Binary nanoparticle superlattices of soft-particle systemsAlex Travesset1

Department of Physics and Astronomy, Ames Laboratory, Iowa State University Ames, IA 50011

Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved June 26, 2015 (received for review March 9, 2015)

The solid-phase diagram of binary systems consisting of particlesof diameter σA = σ and σB = γσ (γ≤1) interacting with an inversep = 12 power law is investigated as a paradigm of a soft potential.In addition to the diameter ratio γ that characterizes hard-spheremodels, the phase diagram is a function of an additional parame-ter that controls the relative interaction strength between thedifferent particle types. Phase diagrams are determined from ex-tremes of thermodynamic functions by considering 15 candidatelattices. In general, it is shown that the phase diagram of a softrepulsive potential leads to the morphological diversity observedin experiments with binary nanoparticles, thus providing a generalframework to understand their phase diagrams. Particular empha-sis is given to the two most successful crystallization strategies sofar: evaporation of solvent from nanoparticles with grafted hydro-carbon ligands and DNA programmable self-assembly.

phase separation | superlattices | crystalline phases | stoichiometry

Arrangement of nanoparticles (NPs) into structures withlong-range order encompasses a fundamental new type of

materials with potential revolutionary applications in optics, pho-tonics, catalysis, or novel fuel energy sources, just to name a few.Over the recent years there has been a spectacular success in theassembly of nanoparticle superlattice (NPS), with the two mostsuccessful strategies consisting of evaporation of a solvent fromNPs with grafted hydrocarbon chains (1–4) [solvent evaporation(SE) systems] or the programmable self-assembly of DNA graftedNPs (5–7) in water (DNA systems).Although there are different models available to investigate

DNA systems (8–13), studies of SE systems have been almost(except, for example, in ref. 14) exclusively based on the hard-sphere (HS) model, following the pioneering work of Murrayand Sanders on micrometer-sized colloidal systems (15). How-ever, HS models do not provide a satisfactory description ofexperiments, as clear from the fact that crystals isomorphic toMgZn2, CaCu5 (1), body centered cube AB6 (bcc-AB6) (4) [alsoknown as Cs6C60 (7)], or quasi-crystals (16), just to name a few,have not been reported as equilibrium phases for HS (17, 18). Ithas also been observed that different binary systems with thesame NP hydrodynamic radius (with different hydrocarbon chainlength, for example) do not exhibit the same equilibrium phase(19), clearly pointing to a phase diagram that depends on otherparameters besides the ratio of the two NPs diameters, whichcompletely determines the phase diagram of the HS system (15).The interaction between two NPs is far more complex than a

HS because the polymer shell (consisting of grafted hydrocar-bons or DNA) is flexible. In the limit where the grafted polymersare infinitely long (f-star limit) such potential is known analyti-cally (20) and does reveal a very soft tail. In SE evaporated sys-tems, however, the grafted hydrocarbon chains contain between 9and 18 hydrocarbons (3), which are too short to be described bythe f-star limit.Motivated by the partial success of HS models, namely the

imperfect but clear correlation between experimental equilib-rium structures and those with high packing fraction (PF) andthe need, for the reasons exposed (see also ref. 19), to consider a

soft potential, we examine particles of different diameters inter-acting with an inverse power law (IPL):

Vj,hðrÞ= «AA«jh

�σjhr

�p, [1]

where σjh = 1=2ðσj + σhÞ and «jh is a dimensionless number. Forp=∞, the HS model is formally recovered, but at any finite p themodel is continuous (in r) and thus provides a generic exampleof a soft interaction; Fig. 1. In this paper, only the p= 12 casewill be considered; this is a common approximation to describe asoft core, as done, for example, in Lennard-Jones systems. Thephase diagram for other sufficiently short-range interactions fol-lows similar qualitative trends, but with longer range potentials(p= 6) significant differences, which will be discussed elsewhere,are found.IPL potentials with p in the range 3–12 have been used in

models of soft colloids before, such as, for example, in char-acterizing the phase diagram of microgel particles (see reviewin ref. 21), thus showing that IPL potentials provide a genericdescription of soft particles, although, admittedly, more re-fined potentials are needed for more rigorous quantitativestudies (22).

