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Colloidal Cluster Assembly into Ordered Superstructures via Engineered Directional Binding Mehdi B. Zanjani, Ian C. Jenkins, John C. Crocker, and Talid Sinno* Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States S Supporting Information *

ABSTRACT: Recent experimental studies have demonstrated a facile route for fabricating large numbers of geometrically uniform colloidal clusters out of submicron DNA-functionalized spheres. These clusters are ideally suited for use as anisotropic building blocks for hierarchical assembly of superstructures with symmetries that are otherwise inaccessible with simple spherical particles. We study computationally the self-assembly of cubic, tetrahedral, and octahedral clusters mediated by “bond spheres” that dock with the clusters at specific preferential sites, providing robust and well-defined directional bonding. We analyze the assembly process with a combination of direct molecular dynamics simulations of superstructure growth and state-of-the-art umbrella sampling techniques to compute nucleation free energy profiles. The simulations confirm the versatility and robustness of hierarchical cluster assembly but also reveal potential obstacles in the form of energetically accessible defect states. We find and study solutions for bypassing these defects that rely on appropriate selection of particle size and interparticle interaction as a function of building block shape and, therefore, provide operational guidelines for future experimental demonstrations. KEYWORDS: hierarchical self-assembly, colloidal clusters, directional binding, DNA-mediated assembly, nucleation and growth

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not the only route for increasing the scope of self-assembly certain spherically symmetric interactions may also produce low-packing fraction assemblies24−31 it does appear to be the most practical one. Experimental efforts aimed at realizing directional bonding between colloidal particles have focused on nonspherical particles,32−37 patchy particles with localized functionalization and binding,38−42 or colloidal clusters composed of spherical particles in well-defined geometric arrangements.43−46 While all of these avenues have been shown theoretically to lead to a host of interesting assembly phenomena,47−60 the cluster approach is arguably the most experimentally practical, particularly at the micron scale where the opportunities lie for realizing photonically active assemblies. For example, anisotropic particle assembly has been highly successful at the nanoscale,32−35 but only a few studies have demonstrated the ability to systematically tune particle shape at the micron scale.36 The promise of patchy particles also has been slow to materialize because of experimental challenges; here again, relatively few studies have been able to fabricate particles with sufficient flexibility and uniformity.39−41

he DNA-mediated self-assembly of nano- and microscale particles into a variety of crystalline configurations has now been demonstrated experimentally in numerous studies.1−7 Example assemblies include a large variety of binary lattices ranging from face-centered cubic (CuAu) and body-centered cubic (CsCl) crystals,8−15 to more exotic structures (e.g., AlB2 and Cr3Si).16−18 The vast majority of DNA-driven colloidal and nanoparticle crystallization demonstrations have been based upon spherically symmetric interactions between multiple sup-populations of spherical particles that differ only in particle size and/or grafted DNA sequence. While this assembly paradigm has been very successful in terms of its accessible structural diversity, efficient nucleation and growth kinetics, and high resultant crystal quality, it is apparent that assembly of useful designer configurations will require additional control parameters beyond particle size and interaction strength. Notably, the long-standing goal of creating a self-assembled colloidal crystal structure with photonically or phononically active properties19−22 seems to be out of reach for spherically symmetric systems. Motivated by the vast majority of atomic and molecular materials found in nature,23 work aimed at overcoming these limitations has focused largely on the engineering of directional bonding between individual building blocks. Although this is © 2016 American Chemical Society

Received: September 22, 2016 Accepted: December 8, 2016 Published: December 8, 2016 11280

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particle (red sphere, “A”) surrounded by halo particles (blue spheres, “B”). The center particles represent the impurity species in the crystal templating approach and are irreversibly ligated to the halo particles. Bond particles (yellow spheres, “C”) may interact with either halo and/or center particles (if accessible) and join two clusters together as shown in Figure 1a−c. Each cluster species is characterized by a distinct number of preferred binding sites that correspond to maximum valency between a cluster and a bond sphere. For cubes, the maximum cluster-bond sphere valency is four and occurs when a bond sphere docks at the center of each of the cluster facesthere are six such sites on each cube. For octahedra and tetrahedra, there are eight and four preferred binding sites, respectively, each with a maximum valency of three. Note that the probability of a bond sphere binding to a cluster at other (lower valency, and thus defective) locations is not zero. Rather, it is dictated by the free energy difference between optimal and defective binding. Subject to these considerations, we focus on the assembly of three distinct superstructures, as shown in Figure 1d−f, which are defined based on the lattices created by the cluster center particles. Specifically, octahedral clusters are expected to form body-centered cubic (bcc) superstructures, cubic clusters should form simple cubic (sc) lattices, while tetrahedra should assemble into a superstructure with diamond symmetry. Parametric Space Considerations. Despite the constraints inherent in the cluster-bond sphere construct, the three particle species (A, B, and C) introduce several size and interaction energy parameters, which, even in simulation, would be difficult to span comprehensively without additional constraints. We, therefore, consider how this large parameter space may be reduced without omitting important combinations. First, the ratio of the halo sphere to center particle diameters, rAB ≡ σA/σB, is identified as a key geometric parameter for describing the cluster binding surface. As shown in Figure 2,

