Colloids with valence and specific directional bonding

The ability to design and assemble three-dimensional structures from colloidal particles is limited by the absence of specific directional bonds. As a result, complex or low-coordination structures, common in atomic and molecular systems, are rare in the colloidal domain. Here we demonstrate a general method for creating the colloidal analogues of atoms with valence: colloidal particles with chemically distinct surface patches that imitate hybridized atomic orbitals, including sp, sp2, sp3, sp3d, sp3d2 and sp3d3. Functionalized with DNA with single-stranded sticky ends, patches on different particles can form highly directional bonds through programmable, specific and reversible DNA hybridization. These features allow the particles to self-assemble into ‘colloidal molecules’ with triangular, tetrahedral and other bonding symmetries, and should also give access to a rich variety of new microstructured colloidal materials.

The ability to design and assemble three-dimensional structures from colloidal particles is limited by the absence of specific directional bonds. As a result, complex or low-coordination structures, common in atomic and molecular systems, are rare in the colloidal domain. Here we demonstrate a general method for creating the colloidal analogues of atoms with valence: colloidal particles with chemically distinct surface patches that imitate hybridized atomic orbitals, including sp, sp 2 , sp 3 , sp 3 d, sp 3 d 2 and sp 3 d 3 . Functionalized with DNA with single-stranded sticky ends, patches on different particles can form highly directional bonds through programmable, specific and reversible DNA hybridization. These features allow the particles to self-assemble into 'colloidal molecules' with triangular, tetrahedral and other bonding symmetries, and should also give access to a rich variety of new microstructured colloidal materials.
The past decade has seen an explosion in the kinds of colloidal particles that can be synthesized 1,2 , with many new shapes, such as cubes 3 , clusters of spheres [4][5][6] and dimpled particles 7,8 reported. Because the self-assembly of these particles is largely controlled by their geometry, only a few relatively simple crystals have been made: face-centred and body-centred cubic crystals and variants 9 . Colloidal alloys increase the diversity of structures [10][11][12] , but many structures remain difficult or impossible to make. For example, the diamond lattice, predicted more than 20 years ago to have a full three-dimensional photonic bandgap 13 , still cannot be made by colloidal self-assembly because it requires fourfold coordination. Without directional bonds, such low-coordination states are unstable.
Unlike colloids, atoms and molecules control their assembly and packing through valence. In molecules such as methane (CH 4 ), the valence orbitals of the carbon atom adopt sp 3 hybridization and form four equivalent C-H bonds in a tetrahedral arrangement. In the colloidal domain, the kinds of structures that could be made would vastly increase if particles with controlled symmetries and highly directional interactions were available. What is needed are colloids with valence 14 .
One approach is to decorate the surface of colloidal particles with 'sticky patches' made of synthetic organic or biological molecules (for example) and assigned to specific locations [15][16][17][18][19] . Bonding between particles occurs through patch-patch interactions, so that in principle the location and functionality of the patches can endow particles with bonding directionality and valence. This approach is conceptually simple, yet challenging to realize. For example, so-called Janus particles with asymmetrically functionalized surfaces can be made, but normally have only a single patch [20][21][22] . Triblock Janus particles have also been fabricated by glancing-angle deposition and assembled into a kagome lattice, the twodimensional analogue of a diamond crystal 23 . However, only two patches are made using this method, and low quantities of particles are obtained. Other strategies have used faceted particles 24 , particles with protrusions 25 or coordinated patches [26][27][28] , but three-dimensional directional bonding and assembly have yet to be demonstrated 29 .
Here we demonstrate the synthesis and assembly of colloidal particles with directional interactions that mimic those of atoms with either monovalent s or p orbitals, or multivalent sp, sp 2 , sp 3 , sp 3 d, sp 3 d 2 or sp 3 d 3 hybridized orbitals. We do so by making particles with various numbers of patches, n 5 1-7 and higher, that adopt spherical, linear, triangular, tetrahedral, trigonal dipyramidal, octahedral or pentagonal dipyramidal symmetries. The patches are then site-specifically coated with oligonucleotides, enabling a reversible and controllable attraction between patches on different particles. Using these colloidal 'atoms', we demonstrate that a vast collection of colloidal molecules and macromolecules are readily accessible through selfassembly schemes that are analogous to chemical reactions.