Thermodynamic ParametersThe potential Eq. 1 is parameterized by

γ =σ2σ1

≡σBσA

≤ 1, «AB ≡ «12, «BB ≡ «22 > 0, [2]

with A being the particle with the largest diameter and B thesmallest. These parameters are the ratio of the two diametersand the strength of the A–B and B–B interactions in units of theA–A interaction («AA ≡ 1). An important advantage of power-lawpotentials is that all excess quantities do not depend on (T,V) or(T,P) but rather on a single parameter, either ξp or its conjugate

Significance

The phase diagram of a system of two species of particles withdifferent diameters interacting with a soft (1=r12) potential isprovided. The results provide a general framework to predictthe crystalline phases observed in nanoparticle superlattices.The approach is particularly relevant for nanoparticles with hy-drocarbon capping ligands where theoretical descriptions havebeen based on hard-sphere systems and is general enough toconsider supercrystals obtained by DNA programmable self-assembly, thus providing a conceptual unification for generalnanoparticle superlattice systems.

Author contributions: A.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: trvsst@ameslab.gov.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1504677112/-/DCSupplemental.

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λp (23–25) as well as on the stoichiometry xB (fraction of Bparticles to the total number of particles),

ξp = T−3=pρ, λp ≡ PT−ð1+3=pÞ, xB =NB=N. [3]

The dimensionless variables are T = kBT=«AA, ρ=Nσ3A=V . Theparameter NB is the number of particles of type B and N =NA +NB the total number of particles.It should be noted, however, that, without loss of generality it

can be set that «BB ≡ 1 [by defining γ′= γð«BBÞ1=p and «AB′ =«ABðð1+ γÞ=1+ γ′Þp]. So, in what follows, we will assume «BB ≡ 1.In summary, the phase diagrams will be presented in terms of thefollowing four parameters:

λp ≡Reduced  pressure

xB ≡ Stoichiometry  of   B  vs ðA+BÞγ ≡Ratio  of  NPs  radii

«AB ≡Relative  strength  of  A  vs  B. [4]

All binary lattices considered (Table 1) were chosen followingtwo criteria: high PF (see Discussion below and plot of the re-spective PF in SI Appendix, Fig. S1) and being experimentallyrelevant. All lattices are parameterized in terms of an equivalentbinary HS model d1 ≡ dAA and d2 ≡ γLdAA. We note that the di-ameters ðd1, d2Þ parameterizing the lattice do not need to be thesame as ðσA, σBÞ in Eq. 2, but test calculations showed that allowing(γL, d1) to differ from (γ, σAÞ did not lead to any significant newprediction. All calculations presented are for d1 = σA and γL = γ.The fact that the underlying HS radii parameterizing the lat-

tice match so well with the two radii defined by the potential Eq.2, as well as the strong correlation between stability of the phaseand HS PF, shows that the parameter γ has a clear physical in-terpretation as the ratio of the two soft-particle radii.

ResultsThe general phase diagram is a function of four coordinates,defined in Eq. 4: two that parameterize the interaction (Eq. 2)and two purely thermodynamic variables (Eq. 3). The excess freeenergy per particle in units of kBT for a given crystal phase in termsof the variable ξp (described in Eq. 3) can be written as (25)

f x�ξp, γ

�= aðp, γÞ�ξp�p=3 +

�p2+ 1

�log

�ξp�+ bðp, γÞ

+1�

ξp�p=3 f anhðξ−1p Þ.

