carefully opened and the sample is flushed gently on to a clean glass
coverslip with PBS solution (PBS, 10 mM, pH = 7.4, 280 mM NaCl, Plu-
ronic F-127 1 wt%). The sample is washed with deionized water and left
on the bench and dried overnight.
Photonic bandgap calculation
We performed a series of photonic bandstructure calculations using
the MIT Photonic Bands software, determining the bands along the
high-symmetry points of the Brillouin zone. The direct and inverse
cluster diamond lattices both have complete photonic bandgaps
between the second and third bands for a range of compression
ratios. Photonic band calculations were performed for each of the
five lowest energy bands at 223 k-vectors, with the primitive unit cell
discretized on a 32 × 32 × 32 grid. The cluster diamond is constructed
from the face-centred-cubic Bravais lattice vectors, with a two-cluster
basis. The only bandgap that appears among the 50 bands calculated
is the one shown, which occurs between the second and third bands.
Finite-difference time domain (FDTD) transmission simulations were
performed to determine whether the bandgap of the inverse cluster
diamond is robust to the disorder caused by polydispersity in the size
of each sphere that comprises a cluster. Transmission of a plane-wave
source was simulated through 8-unit-cell-thick crystals with polydis-
persity in particle diameter up to ±5%, exceeding any polydispersity
used in experiment. Even the largest disorder studied has only a very
minor effect on the transmission, preserving the bandgap.
Simulation
The tetrahedral particles generated in the laboratory can be described
as four partially overlapping spheres arranged with tetrahedral symme-
try, with patches located in the middle of the facets. In the experiment,
the patches are coated with DNA strands, giving rise to an attractive
interaction between patches, while the rest of the particle is repulsive.
We model the attractive patches as being the exposed surface of a
sphere whose centre is located at the centre of mass of the tetrahedral
cluster. Two parameters describe the tetrahedral patchy particles: the
compression ratio dcc/(2a) and the size ratio b/a (Fig. 1 ). Experimen-
tally, it is possible to vary dcc/(2a) substantially, as well as b/a to a lesser
extent, which defines a substantial phase space. To refine the domain
over which the diamond lattice self-assembles, we ran computer simula-
tions using the HOOMD-blue simulation package^36 ,^37. The simulations
are performed using a short-range attractive Lennard–Jones potential
Up between patches, given by Up(r) = 4ε[(σ/r)^2 n − (σ/r)n] with n = 24, where
r is the distance between the centres of the spheres and σ is the radius of
the spheres. This places the minimum in the potential at r = 21/nσ = 1.03σ.
The energy scale is set by the well depth ε, which we arbitrarily set to
be 10 for the attractive interaction between the DNA-coated patches.
To capture the weak attractive depletion interaction between lobes
due to the presence of F-127 micelles and a small amount of DNA, we
use a Lennard–Jones potential with n = 24 and ε = 3. The interaction
between the lobes and DNA-coated patches (which suppresses the
depletion interaction) is modelled by a short-range repulsive WCA
(Weeks–Chandler–Andersen) potential Uc, given by Uc(r) = 4ε[(σ/r)^2 n −
(σ/r)n + 1/4] for r ≤ 21/nσ and Uc(r) = 0 for r > 21/nσ, with n = 24 and ε = 10.
For these parameters, the melting temperature Tm in energy units is in
the range 1.6–1.7. We explored the phase diagram for a shorter-range
potential with n = 48. In a typical simulation run, 216–8,000 particles in
a periodic box at 5 vol% are slowly cooled in the vicinity of the aggrega-
tion temperature. The final system is analysed using the open-source
visualization tool OVITO to discriminate random aggregates from the
formation of any crystalline phases^51. The results of the simulations are
summarized in Extended Data Fig. 3.Microscopy
Bright-field optical images and videos are obtained using a Nikon
Eclipse TiE microscope equipped with a CCD camera. Fluorescent
images and videos are taken using a Leica SP8 confocal fluorescence
microscope. An oil with n = 1.59 (purchased from SPI supplies, Cargille-
Brand Refractive Index Fluid Series A) was added to dried samples,
which were used for the z-stack confocal microscopy. SEM images are
taken using Merlin field-emission SEM. All crystals appeared to be
isotropic. In some cases, however, the crystals cracked along grain
boundaries during drying.Data availability
All data that support the findings are available from the corresponding
authors on reasonable request.Code availability
The HOOMD-blue simulation package used to determine the phase
diagram of the tetrahedral patchy particles^36 ,^37 and the MIT Photonic
Bands software used to calculate the photonic bandstructure calcula-
tions^38 are publicly available.- van der Wel, C. et al. Preparation of colloidal organosilica spheres through spontaneous
emulsification. Langmuir 33 , 8174–8180 (2017). - Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO–the
open visualization tool. Model. Simul. Mater. Sci. Engin. 18 , 015012 (2010).
Acknowledgements This research was primarily supported by the US Army Research Office
under award number W911NF-17-1-0328. Additional funding was provided by the National
Science Foundation under award number DMR-1610788. G.-R.Y. acknowledges support from
the NRF (South Korea) under award number 2017M3A7B8065528. We acknowledge the use of
shared facilities provided through the Materials Research Science and Engineering Center
(MRSEC) programme of the National Science Foundation under award number DMR-1420073.
The computational work was supported in part through the NYU IT High Performance
Computing resources, services and staff expertise.Author contributions M.H. designed the synthetic protocol, synthesized and crystallized
the patchy colloidal clusters, dried the crystals, and performed the optical and electron
microscopy. M.H. and Z.G. synthesized the spherical patchy colloidal particles that led to
the patchy colloidal cluster idea. É.D. performed the simulations and J.P.G. performed the
photonic bandgap calculations that contributed to the design of the photonic crystals. D.J.P.
and S.S. conceived the study and supervised the research, with the help of G.-R.Y. The
manuscript was written by D.J.P., M.H. and J.P.G. All authors discussed the results and
commented on the manuscript.
Competing interests The authors declare no competing interests.Additional information
Supplementary information is available for this paper at https://doi.org/10.1038/s41586-020-
2718-6.
Correspondence and requests for materials should be addressed to S.S. or D.J.P.
Reprints and permissions information is available at http://www.nature.com/reprints.