Self-Assembly of Two-Dimensional Binary Quasicrystals: a Possible Route to a DNA Quasicrystal Aleks Reinhardt,1 John S

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Self-assembly of two-dimensional binary quasicrystals: A possible route to a DNA quasicrystal Aleks Reinhardt,1 John S. Schreck,2 Flavio Romano,2, a) and Jonathan P. K. Doye2 1)Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, United Kingdom 2)Physical and Theoretical Chemistry Laboratory, Department of Chemistry, University of Oxford, South Parks Road, Oxford, OX1 3QZ, United Kingdom We use Monte Carlo simulations and free-energy techniques to show that binary solutions of penta- and hexavalent two-dimensional patchy particles can form thermodynamically stable quasicrystals even at very narrow patch widths, provided their patch interactions are chosen in an appropriate way. Such patchy particles can be thought of as a coarse-grained representation of DNA multi-arm ‘star’ motifs, which can be chosen to bond with one another very specifically by tuning the DNA sequences of the protruding arms. We explore several possible design strategies and conclude that DNA star tiles that are designed to interact with one another in a specific but not overly constrained way could potentially be used to construct soft quasicrystals in experiment. We verify that such star tiles can form stable dodecagonal motifs using oxDNA, a realistic coarse-grained model of DNA.

PACS numbers: 61.44.Br, 47.57.-s, 81.16.Dn

I. INTRODUCTION phases: the quasicrystal was found to be a robust feature of the system and it persisted as the thermodynamically stable Since their discovery was reported in 1984,1 quasicrys- phase over a range of parameterisations of the model and tals have been extensively studied, and many have unusual occupied significant portions of the phase diagrams we com- electronic2 and surface3 properties. While most quasicrys- puted. tals reported thus far have been metallic alloys,4,5 such In the light of these results, we anticipated that it might structures have also increasingly been seen in soft matter not be overly difficult to self-assemble such quasicrystalline systems,6–8 for example in colloidal9 and micellar10,11 sys- structures in experiment. Whilst a true ‘patchy particle’ tems and polymer melts.12–14 Moreover, quasicrystals have system44–55 confined in two dimensions, perhaps by density been observed in a number of computer simulations.15–37 A mismatching,52 would perhaps be the most obvious candi- mean-field approach has shown that dodecagonal quasicrys- date, one attractive alternative might be to make use of DNA tals are often thermodynamically more stable than other types multi-arm motifs,56–61 which have been shown to be able of quasicrystal,38 and a theoretical approach suggests that to self-assemble into a range of effectively two-dimensional quasicrystals in soft matter are likely to be dodecagonal.39 structures. DNA multi-arm motifs, or ‘star tiles’, are DNA We have recently observed an example of such a dodecago- structures which form a star shape with a certain number of nal soft quasicrystal when studying the self-assembly be- protrusions called ‘arms’. The DNA strands in these struc- haviour of a two-dimensional patchy-particle system.30 We tures are complementary such that the bulk of the structure studied a system of particles with an angular dependence, is fully bonded, but the very ends of each of the arms con- such that each particle had five attractive ‘arms’, or patches, tain unpaired strands which can bond with other star tiles. which could bond with other particles. For particles with Since the bonding between the tiles is mediated by DNA, fairly narrow patches, the system forms a crystal where each one important advantage of this approach is that the bond- particle has a co-ordination number of five, albeit in an ar- ing can be chosen to be as generic or as specific as we wish, rangement that must deviate from perfect five-fold symmetry. simply by selecting appropriate DNA sequences. A simi- For sufficiently wide patches, a competition is set up be- lar approach involves the construction of multi-arm motifs tween such a non-uniform pentavalent co-ordination and the using DNA origami.62,63 Using DNA tiles to construct qua- hexagonal co-ordination characteristic of crystals of spheri- sicrystals would be broadly similar to the recently observed cally symmetric particles. These hexa- and pentavalent en- lanthanide-directed self-assembly of quasicrystals,64 but the vironments form square–triangle tilings, which are known underlying framework is different in the sense that DNA star to be capable of forming dodecagonal quasicrystals.40,41 In- tiles are effectively ‘patchy particles’ with varying numbers deed, we observed precisely such quasicrystals in brute-force of arms, whereas the basic units in the lanthanide-directed arXiv:1607.06626v1 [cond-mat.soft] 22 Jul 2016 simulations.30 In order to confirm whether such quasicrys- self-assembly approach are point vertices (the metal) and tals are stable rather than just kinetic products, we have also separate edges (molecular linkers). computed explicit phase diagrams for this one-component The simple patchy-particle model we have previously intro- 31 patchy-particle system. To do this, we used Frenkel–Ladd duced describes much of the fundamental physical behaviour 42 43 integration and direct-coexistence simulations to com- of the DNA star tiles; however, unlike colloidal patchy par- pute the free energies of the quasicrystal and its competing ticles with wide patches, DNA star tiles have a well-defined valence, determined by the number of arms, and so a five- arm star tile cannot bond with six neighbours. Therefore, the DNA tiles best map onto patchy particles with a narrow a)Current address: Dipartimento di Scienze Molecolari e Nanosistemi, Uni- patch width, and there is no parameter equivalent to the patch versità Ca’ Foscari, Via Torino 155, 30172 Venezia Mestre, Italy. width that could be varied in order to facilitate five-arm DNA 2 tiles to form quasicrystals. Indeed, experimentally, five-arm 1 1 2 2 2 2 DNA star tiles have been observed just to form the same two- σ dimensional crystalline arrays with pentavalent co-ordination 1 H 1 2 Z 2 59 1 1 as the five-patch particles do at narrow patch width. 