Generic phase diagram of binary superlatticesAlexei V. Tkachenkoa,1

aCenter for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973

Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved July 20, 2016 (received for review December 22, 2015)

Emergence of a large variety of self-assembled superlattices is adramatic recent trend in the fields of nanoparticle and colloidalsciences. Motivated by this development, we propose a model thatcombines simplicity with a remarkably rich phase behavior appli-cable to a wide range of such self-assembled systems. Those sys-tems include nanoparticle and colloidal assemblies driven byDNA-mediated interactions, electrostatics, and possibly, controlleddrying. In our model, a binary system of large and small hardspheres (L and S, respectively) interacts via selective short-range(“sticky”) attraction. In its simplest version, this binary sticky spheremodel features attraction only between S and L particles. We showthat, in the limit when this attraction is sufficiently strong comparedwith kT, the problem becomes purely geometrical: the thermody-namically preferred state should maximize the number of LS con-tacts. A general procedure for constructing the phase diagram as afunction of system composition f and particle size ratio r is outlined.In this way, the global phase behavior can be calculated very effi-ciently for a given set of plausible candidate phases. Furthermore,the geometric nature of the problem enables us to generate thosecandidate phases through a well-defined and intuitive construction.We calculate the phase diagrams for both 2D and 3D systems andcompare the results with existing experiments. Most of the 3D super-lattices observed to date are featured in our phase diagram, whereasseveral more are predicted for future discovery.

self-assembly | sticky spheres | colloids | superlattices

In recent years, the fields of nanoparticle (NP) and colloidalsciences have been transformed by a dramatic expansion of the

variety of self-assembled superlattices (1–13). These periodicstructures have been reported for a broad range of particle sizesand their interactions. These experiments include electrostati-cally driven and DNA-mediated as well as drying-induced as-semblies. The overall morphological diversity of such superlatticesis remarkable: even relatively simple binary systems have beenshown to self-assemble into more than 10 different crystallinestructures. On a theoretical side, some of the behavior has beencaptured with the system-specific models (14–22). However, mostof the experimental systems are conceptually similar and oftenexhibit the same morphologies on self-assembly, which suggests apossibility of a unified theoretical description. With that goal inmind, in this paper, we study the equilibrium phase behavior of abinary mixture of mutually attractive sticky spheres. This simplegeneric model is a natural starting point for developing a moreversatile theory. Thanks to its elegance and simplicity, this modelalso has a great conceptual value of its own.A common general argument used to explain the emergence

of self-assembled superlattices is based on their high packingdensity (1–4). It implicitly relies on the classical behavior of thehard sphere systems. In that case, the phase transformations aredriven solely by entropy, and therefore, under strong enoughcompression, the densest packing would automatically correspondto the equilibrium structure. Although this model is certainly ap-plicable to a number of colloidal and nanoparticle systems withshort-range repulsive potential, its relevance for the attractive caseis not well-justified. Furthermore, the phase diagram of the binaryhard sphere system is well-known, and it only features three crys-talline phases (23, 24) in a sharp contrast to a much greater struc-tural diversity of the experimentally observed binary superlattices.

It is certainly more natural to describe the colloids and nano-particles with a short range attraction as sticky spheres. As will beshown below, while preserving the generality and simplicity similarto the hard sphere systems, the binary sticky sphere model has asurprisingly rich phase behavior.

ResultsBinary Sticky Sphere Model. We consider a binary system of large(L) and small (S) spheres, in which different types (L and S) havevery short-range mutual attraction, whereas same-type particlesrepel each other with a hard core potential. If the binding energyper single LS contact is −e, the overall Hamiltonian of the systemcan be written in the form

H =−eZðNL +NSÞ. [1]

Here, NS and NL are the total numbers of particles of each type,and Z is the average number of LS bonds per particle. As usualfor statistical mechanical models, this Hamiltonian is technicallya free energy of a specific physical system integrated over its internaldegrees of freedom, except for the particle positions.In this work, we are interested in constructing the equilibrium phase

diagram as a function of system composition, f ≡NS=ðNS +NLÞ, andthe particle size ratio, r= rS=rL. It is convenient to characterizevarious structures made of L and S particles with the two co-ordination numbers ZL and ZS, respectively. ZL is defined as theaverage number of contacts that a large sphere has with the smallones, whereas ZS is the number of such LS contacts per smallparticle. By definition, ZSNS = ZLNL; therefore, the compositioncan be expressed in terms of these two numbers: f =ZL=ðZL +ZSÞ.We will focus on the regime when self-assembled superlattices

coexist with a very dilute “gas” of particles or particle clusters. Thisregime corresponds to the limit of strong LS binding, e � kT,because the binding energy must balance the large loss in trans-lational entropy. In this limit, the thermodynamic stability of theself-assembled phases is determined primarily by energy (Eq. 1)