By contrast, recent efforts have proven that colloidal clusters can be fabricated in large quantities, with a high degree of uniformity, and are relatively easy to separate out from unwanted species based on size fractionation. For example, colloidal clusters with a diversity of symmetries have been realized experimentally via a “crystal templating” technique.43,44 Briefly, in this approach, template crystals composed of DNAfunctionalized spheres are grown to include a small amount of an “impurity” species that is irreversibly ligated to its neighbors, which upon melting of the template leaves behind clusters each containing a single impurity sphere. By adjusting the size of the impurity spheres, the structure of the template, and/or the ligation process, clusters with tetrahedral, octahedral, and cubic symmetries, among others, have been generated. The directional nature of the binding surfaces exhibited by these building blocks also has been demonstrated experimentally. Here, we study computationally the hierarchical assembly of colloidal clusters into ordered superstructures and demonstrate the versatility and overall promise of this emerging pathway for realizing complex ordered configurations. Our study is distinguished by several important elements. First, the colloidal clusters we investigate correspond closely to those described previously,43,44 and the interparticle potentials used to drive the simulations are qualitatively realistic.61 Together, these two elements ensure that the predictions resulting from this study will be readily transferable to future experimental studies. Moreover, we focus on a particular hierarchical assembly scheme in which clusters are connected to each other indirectly via “bond spheres” (Figure 1) rather than by direct cluster−

Figure 1. Colloidal cluster building blocks composed of a center sphere (red), surrounded by halo spheres (blue) exhibiting directional binding mediated by spherical bond particles (yellow). (a−c) Single cluster building blocks and dimers connected by bond spheres: (a) octahedra, (b) cubes, and (c) tetrahedra. (d−f) Corresponding expected ordered superstructures: (d) body-center cubic (bcc), (e) simple cubic (sc), and (f) diamond. In each case, the notional superstructure symmetry is given by the sublattice formed by the cluster center spheres.

Figure 2. Closed, open, and sintered cluster configurations for (a) octahedra, (b) cubes, and (c) tetrahedra are delineated by a critical value (dashed line) of the ratio of diameters of the center (red) and halo (blue) spheres. Closed configurations correspond to contacting halo spheres at the critical ratio and inaccessible center particles, while open clusters allow for binding with the center particle. Sintered configurations correspond to overlapping, or merged, halo particles.

cluster interactions. The advantage of this construct is that it simplifies the nature of the directional bonding exhibited by the clusters, making it easier to predict outcomes, control the nature of the binding landscape, and ultimately engineer the desired superstructures.

three distinct cluster configurations may be identified on the basis of this parameter. At a critical value, rcrit AB , denoted by a dashed line for each cluster type, the halo B particles form a close-packed assembly and shield the center A particles from contact with bond spheres; these clusters are referred to as being “closed”. Above this critical diameter ratio, the A particles are sufficiently large to prevent B particles from touching (on average). In this “open” configuration, the center A particles are exposed to potential interactions with bond spheres. As shown in Figure 2 the critical size ratios that separate the two crit configurations are rcrit AB = √2 − 1 for octahedra, rAB = √3 − 1

RESULTS AND DISCUSSION The cluster and bond sphere constructs are shown schematically in Figure 1 for (a) octahedra, (b) cubes, and (c) tetrahedra, all of which have been demonstrated experimentally;43,44 all atomic visualizations were created using the Ovito software package.62 In each case, the cluster consists of a center 11281