2 0 2 2 In this work, we suggest a potential means to enable DNA star tiles to self-assemble into a variety of structures at low FIG. 1. Neighbour classification of the σ, H and Z environments. temperatures, including a quasicrystalline phase. Rather than The nearest neighbours of the central particle are shown in light grey, and where applicable, the second-nearest neighbours in dark rely on a competition between hexavalent and pentavalent grey. The number given for each nearest neighbour specifies how environments corresponding to patchy particles with five many neighbours that particle shares with the central particle. patches, as we have done in previous work,31 here, we simulate a two-component mixture of patchy particles with five and six patches of the appropriate composition. We show potential, that such mixtures continue to exhibit stable quasicrystals. ( LJ Although the behaviour of patchy particles maps onto star V (rij) rij < σLJ, 65,66 V(푟ij,ϕi,ϕj) = (1) tiles perhaps surprisingly well, this level of abstraction VLJ(r )VA(푟 ˆ ,ϕ ,ϕ ) σ ≤ r , may seem rather extreme. Furthermore, given that quasicrys- ij ij i j LJ ij tals to the best of our knowledge do not appear to have been where 푟 is the interparticle vector connecting the particles observed with DNA star tiles in experimental work,67 it is ij i and j, rij is the magnitude of this vector, and ϕi and ϕj are important to investigate whether the simple coarse-grained the orientations of the particles i and j, respectively. The model we have considered here is sufficient to capture the un- Lennard-Jones potential is given by derlying physics of the DNA star tile self-assembly process. Unfortunately, it would be prohibitively expensive to perform " 12 6# LJ σLJ σLJ such simulations using a brute-force all-atom approach, since V (rij) = 4ε – , (2) both the time and the length scales involved are far too large. rij rij As a compromise, we use what is still a coarse-grained, but much more realistic model of DNA, oxDNA,68,69 to make and we use a potential cutoff of rcut = 3σLJ and shift the further progress. Even though a number of features of DNA potential so that it equals zero at rcut. The angular modula- have already been coarse-grained within this model, study- tion term in the potential is given by a product of gaussian ing the formation of quasicrystals and phase behaviour with functions, oxDNA is still too computationally intractable to be feasible ( " –θ 2 # " –θ 2 #) to attempt in full. However, what we can study is the be- A 푟 ϕ ϕ kij lji V ( îj, i, j) = max exp 2 exp 2 , (3) haviour of the basic quasicrystalline motifs that we observe k,l 2σpw 2σpw in the patchy-particle simulations. We have confirmed from these simulations that the structures predicted by our ‘toy’ where σpw is a parameter reflecting the patch ‘width’ and θkij patchy-particle model are reasonably well behaved and the is the angle between the patch vector of patch k on particle patchy model does appear to capture the necessary funda- i and the interparticle vector 푟ij. The product of gaussian mentals of the physical system. This is a very exciting result functions is evaluated over all possible patch pairs {k, l}, and because it gives us a considerable degree of confidence that the optimum combination is chosen, i.e. a pair of particles it might be possible in experiment to self-assemble a soft can only interact via that pair of patches which is most ener- quasicrystal using DNA molecules. getically favourable. In simulations with multiple patch types, the angular modulation of Eqn (3) is modified to include a prefactor that II. PATCHY-PARTICLE SIMULATIONS depends on the interaction matrix of the two patches k and l considered. In all simulations considered here, the matrix elements of this matrix are either zero or unity, i.e. all patches A. Model and methods that interact have the same strength, and patches that do not interact do not contribute at all to the energy in the attractive Patchy-particle models have been used extensively part of the Lennard-Jones potential. The most energetically to study a wide range of behaviours in computer favourable pair of patches is still chosen in the computation simulations,30,31,65,70–84 including self-assembly and crys- of the angular modulation in Eqn (3). tallisation, and represent one of the simplest types of ‘toy In order to characterise the structures we observe in our model’ which can account for the complexity of behaviour simulations, we classify each particle according to its nearest- seen in experiments on a number of colloidal systems. neighbour environment.30,31 To do this, we determine the In our simulations, we use the Metropolis Monte Carlo neighbours of each particle, using a simple spherical cut- 85 86,87 30 scheme with volume moves and periodic boundary off of 1.38σLJ, and then determine how many neighbours conditions. In simulations with multiple particle types, we each neighbouring particle shares with the particle we are furthermore allow moves in which two particles of distinct classifying. We classify particles into three distinct types types are exchanged with one another in order to help facili- of environment, σ, H and Z, with common neighbour sig- tate equilibration. natures of {21111}, {22110} and {222222} respectively, as We model particles with attractive patches using a simple illustrated in Fig.1. This labelling corresponds to the equiv- angular modulation of the attractive part of the Lennard-Jones alent Frank–Kasper phases.88,89 In all simulation snapshots 3

shown in the next section, particles are coloured according (a) Type A Type B to their classiﬁcation following the colour-coding shown in Fig.1, namely cyan ( σ), violet (H) and red (Z), with particles whose environments give any common neighbour signature not listed above depicted in green. In simulations with multiple particle types, the base particle colour corresponding xA = 1/13 xB = 12/13 to Fig.1 is mixed with varying amounts of black for parti- (b) cles of types A and C (as deﬁned below) in order to help to distinguish them. 2 y q