Significance

Self-assembled superlattices represent one of the most prom-ising trends in current nano- and colloidal sciences. Their po-tential applications range from photonics and plasmonics tocatalysis and biomedical devices. This paper introduces a simplebut general and remarkably rich theoretical model that pro-vides an explanation for the observed morphological diversityand high degree of universality in superlattice self-assembly.The model describes binary system of mutually attractive stickyspheres of different sizes. An intuitive geometric procedure isused for constructing the phase diagram at variable composi-tion and size ratio. The great conceptual value of this model iscombined with its potential for predicting phase behavior for abroad range of self-assembly scenarios: DNA mediated, elec-trostatically driven, or induced by controlled drying.

Author contributions: A.V.T. designed research, performed research, analyzed data, andwrote the paper.

The author declares no conflict of interest.

This article is a PNAS Direct Submission.1Email: oleksiyt@bnl.gov.

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rather than entropy. We can write the composition-averagedchemical potential of a specific structure as

μ≡ ð1− f ÞμL + fμS =−eZLZS

ZL +ZS− kTδ. [2]

Here, the first term is the total binding energy per particle (ofeither type), and δ is an entropic correction. This entropy term isnormally subdominant and expected to be important only in twocases: (i) when the ground state of the Hamiltonian is degener-ate and (ii) when the structures of interest are either singleparticles or small clusters with a substantial translational en-tropy. When this entropic correction is neglected, the problembecomes purely geometrical: the ground state of the Hamilto-nian (Eq. 1) corresponds to the maximum possible number of LScontacts for a fixed overall composition f subject to the excludedvolume constraints.

Constructing the Phase Diagram. It is a common practice in the fieldof nanoparticle and colloidal assemblies to use chemical equiva-lent to designate a structure. For instance, CsCl would correspondto body-centered cubic (BCC) arrangement with two particle typesforming two cubic sublattices, similar to Cs and Cl ions. We willalso use this convention when applicable but in addition, introducean alternative notation for binary structures:

LnSm   is  a  structure  with  ZL =m  and  ZS = n. [3]

This notation is similar to the one based on the overall com-position (e.g., AB8 or AB6) but contains additional informationabout the connectivity. For instance, our notation can differen-tiate between dimers (LS), NaCl (L6S6), and CsCl (L8S8) that allhave 1:1 composition. The classification is certainly not complete,but it is compact, is intuitive, and can be applied to binary structuresthat do not have direct chemical equivalents. Finally, from the pointof view of this work, the formula LnSm encodes not only thecomposition and topology of a structure but also, its energetics:according to Eq. 2, the dominant contribution to μ is the harmonicaverage of numbers ZL and ZS times the constant prefactor −e.In a general case, the minimal free energy of a multicompo-

nent system with composition f may be achieved by forming ei-ther a single phase (e.g., crystal) or two coexisting phases. Notethat, in either case, the aggregated structures also coexist with agas of free S and L that has exponentially low-volume fraction. Ifthe phases “1” and “2” coexist, the chemical potentials of eachparticle type can be found from Eq. 2: μL = ðf1μ2 − f2μ1Þ=ðf1 − f2Þand μS = ðμ2 − μ1Þ=ðf1 − f2Þ+ μL. Therefore, the phase with anintermediate composition f p (f1 < f * < f2) is stable with respect tothe phase separation onto phases 1 and 2 when its chemicalpotential satisfies the following condition:

μ<�f * − f2

�μ1 +

�f1 − f *

�μ2

f1 − f2. [4]

