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Self-assembly of emulsion droplets through programmable folding

Abstract

In the realm of particle self-assembly, it is possible to reliably construct nearly arbitrary structures if all the pieces are distinct1,2,3, but systems with fewer flavours of building blocks have so far been limited to the assembly of exotic crystals4,5,6. Here we introduce a minimal model system of colloidal droplet chains7, with programmable DNA interactions that guide their downhill folding into specific geometries. Droplets are observed in real space and time, unravelling the rules of folding. Combining experiments, simulations and theory, we show that controlling the order in which interactions are switched on directs folding into unique structures, which we call colloidal foldamers8. The simplest alternating sequences (ABAB...) of up to 13 droplets yield 11 foldamers in two dimensions and one in three dimensions. Optimizing the droplet sequence and adding an extra flavour uniquely encodes more than half of the 619 possible two-dimensional geometries. Foldamers consisting of at least 13 droplets exhibit open structures with holes, offering porous design. Numerical simulations show that foldamers can further interact to make complex supracolloidal architectures, such as dimers, ribbons and mosaics. Our results are independent of the dynamics and therefore apply to polymeric materials with hierarchical interactions on all length scales, from organic molecules all the way to Rubik’s Snakes. This toolbox enables the encoding of large-scale design into sequences of short polymers, placing folding at the forefront of materials self-assembly.

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Fig. 1: Colloidomer design and folding.
Fig. 2: Folding pathways of a heptamer.
Fig. 3: Predicted and experimentally realized foldamers.
Fig. 4: Colloidomer sequence design and supracolloidal assembly.

Data availability

The data that support the findings of this study are available from the corresponding author upon request. The data includes experimental as well as computational datasets, Matlab scripts for experimental video analysis and Python scripts for computational dataset analysis.

Code availability

The custom computer codes to build folding trees, to identify foldamers and the Dissipative Particle Dynamics (DPD) code to simulate folding of colloidomers are available from the corresponding author upon request. We have provided pseudo-code for the enumeration code in Extended Data Figure 2. DPD is also available in open-source packages such as HOOMD and LAMMPS.

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Acknowledgements

We thank D. Pine, A. Grosberg, P. Chaikin, S. Hilgenfeldt, E. Clément, O. Rivoire and L. Leibler. This work was supported by the Paris Region (Région Île-de-France) under the Blaise Pascal International Chairs of Excellence. This work was also supported by the MRSEC programme of the National Science Foundation under grants No. NSF DMR-1420073, No. NSF PHY17-48958 and No. NSF DMR-1710163, the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754387, as well as by the city of Paris EMERGENCE(S) grant.

Author information

Authors and Affiliations

Authors

Contributions

A.M. designed the materials, synthesized the droplet system, performed the folding experiments and developed data analysis tools to extract time-dependent yields of foldamers. M.M.B. and Z.Z. constructed the theoretical model for generating folding trees and rigid states. M.M.B. developed the algorithm for enumerating foldamers, wrote the molecular dynamics codes and performed numerical simulations. J.B. and Z.Z. conceived the study and supervised the research. The manuscript was written by J.B. and Z.Z. together with A.M and M.M.B.

Corresponding authors

Correspondence to Zorana Zeravcic or Jasna Brujic.

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The authors declare no competing interests.

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Nature thanks Ning Wu and Menachem Stern and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Chain length distribution.

This panel shows a distribution of chain lengths from a typical chaining experiment, on a semi-log scale. Observed chains in the range N = 3−14 are exponentially distributed.

Extended Data Fig. 2 Flowchart of the foldamer search algorithm.

The top panel shows the ingredients required to run the algorithm: the ensemble of Hamiltonian paths {Hi,N}, a sequence, and a protocol. The input is a set of colored Hamiltonian paths embedded on the geometries, as shown for N = 7 and an alternate ABABABA sequence. The bottom panel outlines the main steps of the algorithm. In the figure, LM stands for Local Minimum, TS for Transient State and GM for Global Minimum.

Extended Data Fig. 3 Foldamer yields for an alternating ABAB sequence with length N = 6−13.

From left to right, we show the results for single, two, and three-step protocols. All yields are given as relative yields, in which the number of foldamers is normalized by the total number of rigid structures observed at the end of the corresponding protocol. The experimental number of observations is n6[ladder, triangle, chevron] = (67, 19, 86), n7[rocket#1, rocket#2, flower] = (175, 25, 7), n8[hourglass] = 8, n9[poodle] = 24 and n10[crown] = 8. ‘ND’ stands for ‘No Data’. Simulation yields YSimDownhill and YSimStepThermal come from two different protocols, the pure downhill and the step-thermalized quench, respectively, and result from averaging over > 2000 different initial conditions (sampling error is negligible). The significant difference between the two simulation protocols arises because the finite unbinding probability in the step-thermalized case is optimized to allow the colloidomer to escape kinetic traps, i.e., local minima, and fold correctly (see Supplementary Video 8 and the Extended Data Fig. 4(a)), whereas this is not possible in the downhill case. Exceptions are the N = 7 flower and N = 10 bed, foldamers that undergo geometric frustration, as the irreversibility of the locking bonds in the downhill protocol actually improves the yield. The two simulation methods give the range of yields one can access by folding strictly in 2D. Experiments undergo a finite temperature quench, which is better mimicked by the thermalized simulation protocol in all cases. Note that all but one (triangle) experimental yield fall within the range predicted by simulations. This experimental yield exceeds that of the optimized simulation, owing to the fact that local minima can be escaped by rare 3D excursions only possible in experiments.

Extended Data Fig. 4 Optimized foldamer yields for alternating and designed sequences.

(a) Relative foldamer yield YR(%) as function of bond strength ϵ/kBT for three foldamer structures: the ladder (blue), hourglass (orange) and crown (green). These structures exemplify foldamers resulting from 1-step, 2-step and 3-step protocols, respectively. The results were obtained by waiting τ = 103 time units (t. u.) with the first interaction on, and setting τ = 102 t. u. for subsequent interactions. We note that ϵ/kBT refers only to the binding strength during the first step in the protocol; subsequent steps in the protocol are treated as pure downhill folding. (b) Relative yields of the three foldamers in Fig. 4(b). As discussed in Extended Data Fig. 3, YSimDownhill corresponds to pure downhill folding, where the chain is likely to get trapped in a local minimum along the pathway. YSimStepThermal yields are the result of optimized binding strengths ϵ and quench lengths τ. Results in the table were obtained by setting τ = 103 t.u., and choosing ϵ/kBT = 8 for the N = 8 and N = 13 chains, and ϵ/kBT = 7 for the N = 11 chain.

Extended Data Fig. 5 Non-compact clusters in a hexagonal lattice in 2D.

The first non-compact cluster arises at N = 13; for N = 14 there are 6 different clusters, and for N = 15 we identify 41. All non-compact clusters contain a single hexagonal hole. For those structures colored in green, there exists at least one protocol in random sequence space that can fold them uniquely.

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McMullen, A., Muñoz Basagoiti, M., Zeravcic, Z. et al. Self-assembly of emulsion droplets through programmable folding. Nature 610, 502–506 (2022). https://doi.org/10.1038/s41586-022-05198-8

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