COLLOIDAL MATERIALS

Particle analogs of electronsin colloidal crystalsMartin Girard1,2,3*, Shunzhi Wang3,4*, Jingshan S. Du1,3*,Anindita Das3,4*, Ziyin Huang1,3, Vinayak P. Dravid1,3, Byeongdu Lee5,Chad A. Mirkin1,3,4†, Monica Olvera de la Cruz1,2,3,4†

A versatile method for the design of colloidal crystals involves the use of DNA asa particle-directing ligand. With such systems, DNA-nanoparticle conjugates areconsidered programmable atom equivalents (PAEs), and design rules have beendevised to engineer crystallization outcomes. This work shows that when reducedin size and DNA grafting density, PAEs behave as electron equivalents (EEs),roaming through and stabilizing the lattices defined by larger PAEs, as electrons doin metals in the classical picture. This discovery defines a new property of colloidalcrystals—metallicity—that is characterized by the extent of EE delocalization anddiffusion. As the number of strands increases or the temperature decreases, theEEs localize, which is structurally reminiscent of a metal-insulator transition. Colloidalcrystal metallicity, therefore, provides new routes to metallic, intermetallic, andcompound phases.

The interactions among electrons and atomsto form molecules and materials are foun-dational in physics and chemistry. How-ever, in the science of colloidal crystals, inwhich particles are often analogized with

atoms, a particle analog to electrons has notbeen invoked, despite the synthesis of hundredsof colloidal crystals (1–24) and the developmentof certain approaches into elaborate forms ofcrystal engineering (9–21). In particular, colloi-dal crystal engineering with DNA has led to thedesign of structures with diverse symmetries,lattice parameters, and crystal habits (12–22).However, to date, the particles modified withDNA that define such structures behave as pro-grammable atom equivalents (PAEs) and havefixed particle positions at set stoichiometric ra-tios. These systems are governed by a set of de-sign rules and the complementary contact model(CCM) (14), the premise that they organize them-selves to maximize contacts that lead to hybrid-ization and structures such as ionic compounds.We report on an electron-atom duality analog

in colloidal crystal engineering with DNA, inwhich the resulting colloidal assemblies are bet-ter classified as “metallic” structures. In such struc-tures, small DNA-functionalized NPs becomemobile and “electron-like” [or electron equiv-alents (EEs)] and are essential for maintainingthe positionsof the larger PAE “atom” components.

Mixtures of complementary DNA-functionalizednanoparticles (NPs) that vary in size and DNAsurface density were assembled and character-ized by means of electron microscopy, synchro-tron small-angle x-ray scattering (SAXS), andscale-accurate molecular dynamics (MD) sim-ulations with explicit hybridization (17, 25, 26).Through a combination of theory, simulations,and experiments, we show that small particlesgrafted with low numbers of DNA strands (forexample, <6), when mixed with complementaryfunctionalized NPs (Fig. 1A), form crystals butdo not occupy specific lattice sites and diffusethrough the crystal in a manner reminiscent ofclassical electrons in metals, as described bythe original Drude model. The PAEs alone willnot form crystals because they are almost solelyrepulsive. The delocalized EEs that move freelythrough the lattice are responsible for stabiliz-ing it, a type of bonding more reminiscent ofmetals than ionic compounds (Fig. 1B).Furthermore, when the interactions are tailored

by increasing the number of potential DNAbonds or lowering the temperature, these EEscondense into specific locations, yielding a tran-sition akin to a metal-insulator transition. Last,by taking advantage of this duality and the struc-tural features of the DNA-modified particlesthat govern it, we realized three polymorphiccrystal phases—body-centered cubic (bcc), face-centered cubic (fcc), and Frank-Kasper A15—and analyzed the distribution and diffusion ofthe particles (EEs and PAEs) within them as afunction of temperature and number of link-ers per EE.In a typical set of experiments, 10-nm-diameter