LJ 0 σ B. Results and discussion

2 − 2 0 2 In all simulations reported here, the patch width was cho- − σLJqx sen to be σpw = 0.3rad. For the patchy potential introduced in ln Re(S(q)) | | 65 1.5 4 Eqn (1), this is quite a narrow patch width, making interac- ≤ ≥ tions very angularly dependent. Such a narrow patch width allows us to account for the relatively highly directional nature of DNA multi-arm motif structures and prevent competition (c) between environments of different valencies. However, for precisely the same reason, this choice of patch width leaves 2 us in a region of parameter space where, for pentavalent par- y

30,31 q

ticles we considered previously, the quasicrystal was not LJ 0 thermodynamically stable, as the patches are so narrow that σ it is not possible for six neighbours to bond competitively: 2 instead, the σ phase was stable for pentavalent particles under − 2 0 2 − σLJqx these conditions.31 ln Re(S(q)) We note that in the quasicrystals we studied | | 30,31 1.5 4 previously, the most common structural feature ≤ ≥ was a series of edge-sharing dodecagonal motifs; one of the most common such motifs is shown in Fig.2(a). In the ‘unit cell’ of the approximant crystal corresponding to this FIG. 3. Non-specific patchy particles assembling into a quasicrystal. motif, also illustrated in Fig.2(a), there are two particles (a) Particle types and the mole fraction of each type used in simulations. (b) An equilibrated quasicrystalline approximant correspond- in a hexagonal (Z) environment and 24 particles in a σ ing the structure of Fig.2(a). 1040 particles in total. kBT/ε = 0.1, environment. Since the former correspond to particles having 2 σLJβP = 1.5. (c) A dodecagonal quasicrystalline configuration that six neighbours and the latter to only five, we can surmise was obtained from a cooling run, starting from a liquid state, as the 2 that in order to achieve our goal of assembling quasicrystals temperature was gradually decreased to kBT/ε = 0.16. σLJβP = 1.5. with particles with a narrow patch width, including both 2496 particles in total. In (b) and (c), the diffraction pattern com- hexa- and pentavalent patchy particles in a simulation box in puted for the configuration depicted is also shown. S(푞) is the a ratio of 1 : 12 would be a sensible choice. structure factor evaluated at the reciprocal space vector 푞 = (qx, qy). We have run simulations with a mixture of hexa- and

pentavalent patches in this ratio [Fig.3]. A quasicrystal (a) (b) phase forms spontaneously when the liquid phase is cooled [Fig.3(c)]; its quasicrystallinity is confirmed by the diffraction pattern shown in Fig.3(c), which clearly exhibits dodecagonal symmetry: since this dodecagonal order is co- herent through the whole of the simulation box, this is a single quasicrystal. There are several features of this quasicrystalline configuration worth noting. The large majority of hexavalent patches are at the centres of dodecagons, and many of these dodecagons are arranged locally in the motif of Fig.2(a). However, a common alternative motif for a FIG. 2. Two of the most common local structures in the dodecago- quasicrystalline approximant is depicted in Fig.2(b), based nal quasicrystals studied previously,30,31 with (a) edge-sharing and on overlapping, rather than edge-sharing, dodecagonal mo- (b) overlapping dodecagonal motifs. For clarity, the individual do- tifs. We can see that there are sections of the structure in decagonal motifs are shaded in distinct colours to emphasise their Fig.3(c) which are locally like both the edge-sharing and overlap. These individual motifs are rotated by 90◦ in (b). Each the overlapping approximants, both of which form a trian- particle is colour-coded based on its classification as per Fig.1.A gular lattice with longer and shorter distances between the possible unit cell for each approximant crystal is outlined in violet. ‘vertices’ of the lattice. However, it is worth noting that there 4

precise coexistence point between the approximant and the hexagonal plastic crystal quasicrystal depends quite strongly on how well equilibrated 101 the quasicrystal is: the lower temperature limit thus gives the minimum region of stability of the quasicrystal, and its

