Minimal Positive Design for Self-Assembly of the Archimedean Tilings

Abstract

A challenge of molecular self-assembly is to understand how to design particles that self-assemble into a desired structure and not any of a potentially large number of undesired structures. Here we use simulation to show that a strategy of minimal positive design allows the self-assembly of networks equivalent to the 8 semiregular Archimedean tilings of the plane, structures not previously realized in simulation. This strategy consists of identifying the fewest distinct types of interparticle interaction that appear in the desired structure, and does not require enumeration of the many possible undesired structures. The resulting particles, which self-assemble into the desired networks, possess DNA-like selectivity of their interactions. Assembly of certain molecular networks may therefore require such selectivity.Molecular self-assembly is the spontaneous organization of molecules or other small particles into structures [1][2][3][4][5][6][7][8][9][10]. Despite many successes in the laboratory we lack complete understanding of how to design particles that will self-assemble into a desired structure: sometimes the outcome of self-assembly is an undesired structure, which might be metastable or kinetically trapped [11]. Simulation can help us understand how particle design influences the process and outcome of self-assembly [12][13][14][15][16][17]. Ideas derived from such studies include the notion of positive and negative design, used by Doye, Louis and Vendruscolo [12] to describe the design of particles to promote a desired structure or to suppress undesired structures. We show here that a particular type of positive design, which we shall call minimal positive design, allows the self-assembly of network structures equivalent to the Archimedean tilings of the plane. The concept is simple to implement: we drew pictures of the desired structures and labeled the interparticle interactions that arise, and then (on the computer) made particles with those interactions and no others. Those particles self-assembled, under a simple cooling protocol, into the desired structures, which had not previously been realized in simulation [18]. The field of DNA nanotechnology makes widespread use of the principle of chemical selectivity [19][20][21][22], allowing, for instance, self-assembly of structures in which particles are of many distinct types [7,23,24]. Here we show that chemically selective interactions are also necessary to assemble certain complex single-component structures, which a priori do not appear to require this capability. By formalizing the idea of minimal positive design we hope to provide a way of thinking about chemical selectivity and its role in self-assembly.We illustrate the concept of minimal positive design by considering the self-assembly of networks equivalent to tilings of the plane [25,26]. Such networks are complex geometrically, and so pose a challenge for design, and their experimental realizations, via the self-assembly of real molecules at surfaces, have useful properties [27- * swhitela...

Programming patchy particles to form complex periodic structures

“…Previous work on patchy particles and crystallization has mainly focussed on particles without torsions. A significant fraction of that work has been on 2D crystals, [25][26][27][28][29][30][31] and those studies focussing on 3D crystals have mainly been with one-component systems. 21,[32][33][34][35][36][37] By contrast, most work using torsionallyspecific patchy particles has been on the assembly of finite structures (analogous to protein complexes), 38,39 an exception being the study on diamond mentioned above.…”

Irregular model DNA particles self-assemble into a regular structure

“…2. When φ 0 ≈ 90 • we observe assembly of a network equivalent to the 4.8.8 Archimedean tiling [6], signaled on the plots by the roughly equal numbers of 4-gons and 8-gons. When φ 0 ≈ 120 • we observe assembly of a network equivalent to the 6.6.6 tiling.…”

“…5. Regions of parameter space in which the indicated phase is lowest in rotational free energy, according to the equations (5), (6), and (8). We show only phases whose rotational free energies lie within kBT of ln 2w: thus in the white region all phases are strained (this strain can be inferred from simulation: see e.g.…”

“…The angles 90 • and 135 • are equal to the internal angles of the square and the octagon, making clear the connection between the particle and the assembly. Notably, the sticky patches of the particle must be chemically specific, in the sense that only two types of complementary interaction are possible [5,6]. In the absence of chemical specificity, if all particles stick to all other particles, then there exist too many competing structures to allow self-assembly of the 4.8.8 tiling.…”

Exotic Vortex Lattices in Binary Repulsive Superfluids

“…These regular structures are important for their resulting mechanical, optical, electrical, and chemical properties. Appreciating how certain systems self-assemble into specific regular tessellations is relevant both for the fundamental understanding of the systems themselves as well as for the design of new metamaterials [10][11][12].…”