AuNPsweredenselymodifiedwith single-strandedpropylthiolated DNA to yield conjugates with~160 strands per NP. These modified NPs werehybridized with a complementary strand toform a rigid duplex region (18 bases) with a six-

base single-stranded overhang (fig. S1). NPswith average diameters of 10, 5, 2, and 1.4 nm,respectively, were modified in a similar mannerbut with a second type of complementary DNAoverhang (Fig. 1A). The average number of DNAoverhanging strands available for bonding (num-ber of linkers per EE) is a function of input linkerconcentrations in the solution (Fig. 1F). For thefirst three combinations of NPs (10 + 10, 10 + 5,and 10 + 2 nm), all formed the expected CsCllattice (space group Pm�3m) (27) based on theconventional CCM model and the descriptionof them as ionic compound analogs (14).However, with the 10 + 1.4 nm combination,

the 10-nmNPs assumed a bcc lattice (space groupIm�3m), but the 1.4-nmNPswere invisible to SAXS(Fig. 1C). The lattice assignments were all verifiedby means of electron microscopy with low-angleannular dark field (LAADF) imaging after thestructures were encased in silica (28), whereasthe 1.4-nm NPs in the 10 + 1.4 nm combinationdid not appear at specific lattice sites (Fig. 1D).For the first three combinations, the NP posi-tions were determined by the length of the DNAbonding elements that define them (Fig. 1E).However, for the 10 + 1.4 nm combination, therewas a marked decrease in the interparticle dis-tance compared with the expected value based onCCM prediction (14).To visualize the positions of the 1.4-nm NPs,

we performed cryogenic transmission electronmicroscopy (cryo-TEM) on the as-synthesizedbcc lattice formed from the 10 + 1.4 nm NPs andthen stacked the repeating “unit cell” images andEE locations along the [111] zone axis (Fig. 2A).Cryo-TEM showed that the large NPs (PAEs)assumed a bcc lattice, and the small NPs (EEs)were randomized throughout that lattice (Fig.2B), which is in agreement with the MD sim-ulations (Fig. 2D). In the simulations, crystallinestructures were obtained for mixtures of comple-mentary DNA–functionalized Au NPs at a fixedsize ratio (10- to 2-nm diameter) but with var-iable EE:PAE ratios from 4:1 to 12:1 and num-ber of linkers per EE from four to eight (Fig. 2,D and E). Because the MD simulations wereperformed in the isobaric-isothermal ensemble(NPT) with pressure near zero (supplementarymaterials), these complementary EEs, which aredelocalized from specific lattice sites (Fig. 2D),were responsible for the attraction that holdsthe large PAEs in crystalline positions in thesemetal-like assemblies.We determined the degree of delocalization of

EEs in the MD simulations by discretizing theunit cell volumes Vcell into (128)3 voxels of equalvolume a3 so that Vcell = (128a)3 and then count-ing the EE visitation frequency in each voxel. Thisfrequency gives a probability distribution fk ineach voxel, k. A quantitative measure of clus-tering tendency, Scl, is defined as

Scl ¼ �X

kfklnð fkÞ þ lnðVcell=a

30Þ ð1Þ

where a0 is the average of a over all simulationsand used as a constant value to normalize thevolume. The quantity Scl can be associated with

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1Department of Materials Science and Engineering,Northwestern University, Evanston, IL 60208, USA.2Department of Physics and Astronomy, NorthwesternUniversity, Evanston, IL 60208, USA. 3InternationalInstitute for Nanotechnology, Northwestern University,Evanston, IL 60208, USA. 4Department of Chemistry,Northwestern University, Evanston, IL 60208, USA.5X-ray Science Division, Advanced Photon Source,Argonne National Laboratory, Lemont, IL 60439, USA.*These authors contributed equally to this work.†Corresponding author. Email: m-olvera@northwestern.edu(M.O.d.l.C.); chadnano@northwestern.edu (C.A.M.)