P region of stability is expected to be somewhat larger still β 2

LJ in practice, since, unlike for pure pentavalent particles, the σ 100 fluid quasicrystal could largely be fully bonded, and so we expect little difference in energy between such a conﬁguration and

quasicrystal the approximant. An approximate phase diagram is shown in Fig.4. 90 Interestingly, the quasicrystalline phase is stable approximant crystal over a wider range of temperatures than for the pure pentava- 10 1 − 0.1 0.2 0.3 0.4 0.5 0.6 0.7 lent patchy particle system we previously studied;31 this is kBT/ε because the energy difference between the quasicrystal and the approximant in the current system is smaller than that FIG. 4. An approximate phase diagram for the non-specific between the quasicrystal and the σ phase for the pentavalent quasicrystal-forming system. The approximant–quasicrystal coexistence data points come from free-energy calculations; the remaining particles. points all come from direct-coexistence or brute-force simulations. Unlike in the single-component phase diagram considered The dotted lines are guides to the eye only. The boundary between in Ref. 31, the low-temperature phase, at which the configu- the quasicrystal and the quasicrystalline approximant is likely to be rational entropy afforded by the quasicrystal is no longer as an overestimate in temperature, and the true region of stability for important as it is to maximise the bonding, is the approximant the quasicrystal is likely to be somewhat larger than shown here. crystal rather than the σ phase, since the presence of hexavalent particles means that bonding cannot be maximised in the σ geometry for all particles. However, it would alternatively are also a number of other motifs of dodecagon centres, such be possible that the mixture may phase separate into a σ and as rectangular and isosceles triangular ones, which feature in a Z phase. We have confirmed that the zero-temperature en- Fig.3(c). thalpy of the approximant phase is lower than the enthalpy As we decrease the temperature further still, the quasicrys- of a 1 : 12 mixture of Z and σ phases comprising solely hexa- talline approximant is expected to become the stable phase, and pentavalent particles respectively at all pressures consid- as the configurational entropy of the quasicrystal becomes ered, even with no interfacial energy penalty imposed. The relatively less important compared to the enthalpic stability approximant crystal is therefore stable with respect to phase of the approximant.31 In order to check that the quasicrys- separation at zero temperature. tal is thermodynamically stable at intermediate temperatures, The stability of the quasicrystal over a range of conditions rather than simply a kinetic product, we have computed the demonstrates that it is possible to set up a competition be- free energies of the competing phases. To find the free en- tween penta- and hexavalent co-ordination by means other ergy of the edge-sharing approximant [Fig.3(b)], we used than having a large patch width. In particular, in previous 42 2 Frenkel–Ladd integration at kBT/ε = 0.1 and σLJβP = 0.5 work, the quasicrystal was never found to be stable for patch 31 (where β = 1/kBT), and, for consistency checking, also at widths below σpw ≈ 0.45. By contrast, we have shown here 2 σLJβP = 1.5, and then integrated this free energy along iso- that a quasicrystal phase is thermodynamically stable even 43 (βP) curves. We found the free energy of the fluid phase if the patch width is considerably narrower (i.e. σpw = 0.3). by integrating from the ideal gas, bearing in mind that it is a This is very encouraging if our aim is to construct quasicrys- multicomponent gas. The free energy of the quasicrystal was tals using DNA multi-arm motifs. However, at this stage, then set by equating it to the free energy of the fluid at the the set-up we have considered completely ignores one of the point at which the quasicrystal and the fluid phase are at equi- most important reasons why DNA is so popular in exper- librium; we determined this condition by direct-coexistence iment: using DNA makes it very easy to design mutually simulations.31 By finding where the free energy curves of the orthogonal interactions. Indeed, the multi-arm DNA tiles approximant and the quasicrystal cross, we can obtain the equivalent to the patchy particles that we have considered relevant coexistence curve between the two phases. Finally, thus far, where all the sticky ends at the end of the arms have at very high pressures, the hexagonal (Z) plastic crystal phase a favourable interaction with all other sticky ends, might be dominates because its density is larger,31 whether or not all rather more difficult to self-assemble than envisaged because neighbour–neighbour interactions can be satisfied. However, growth might be arrested as multiple undesigned bonds are bearing in mind that the corresponding DNA systems are formed, and so it is prudent to investigate whether it remains very dilute solutions, if we work at reasonable pressures, the possible to form quasicrystals if the patches are made to be Z phase does not need to be considered further. more specific. However, while the kinetics of self-assembly 2 < At such reasonable pressures (i.e. for σLJβP ∼ 10), the may be less frustrated as the interactions are made more spe- quasicrystal is thermodynamically stable relative to the fluid cific, it is important to bear in mind that it is in the nature of at temperatures below about kBT/ε = 0.2. The approximant quasicrystals that they are not completely ordered (indeed, crystal takes over below about kBT/ε = 0.1. However, it this is what affords them their additional entropic stability): is worth bearing in mind that it is considerably more diffi- it is not possible, by construction, to have a set of interac- cult to equilibrate the quasicrystal at low temperatures when tions for which the fully bonded configuration is uniquely there is a mixture of hexa- and pentavalent components in determined to be the quasicrystal. the system than in the previously considered work, and the As a first step in exploring the factors that govern qua- 5

(a) Type A Type B (a) Type A Type B B A A C A A B* A A A* C C A A A* A A B* C* xA = 1/13 xB = 6/13 A A B Type C xA = 1/7 xB = 6/7 E E* (b) C* 2 C* D=D* y xC = 6/13 q

LJ 0 (b) σ

2 2 − 2 0 2 − σLJqx y

q ln Re(S(q))