[5]

The first term on the right-hand side is the free energy at zerotemperature (lattice sums), the second terms are the harmonic

dynamical lattice theory (DLT) corrections, and the last termencompasses the anharmonic contributions. How to obtain theexplicit values of aðp, γÞ and bð p, γÞ (dependence on eAB is omitted)from Eq. 11 has been extensively discussed in ref. 25. Concretevalues for these coefficients are given in SI Appendix, Tables S2and S3). The symbol x refers to the reference state (as describedin SI Appendix, section SII, Eq. S14). The anharmonic functionf anhðxÞ is finite at x= 0 and can be computed from a thermody-namic integration (25, 26). This calculation is beyond the scopeof this paper and f anhðxÞ≡ 0 is assumed. As discussed in Materialsand Methods as well as in ref. 25, the error introduced by thisapproximation is less than 0.1% right at the melting temperature,becoming virtually negligible deep in the solid phase, which is themain region of interest in this paper.The equation of state relates λp and ξp

λp�ξp�=p3aðp, γÞ�ξp�p=3+1 + p+ 2

2ξp. [6]

The chemical potential is given from Eqs. 5 and 6 as

μxL�λp�≡�1+

p3

�aðp, γÞ�ξp�λp��p=3

+ bðp, γÞ+�p2+ 1

�log

�ξp�λp��

+p+ 22

.

[7]

In the limit of low temperatures and/or high density (ξp →∞),the chemical potential has the analytical form

μxL�λp�=�1+

p3

��3aðp, γÞp

p

�3=ðp+3Þλp=ð3+pÞp . [8]

Phase diagrams for large ξp, as determined from this equation,provide exact limiting results that guide the phase diagrams ofthe more general cases.The calculation of a phase diagram at a given value for the

potential parameters Eq. 2 proceeds by first establishing thestable phases, determined by the condition that all eigenvalues ofthe dynamical matrix (DM) except for the three acoustic modes,are positive. Then, the aðp, γÞ and bðp, γÞ coefficients are calcu-lated for all these stable phases (examples are provided in SIAppendix, Tables S2 and S3), and with it, the chemical potentialEq. 7. The stability range for each phase (for the phase diagramsreported in this study) is provided in SI Appendix, Table S1 for allcases considered; a given lattice is not always stable, but if it is, it

Fig. 1. Plot of the soft potential, Eq. 1. σAB = ðσA + σBÞ=2. A denotes thelarge particle and B the smallest. The HS case is provided for comparison.

Table 1. The 15 lattices considered

N ST SG S P A

fcc A or B Fm-3m A1 cF4hcp A or B P63/mmc A3 hP2NaCl AB Fm-3m B1 cF8AuCu AB P4/mmm L10 cP4CsCl AB Pm-3m B2 cP2ZnS AB F-43m B3 cF8AlB2 AB2 P6/mmm C32 hP3CaF2 AB2 Fm-3m C1 cF12MgCu2 AB2 Fd-3m C15 cF24 LavesMgZn2 AB2 P63/mmc C14 hP12 LavesAuCu3 AB3 Pm-3m L12 cP4CaCu5 AB5 P6/mmm D2d hP6CaB6 AB6 Pm-3m D21 cP7bcc-AB6 AB6 Im-3m cI14 Cs6C60

NaZn13 AB13 Fm3c D23 cF112

A, alternative name; N, name; P, Pearson symbol; S, Strukturberichtsymbol; SG, space group; ST, stoichiometry.

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is always around a finite interval around its highest PF (η) andbecomes unstable around η∼ 0.5.The next step consists of establishing whether the stable pure

AkBq phase, which exists only at stoichiometric value xB = q=ðk+ qÞ,does not phase separate into two coexisting ArBs and AtBu phases.The chemical potential for these coexisting phases follows frombasic thermodynamics (for completeness, a derivation is provided inSI Appendix, section S1):

μ�λp, xB

��r, s, t, u�= ðr+ sÞ−u+ ðt+ uÞxBst− ur

μr,s�λp�

+ ðt+ uÞ s− ðs+ rÞxBst− ur

μt,u�λp�.