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an information entropy (29). This quantity isminimized if the EEs are localized and maxi-mized if they are delocalized (when they have auniform distribution). The resulting Scl shows astrong correlation with both EE:PAE ratio andnumber of linkers per EE (Fig. 2F): Either anincrease in the number of EEs in the lattice or adecrease in the number of duplexed DNA link-ers on the EEs resulted in a more randomizeddensity distribution of EEs in the lattice.The EEs in the PAE-EE assemblies are classical

particles and as such follow a Boltzmann dis-tribution. Thus, the movement of EEs in time isrelated to free-energy barriers that allow for anexponential relationship between their trappingtime and temperature (Fig. 2G), which is directlyrelated to the degree of EE delocalization. To vi-sualize the spatial distribution of EEs from MDsimulations, we calculated the cumulative den-sity of EEs by integrating fk from its maxima(where the density of EEs is the highest) anddrew isosurfaces that separate the unit cell intoequally probable accessible volumes for the EEs,termed Boltzmann volumes (supplementarymaterials). The Boltzmann volumes for an as-sembly with a low number of DNA linkers perEE were widely dispersed across the volumeof the crystal (Fig. 2H). Increasing the num-ber of linkers per EE resulted in the localiza-tion of EEs into a group of locations near theB sites in AB6-type binary compounds (Fig. 2Iand Table 1). This localized state is similar tothe electron charge-density distributions insemiconductors and insulators (30). Further-more, the Boltzmann volumes became more dis-persed as the EE:PAE ratio increased (Fig. 2Jand fig. S33). Such a response was also experi-mentally observed by comparing the projectedEE locations determined with cryo-TEM onsamples of bcc assemblies with varying inputEE:PAE ratios and EE linker concentrations inthe solution. The EE local density in the crystalwith a low EE:PAE ratio and high linker con-centration (Fig. 2C) around the predicted lo-calization sites was substantially higher thanthe uniform distribution baseline and than thelocal densities in the assemblies with higherEE:PAE ratios and lower linker concentrations(Fig. 2B and fig. S17).Compared with PAEs that reside on relatively

fixed lattice sites, EEs in a PAE-EE assembly aremacroscopically mobile beyond local vibrations.Time-series MD simulation snapshots showedthat the EEs could diffuse between unit cells(Fig. 3A, fig. S39, andmovie S4). To further probethe diffusion of EEs in experimentally realizedassemblies, 10-nm PAEs and Cy5-DNA–labeled1.4-nm EEs were assembled in the bcc structure.Subsequently, these crystals were incubated witha solution of Cy3-DNA–labeled EEs. To track anyexchange of EEs between the crystalline assem-bly and EEs in solution, the ultraviolet-visible(UV-vis) extinction spectra of the supernatantwere measured over time (Fig. 3B, top). A simul-taneous increase of Cy5 signal and reductionof Cy3 signal suggested that the EEs were highlymobile and could diffuse macroscopically be-

Girard et al., Science 364, 1174–1178 (2019) 21 June 2019 2 of 5

Fig. 1. Transition from PAE-PAE systems to PAE-EE systems. (A) Illustrations of DNA-functionalized Au NPs behaving as programmable atom equivalents (PAEs) or electronequivalents (EEs) used in the MD simulation. (B) Snapshots from the MD simulation depicting“ionic” bonding behavior shown by PAE + PAE assemblies, and “metallic” bonding behaviorshown by PAE + EE assemblies, where roaming EEs hold the crystal of repulsive PAEstogether. (C and D) Four crystalline lattices assembled from 10-nm PAEs and complementaryDNA–functionalized Au NPs (nominal core diameters of 10, 5, 2, and 1.4 nm, respectively).Shown are (C) SAXS spectra, (D) models, and cross-sectional LAADF images ofsilica-encapsulated samples. Scale bar, 25 nm. The 1.4-nm Au NPs in (D) (yellow arrowsindicate visually identified ones) are dispersed randomly in the lattice and do not occupyspecific lattice sites. (E) SAXS-determined distance between bonding 10-nm PAE pairs(same DNA type, defined in the inset according to CCM assumptions). (F) Quantification oflinker DNA strands duplexed on 1.4-nm Au NPs (EEs) as a function of the input number oflinkers per EE in the solution.

Table 1. Symmetry of PAE-EE assemblies in the fully localized state.Wyckoff positions insquare brackets have higher energies than the ground-state configurations.

PAE lattice Space group Localized EE Wyckoff positions

bcc Im�3m 12d. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .

fcc Fm�3m 8c, 32f. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .

A15 Pm�3n 6c, 16i, 24k, [12f], [24j]. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .

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Girard et al., Science 364, 1174–1178 (2019) 21 June 2019 3 of 5

Fig. 2. Spatial probability distribution ofEEs in PAE-EE assemblies in the bccstructure. (A) Workflow for obtaining EElocation-labeled “unit cell” images fromcryo-TEM by using image segmentation.ACF, auto-correlation function. (B and C)Overlay of averaged-intensity TEM imagesand identified EE locations in “unit cells”along the [111] direction. The input parame-ters refer to the ratio of each substanceadded to the solution, not within a crystal.(D and E) MD simulation snapshots ofPAE-EE assemblies. (F) Measure ofclustering tendency (Scl) of EEs in MDsimulations. (G) Temperature-dependenttrapping time (t) of EEs in MD simulations.kB, Boltzmann constant. (H to J) SimulatedBoltzmann volumes of EEs viewed alongthe [111] direction with EE:PAE = 4:1 and (H)4 or (I) 8 linkers per EE, or with (J) EE:PAE =9:1 and 8 linkers per EE. Orange dashesapproximate a repeating “unit cell” used incryo-TEM image analysis.

Fig. 3. Diffusion of EEs in PAE-EE assemblies in the bcc structure.(A) Trajectory of one EE over time in a lattice of PAEs (yellow) from the MDsimulation. The color of the EE positions (red to green to blue) representsdiferent time points. (B) The exchange of dye-labeled particles betweencrystalline lattices and solution was monitored by the change of light extinctionin solution by means of UV-vis spectroscopy. Cy5-DNA–labeled EEs were

exchanged from “metallic” PAE-EE assemblies (10 + 1.4 nm, bcc) by Cy3-DNA–labeled EEs in the supernatant over 24 hours (top), whereas no appreciableexchange was observed for PAE-PAE assemblies (10 + 10 nm, bcc) (bottom).(C and D) Predicted colloidal crystal metallicity (Mcc) in bcc assemblieswith different number of linkers per EE. The Mcc value has a minimum againstthe EE:PAE ratio at 6:1 (C) but is monotonic against temperature (D).

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tween the colloidal assemblies and solution. Incomparison, no appreciable exchange eventswere observed for the bcc assemblies formedby 10 + 10 nm PAEs modified with identicaldye-labeled DNA (Fig. 3B, bottom), suggestinga much weaker diffusion of PAEs as comparedwith that of EEs. In both cases, the crystallinityof the assemblies was preserved after the ex-change (fig. S18).The tunable spatial density distribution of

EEs (based on number, temperature, and linkerdensity) and their macroscale diffusion insidea crystalline PAE framework establish the con-cept of colloidal crystal metallicity, Mcc, as

Mcc = Scl – ln(Ncell) (2)

where Ncell is the number of EEs per unit celland is used to make the metallicity indepen-dent of crystal unit cell and to reduce to the idealentropy of a gas in the limit of noninteractingparticles. In Eq. 1, the EEs are considered as agroup (Fig. 2F), but in Eq. 2, Mcc is a quantita-tive measure of the average degree of delocal-ization per EE, which can decrease when theEE:PAE ratio increases (Fig. 3C) even if Scl in-creases or remains nearly the same (Fig. 2F). Ametallicity minimum was attained at EE:PAE =6:1, suggesting that the metal-like bonds in thecolloidal system become saturated. Increasingthe number of duplexed strands on EEs led tocrystals with lower Mcc values (Fig. 3C) becausethe EEs were more localized or trapped as thefrequency of DNA binding events increased.When the system temperature increased, Mcc

also increased (Fig. 3D) because of the decreasein the number of hybridized sticky ends. Theseresults show that the EEs can undergo a clas-sical (nonquantized) process analogous to ametal-insulator transition observed in atomicsolids (31) because the degree of EE delocal-ization and diffusion drastically changed wheneither the temperature, the number of linkersper EE, or the EE:PAE ratio was varied.Colloidal crystal metallicity depends on the