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3 2

− 1 for tetrahedra. The third type of

crit . This cluster configuration corresponds to r AB < rAB configuration, which we do not address in the present work, reflects clusters in which the halo particles have been merged, or sintered, during processing; see ref 45 for examples. The ratio of the bond and halo sphere diameters, rCB ≡ σC/ σB, represents the only other independent geometric parameter. Again, this size ratio may be used to identify constraints for hierarchical assembly. Generally, if this ratio is smaller than a lower critical value, bond spheres will be unable to establish separation between two clusters and direct cluster−cluster interactions become relevant (constraint 1). Conversely, if the ratio is above some upper critical value, a single bond sphere may be able to bind with three or more clusters, generally leading to disordered configurations (constraint 2). Additional constraints on rCB may apply for open clusters where A−C interactions are operational. Algebraic expressions for the constraints on allowable values of rCB are summarized in Table 1 for each cluster type and are discussed in more detail later.

Figure 3. (a) Body-centered cubic superstructure ({111} view) composed of closed octrahedral clusters (blue/red) and bond spheres (yellow). (b) Corresponding interaction matrix; empty elements denote inaccessible interactions, “0” denotes short-range repulsion due to noncomplementary DNA strands, and “+” denotes attractive interactions. (c) MD simulation snapshot (107 ms) of a crystallite grown from a cubic bcc superstructure seed initially containing 91 octahedral clusters and 364 bond spheres (rAB = 0.41, rCB = 0.875, ϕ = 0.19, and βEBC = 4). The center (A) particles of clusters identified as belonging to the superstructure are denoted by large spheres, while B and C particles of the superstructure, as well as all fluid particles, are reduced in size for clarity. Superstructure particles were identified based on a bond-orientational order parameter analysis (see Methods for details). Coloring scheme is the same as for (a).

Table 1. Summary of Geometric Constraints for Different Types of Clusters (c and o Designate Closed and Open Building Blocks, Respectively) constraint

octahedra (c)

cubes (o)

tetrahedra (o)

1 2

rCB < 0.72 rCB > 1.08

rCB < 2.57 + 0.16rAB rCB > (0.67(1 + rAB)2 + 1)1/2 − 1

rCB < 0.88(1 + rAB) rCB > 1.11 − 0.5rAB

3

rCB ≤

0.85rAB(1 + rAB) 3.16 − 0.85rAB

rCB ≤

2 (rAB + rAB) (2 − rAB)

Finally, note that the absolute particle sizes do not play a primary role and only impact subtly the interparticle interactions as a function of separation. Bearing this in mind, the clusters demonstrated in refs 43 and 44 were created out of particles with diameters 100−400 nm, and the simulations reported here are matched to this range. For DNA-driven interactions that are about 20 nm in range, these particles may be considered near the “sticky-sphere” limit. The interparticle interaction parameter space is subject to the following additional constraints. First, because halo and center particles are assumed to be irreversibly bound, no A−B interactions are considered. Also, A−A interactions are ignored because A particles are located at the center of clusters. Next, in all cases studied here, bond spheres are assumed to be functionalized with noncomplementary DNA single-strands and, therefore, only interact with each other through entropic repulsion induced by the short-ranged oligomer brushes on their surfaces. Similar considerations apply to the halo particles, although B−B interactions may be useful in certain situations. Overall, B−C and A−C interactions represent the key interaction parameters. In the following simulations, all particle interactions, whether attractive or repulsive, are modeled using the validated DNA-brush force-field model introduced in ref 61 and applied to our prior work.8,63−65 Assembly of Closed Octahedral Clusters. The assembly of closed octahedral clusters (rAB = 0.41) into an ordered superstructure represents the simplest case of the three cluster geometries. Shown in Figure 3a is the expected bcc superstructure resulting from the crystallization of closed octahedra and bond spheres (4 bond spheres per octahedral cluster). The corresponding interaction matrix is shown in Figure 3b: only B−C interactions are attractive, while other