A graphic interpretation of this condition is well-known instatistical physics. Let us represent all geometrically plausiblephases as points in ðf , μÞ space (Fig. 1C). On such a plot, a subsetof thermodynamically stable structures that appear sequentiallyon changing the composition f can be connected to define a“trajectory” of the system. The line must be concave up, and nocandidate phase should be below it. We, therefore, can constructthe overall ðr, f Þ phase diagram in three steps: (i) generate thecandidate phases, (ii) select a subset of such phases that areallowed geometrically for a given size ratio r, and (iii) find theconcave-up trajectory in ðf , μÞ space.Although brute force search over a large number of structures

is certainly possible, here, we present a prescription that allows

for generating the candidate phases in a more rational and sys-tematic manner. In this model, the contact between L and Sparticles is only possible when their center to center distance isstrictly fixed. This condition means that, for a given L particle,the centers of all of its S neighbors must be on a sphere of radiusrS + rL (or circle in the 2D case).However, if two L particles share the same S neighbor, its

center must belong to the plane equidistant from both of them. Ifthe arrangement of all L particles is known, such equidistantplanes form faces of the Voronoi cells. In a case where L par-ticles form a crystal lattice, it would correspond to the Wigner–Seitz cell. Therefore, any S particle with at least two L neighborsmust be positioned at the intersection of a surface of a Wigner–Seitz cell and a sphere of radius rS + rL with the same center. Wewill use this geometric property to generate families of binarystructures starting with various single-particle lattices.

2D Phase Diagram.The procedure of generating the candidate phasesin 2D is illustrated in Fig. 1 A and B. We consider the simplest casewhere L particles form a square lattice (sq). Because the corre-sponding Wigner–Seitz cell is square-shaped, there are three to-pologically distinct possibilities of its intersection with a circle ofradius rS + rL, as shown in Fig. 1A. As a result, the sq generates afamily of three binary structures with the same symmetry: L4S

ðsqÞ4 ,

L2SðsqÞ8 , and L2S

ðsqÞ4 . Similarly, one can generate a family of struc-

tures based on another high-symmetry 2D lattice [i.e., hexagonal(hex)]: L3S

ðhexÞ6 , L2S

ðhexÞ6 , and L2S

ðhexÞ12 , as shown in Fig. 1B. It should

be reminded that only centers of S particles with two or moreL neighbors have to be located at the intersections of the Wigner–Seitz boundary with the circle. The S particles that have only a singlesticky bond may be positioned elsewhere on the same circle. Thiscondition opens a possibility of loading the structures L2S

ðsqÞ4 and

L2SðhexÞ12 with additional S particles. In particular, the latter generates

a subfamily with a general formula L1+1=nSðhexÞ6ð1+ nÞ, shown in Fig. 1B.

Each candidate phase is characterized by two coordinationnumbers, ZL and ZS, and a range of an allowed size ratio,rmin < r< rmax. This range can be determined by enforcing therequirement that the distances between same-sized pairs, S–Sand L–L, must be greater than 2rS and 2rL, respectively. For aspecific value of size ratio r, only a subset of all candidate phasesis geometrically possible. We can now determine a sequence ofequilibrium phases for a given r and variable composition f byusing the ðf , μÞ plot introduced earlier. For instance, Fig. 1Cshows the ðf , μÞ plot for r= 0.5.Of all candidate phases generated earlier, five lattices are

geometrically allowed at r= 0.5 (they are represented by filledsymbols in Fig. 1C). By constructing the concave-up trajectoryenclosing all of those phases, we find that only three of thesuperlattices will actually appear in an equilibrium phase dia-gram: L4S

ðsqÞ4 , L3S

ðhexÞ6 , and L2S

ðsqÞ8 . In addition, one has to con-

sider the phases that contain gases of single S and L particles aswell as “flower-shaped” clusters with S or L particles in thecenter (i.e., LnS and LSn, respectively). These gas phases requirespecial treatment, because the individual particles and clustershave significant translational entropy. The average chemicalpotential of a particle that belong to an LnS or LSn cluster isgiven by the following modification of Eq. 2:

μ=−en

n+ 1+kT   log Cv0

n+ 1 . [5]

Here, C is the concentration of those clusters, and v0 is acharacteristic microscopic volume defined by the range of thesticky potential. This equation also describes gases of single L orS particles that correspond to n= 0. Because of the presence ofthe translation entropy term, the points that represent thesediscreet species are shifted down in Fig. 1C compared with theenergy-only result used for superlattices. This shift is significantly

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stronger for singles than for multiparticle clusters. Thanks to thiseffect, one can resolve the degeneracy problem that occasionallyemerges in this analysis. For example, three plausible structures inFig. 1C, L4S