LJ 0 | |

σ 1.5 4 ≤ ≥

2 − 2 0 2 − σLJqx (c) ln Re(S(q)) | | 2 1.5 4 ≤ ≥ y q

LJ 0 (c) σ

2 2 − 2 0 2 − σLJqx y q ln Re(S(q)) LJ 0 | | σ 1.5 4 ≤ ≥

2 − 2 0 2 − σ qx LJ FIG. 6. Fully specific patchy particles designed to form the qua- ln Re(S(q)) sicrystalline approximant of Fig.2(b). (a) Particle types and the | | 1.5 4 mole fraction of each type used in simulations. Patches only in- ≤ ≥ teract with complementary patches, indicated by an asterisk. The basic motif of overlapping dodecagons with explicit patch–patch FIG. 5. Fully specific patchy particles designed to form the qua- interactions is also shown. (b) An equilibrated approximant crystal corresponding to a structure in which all bonding interactions are sat- sicrystalline approximant of Fig.2(a). (a) Particle types and the 2 mole fraction of each type used in simulations. Patches only in- isfied [Fig.2(b)]. 980 particles in total. kBT/ε = 0.16, σLJβP = 1.5. teract with complementary patches, indicated by an asterisk. The (c) A crystalline configuration obtained from a cooling run, starting from the fluid, in which the temperature was gradually decreased to basic motif of edge-sharing dodecagons with explicit patch–patch 2 interactions is also shown. (b) An equilibrated approximant crystal kBT/ε = 0.16. σLJβP = 1.5. 2492 particles in total. For (b) and (c), corresponding to a structure in which all bonding interactions are sat- the corresponding diffraction patterns are also shown. 2 isfied [Fig.2(a)]. 1040 particles in total. kBT/ε = 0.1, σLJβP = 1.5. (c) A crystalline configuration obtained from a cooling run, starting from the fluid, in which the temperature was gradually decreased to 2 two different types, as depicted in Fig.5(a). In this set-up, kBT/ε = 0.16. σLJβP = 1.5. 2496 particles in total. For (b) and (c), the corresponding diffraction patterns are also shown. patches only interact with complementary patches denoted by the same letter and an asterisk; other patch pairs do not interact at all. The unit cell of the edge-sharing approximant of sicrystal self-assembly in binary mixtures corresponding to Fig.2(a) can readily be identified in the approximant shown DNA star tiles, we consider the interaction set required to in Fig.5(b). However, whilst the approximant is stable in form two of the quasicrystalline approximants considered roughly the same conditions as it was before, the quasicrystal above [Fig.2], before considering how these interactions is not expected to form with such specific interactions. Even could be relaxed to allow the variety of environments typical when cooled very slowly, kinetic products such as that shown of our target dodecagonal quasicrystal to form. Let us first in Fig.5(c) are obtained. Whilst the underlying approximant consider how we can make every distinct type of interaction crystal ordering can certainly be identified in this figure, the that can be identified in the unit cell of Fig.2(a) different. fact that these dodecagonal motifs are not orientated in the This can be achieved by making the pentavalent particles of same way throughout the simulation box means that large 6 gaps must be left in order to reduce the strain in the sys- (a) Type A Type B tem, and since the bonding is so specific, no particles can B* be used to ‘glue’ the different regions together. The result- A A C=C* ing structure is therefore, unsurprisingly, full of defects: it A* is not a single crystal, but has several crystalline domains A A C=C* with grain boundaries between them. The fact that there are A A B multiple crystallites is confirmed by the smeared out diffrac- xA = 1/13 xB = 6/13 tion pattern that involves a superposition of patterns from the Type C different crystallites [Fig.5(c)]. C=C* The alternative approximant motif of Fig.2(b) is based on C=C* overlapping rather than edge-sharing dodecagonal motifs. In C=C* the ‘unit cell’ of this alternative approximant crystal, there C=C* are 2 particles in a hexagonal (Z) environment and 12 parti- C=C* cles in a σ environment, necessitating a ratio of 1 : 6 of hexa- xC = 6/13 and pentavalent patchy particles in the simulation box. If the (b) bonding interactions are made fully specific with this alternative set-up, as shown in Fig.6(a), we can again stabilise 2 the approximant crystal [Fig.6(b)]. However, similar to the y edge-sharing fully specific system, on cooling, multiple nu- q

LJ 0 cleation events occur, leading to multiple crystallites of the σ overlapping approximant separated by highly defective grain boundaries. 2 − 2 0 2 In order to improve the kinetics whilst retaining enough − σLJqx plasticity in the interactions to allow quasicrystals, rather ln Re(S(q)) | | than just crystals, to form, we can aim to strike a balance 1.5 4 between the full speciﬁcity considered in simulations illus- ≤ ≥ trated by Figs5 and6 on the one hand and the completely non-speciﬁc bonding of Fig.3. To this end, we have consid- (c) ered an alternative set-up in which we allow a competition between the overlapping and edge-sharing dodecagonal mo- 2 tifs to be set up. As illustrated in Fig.7(a), by permitting y the outlying ‘type C’ particles to bond freely with two of the q