combinations are either not possible (A−A and A−C) or repulsive (B−B and C−C). As shown in Table 1, only two constraints are present on rCB because A−C interactions are not relevant for closed clusters. Direct NVT ensemble molecular dynamics (MD) simulations reveal the facile nature of bcc superstructure crystallization; see Methods. In these simulations, a cubic bcc superstructure seed was immersed in a supersaturated fluid composed of a stoichiometric mixture of octahedral clusters and bond spheres (4 bond spheres per cluster) and allowed to evolve freely. The simulation snapshot in Figure 3c shows the defect-free bcc superstructure after 107 ms of simulation time. Note that only the center spheres of clusters identified as belonging to the superstructure are represented by large spheres, whereas bond spheres and halo particles in the superstructure, as well as all fluid particles, are reduced in size for clarity. The grown crystallite exhibits a compact structure with reasonably well-defined facets. In addition to the direct MD simulations of superstructure growth, umbrella sampling simulations (see Methods) were performed to compute nucleation free energy barriers for bcc assemblies of closed octahedra. The nucleation barrier results are summarized in Figure 4a−c for several combinations of B− C interaction strength (EBC), total particle volume fraction (ϕ), and particle size ratio (rCB). The nucleation barriers, while only one component of the overall nucleation rate, may be considered as a meaningful proxy for the nucleation kinetics subject to the assumptions that (1) the prefactor is controlled by diffusion that is fairly constant across different cases, and (2) no large entropic barriers exist for attachment and detachment 11282

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matrix, is shown schematically in Figure 5a, highlighting three configurationally different, but energetically degenerate, binding

Figure 5. (a) Simple cubic superstructure ({100} view) composed of closed cubic clusters (blue and red) and bond spheres (yellow). The arrows denote a “correct” site and two types of defective sites for bond sphere docking on the ({100}) surfaces of the sc superstructure. Corresponding interaction matrix is shown on the right. (b) MD simulation snapshot (55 ms) of a crystallite grown from a cubic sc superstructure seed initially containing 64 closed cubic clusters and 192 bond spheres (rAB = 0.73, rCB = 0.875, and βEBC = 4). All particles identified as belonging to the superstructure are denoted by large spheres, while fluid particles are reduced in size for clarity. Superstructure particles were identified according to their connectivity (see Methods). Coloring scheme is the same as for (a).

Figure 4. Nucleation free energy profiles for the bcc superstructure of octahedral clusters and bond spheres, as a function of (a) B−C interaction strength, EBC (ϕ = 0.19 and rCB = 0.875), and (b) fluid density, ϕ (EBC = 4kBT and rCB = 0.875). (c) Nucleation free energy barrier height as a function of rCB (βEBC = 4 and ϕ = 0.19).

sites for bond spheres on the {100} surfaces of the sc superstructure. The “correct” sites represent locations on which bond spheres are tetravalently bound to a single cluster, while “defective” sites, which are also tetravalent, reflect binding to multiple clusters. The energetic degeneracy of correct and defective sites precludes perfect growth of the sc superstructure. As expected, direct MD simulation of the growth of a sc superstructure with closed cubic clusters leads to highly defective morphologies; see Figure 5b. It is notable that the defective sites lead to both microscopic and macroscopic effects. The sc superstructure is disturbed locally in multiple places, and the overall shape of the crystallite is nonuniform, presumably due to the presence of surfaces with highly variable growth propensities. In other words, surfaces that accumulate a number of bond spheres in defective configurations are expected to become slow-growing as they become limited by particle detachment from the defective sites. A wide distribution of surface growth propensities then lead to the irregular crystallite morphology observed in Figure 5b. To break the defect degeneracy, we utilize open clusters and enable A−C interactions between the center and bond spheres; see Figure 6a. Here, the additional A−C interaction present on the correct sites effectively introduces a tunable energetic penalty for bond spheres located on the defective sites. The MD snapshot shown in Figure 6b confirms the success of this

processes on the superstructure nucleus. Confirmation of this hypothesis is provided in the Supporting Information, which demonstrates that homogeneous nucleation of a bcc superstructure is readily accessible in MD simulation for moderate nucleation barrier height. As expected, the interaction strength is critically important for assembly; at ϕ = 19% and rCB = 0.875, the nucleation barrier is essentially zero when βEBC = 5 and rises to about 3 for βEBC = 4. At βEBC = 3, the bulk fluid free energy becomes lower than the crystal free energy and no nucleation barrier can be defined. The overall particle volume fraction also plays an important roleas the volume fraction is increased from 12 to 29%, the nucleation barrier height, βΔGmax, decreases from about 12 to 2 (βEBC = 4 and rCB = 0.875). Interestingly, the nucleation barrier is lowest at the midway point between the lower and upper critical value of rCB, corresponding presumably to a maximum in configurational entropy. In summary, closed octahedral clusters are expected to be readily assembled into a bcc superstructure with nucleation and growth dynamics that are easily tuned with accessible parameters. Assembly of Cubic Clusters. In contrast to the octahedral case, the assembly of closed cubic clusters into a simple cubic superstructure was found to be significantly more challenging. The closed-cube sc superstructure, along with its interaction 11283

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Figure 7. (a) Feasibility diagram for assembling the simple cubic superstructure with open cubic clusters with respect to geometric parameters rAB and rCB. Constraint 1 is denoted by the yellow line, constraint 2 by the purple line, and constraint 3 by the blue line; see Table 1. (b) Nucleation free energy profiles for the sc superstructure of open cubic clusters and bond spheres as a function of A−C interaction strength, EAC (ϕ = 0.17, rAB = 1.39, rCB = 1.22, and βEBC = 2).