Synthesis
The fabrication of patchy particles, summarized in Fig. 1a, starts with cross-linked amidinated polystyrene microspheres, 540 nm or 850 nm in diameter 30 . Small clusters of these microspheres are assembled using an emulsion-evaporation method 5 that produces so-called 'minimal-moment' clusters with reproducible symmetries and configurations: spheres, dumbbells, triangles, tetrahedra, triangular dipyramids, octahedra and pentagonal dipyramids, for clusters of n 5 1-7 particles (Fig. 1b).
Patchy particles are formed from the clusters using a two-stage swelling process followed by polymerization 31 . First, a low-molecularmass, water-insoluble organic compound (1-chlorodecane) is introduced into the colloidal clusters that are suspended in water with surfactant (sodium dodecyl sulphate, SDS). Adding a small amount of acetone to the suspension aids in the transport of the 1-chlorodecane into the colloidal clusters. We also introduce an oil-soluble initiator, benzoyl peroxide (BPO), and 1,2-dichloroethane, which dissolves BPO and is miscible with 1-chlorodecane. Subsequent stripping of the acetone and 1,2-dichloroethane from the solution traps both the 1-chlorodecane and the BPO in the polymer particles. The clusters are then swollen by the styrene monomer. The 1-chlorodecane introduced earlier acts as an osmotic swelling agent that increases the amount of monomer that can be absorbed by the clusters 32 . Because each cluster of a given number of particles contains the same amount of swelling agent, chemical equilibrium assures that clusters of the same size all swell by the same amount, with the total amount of swelling controlled by the quantity of added monomer.
After swelling, we polymerize the styrene by thermally degrading the BPO previously introduced into each cluster. Swelling is controlled so that the extremities of the original clusters are not encapsulated, but are left as patches. Clusters of the same order n are encapsulated to the same extent, leading to uniform patch configurations, as seen in Fig. 1c, which shows scanning electron microscope (SEM) images of particles with 1 to 7 patches (see Supplementary Fig.  1a for higher-order patches). Using BPO as the initiator ensures that there are no functional groups introduced, so the surface created by swelling the clusters-the 'anti-patch' surface-is chemically inert and different from the patches: only the patches have the functional amidine groups.
Patch size is controlled during the swelling process by adjusting the amount of monomer that is introduced: the more monomer that is added, the smaller the patches are. Figure 2 shows that considerable variation in patch size can be achieved in this way. Small patches favour greater directionality, and larger patches permit multiple links per patch, as we discuss below.
A key design feature of our method is the use of clusters as intermediates. Their diversity in particle number and symmetry is translated directly to the number and symmetry of the particle patches. In contrast to the planar symmetry of Janus particles 23,33 , the symmetries of these patchy particles are fully three-dimensional.
Our method converts essentially all the starting colloidal particles into particles with one or more patches. Each sample produced contains large scalable quantities of particles having different valence (numbers of patches), the relative distribution of which can be changed by adjusting the emulsification conditions used when making the clusters 34 . Using a higher shear rate, for example, makes smaller emulsion droplets, which skews the distribution towards lower-valence particles. We fractionate the particles through density gradient centrifugation, obtaining up to 12 clear bands corresponding to particles with different valence (see Supplementary Fig. 1b, c). Table 1 summarizes the fraction of particles obtained in each band for two different shearing conditions. For the lower-shear preparation, each of the four upper bands, which correspond to particles with 1 to 4 patches, contains 10 8 to 10 9 identical particles. For the higher-shear preparation, greater quantities are produced in the upper bands and lower quantities are produced in the lower bands. In most cases, we use conditions (see Methods) that produce the most 2-, 3-and 4-patch particles, which are the ones most useful for making analogues of common molecules. If pure samples containing patchy particles of identical valence are desired, then it is the fractionation step that ultimately limits the quantity available. Typically, we collect the same valence from up to 40 separations run in parallel, accumulating 10 9 to 10 11 particles.
