Abstract
Encoding Archimedean and non-regular tessellations in self-assembled colloidal crystals promises unprecedented structure-dependent properties for applications ranging from low-friction coatings to optoelectronic metamaterials1,2,3,4,5,6,7. Yet, despite numerous computational studies predicting exotic structures even from simple interparticle interactions8,9,10,11,12, the realization of complex non-hexagonal crystals remains experimentally challenging13,14,15,16,17,18. Here we show that two hexagonally packed monolayers of identical spherical soft microparticles adsorbed at a liquid–liquid interface can assemble into a vast array of two-dimensional micropatterns, provided that they are immobilized onto a solid substrate one after the other. The first monolayer retains its lowest-energy hexagonal structure and acts as a template onto which the particles of the second monolayer are forced to rearrange. The frustration between the two lattices elicits symmetries that would not otherwise emerge if all the particles were assembled in a single step. Simply by varying the packing fraction of the two monolayers, we obtain not only low-coordinated structures such as rectangular and honeycomb lattices, but also rhomboidal, hexagonal and herringbone superlattices encoding non-regular tessellations. This is achieved without directional bonding, and the structures formed are equilibrium structures: molecular dynamics simulations show that these structures are thermodynamically stable and develop from short-range repulsive interactions, making them easy to predict, and thus suggesting avenues towards the rational design of complex micropatterns.
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Data availability
The data that support the findings of this study are available at https://doi.org/10.3929/ethz-b-000402331 under Creative Commons Attribution-NonCommercial 4.0 International license.
Code availability
Numerical simulations and analysis code that support the findings of this study are available at https://doi.org/10.3929/ethz-b-000402331 under Creative Commons Attribution-NonCommercial 4.0 International license.
Change history
08 July 2020
The online publication date in the printed version of this article was listed incorrectly as 10 June 2020; the date was correct online.
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Acknowledgements
We thank E. Zaccarelli, F. Camerin, W. Steurer and T. Weber for discussions. We also thank M. R. Bailey for proofreading the manuscript. L.I. and M.A.F.-R. acknowledge financial support from the Swiss National Science Foundation Grant PP00P2-172913/1.
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Author contributions are defined based on the CRediT (Contributor Roles Taxonomy) and listed alphabetically. Conceptualization: M.-N.A., M.A.F.-R., F.G. and L.I. Data curation: F.G. Formal analysis: M.-N.A., M.A.F.-R. and F.G. Funding acquisition: L.I. Investigation: M.-N.A., M.A.F.-R., D.G. and F.G. Methodology: M.-N.A, M.A.F.-R., F.G. and L.I. Project administration: L.I. Software: M.A.F.-R. and F.G. Supervision: M.A.F.-R. and L.I. Validation: M.-N.A., M.A.F.-R., D.G. and F.G. Visualization: M.-N.A., M.A.F.-R., F.G. and L.I. Writing original draft: M.A.F.-R., F.G. and L.I. Writing review and editing: M.-N.A., M.A.F.-R., D.G., F.G. and L.I.
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Extended data figures and tables
Extended Data Fig. 1 Compression of individual monolayers.
Surface pressure Π versus area per particle Ap and representative AFM images obtained by the simultaneous compression and deposition of a monolayer of 3CS0 microgels from a water–hexane interface onto a silicon wafer. Scale bar, 3 µm.
Extended Data Fig. 2 Agreement between measured and extrapolated values of ϕ1 and ϕ2.
The deposition process implies that ϕ1 and ϕ2 are expected to vary only along x and y, which correspond to the respective compression directions. This is particularly advantageous because ϕ1(x) and ϕ2(y) can be reliably estimated from AFM images taken along the regions where only a single hexagonal monolayer was deposited (y < 0 or x < 0). To test whether extrapolating ϕ1(x) and ϕ2(y) from such regions to the entire Si wafer provides a good estimate, we systematically compared extrapolated (ϕ1,ϕ2) pairs with the values measured from AFM images taken in regions where two monolayers were deposited onto each other (x, y > 0), where a slight difference in height between the particles from the two depositions could be leveraged to measure ϕ1 and ϕ2 independently. The dots denote the different measurements based on AFM images taken for x and y > 0. RMSE, root-mean-square error.
Extended Data Fig. 3 Depositing two monolayers onto the same substrate leads to in-plane packing.