Figure 6. (a) Simple cubic superstructure ({100} view) composed of open cubic clusters (blue and red) and bond spheres (yellow). In this construct, “correct” sites immediately above the center particles (red) are energetically favored by bond spheres via A−C interactions. Corresponding interaction matrix is shown on the right. (b) MD simulation snapshot (192.5 ms) of a crystallite grown from a sc seed initially containing 64 open cubic clusters and 192 bond spheres (rAB = 1.39, rCB = 1.22, βEBC = 2, and βEAC = 6). Superstructure particles were identified based on a bond-orientational order parameter analysis (see Methods). (c) MD simulation snapshot (75 ms) of a crystallite grown from a sc seed (rAB = 2.08, rCB = 3.33, βEBC = 2, and βEAC = 4.5). For (b) and (c), all particles identified as belonging to the superstructure are denoted by large spheres, while fluid particles are reduced in size for clarity. Coloring scheme is the same as for (a).

feasible region (rAB = 1.39 and rCB = 1.22) are shown in Figure 7b; the values of the other system parameters are given in the figure caption. Once again, the nucleation barriers, and therefore the nucleation rates, are very sensitive to the interaction energies. Although a more comprehensive scan of the two interaction energies would be required to optimize the system, it is apparent (data not shown) that increasing the A−C binding energy relative to the B−C one increases the effective penalty for defect formation and lowers the nucleation barrier. We may also consider a second, potentially experimentally accessible, approach for assembling sc superstructures that does not require the use of open cubic clusters or A−C interactions. As shown in Figure 8a, two types of closed cubic clusters are employed in this approach: the first is the standard (empty) cubic cluster shown previously in Figure 5a, while the second is prepassivated with six bond spheres that are engineered to interact only with the halo particles on the empty clusters. This construct removes the bond sphere docking degeneracy described in Figure 5 and may be considered as a variant of the “lock-and-key” concept proposed in previous studies.45,66 Direct MD simulation of superstructure growth of the sc lockand-key superstructure is shown in Figure 8b. Of course, the feasibility of this approach is dependent on the ability to fabricate the prepassivated clusters. The preceding analyses establish boundaries for growing high-quality sc superstructures. It should be noted that within these boundaries, nucleation and growth dynamics are expected to play important roles. In particular, high driving conditions, such as high particle volume fraction and/or binding energies, will increase nucleation and growth rates but may also lead to

approachthe sc crystallite now grows with essentially perfect ordering. Interestingly, the crystallite is now also morphologically perfect, with layer-by-layer growth leading to the preservation of the initially cubic seed bounded entirely by {100} surfaces. By contrast, the effect of violating constraint 2 (bond spheres that are large enough to allow three or more clusters to dock) is shown in the MD snapshot in Figure 6c. Here, crystallite growth is highly polycrystalline, forming tendrils of locally good crystallinity but with random orientation. As noted above, the open cluster construct also introduces a third constraint on the value of the size ratio rCB in addition to the two that are always present, namely, that the bond spheres must be able to access the center particles (constraint 3); see Table 1. Shown in Figure 7a is a “phase diagram” that establishes the feasibility region (highlighted in green) for sc superstructure growth in terms of the two geometric parameters, rAB and rCB. Example nucleation free energy barriers for sc superstructures built with open cubic clusters in the geometrically 11284

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Figure 9. (a) Schematic of diamond superstructure composed of closed tetrahedral clusters and bond spheres showing a correct bond sphere binding site and a defective site. Corresponding interaction matrix is shown on the right. Below is an MD simulation snapshot (29.5 ms) showing a disordered superstructure grown from a diamond seed initially containing 80 closed tetrahedral clusters and 160 bond spheres (rAB = 0.23, rCB = 0.41, βEBC = 3, and βEBB = 2). A (red) and C (yellow) particles identified as belonging to the superstructure are denoted by large spheres, while B particles belonging to the superstructure, as well as all fluid particles, are reduced in size for clarity. (b) Diamond superstructure composed of open tetrahedral clusters and bond spheres, with A−C interactions that break the defective site degeneracy. Corresponding interaction matrix is shown on the right. MD simulation snapshot (587.5 ms) shows a crystallite grown from a diamond seed initially containing 80 open tetrahedral clusters and 160 bond spheres (rAB = 0.67, rCB = 0.83, βEBC = 4, and βEAC = 8). Particle identification scheme is the same as in (a). Superstructure particles were identified based on a bond-orientational order parameter analysis (see Methods).