LJ 0 patches of ‘type B’ particles, we can assemble either struc- σ tures analogous to those of Fig.5, involving all three types of 2 particle, or of Fig.6, involving only particles of types A and − 2 0 2 B, with the excess of particles of type C forming regions of − σLJqx σ-environments that can fill the gaps between ‘approximant’ ln Re(S(q)) | | 1.5 4 motifs that are orientated in different ways. We have chosen ≤ ≥ the composition of particle types in this set-up to be the same as the one we considered for the edge-sharing approximant system of Fig5. FIG. 7. Patchy particles of ‘intermediate’ specificity that are de- With this set-up, cooling a liquid again results in a qua- signed to allow quasicrystal formation. (a) Particle types and the sicrystalline phase, such as that shown in Fig.7(c), with the mole fraction of each type used in simulations. Patches only interact corresponding diffraction pattern confirming its dodecagonal with complementary patches, indicated by an asterisk. The basic motifs of both edge-sharing and overlapping dodecagons with explicit symmetry. As in the non-specific case of Fig.3, there is again patch–patch interactions are also shown. (b) An equilibrated edge- a variety of ways in which the dodecagons pack in addition sharing approximant crystal. 1040 particles in total. kBT/ε = 0.1, to the two triangular lattice patterns of the approximants of 2 σLJβP = 1.5. (c) A quasicrystal resulting from a cooling run in Fig.2. To verify that the quasicrystal is thermodynamically which the temperature was gradually decreased to kBT/ε = 0.16. stable, we have repeated the calculations considered above σ 2 βP = 1.5. 2496 particles in total. For (b) and (c), the correspond- 2 LJ for the non-specific case at σLJβP = 1.5. In particular, we ing diffraction patterns are also shown. have computed the free energies of the edge-sharing and overlapping approximants as well as the σ phase using Frenkel– Ladd integration. The free energy of the edge-sharing ap- of temperatures. proximant matches the free energy of a system combining The fact that the quasicrystalline phase is thermodynami- 7 parts of the overlapping approximant and 6 parts of the σ cally stable for this system of ‘intermediate’ specificity sug- phase; this ratio accounts for the excess of ‘type C’ particles gests that a DNA star tile system with interactions chosen when an overlapping approximant is formed at the considered in this way would be the most likely to result in a two- composition of particles of types A, B and C [cf. Fig.7(a)]. dimensional soft DNA-based quasicrystal. However, it is At temperatures above kBT/ε ≈ 0.1, the quasicrystal’s free certainly the case that we have ignored a number of consid- energy is lower than that of the approximants, confirming erations when abstracting the system to the toy-model level that it is the thermodynamically stable phase across a range considered here. For example, it is not at all clear a priori 7 that dodecagonal motifs comprising DNA star tile would be nucleotides (the largest has 50 tiles and 22704 nucleotides), sufficiently planar to permit the growth of suitably large two- to make the simulations feasible on a reasonable time scale, dimensional quasicrystalline structures, although this consid- they are run on GPUs using a specially developed code.101 eration may be mitigated to a large extent by performing the We consider systems of tiles both when free in solution, as self-assembly on a surface. In order to address this point, we is typical during the assembly process, and when adsorbed turn briefly to a more realistic potential of DNA molecules on a surface, as is the case when visualised by some type of themselves. microscopy (e.g. AFM). The interaction of the nucleotides with the surface is modelled with a simple one-dimensional Lennard-Jones interaction that depends only on the distance III. SIMULATING DNA TILE ARRAYS WITH A of a nucleotide from the surface. REALISTIC MODEL