[9]

In terms of the chemical potential μk,qðλpÞ of the pure AkBqphase, such phase is unstable toward phase separation ifμk,qðλpÞ> μðλp, xB = q=ðk+ qÞjr, s, t, uÞ.Two illustrative examples are provided in Fig. 2. The first

consists of eAB = 0.2 at γ = 0.5275: out of the 15 possible phases,only 5 are stable: fcc/hcp, NaCl, AlB2, and NaZn13. The secondconsists of eAB = 0.8 at γ = 0.558, and here six stable phases: fcc/hcp,CsCl, AlB2, CaCu5, and NaZn13. Both pure A and B phases arefcc. The resulting phase diagram is shown in Fig. 3: In the firstcase all stable phases are equilibrium at least within a limitedpressure range. In the second case the CsCl is not an equilibriumphase and CaCu5 is only stable over a very limited pressure range.In both cases, the phase diagram is different from the one ob-tained by direct application of the large ξp formula, Eq. 8.

DiscussionIn this way, by performing the calculations outlined in the pre-vious section, the phase diagram for all γ-values at 12 differentvalues of eAB are compiled. Four representative phase diagrams(the rest are found in SI Appendix, Figs. S2–S13) are shown inFig. 4. The interval where each phase is not just stable, butan equilibrium phase is displayed, with its corresponding PF

highlighted. At a given γ, many phases can exist, either becausethey occur with different stoichiometry (such as NaCl and AlB2in Fig. 3) or because there is a phase transition at a given λp(such as the NaZn13 phase). If this latter case occurs, some lossof information results from the plots, because a full phase dia-gram like the one in Fig. 3 cannot be fully reconstructed fromthose plots as the critical pressure λp (for the corresponding γ) isnot provided.Rather interestingly, phase diagrams for eAB > 1.1 are trivial

(consist of A + B). As an example in Fig. 4, the case eAB = 1 hasonly MgCu2 as the only nontrivial equilibrium phase. Nontrivialphases are found for eAB < 1 (Fig. 4). In almost all cases, theequilibrium phases exist around the maximum of the PF; theopposite is not true: there are many examples where a high PF isnot enough to ensure that a phase will be in equilibrium. TheZnS phase, for example, is very stable and has high PF atγ ∼ 0.22, but was never found to be an equilibrium state.Fig. 5 summarizes all phase diagrams. In this plot, the regions

in eAB where each phase is an equilibrium state at the corre-sponding γ-values are shown.

Implications for NPSIn addition to the diameter ratio γ that fully characterizes thephase diagram of HS models, IPL potentials have one additionalparameter eAB, which allows one to relate them to the experi-mental systems mentioned.

DNA Systems. NPS in DNA systems are driven by hybridizationsbetween A and B particles. Such hybridizations can be describedby the interaction energy between A–B particles «AB being muchsmaller than those for A–A («AA = 1) and B–B («BB = 1). In thisway, association of A and B particles is energetically favored overAA or BB, thus mimicking the role of hybridizations. Therefore,DNA systems should be described by the region «AB � 1 (seeFig. 4 for an example).As is clear from Fig. 5, those systems display the CsCl, AlB2,

bcc-AB6 (known as Cs6C60 in this context) found in experiments

Fig. 2. Difference of chemical potential between a pure crystal phase and amixture of A+ B, as calculated from Eq. 9 for (γ = 0.5275, eAB = 0.2) and (γ = 0.558, eAB = 0.8).

Fig. 3. Phase diagram as calculated from Eq. 9 for (γ = 0.5275, eAB = 0.2) and (γ =0.558, eAB = 0.8). The shadowed area is in the liquid phase, as estimated fromthe Lindemann criterion.