The amidine groups on the colloid surface are crucial to the patchy particle fabrication process. First, the positive charge created from the dissociation of amidine hydrogen chloride salt (-C(NH)NH 3 Cl), along with the SDS surfactant, stabilizes the microspheres as well as the clusters by electrostatic repulsion. Second, when the clusters are swollen and encapsulated, the positive charges make the patches of the cluster more hydrophilic than the monomer-water interface, which is stabilized only by SDS. This difference in interfacial energies leads to finite contact angles and well-defined patches. Most importantly, the amidine groups can be easily functionalized in aqueous solution.
We functionalize the amidinated patches with biotin, and then use a biotin-streptavidin-biotin linkage to attach DNA with High shear 61% 15% 4% 1% 0.2%* 0.02%* 0.001%* Low shear 7% 16% 25% 15% 8% 5% 3% Density gradient centrifugation is used to fractionate the patchy particles. The fraction of identical particles obtained from a single centrifuge tube is shown. Fractions of 10%-20% correspond to 10 8 -10 9 particles in a single fractionation. * For these higher-valence particles, the fractions were estimated from their number ratio relative to lower-valence particles observed under a microscope (see Methods). particles with DNA-functionalized patches having well-defined symmetries. A four-patch particle is shown as an example. 1, A cluster of four amidinated polystyrene microspheres, prepared by the method of ref. 5, is swollen with styrene such that the extremities of the cluster-a tetrahedron in this caseprotrude from the styrene droplet. The styrene is then polymerized and the protrusions from the original cluster become patches. 2, Biotin is sitespecifically functionalized on the patches. 3, Biotinated DNA oligomers are introduced and bind to the particle patches via a biotin-streptavidin-biotin linkage. b, Electron micrographs of amidinated colloidal clusters, showing the particle configurations for clusters of n 5 1-7 microspheres. c, Electron micrographs of amidinated patchy particles after encapsulation. The patches inherit the symmetries of their parent clusters. d, Confocal fluorescent images of corresponding patchy particles, verifying that only the patches are functionalized with DNA. The fluorescence comes from the dye-labelled streptavidin that links DNA with the patches. Scale bars, 500 nm. showing that the sizes of patches can be adjusted by changing encapsulation conditions. a, Particles with relatively large patches are fabricated when clusters are swollen with 1.0 ml of styrene monomer. Primary spheres are 540 nm in diameter. b, Under identical conditions, smaller patches are obtained when more monomer (1.2 ml) is added. c, Smaller patches, relative to particle size, are obtained using primary microspheres 850 nm in diameter. Using larger particles facilitates observation under an optical microscope. Divalent, trivalent and tetravalent particles from this batch were used in colloidal molecule formation, and the monovalent particles were used in kinetics study, as discussed below. Arrow indicates increasing patch size. Scale bars, 500 nm.

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single-stranded 'sticky' ends. The first step uses sulpho-NHS-biotin (biotinamidohexanoic acid 3-sulpho-N-hydroxysuccinimide ester sodium salt, a water-soluble biotin derivative) and is carried out in phosphate-buffered saline (PBS) (pH 5 7.42), where the Nhydroxysuccinimide ester (NHS) can react with amidine groups and covalently link the biotin to the patches.