A. OxDNA model and methods B. Results and discussion

OxDNA69,91 is a coarse-grained DNA model at the nu- Our aim here is use oxDNA to check whether there might cleotide level that allows the simulation of systems of large be any structural reasons why the dodecagonal quasicrystals numbers of nucleotides. The nucleotides are modelled as that we have seen for the above patchy particles might not be rigid bodies interacting with a series of effective interactions realisable using DNA star tiles. Previous experiments do not (the solvent is not explicitly modelled) that account for hydro- suggest any obvious hindrances. For example, both five-arm gen bonding between Watson–Crick base pairs, stacking be- and six-arm tiles have been produced and found to assem- tween bases, electrostatic repulsion between the phosphates, ble into two-dimensional crystalline arrays, forming σ59 and excluded volume and chain connectivity. These interactions hexagonal58 crystals, respectively. Furthermore, mixtures of have been fitted to reproduce the thermodynamics of hybridi- three- and four-arm tiles with specifically designed interac- sation and the structural and mechanical properties of both tions have been shown to be able to produce more complex single-stranded and double-stranded DNA. Here, we use the crystal structures.60,61 One possible complication is that, if second version of the model (‘oxDNA2’) that includes fine- the tiles are not flat, but rather the arms possess an intrinsic tuned properties to reproduce better the properties of large preference to bend in a given direction, then if all tiles face in nanostructures – in particular DNA origamis.69 the same direction, this has the potential to lead to curvature The oxDNA model is the most widely used coarse-grained in the resulting structure that could hinder the assembly of model of DNA at the nucleotide level, and has been used to an extended two-dimensional structure. One solution is to study the biophysical properties of DNA, a wide variety of design the tiles so that they alternate in orientation, and any DNA nanotechnology systems and applications in soft matter curvature cancels out,57 but this approach is not available materials.92 These applications have confirmed the model’s for structures that possess polygons with an odd number of robustness and quantitative accuracy (e.g. Ref. 93’s repro- edges. However, in the examples above,58–61 conditions and duction of six orders of magnitude variation in the kinetics designs were still found for which the assembly of extended of strand displacement). Particularly relevant to the current two-dimensional arrays dominated over the formation of fi- application is the model’s ability to account for the structural nite closed objects (e.g. icosahedra for five-arm star tiles59). properties of both bulged duplexes,94 a motif that is crucial Furthermore, self-assembly of DNA star tiles on a surface to the properties of DNA star tiles, and polyhedral assemblies has also been shown to be possible.102 of star tiles,95 and to rationalise how the kinetics of star tile Example five- and six-arm star tiles are illustrated in self-assembly can be controlled through the size of the bulges Figs8(a) and8(b). The tiles consist of a long central strand, they contain.96 five or six medium-length strands that bridge two arms and Our aim here is to use oxDNA to explore the structural five or six short strands which bind at the ends of each arm. stability of quasicrystalline arrays made out of DNA star The bulges on the long strand between the arms provide flex- tiles. To achieve this, we first need to generate starting initial ibility, allowing the arms to bend back on themselves. In the structures for these arrays. We do this by first designing the current examples, there are four nucleotides in the bulges, arrays using vhelix,97 a recent DNA nanostructure design pro- the same as was used experimentally to produce extended gramme that allows free placement of the component DNA structures with five- and six-arm tiles.58,59 helices in space rather than on a lattice.98 We then convert the We have constructed DNA analogues of three example vhelix design into a starting geometry for the oxDNA model. motifs that are important for the quasicrystalline structures This geometry is not yet a suitable starting point for molecu- observed in our patchy-particle simulations, namely a do- lar dynamics simulations, as it may have particle overlaps or decagon, three overlapping dodecagons, and three edge- extended bonds that give rise to unreasonably large energies sharing dodecagons. The simulations of these structures and very large forces. We therefore first relax the structure showed that they are all stable at room temperature with the using a steepest-descent-like minimisation technique. The correct topology of the network maintained throughout the details of these procedures will be described elsewhere.99 simulation. Example configurations that have been adsorbed The structures are then simulated with a molecular dynam- on a surface [Figs8(c)–(e)] clearly show the expected struc- ics algorithm employing an Andersen-like thermostat100 both tures. Due to the flexibility of the star tiles, the quadrilaterals to keep the temperature constant and to generate diffusive in the network need not be perfectly square. Furthermore, motion of the nucleotides, as is appropriate for molecules bending is not always localised to the bulge regions of the in solution. As the systems we study contain thousands of tiles, and can sometimes occur at the four-way junctions in 8

(a) (b) (c)

(d) (e) (f)