Figure 8. (a) “Lock-and-key” approach for sc superstructure growth based on alternating empty and prepassivated cubic clusters. Corresponding interaction matrix shown on the right. (b) MD simulation snapshot (215 ms) of a crystallite grown from a sc “lockand-key” seed (rAB = 0.73, rCB = 0.875, and βEBC = 3.5). Superstructure particles were identified based on a bond-orientational order parameter analysis.

the formation of quenched-in crystallographic defects.67 The connection between defect tolerance and annealing behavior of a particular superstructure to the interaction matrix and other system parameters, therefore, will need to be investigated carefully to optimize this technology. Assembly of Tetrahedral Clusters. Finally, we consider the assembly of superstructures based on tetrahedral clusters. Tetrahedral binding symmetry at the colloidal scale has been of particular long-standing interest because it provides potential access to the diamond lattice and a route for producing novel, photonically active structures.21,53 Like the hierarchical assembly of the sc superstructure from cubic clusters, the assembly of closed tetrahedra into a diamond configuration, that is, one in which the cluster center particles form a diamond lattice, is precluded by the presence of energetically degenerate defective bond sphere docking sites. Shown in Figure 9a is a {111}-faceted diamond arrangement of closed tetrahedral clusters. The highlighted binding sites represent examples of correct and defective bond sphere docking sites. Note that, as in the cubic cluster case, the correct sites are located over a single tetrahedral cluster, while the defective sites straddle two clusters. Unfortunately, this superstructure configuration is impossible to realize without additional constraints as demonstrated by direct MD simulations of superstructure growth in Figure 9a. As in the cubic cluster case, we investigate breaking the degeneracy described above by using open clusters to enable A−C contacts, as shown in Figure 9b. The open diamond construct is also subject to three constraints on the size ratio rCB, which are summarized in Table 1 and shown graphically in Figure 10a by the green-shaded region. Example nucleation barriers that confirm the feasibility of growing diamond lattice configurations using open tetrahedral clusters are shown in Figure 10b for rAB = 2/3 and rCB = 5/6 and three different binding energy combinations at ϕ = 0.17. As shown in Figure 9b, direct MD simulations confirm the facile growth of the diamond superstructure using open tetrahedra.

Finally, we note that the lock-and-key approach described for the sc superstructure case (Figure 8) also may be applied to the diamond superstructure. The lock-and-key route for diamond assembly is essentially identical to the one described for cubes, except that the prepassivated tetrahedral clusters now are attached to four bond spheres instead of six. Sublattice Identification. Throughout the preceding discussion, superstructure classification has been made solely on the basis of the sublattices formed by A particles. This choice was one of expediency: there is nothing uniquely meaningful about these particular sublattices and other sublattices may be defined by considering different combinations of particles within each superstructure. Shown in Figure 11 are some sublattice symmetries created by different combinations of particle types for each cluster shape. For example, the superstructure formed by octahedra is bcc with respect to A particles but also sc with respect to C particles and has P23 symmetry when considering both A and C particles. Interestingly, the superstructure formed with tetrahedra, exhibits β-tridymite (SiO2) structure based on the combination of A and C particles. It is important to note that many sublattices do not represent “connected” structures in which the particles form a continuous network. Consequently, these configurations may not be realized as stable structures, for example, by selectively removing the other particles. Nonetheless, they may still be 11285

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symmetry of the A particle sublattice may be “exposed” by making the A particles out of one (active) material and all other particles out of another (inactive) one. Clearly, future experimental studies will be required to more comprehensively assess the scope of symmetries accessible with cluster-based hierarchical assembly.