The DNA oligomer is prepared separately. It has three parts. At the 59 end, it has a biotin as an anchoring molecule. In the middle, there is a 49-base-pair double helix that acts as a spacer. Finally, at the 39 terminus, a single strand of 11 complementary or 8 palindrome base pairs forms the sticky end (for sequences, see Methods). Mixing DNA with streptavidin in a 1:1 ratio yields a streptavidin-DNA complex, which is then added to the biotin-functionalized patchy particles to produce DNA-functionalized patchy particles. The streptavidin contains a fluorescent tag for visualization by confocal microscopy. Figure 1d shows that only the patches of the particle are fluorescent, indicating that the streptavidin-DNA complex successfully coats the particle patches and that the amidine-NHS chemistry used for biotin functionalization works as designed.
The binding between patches on different particles is realized by hybridizing DNA oligomers on different patches. The oligomers, about 18 nm in length, provide short-range attractions and thus enforce the directionality defined by the particle patches. DNA is widely used for linking nanoparticles because it can be synthesized with control over the length and sequences of the base pairs, which, in turn, controls the specificity and the strength of interaction [35][36][37][38] . Hybridization of the complementary strands is fully reversible with temperature so that particle assembly can be controlled by varying temperature.
The oligomers we use to functionalize purified patchy particles are complementary DNA strands designated R (red) or G (green) and designed to bind selectively only to each other, or a palindrome P strand that only binds to other P strands. To differentiate the particles under the confocal microscope, red fluorescent (Alexa 647) streptavidin is used with R particles, and green fluorescent (Alexa 488) streptavidin is used with G particles (see Supplementary Fig. 2).

Assembly of colloidal molecules
With our collection of R, G and P patchy particles, we can build colloidal assemblies that mimic not only the geometry, but also the chemistry of molecules. Figure 3a (left panel) shows the formation of AB-type colloidal molecules from two 1-patch particles with complementary sticky ends. The system produces colloidal dumbbells without the random aggregation observed using spherical particles uniformly coated with DNA, and consistently with there being only one patch per particle. The confocal fluorescent image in Fig. 3a (middle panel) shows only complementary R-G particle pairs and no R-R or G-G pairs, confirming that DNA hybridization drives particle assembly. The resulting dumbbells are the colloidal analogues of AB-type molecules such as hydrogen chloride (Fig. 3a, right panel). Here, in contrast to hydrogen and chlorine, the sizes of the two atoms are the same, although they need not be. Patchy particles of different sizes can be fabricated and DNA bonds of various strengths can be used, so colloidal molecules of different size ratio and bond strength can be obtained. Figure 3b shows linear AB 2 -type colloidal molecules, the colloidal analogues of molecules like carbon dioxide (CO 2 ) or beryllium chloride (BeCl 2 ), that are obtained when green divalent (2-patch) particles are mixed with red monovalent particles. Triangle-like AB 3 (Fig. 3c) and tetrahedron-like AB 4 (Fig. 3d) colloidal molecules are similarly obtained by mixing trivalent (3-patch) particles and tetravalent (4-patch) particles, respectively, with monovalent particles (see Supplementary Video 1 for all colloidal molecules).
Bonding specificity is critical for the formation of all of the AB n structures. It promotes the formation of AB bonds while prohibiting the formation of AA and BB bonds, ensuring that the divalent, trivalent and tetravalent particles can act as the central atoms and the bonding interactions mimic that of atomic orbitals in geometry and valence. Complementary monovalent particles then serve as ligands that form bonds with the central atom. The confocal images in Fig. 3 (middle panels) show the directionality and specificity of the interactions between the central atoms and their monovalent particle ligands.
Other structures that can be made include colloidal analogues of alternating copolymers, formed using complementary divalent particles. Figure 3e shows that only green and red divalent particles bind to each other.
If particles have patches big enough to accommodate more than one complementary particle, molecular isomers and branched polymers are obtained. Figure 3f shows two isomers of a nonlinear AB 2type assembly that mimic the cis-and trans-conformations of molecules with a double bond. Such isomers may behave quite differently from one another in diffusion, rotation and reactivity. Additional monovalent particles can bind to the isomers and form ethylene-like structures (Fig. 3f, bottom panels). In the assembly of colloidal polymers from divalent particles, particles with bigger patches lead to branched chains and cross-linked networks (see Supplementary Fig.  3a). These results highlight the importance of controlling the patch size and in particular the ability to make patches sufficiently small that steric hindrance prevents more than one particle from attaching (see Supplementary Fig. 4 for SEM pictures of patchy particles used in colloidal molecules and geometry analysis for hindrance).