ðsqÞ4 superlattice, L4S flower cluster, and single L

particles, would be on the same straight line if the entropic cor-rections were not included. The entropic shift results in a sup-pression of the cluster structure. In other words, the system wouldprefer coexistence of L4S

ðsqÞ4 lattice (where L particles have the

same connectivity as in those clusters) and single L particles thathave substantial translation entropy. Naturally, the system alwayscontains a finite concentration of both types of single particlesas well as various small clusters. Here, we only determine the

dominant specie among them. All others will be present at ex-ponentially low fractions.Based on our analysis, the complete phase diagram in 2D can

be constructed. As shown in Fig. 1D, it features two crystallinephases with square symmetry [L4S

ðsqÞ4 and L2S

ðsqÞ8 ] as well as

multiple hex phases: L3SðhexÞ6 , L2S

ðhexÞ12 , and other representatives of

L1+1=nSðhexÞ6ð1+ nÞ family. Note that, according to the above argument,

clusters LnS for n≤ 4 and LS6 get suppressed because of compe-tition with the corresponding crystal phases.

3D Phase Diagram. We now apply the same procedure to obtainthe generic phase diagram of binary sticky spheres in 3D. As in the

Fig. 1. Constructing the phase diagram in 2D. (A and B) Generating candidate phases. (C) Identifying the phase sequence for a specific particle size ratior = 0.5. (D) The final phase diagram as a function of composition f and size ratio r.

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2D case, we start by arranging L particles into high-symmetrylattices: three of them cubic [simple cubic (SC), face-centeredcubic (FCC), and BCC] and one hexagonal. The shapes of thecorresponding Wigner–Seitz cells are shown in Fig. 2, with dif-ferent symbols representing various locations of their intersec-tions with a sphere. According to our construction procedure,these intersection points correspond to the distinct binary struc-tures generated by a given L particle lattice. For instance, FCCcrystal generates two binary structures that have direct chemicalanalogs [L6S

ðfccÞ6 ðNaClÞ and L4S

ðfccÞ8 ðCaF2Þ] as well as several

nonclassical lattices [L3SðbccÞ24 , L3S

fcc48 , and L2S

ðfccÞ12n ].

To determine the range of the geometric parameter r for eachstructure, we need to enforce the hard core constraint. Namely,the minimal distance between any pair of same-sized particles, dSSand dLL, should be greater than 2rS, and 2rL for S and L particles,respectively. This procedure in 3D is not as straightforward as thatfor symmetric 2D lattices. Instead, we need to calculate two geo-metric parameters, α= dLL=dSL and β= dSS=dSL, for each structure.After that, the range of r is determined as

2α− 1< r<

β2− β

. [6]

Parameters, α and β, and the resulting values of minimal andmaximal size ratios, rmin and rmax, are presented in Table 1 for our 3Dcandidate phases [note that some of the structures, L4S

ðscÞ24 ðCaB6Þ

and L3SðbccÞ72 , are automatically disqualified on pure geometrical

ground, because rmin > rmax].We can now follow the general procedure outlined earlier to

construct the overall phase diagram of the binary sticky spheresin 3D. The result is presented in Fig. 3. Interestingly, among allof the phases analyzed, only one pair is characterized by thesame combination of coordination numbers ðZL,ZSÞ: Cr3Si andAuCu3 are both represented as L4S12 in our notations; however,the former belongs to BCC family, and latter belongs to SCfamily. Therefore, pure energetic analysis cannot determine thedominant structure between them, and entropy has to be takeninto account. Although such an analysis is significantly morecomplicated, its result can be implied from geometric consider-ations. Namely, the range of size ratios consistent with AuCu3 isshifted upward compared with Cr3Si. This observation impliesthat there is an entropy-driven transition between the two phasesat a certain intermediate value of r (shown as a dashed line in thephase diagram in Fig. 3). Similar to the 2D case, we expect thatthe cluster-dominated phases get suppressed for “magic” com-positions: LS12 and LnS, with n≤ 6 because of competition with thecorresponding crystal structures. Below, in Discussion, we compareour phase diagram with a number of experimental systems and showthat it does feature many of the previously reported superlattices. Inaddition, we predict several of structures that are yet to be discov-ered. One of them is L3S

fcc48 [space group Fm− 3m; Wyckoff particle

positions L: 4að0,0,0Þ and S: 32f ð1± x=3, 1∓x=6, 1∓x=6Þ, where x is

Fig. 2. Construction of candidate phases in 3D. The four polyhedra are the Wigner–Sietz cells of most symmetric one-particle structures: FCC, BCC, SC, andhex. Different symbols represent possible intersection points of these polyhedra with a sphere, which in turn, correspond to positions of S particles around L.