AFM images of particle monolayers with similar area per particle (a, 0.3 µm2; b, 0.4 µm2) but obtained after single (a, and red curve in c) and double (b, and blue curve in c) deposition, respectively. c, The height profiles along two representative line scans show that particles from both single and double depositions sit at the same height. The patterns arising from double depositions are therefore not the result of an out-of-plane stacking, but originate from the re-arrangement of the particles of the second monolayer into in-plane interstitial positions. Furthermore, the maximum height of the particles from the second deposition (second, fourth and sixth peak of the blue curve) is systematically higher (by about 10 nm) than the ones from the first deposition (first, third and fifth peak of the blue curve), owing to the in-plane compression exerted by the previously deposited particles. This fact was leveraged to estimate ϕ1 and ϕ2 from individual images, as particles arising from different depositions can be singled out via a height threshold (see Extended Data Fig. 2). Scale bar, 3 µm.
Extended Data Fig. 4 Effect of α on the shape of the interaction potential and on bond-orientation order.
Top, generalized Hertzian potential U(r) (left) and corresponding interparticle force F(r) (right) as a function of the normalized separation distance r/σ for different values of α. Bottom, ψ3, ψ4 and ψ5 of the simulated ground-state structures as a function of ϕ1 and ϕ2 for different values of α.
Extended Data Fig. 5 Distribution of the observed bond orientational order.
ψk (with k ∈ 3,.., 12) is represented through box and whisker plots for 3CS0 (a) and 3CS1 (b). The bottom and top edges of the box indicate the 25th and 75th percentiles, the central red line indicates the median, and the lower and top whiskers indicate values equal to the median ±1.5 times the interquartile range. The inset shows the interquartile range (IQR) of the observed ψk. c, Measured values of ψ3, ψ4, and ψ5 for 3CS0 microgels as a function of ϕ1 and ϕ2.
Extended Data Fig. 6 Experimental and simulated intermediate superstructures.
AFM images of intermediate superstructures obtained with 3CS0 microgels and their corresponding simulation snapshots. The values in brackets indicate the respective (ϕ1,ϕ2) pairs. The simulations were run with α = 1.9. The scale bars in the AFM images are 5 µm.
Extended Data Fig. 7 Estimating ground-state structures by molecular dynamics simulations.
Analysis of molecular dynamics simulations showing the structural evolution towards minimum energy configurations as a function of time and reduced temperature kT/ε for two different (ϕ1,ϕ2) pairs, namely (1.4,1.4) in panels a and c, and (0.9,1.3) in panels b and d. For illustration purposes, the disks in panels a and b are drawn with a diameter of 0.6σ. The FIRE algorithm41 was used to minimize the potential energy while bringing kT/ε from 10−4 to 0. The insets in c and d show simulations snapshots taken after 2 × 107 steps.
Extended Data Fig. 8 Relaxing the constraint on the mobility of the first monolayer leads to simpler structures.
Simulated ground-state structures obtained at different packing fractions ϕ1 + ϕ2 for α = 1.9 when all the particles are allowed to re-arrange freely. For illustration purposes, the particles are drawn with a diameter of 0.6σ.
Extended Data Fig. 9 Realization of target structures over centimetre-scale substrates.
AFM images of honeycomb lattices obtained from double depositions of 5CS1 (ϕ1 ≈ ϕ2 ≈ 1.2) (a), 3CS0 (ϕ1 ≈ ϕ2 ≈ 1.4) (b) and 3CS1 (ϕ1 ≈ ϕ2 ≈ 1.5) (c) microgels. d–i, Optical microscope images of two different superstructures deposited over two different 2 cm × 2 cm substrates by keeping the surface pressure constant throughout each deposition. The optical images were taken at random locations that were more than 1 mm apart from each other. The scale bars are 10 µm for the optical microscopy images and 2 µm for the AFM image in the insets. The surface pressures during the sequential depositions were Π1 = Π2 = 18 mN m−1 for d–f and Π1 = 26 mN m−1 and Π2 = 9 mN m−1 for g–i.
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Grillo, F., Fernandez-Rodriguez, M.A., Antonopoulou, MN. et al. Self-templating assembly of soft microparticles into complex tessellations. Nature 582, 219–224 (2020). https://doi.org/10.1038/s41586-020-2341-6
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DOI: https://doi.org/10.1038/s41586-020-2341-6
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