Self-complementary palindrome strands can also be used for selfassembly of mono-and divalent particles. Monovalent particles with palindrome sticky DNA yield A 2 -type colloidal molecules, analogous to H 2 or Cl 2 , whereas divalent particles yield homopolymers. One can also envision higher-order palindrome particles that might assemble into extended open structures like a diamond lattice 39 .

Colloidal reactions
The self-assembly of patchy particles into a target structure can be viewed as a 'colloidal reaction' or more generally as ''supracolloidal chemistry'' 40 . As in conventional chemical reactions, colloidal particles with a particular morphology and binding capacity are used as reagents and mixed together stoichiometrically. For example, four equivalents of monovalent and one equivalent of complementary tetravalent particles assemble into AB 4 colloidal molecules. The yield is about 50% after a few days; that is, half of the tetravalent particles have four monovalent particles attached, with the remainder consisting of incomplete structures like AB 3, AB 2 and AB. Using an excess of the monovalent particles increases the yield of the final AB 4 product, just as for conventional chemical reactions. A fivefold excess of monovalent particles, for example, increases the yield of AB 4 to 80% (see Supplementary Fig. 3b). It should also be possible to increase the yield by increasing the strength of the DNA-mediated patch binding, which can be done by increasing the length of the DNA sticky ends or by increasing the density of the DNA attached to the patches 41 . An obvious and important difference between the molecular and colloidal domains is the size of the constituents. The much larger colloids exhibit much slower dynamics and reaction kinetics, and thus can be observed in situ under an optical microscope. As shown in Fig. 4a, the formation of an AB 4 molecule proceeds by the central tetravalent particle picking up monovalent particles, one at a time, over a period of about 30 min (see Supplementary Video 2). In the case of divalent particle chain formation, the 'polymerization' also follows a step-growth mechanism. Figure 4b illustrates how a polymer chain can be extended by adding divalent particles one by one at the end (see Supplementary Video 3). Alternatively, two polymer chains can fuse into a longer chain.
We can understand the stepwise growth mechanism by examining the kinetics of formation of the AB 3 molecules (see Supplementary  Fig. 5 and Supplementary Video 4). We first heat a trivalent and monovalent particle mixture, with monovalent particles in large excess, to 55 uC, well above the melting temperature T m (50 uC) of the DNA, thus causing the particles to dissociate. The system is then quenched to room temperature, well below T m , so that the reaction kinetics is controlled by diffusion and by the size of the sticky patches. The collision frequency between monovalent and trivalent particles can be estimated from the well-known Smoluchowski equation 42 : where b m 5 0.49 mm is the radius of the monovalent particle, b t 5 0.91 mm is the radius of the trivalent particle, and C m is the number concentration of the monovalent particles, estimated to be one particle per 50 mm 3 from direct observation. The diffusion coefficients are about D m 5 0.50 mm 2 s 21 and D t 5 0.28 mm 2 s 21 for the monovalent and trivalent particles, respectively. These values give a collision frequency between trivalent and monovalent particles of 0.27 s 21 or a time between collisions of 3.6 s. Not every collision results in a bond, however, owing to the anisotropic nature of patchy particles. Only collisions between patches with complementary DNA can result in adhesion, with a rate proportional to the patch size. Thus, smaller patches form bonds more slowly than larger patches.