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a free parameter that depends on the actual lattice constant that, inturn, is determined by entropic effects]. Another structure is L3S

ðbccÞ36

[space group Im− 3m; Wyckoff particle positions L: 2að0,0,0Þ andS: 24hð1=2, 1=8, 1=8Þ]. One more predicted phase, L2S

ðfccÞ12n , corre-

sponds to FCC packing of L particles, with S particles allowed to“slide” along the circular orbits indicated as dashed lines in Fig. 2.

DiscussionOur procedure allows us to generate the families of binary struc-tures starting from single-particles lattices. This constructiongreatly simplifies the problem but, strictly speaking, does not pro-vide a complete search over all plausible candidate structures. Inthis paper, we limited our discussion to only the most symmetricsingle-particle arrangements. Nevertheless, our analysis effectivelycovers a significantly broader class of the candidate phases becausemost of the structures can be uniformly deformed without losingSL contacts. In this way, one can break some of the symmetries(e.g., cubic lattice may be transformed to tetragonal). Although thestructures obtained in this way have the same exact binding energy,the lower-symmetry lattice has two disadvantages: (i) narrowerrange of geometrically allowed size ratios and hence, (ii) lowerentropy. We, therefore, expect entropic corrections to favor high-symmetry states. In the future, the predicted phase diagramsshould be tested against a more extensive search in the morpho-logical space (e.g., by using genetic algorithms) (19).

Our key result is a remarkable morphological diversity of thebinary sticky sphere model. In particular, obtained phase dia-gram is significantly richer than the one for traditional binaryhard spheres. There are multiple experimental systems to whichthis simple generic model is potentially applicable, possibly oncertain modification.

Fig. 3. Calculated phase diagram of binary sticky spheres in 3D as a function of composition f and particle size ratio r = rs=rL. Experimental data for DNA–NP(●) (8), DNA–colloids (♦) (12), and electrostatically assembled NP–protein complexes (■) (13) systems are included. Color coding represents different crystals:NaCl (red), AlB2 (black), Cr3Si (green), and L3S

ðfccÞ24 (AB8; magenta).

Table 1. Geometric parameters of the 3D candidate phases

Height structure α β rmin rmax

L6SðfccÞ6 (NaCl)

ffiffiffi2

p ffiffiffi2

p0.414 >1

L4SðfccÞ8 (CaF2)

ffiffiffi6

p=2

ffiffiffi3

p0.225 >1

L2SðfccÞ24 (AB8) 2=

ffiffiffi3

p2

ffiffiffi3

p0.155 0.406

L3SðfccÞ48 7=6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi21+ 8

ffiffiffi6

pp0.167 0.186

L4SðbccÞ24 (K6C60)

ffiffiffiffiffiffi15

p=6

ffiffiffiffiffiffi10

p0.291 >1

L3SðbccÞ36

ffiffiffi6

p=2 6=

ffiffiffi7

p0.225 0.789

L3SðbccÞ72 5=4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12+13

ffiffiffi2

pp0.25* 0.22*

L8SðscÞ8 (CsCl)

ffiffiffi3

p ffiffiffi3

p0.732 >1

L4SðscÞ12 (AuCu3)

ffiffiffi2

p1 0.414 1

L4SðscÞ24 (CaB6) 3=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7+4

ffiffiffi2

pp0.5* 0.39*

L6SðhexÞ12 (AlB2)

ffiffiffiffiffiffiffiffi7=3

p5=2 0.528 0.666

L3.6SðhexÞ18 (CaCu5) 2=

ffiffiffi3

p2

ffiffiffi2

p0.155 0.547

*These structures are geometrically impossible at any size ratio.