number of particles, the number of DNA strandsthat can engage in bonding, and the strength ofthe bonds formed. Because these parameters aredifficult if not impossible to control in purelyelectrostatic systems owing to electroneutralityrequirements, metallicity has been neither ob-served nor defined in conventional ionic col-loidal crystals (crystals formed from oppositelycharged particles) (4, 9). The complementaryDNA design on NPs ensures that a crystal willnot form from PAEs alone (and vice versa fromEEs alone), which distinguishes this systemfrom a description of small particles doped in acrystalline solid formed from larger particles.Moreover, although this concept of metallicitysuperficially resembles “sublattice melting” insuperionic conductors (32), in which one sub-lattice of an ionic compound loses long-rangeorder while the other is fixed (for example,AgI), ion diffusion in such systems is mediatedby Frenkel defects because of the constraint ofcharge balance. Thus, ionic systems have lim-

ited tunability and more strict stoichiometryrequirements compared with those of the col-loidal crystals formed through DNA-directedassembly events.New phases were accessed by adjusting the

input EE:PAE ratio in solution and the totalDNA coverage on EEs. For example, as the inputEE:PAE ratio was progressively increased, an fcclattice (space group Fm�3m) emerged (Fig. 4, Aand C). The increase in EE:PAE ratio resulted instronger cumulative bonding interactions (thereare more DNA bonding connections under suchcircumstances), which was reflected by the in-crease in the crystal melting temperature, Tm,

from 31° (bcc) to 41°C (fcc) (Fig. 4B). In addition,if both the total DNA coverage on EEs (charac-terized by the total number of duplexed andnonduplexed strands) and the input EE:PAEratio in solution increase, the Frank-Kasper A15phase (space group Pm�3n) emerged (Fig. 4, Dand E). The formation of the A15 phase may beassociated with its tendency to minimize thecontact area between repulsive PAEs, which issimilar to the trend observed in dendrimer as-semblies (fig. S27) (33) but different from theCr3Si structure in binary superlattices (14, 34).These phases were also observed in the simu-lations (table S9). The three phases realized in

Girard et al., Science 364, 1174–1178 (2019) 21 June 2019 4 of 5

Fig. 4. Equilibrium phases realized by PAE-EE assemblies. (A) Schematic representationof the equilibrium conditions of bcc, fcc, and A15 phases and (B) their corresponding thermalmelting transitions. (C) SAXS spectra showing the equilibrium phase transition from bcc(red, input EE:PAE = 10:1) to a bcc/fcc mixture (purple, input EE:PAE = 20:1), and then to amajority fcc phase (blue, input EE:PAE = 40:1). (D) Experimental (green) and simulated(black) SAXS spectra of A15 assemblies. (E) Cryo-TEM image of an A15 assembly. (Inset) A15lattice model along the [001] direction. (F and G) Simulated Boltzmann volumes of fcc (F) andA15 (G) phases as a function of the number of linkers per EE and EE:PAE ratio at a constanttemperature (kBT = 1.30). A whole graph is reconstructed by 1/8 of the unit cells from eachcombination. (H and I) Predicted colloidal crystal metallicity (Mcc) in (H) fcc and (I) A15assemblies as a function of the number of linkers per EE and EE:PAE ratio. Two low-metallicityconfigurations (gray shades) are present in (I).

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this system mimic the metal tungsten, in whichbcc, fcc, and A15 structures have all been realizedeither in the bulk or in thin films (35) (table S6).Critically, the MD simulations allowed us to

identify the positions that the EEs reside in atlowMcc values. For example, when the numberof DNA linkers per EE was maximized and theEE:PAE ratio was decreased, the EEs settled intodistinct locations in the fcc and A15 assemblies,as shown by the Boltzmann volumes (Fig. 4, Fand G) and the correspondingMcc values (Fig. 4,H and I). The localized lattice for the fcc struc-ture resembled a fully filled high-temperatureCu2Se lattice (fig. S34, A4B40), whereas the A15structure shows two possible configurations,either clathrate type I (fig. S35A, A8B46) (36) or anunreported lattice (fig. S35B, A8B82). The latterconfiguration, in which Mcc reached a local min-imum, contained all sites in the clathrate struc-ture that were fully occupied, and the additionalEEs that could not occupy the lowest energypositions began to fill the higher-energy 12f and24j positions (Table 1).Taken together, this work makes the case for