We define A as the fractional surface area of a particle occupied by patches (see Supplementary Figs 4 and 5), with values estimated from SEM measurements of A m 5 0.23 for the monovalent particle and A t 5 0.077 for all three patches of the trivalent particle (see Supplementary Table 1). The estimated reaction time for the first monovalent particle to adhere to a trivalent particle is 1/JA m A t or about 3.4 min. The area A for the trivalent-monovalent particle assembly immediately falls to 0.040 because one of the three patches is covered, and because the attached monovalent particle increases the total surface area of the complex. With two monovalent particles attached, A falls to 0.016. Thus the times for the second and third monovalent particles to attach are estimated to be 6.5 min and 16 min, respectively, which is consistent with the times observed experimentally and with the observed stepwise assembly of patchy particles (see also Supplementary Video 2).

Discussion
We expect that our colloids with valence can assemble not only into the molecular analogues shown here, but also into bulk colloidal phases of fundamental and practical interest. Tetrahedrally coordinated glasses, diamond crystals and empty liquids 43 have all proved difficult or impossible to make with existing colloids but should be accessible using our method. However, making scalable quantities of purified divalent and higher-valence colloidal atoms remains a challenge, owing to the limitations of fractionation by density gradient centrifugation. This difficulty might be overcome by large-scale separations 44 , or by developing methods that produce clusters with controlled morphology so that no separation is needed. Indeed, the swelling and functionalization methods we use to make our colloidal particles could readily be adapted for use with clusters made using other recently developed techniques 22,25,45 .
The ability to design colloidal particles with a variety of well-controlled three-dimensional bonding symmetries opens a wide spectrum of new structures for colloidal self-assembly, beyond colloidal assemblies whose structures are defined primarily by repulsive interactions and colloidal shape. Furthermore, the specificity of DNA interactions between patches means that colloids with different properties, such as size, colour, chemical functionality or electrical conductivity could be linked in well-defined sequences and orientations to make new functional materials. Such materials might include photonic crystals with programmed distributions of defects or three-dimensional electrically wired networks. We note that although valence and interaction strength in atomic systems are coupled by underlying quantum mechanical rules, they can be independently varied in our system. This raises the intriguing prospect of our patchy particles not merely mimicking atoms, but functioning as ''designer Schematic images (top panels) and snapshots from videos (bottom panels) show step-by-step reactions between colloidal atoms. Bent arrows point from the colloidal atom to the site where it is going to attach. Straight arrows indicate the time sequence. a, Monovalent particles attach to tetravalent particle, one by one, forming an AB 4 -type colloidal molecule. b, Complementary divalent particles polymerize into a linear chain structure. Scale bar, 2 mm. Videos of these processes are available in the Supplementary Information.

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atoms'' 46 that can undergo reactions, unconstrained by the rules that govern bonding at the atomic scale, to yield structures with no analogues in the atomic or molecular realm 43,47 .

METHODS SUMMARY
Cross-linked amidinated polystyrene microspheres were synthesized using a surfactant-free emulsion polymerization method. The amidinated clusters were prepared as described by ref. 5. Shearing conditions were optimized to control cluster distribution. A two-stage swelling and polymerization method was used to encapsulate the clusters, thereby fabricating mixtures of patchy particles that were separated by density gradient centrifugation. After purification, the separated particles were dispersed in 10 mM PBS (pH 7.42, 100 mM NaCl) containing 0.1% (w/w) Triton X-100 and reacted with sulpho-NHS-biotin to convert the functionalities on the patches from amidines to biotins. 59-Biotin-DNA was mixed with fluorescent streptavidin in a 1:1 ratio and the resulting complex was used to attach DNA to the biotinylated patches. Finally, the DNA-containing particles were washed with and dispersed in an aqueous PBS containing 1% (w/w) Pluronic F127. This suspension was used for all self-assembly experiments. For the self-assembly studies, the mixture of interest was sealed in a hydrophobic capillary tube and imaged using optical microscopy. Particles dried on a silicon wafer were imaged by field-emission SEM. The fluorescent images were obtained using a Leica SP5 confocal fluorescence microscope. Laser lines at 488 nm and 633 nm were used to excite the green and red fluorescence.
Full Methods and any associated references are available in the online version of the paper.