CONCLUSIONS Preassembled, irreversibly bound colloidal clusters represent a potentially facile route for incorporating directional binding into self-assembling systems. In particular, they offer an attractive alternative to engineered particle patchiness, which has been challenging to achieve in experiment. In this paper, we studied computationally the assembly of three topologically distinct, submicron scale colloidal clusters that have been recently realized experimentally with a bulk crystal templating approach based on DNA-mediated interparticle interactions. Specifically, we considered the coaggregation of clusters and bond spheres that preferentially dock at high-valency sites on the cluster faces. This binary construct greatly simplifies the directional nature of cluster−cluster binding and makes more transparent the routes for assembling target superstructures with desired symmetries. Our studies demonstrate that the cluster-bond sphere construct can indeed be designed to assemble crystalline superstructures with anticipated symmetries. However, the constraints for each situation are different and will, without doubt, pose challenges of varying magnitude for experimental realization. For instance, while closed octahedral clusters readily assembled into bcc superstructures, the superstructures corresponding to closed cubic and tetrahedral clusters were found to be inaccessible because of the presence of energetically degenerate defect states. These defect configurations may be bypassed by engineering open clusters in which the halo particles are separated by a large center particle, or by using a lock-and-key approach with clusters that are prepassivated with the correct number of bond spheres. The experimental feasibility of both of these solutions remain to be established. Prepassivating clusters with bond spheres is likely to be very difficult to achieve with the required degree of uniformity. On the other hand, open cubic clusters are realizable by forming binary bcc template crystals with appropriately sized spheres. However, there are constraints on how different the sphere sizes can be before the bcc crystal phase become unstable relative to a competing structurethese constraints will have to be superposed onto the geometric constraints derived in the present work. Moreover, currently, there is no obvious path to make open tetrahedral clusters with the templating approach; this may change in the near future as the diversity of accessible template crystals increases (e.g., by further understanding the various solid−solid transformations that DNA-bound colloidal crystals are able to accommodate65). In closing, we note that the present study is not intended to provide a comprehensive assessment of all possible cluster structures, stoichiometries, binding combinations, and resulting superstructures. For example, it is likely that the sintered configurations such as the ones demonstrated in ref 45, which were not considered here, will expand the range of accessible structures. Rather, our intent was to identify key issues and parameters, provide a computational framework for future screening studies, and establish the feasibility of using experimentally accessible colloidal clusters to expand the scope of DNA-mediated assembly.

Figure 10. (a) Feasibility diagram for assembling the diamond superstructure with open cubic clusters with respect to geometric parameters rAB and rCB. Constraint 1 is denoted by the yellow line, constraint 2 by the purple line, and constraint 3 by the blue line; see Table 1. (b) Nucleation free energy profiles for the diamond superstructure of open tetrahedral clusters and bond spheres as a function of A−C and B−C interaction strengths (ϕ = 0.17, rAB = 2/ 3, and rCB = 5/6).

Figure 11. Sublattices corresponding to various combinations of particles (shown in the “particle” column) for superstructures composed of combinations of bond spheres with (a) octahedra, (b) cubes, and (c) terahedra. (*) Cubic crystal belonging to space group P23, with Wyckoff positions 1a and 1b occupied by A spheres and 4e occupied by C spheres. (**) Cubic crystal belonging to space group P23, with Wyckoff positions 1a occupied by A spheres and 3d occupied by C spheres.

relevant from a technological point of view by choosing appropriately the particle chemistry. For example, the diamond 11286

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employed for all simulations in this work, which was determined empirically by balancing the magnitude of the fluctuations in cluster size with the convergence rate. The umbrella sampling simulations were executed using a hybrid molecular dynamics technique.70−72 Here, trajectories (2000 time steps each), based on the interparticle forces alone, were first generated with NVT ensemble molecular dynamics simulations. Each trajectory was then accepted or rejected in its entirety according to a Metropolis test based on the bias potential:

METHODS Molecular Dynamics Simulation of Superstructure Growth. Molecular dynamics simulations were performed using the LAMMPS software package68 to qualitatively evaluate superstructure growth. All molecular dynamics simulations were performed in the NVT ensemble (constant number of particles, volume, and temperature) using the Nose-Hoover thermostat for temperature control. Given that micronscale particle dynamics in water are in actuality strongly damped, MD simulations (corresponding to zero viscosity) provide an upper bound on the growth rate, and, therefore, on the non-equilibrium defect density, for given system parameters. Nonligated interactions between the various particle types were implemented using the potential model described previously.61 In each case, binding energies referenced in the text represent the depth of the potential wells. The inputs to the pair-potential model, which is evaluated numerically, are the sphere sizes, the temperature, the length of DNA strands, and the DNA density. The latter two parameters are used to control the strength of the interactions. Ligated interactions between the constituent particles of a single cluster were modeled as harmonic springs with a spring constant of 0.01kBT/nm2, chosen to balance time step size limitations with cluster rigidity. A time step of 1 ns and a temperature of T = 300 K were employed in all MD simulations of growth. Each MD run was initialized with a superstructure seed consisting of clusters and bond spheres arranged in the expected superstructure configuration; seed sizes and stoichiometries for the various cases considered in this paper are summarized in Table 2. The fluid

pacc = exp(−ΔUbias/kBT )

