. Once again, the inverse lattice gives slightly better results with a maximum dimensionless band gap δ that is consistently larger than that for the direct lattice.

Experimentally, once the MgCu₂ structure is formed and the diamond lattice removed so that only the pyrochlore lattice remains, it is possible to further increase the volume fraction of the solid pyrochlore lattice by coating more material onto its surface. For example, a pyrochlore lattice made of silica could be coated with more silica using standard sol-gel synthesis methods or atomic layer deposition [28]. The effect is similar to adding a coating of paint. From the standpoint of pure geometry, one can view this as increasing—or swelling—the radius of the spheres in the tetrahedral clusters. Doing this for the direct pyrochlore lattice results in only a very modest improvement in the optical properties in the best of cases. However, adding such a layer to the pyrochlore lattice and then using the resultant structure as a template to make an inverted structure can result in a significant improvement in the inverse pyrochlore optical properties. Figure 5 shows the result of effectively swelling the radius of the spherical holes of the inverse pyrochlore lattice from their original radius r₀ to their over coated radius r_f. For all values of the refractive index contrast m, the dimensionless band gap δ increases. For r_cc/2r₀ = 0.65, δ increases from 0.07 for no swelling to almost 0.14, nearly doubling the width of the band gap. The minimum value of the refractive index contrast m_min required to open a band gap also falls from 2 to about 1.8. Thus, swelling the holes in an inverse pyrochlore lattice can dramatically improve the optical properties of the photonic crystal.

 

Fig. 5 Relative band width as a function of ratio r_f/r₀ for an inverse pyrochlore lattice constructed from clusters with r_cc/2r₀ = 0.65. INSET: rendering for the corresponding inverse lattice with r_cc/2r₀ = 0.65 and r_f/r₀ = 0.85.

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5. Conclusions

The introduction of compressed tetrahedral clusters and DNA-directed self-assembly enables the self-assembly of the MgCu₂ structure consisting of two sublattices that separately have photonic band gaps, one diamond and the other pyrochlore. The use of compressed clusters lowers the optical contrast required to open up a band gap in the pyrochlore structure, dramatically for the inverted crystal. By tuning the range of the DNA-mediated attraction between clusters and spheres, the domain over which the MgCu₂ structure forms is expanded facilitating the formation of whatever structure is desired. The general approach of employing increasingly complex building blocks, in this case tetrahedral clusters, may be a promising one for fabricating optical materials and optimizing their properties.