FIG. 8. OxDNA representations of (a) a five-arm tile, (b) a six-arm tile, (c), (d), (e) quasicrystal-like motifs of increasing size when adsorbed onto a surface and (f) the largest motif when in solution. In (a) and (b), flat configurations have deliberately been chosen to most clearly illustrate the design and topology of the star tiles. The simulations of these structures were all performed at 22 ◦C and a salt concentration of 0.5moldm–3. the arms leading to further distortions from the idealised ge- IV. CONCLUSIONS ometries, even for the triangles. By contrast, when free in solution, although the topology of the network is retained, the motifs are highly fluxional and We have performed simulations of patchy-particle systems no longer look anything like the idealised two-dimensional with a narrow patch width to investigate the phase behaviour target structure [Fig.8(f)]. This is both because of the inher- of particles that can be considered to be a ‘toy model’ for ent flexibility of the star tiles and because all tiles face in the DNA star tiles or analogous systems. We have confirmed same direction, so any tendencies for the arms to bend away our hypothesis, originally proposed in Ref. 31, that mixtures from the plane in a preferred direction are additive. The non- of penta- and hexavalent particles can mutually associate to planarity is probably also exacerbated by the relative small form stable dodecagonal quasicrystals. size of the motifs, as consequently a large number of the arms We have explored two designs which lead to quasicrys- (over 18% even for the largest motif) are on the edge of the tal formation. The first one of these involved no specificity motif and lack the constraint of being connected to another in the patch–patch interactions: every patch could interact tile. with every other patch in the system. In patchy-particle sim- We should emphasise that the flexibility of the structures ulations using this set-up, the quasicrystalline phase formed and the large fluctuations away from planarity do not affect readily and was shown to be thermodynamically stable using the stability of the networks, nor do they mean that further free-energy calculations. However, one might imagine that self-assembly of the networks is necessarily hindered. When the self-assembly of the quasicrystal in a DNA context might a new tile binds to a free arm on the edge of the motif, further be trickier with such non-specific interactions, since particles binding is probably still most likely to occur in the intended cannot ‘detect’ whether they are in the correct environment way. However, the non-planarity may also make allowed, but from their initial interactions. For example, a hexavalent par- not intended, arm binding more likely than when in a planar ticle should ultimately end up at the centre of a dodecagonal geometry because the relevant arms have been brought closer motif, but has no way of ensuring that this will be the case together by the non-planar fluctuations. This again empha- when it is bonded to only a few of its neighbours. In order sises the importance of annealing to facilitate the melting to ensure that the thermodynamically stable phase can form away of incorrect bindings. at experimentally accessible time scales, particles must be 9 allowed to bind and unbind very readily: the self-assembly maintained, but the dynamics of self-assembly may change if process must therefore take place at temperatures at which the patches were to be more flexible. Secondly, there is com- the driving force to form the target phase is very small. Since paratively little excluded volume in DNA structures, whilst non-optimal configurations are less stable, a high tempera- our patchy particles have a Lennard-Jones-style excluded vol- ture means that such motifs are likely to melt off, allowing ume interaction. However, since the structures of interest are the stable phase to form over time. stabilised by attractive rather than repulsive forces, we do The kinetics of the toy patchy-particle systems are very not think that excluded volume effects would be particularly fast and a quasicrystal forms readily in such a set-up. How- important. Thirdly, DNA assemblies can behave in rather ever, while our simulations certainly do not preclude the more complex ways than we have considered here, since they possibility of self-assembly being feasible in an equivalent exhibit a kind of structural co-operativity: curvature emerges DNA system, we can make use of the information content of from the assembly process, rather than being a property of the DNA to be more selective in the interparticle interactions isolated tiles.95 However, several strategies exist for surface- and thus attempt to reduce the likelihood of kinetic traps pre- mediated self-assembly,102,104,105 and surface assembly may cluding successful self-assembly in DNA tile systems. We help to alleviate such problems with curvature. Nevertheless, must however remember that the stability of quasicrystals even if DNA tiles were to be assembled on a surface, rather is largely down to their configurational entropy.31,41 Unlike than simply being adsorbed on a surface at the end of the pro- for the increasingly complex DNA-based crystalline motifs cess to visualise the structure, there is an important difference that have been considered in the literature61,62 and as ex- between a true two-dimensional system that we considered emplified by the quasicrystalline approximants considered and assembly from a three-dimensional dilute solution on a here, it is not therefore possible to design a set of interac- surface. Finally, we used particle swap moves to help hexa- tions for which the fully bound ground-state configuration and pentavalent particles to find their preferred environment is uniquely specified to be a quasicrystal. Instead, we must in patchy-particle simulations. Such Monte Carlo moves design a set of interactions that provides sufficient freedom have no equivalent in real dynamics. However, in this case, that will allow the full variety of motifs that are typical of the the three-dimensional nature of the DNA assembly process quasicrystal to form. In our design, specific interactions en- may actually be beneficial, as in real DNA systems, even sure that hexavalent particles are at the centre of dodecagons, when adsorbed on a surface, the self-assembly is likely to be but do not prescribe how these dodecagons associate. We from a dilute three-dimensional solution, rather than from a have shown that quasicrystals are thermodynamically stable two-dimensional fluid: we can envisage that the swap moves with a design of this kind, while the additional specificity might correspond to adsorption and desorption of appropriate of interactions should permit such structures to be more ki- molecules from solution. netically accessible than the equivalent non-specific system. An additional advantage of using more specific interactions Of course, it may be possible to alleviate some of the is that they would counter against possible phase separation concerns we have listed above by considering a more sophis- into penta- and hexavalent regions that has been seen for ticated model. For example, in order to address the issue with non-specific mixtures of three- and four-arm motifs.103 The flexibility in the patchy-particle model, we could allow the specific interactions are also likely to reduce the competition patches to shift positions slightly in time, whilst maintaining a fixed valency by keeping the patches narrow. Alternatively, with alternative closed polyhedral objects, but choosing the 63 bulge size appropriately to inhibit these assembly pathways a more rigid DNA structure based on an origami approach further is also likely to be important. We hope that the de- might make it easier to design planar tiles in experiment. signs presented here might help to guide experimentalists in However, while there are certainly a number of simplifica- producing a DNA quasicrystal. tions and omissions in our patchy-particle approach, broadly speaking, as confirmed by the oxDNA simulations, the sim- It may also be possible to design dodecagonal DNA crys- ple model we have considered appears to capture a sufficient tals in other ways than the ones we have considered here. For amount of the underlying physics to serve as a good guide example, we may consider the ‘duals’ of the motifs consid- to the self-assembly behaviour of analogous DNA star tiles. ered above, where we interchange the nature of the vertices We hope that our results will help to invigorate experimental and faces in the structure. Since there is a bijective mapping efforts to produce a soft DNA-based quasicrystal. between a structure and its dual,88 the dual of the quasicrystal we considered is also a quasicrystal. The resulting networks would only have tri- and tetravalent vertices. Although it is possible to design particles that would form the equivalents of the two approximant crystals that we consider here, it is rather less clear how to design specific but not overly con- straining interactions which would encourage quasicrystal ACKNOWLEDGMENTS formation with such a set-up. Finally, it is worth bearing in mind that there are several significant differences between the patchy particles we have This work was supported by the Engineering and Phys- considered here and real DNA multi-arm motifs. Firstly, ical Sciences Research Council [Grants EP/I001352/1, DNA molecules are flexible, but have a fixed valency, whilst EP/J019445/1]. We acknowledge the use of the University patchy particles have a fixed geometry and gain their flexibil- of Oxford Advanced Research Computing (ARC) facility in ity in bonding through a patch width. We have made the patch carrying out some of this work [doi:10.5281/zenodo.22558]. width fairly narrow to ensure that the valency condition is Supporting data are available at [TBD]. 10

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