The hybrid molecular dynamics simulations were executed long enough to compute a converged estimate for the average cystallite size, ⟨n⟩, which, as shown in ref 69 is related to the free energy, F(n), of the crystallite according to

dF | n = k bias(n T − n ) dn

building blocks

superstructure lattice

stoichiometry

octahedra (c) cubes (c) cubes (o) cubes (o) cubes (l-k) tetrahedra (o) tetrahedra (c)

bcc sc-def sc sc-poly sc diamond rhcp

4:1 3:1 3:1 3:1 3:1 2:1 2:1

91 64 64 64 64 80 80

rAB

rCB

0.41 0.73 1.39 2.08 0.73 0.67 0.23

0.875 0.875 1.22 3.33 0.875 0.83 0.41

i Q lm =

(4)

where Ylm are spherical harmonic functions in which θ and ϕ are the longitudinal and polar angles calculated based on the center to center vector between particles i and j, ri̅ j, relative to the arbitrary coordinate system of the simulation box. The summation is performed over the ni i nearest-neighbors of each particle i. We used the functions Q6m computed for −6 ≤ m ≤ 6 to obtain a measure of local orientation similarity, Q6Q6ij, between pairs of particles, i and j. Each pair of particles with a Q6Q6ij value greater than 0.5 were considered to be “crystal-like” neighbors; a particle with more than four crystal-like neighbors was assigned as crystalline. Finally, two crystalline particles that were within a cutoff separation (60 nm) were assigned to the same crystallite.

Initial seed size refers to the number of clusters, and stoichiometry refers to the number of bond spheres per cluster in the seed.

ASSOCIATED CONTENT

surrounding the seed was prepared separately by initializing a stoichiometrically consistent system of clusters and bond spheres with random initial positions in a periodic cubic box. The fluid was equilibrated by gradually shrinking the simulation box during MD simulation until the desired fluid volume fraction was achieved. Following this equilibration step, a spherical region of the fluid was removed and replaced by the superstructure seed. The fluid was then re-equilibrated for about 100000 time steps holding the seed fixed. Finally, both fluid and seed were allowed to evolve without constraint. Sparse Umbrella Sampling for Nucleation Free Energy Barriers. A sparse umbrella sampling technique introduced previously69 was used to calculate nucleation free energy profiles for superstructure crystallites as a function of size ratios, interaction energies, and particle volume fractions. In these simulations, superstructure crystallite-in-fluid configurations were prepared as described above and then evolved under the combined action of the interparticle interactions and a bias potential. The bias used in this work was a harmonic function of the crystallite size: 1 k bias(n − n T)2 2

ni

∑ Ylm(θ( rij)ϕ( rij)) j=1

a

Ubias =

(3)

The free energy as a function of crystallite size may then be obtained by repeating the umbrella sampling simulations at several different target crystallite sizes and integrating eq 3 subject to the initial condition, F(0) = 0. The hybrid molecular dynamics approach was employed to evolve the system because of the computational expense associated with identifying the crystallite size. For a given particle configuration, the crystallite size was computed on the basis of Steinhardt bondorientatonal parameters73,74

Table 2. System Properties for MD Simulations of Superstructure Growth (c, o, and l-k Designate Closed, Open, and Lock-and-Key Building Blocks, Respectively)a initial seed size

(2)

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.6b06415. Direct MD simulation of nucleation and growth (PDF)

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Talid Sinno: 0000-0003-2489-1613 Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS We acknowledge support from the National Science Foundation (CBET-1403237 and a Seed Award through MRSEC DMR11-20901).

(1)

where kbias is the bias strength, n is the instantaneous size of the crystallite, and nT is the target size. A bias strength, kbias = 0.5kBT, was 11287

DOI: 10.1021/acsnano.6b06415 ACS Nano 2016, 10, 11280−11289

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ACS Nano

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