describing certain classes of colloidal crystalsin a fundamentally new way, in which, in thecase of mobile particles (EEs), the concept ofmetallicity becomes important. By understand-ing the factors that govern EE diffusion anddelocalization, we have a better understandingof the structures and phases that can be ac-cessed through colloidal crystals, potentially in-cluding metals, intermetallics, and complex metalalloys. It also challenges the colloidal science com-munity to identify exotic new properties that arisefrom the PAE-to-EE transition and structures thatexhibit high degrees of metallicity as well as todevelop theoretical models that capture the ef-fects that lead to metallicity.

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ACKNOWLEDGMENTS

The authors thank H. Lopez-Rios [Northwestern University(NU)] and M. G. Kanatzidis (NU) for helpful discussions,A. M. Geller (NU) for rendering the Boltzmann volume data,E. W. Roth (NU) for ultramicrotomy, and J. Remis (NU) forcryo-TEM tomography. Funding: This material is based on worksupported by the Center for Bio-Inspired Energy Science(CBES), an Energy Frontier Research Center funded by theU.S. Department of Energy (DOE) Office of Basic EnergySciences (DE-SC0000989, for computational studies), the AirForce Office of Scientific Research (FA9550-17-1-0348, forsynthesis, spectroscopy, and electron microscopy), theVannevar Bush Faculty Fellowship program sponsored by theBasic Research Office of the Assistant Secretary of Defensefor Research and Engineering and funded by the Office ofNaval Research (N00014-15-1-0043), the Sherman FairchildFoundation (for electron microscopy and computationalsupport), and the Biotechnology Training Program of NU(for cryo-TEM). This work made use of facilities at the NUANCECenter at NU (NSF ECCS-1542205 and NSF DMR-1720139),the Structural Biology Facility at NU (NCI CCSG P30CA060553), and the DuPont-Northwestern-Dow CollaborativeAccess Team (DND-CAT) of the Advanced Photon Source(APS) Sector 5 (DOE DE-AC02-06CH11357). Authorcontributions: C.A.M. and M.O.d.l.C. directed the research.M.G. performed simulations. S.W. and A.D. performedsynthesis and x-ray scattering experiments. J.S.D., S.W.,and Z.H. performed electron microscopy studies. All authorscontributed to data analysis and manuscript preparation. Competinginterests: The authors declare no competing interests. Data andmaterials availability: All data needed to evaluate the conclusions inthis manuscript are present in the main text or the supplementarymaterials. Additional data or codes are available upon request to thecorresponding authors.

SUPPLEMENTARY MATERIALS

science.sciencemag.org/content/364/6446/1174/suppl/DC1Materials and MethodsSupplementary TextFigs. S1 to S39Tables S1 to S11References (37–52)Movies S1 to S6

30 January 2019; accepted 28 May 201910.1126/science.aaw8237

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Particle analogs of electrons in colloidal crystals

and Monica Olvera de la CruzMartin Girard, Shunzhi Wang, Jingshan S. Du, Anindita Das, Ziyin Huang, Vinayak P. Dravid, Byeongdu Lee, Chad A. Mirkin

DOI: 10.1126/science.aaw8237 (6446), 1174-1178.364Science 

, this issue p. 1174Scienceinteraction resembles the classical picture of electrons in metals.nanometers in diameter) are present. These smaller particles are mobile and diffuse through the lattice, so the bonding

1.5∼diameter) that have mutual repulsive interactions can form a stable lattice only if much smaller conjugate particles (10 nanometers in∼ show that larger particles (et al.compounds. Inspired by molecular dynamics simulations, Girard

assembly through hybridization. The design rules for interactions between pairs of particles resemble those for ionic The crystallization of nanoparticles can be controlled by functionalizing them with DNA strands that direct

Mobile particles in colloidal crystals

ARTICLE TOOLS http://science.sciencemag.org/content/364/6446/1174

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