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Among its most direct applications is the case of the binarycolloidal system in which the two types of micrometer-scale par-ticles are functionalized with mutually complementary ssDNAchains (10–12). The Watson–Creak hybridization results in anattraction between the particles, whereas the range of this inter-action is normally set by the scale of DNA chains involved(∼ 10  nm). Because this interaction is very short ranged comparedwith the particle sizes, the system should be well-described by thebinary sticky sphere model. The DNA-mediated assembly of col-loidal crystals has been a subject of multiple studies over the pastdecade. However, most of the experiments to date were limitedto same-sized particles that assemble into either BCC (CsCl) orFCC (CuAu) lattices (the latter requires additional same-typeattraction, which is beyond the scope of this theory). Only veryrecently, DNA-driven crystallization has been reported for binarycolloids with size asymmetry (12). In particular, two new structureshave been observed: AlB2 [L6S

ðhexÞ12 in our notations] and K6C60

[L4SðbccÞ24 ] for size ratios 0.54 and 0.36, respectively. Both results,

together with the classical symmetric CsCl crystal, are consistentwith our phase diagram, as shown in Fig. 3.The earlier studies of superlattice self-assembly in binary sys-

tems of DNA-functionalized NPs revealed even greater mor-phological diversity (8). In addition to the observation of CsCl(for multiple size ratios, 0.6< r< 1), AlB2 (for 0.37< r< 0.7), andK6C60 (for 0.34< r< 0.45), that study reported self-assembly ofCr3Si [L4S

ðbccÞ12 in our notations] for a number of size ratios be-

tween 0.4 and 0.5. Note that our model is not a priori applicableto the nanoparticle systems, because the deformable DNA shellis comparable with the overall particle size. This deformabilitymay lead to a number of effects not captured by the simple stickysphere model. Nevertheless, all four experimentally observedstructures are present at our phase diagram. Both Cs6C60 andCr3Si were reported at the size ratios completely consistent withour model. However, the predicted ranges of the allowed sizeratios for ScCl and AlB2 structures are narrower than in theexperiment (0.73< r< 1 and 0.52< r< 0.66, respectively). Thisdeviation is most likely a consequence of a relative softness of theDNA shells that does not obey our strict geometric constraints. Apossible expansion of our approach that would be applicable tosoft particles is its combination with a closely related recent modelby Travesset and coworkers (25, 26). The experimental data pointsfor both DNA–NP and DNA–colloidal systems are superimposedwith our phase diagram in Fig. 3.

Another important class of experimental systems for which thismodel is relevant is binary mixtures of oppositely charged colloidsor nanoparticles. Specifically, the sticky sphere approximation isreasonable in the regime when screening length is much smallerthan the particle sizes. In that limit, existing experiments are ingood agreement with the predictions of our model. These ex-periments include studies of electrostatically driven assembly ofcolloidal crystals (5, 17) and more recent reports of super-lattices self-assembled from oppositely charged nanoparticlesand spherical protein complexes (13). Interestingly, both studiesreported formation of LsS

ðfccÞ24 structures (also known as AB8)

that does not have a direct chemical analog but naturallyemerges from our geometric construction and does appear inthe phase diagram.It should be noted that the studies of electrostatically driven

self-assembly conducted to date were limited to a small set ofsize ratios. Nevertheless, a significant morphological diversitywas shown both experimentally and computationally by varyingthe particle charge asymmetry and the screening length. In-terestingly, many of the observed and predicted structures arethe same as in our phase diagram (5, 15–17). Our model cor-responds to a single limiting case of a strong screening but var-iable size ratio. In other words, this theory gives an alternativeprojection of the same multivariable phase diagram. It can serveas a simple baseline model to introduce additional effects, suchas finite screening or arbitrary charge ratio, in a perturbativemanner. Such generalized model promises even greater diversityand tunability of the phase behavior. In a similar way, our modelcan be extended to describe system-specific features of DNA-mediated self-assembly, especially for nanoparticle systems.Furthermore, the remarkable morphological diversity of the bi-nary sticky sphere model makes it a viable alternative to a moretraditional hard sphere model (e.g., for description of super-lattices self-assembled via controlled drying) (1–4). For that classof systems, an attraction between same-type particles that hasbeen neglected here has to be introduced (27).

ACKNOWLEDGMENTS. I thank O. Gang, M. Hybertsen, D. Talapin, and A. vanBlaaderen for discussions. The research was carried out at the Center forFunctional Nanomaterials, which is a US Department of Energy Office of ScienceFacility, at Brookhaven National Laboratory under Contract DE-SC0012704.

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