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(7, 27). There are four additional equilibrium phases in this region:NaCl, AuCu, AuCu3 and NaZn13. Explicit calculations (7) (see alsoref. 28) have ruled them out as they do not optimally hybridize theDNA shells, but those calculations do not consider the role ofentropy and the possibility of coexistence. As the results presentedhere reveal, they could become equilibrium phases, at either smallγ (NaCl), γ ∼ 1 (AuCu3), or when there is an abundance of Bparticles, that is, xB ∼ 1 (NaZn13). Another possibility to targetthese phases is to modify the DNA shell to include additional ABrepulsion forces that compete against hybridizations (28–30), whichis achieved by inclusion of neutral (those that do not form hy-bridizations) DNA strands into the NP shell (28, 30). In thosesystems, AuCu phases have been found (28, 30).

SE Systems. Experimentally, the NP radius is defined as half thesmallest NP–NP in a 2D hexagonal lattice of same NPs (2). ThePF broadly used to interpret experimental results corresponds tothe value γ obtained by the ratio of the smallest to the largestradii obtained this way. Here, I introduce the softness asymmetry(SA) parameter as

sAB =effective  thickness  of   the  ligand  shell  for  the  small NPeffective  thickness  of   the  ligand  shell  for  the  large NP

.

[10]

The effective thickness is obtained by subtracting the NP coreradii from the total radii. Data compiled from refs. 2, 4, 19comprising 36 experiments are plotted in Fig. 6. The NP radius(defined from the 2D hexagonal separation) was taken from ref.2, table 1 or by the Optimal Packing Model formula (19, 31),depending on whether the hydrocarbon capping ligands are un-saturated or saturated.The plots in Fig. 6 reveal a clear correlation both vertically

(PF) and horizontally (SA). In fact, such correlation is so re-markable that rescaling the SA by a factor c= 0.6 allows a semi-quantitative agreement between theory and experiments. Thereare some differences, namely: the Laves MgCu2 is predicted tobe the equilibrium phase over the competing MgZn2, which isthe one experimentally reported. The calculations show a pre-ponderance of CsCl over the AuCu competing phase, and finally,the CaB6 are not found to be equilibrium phases (and are stableover a very limited PF).The difference in free energy between MgZn2 and MgCu2, as

clear from the a, b coefficients provided in SI Appendix, TablesS2 and S3), is extremely small (but always favorable to the lat-ter). The reasons for the preponderance of CsCl against AuCuseem less clear, although for the values of γ reported, bothphases are structurally very close (they become identical forγ < 0.73). Furthermore, in the very narrow range where CaB6 isstable, the free-energy difference with the bcc-AB6 is very small.A closer analysis of Fig. 6 shows that the results of ref. 19

explore a wide range of the parameter sAB. In this case, theagreement with the theoretical result is remarkable. The resultsof ref. 2 explore a more limited range of SA (sAB ∼ 1) but there isclear correlation. The only point not correctly described corre-sponds to [5.8 nm PbSe (oleic acid), 3.0 nm Pd (dodecanethiol)],which displays the NaCl, Fe4C, and NaZn13 phases. The effectivethickness used for the PbSe ligand was taken as 0.85 nm (ref. 2,table 1). The results of ref. 4, 14 nm Fe3O4 (oleic acid), 4.6 nm Au(dodecanethiol) with the effective thickness of oleic acid esti-mated as 0.9 nm (by analogy with γ-Fe2O3, also from ref. 2, table 1),are also outside the range of the theoretical result. In both cases,

Fig. 4. Four representative phase diagrams at fixed «AB. The arrows indicatethe range of γ where the phase is found to be an equilibrium state, with thecorresponding PF highlighted.

Fig. 5. Summary of all equilibrium phases as a function of γ within ð0.1, 0.9Þand eAB ≥ 0.1.

Fig. 6. Data compiled from the three references (including 36 differentexperiments and phases not analyzed in this study) with γ and sAB calculatedas described in the text, superimposed to the theoretical result (Fig. 5). Thecoefficient is c= 0.6. In those cases where many phases exist at a point, thesymbols are split horizontally for visualization.

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however, should a longer effective thickness for oleic acid beused, the results would be within consistency. In any case, giventhe simplicity of the model, the overall agreement with thetheoretical phase diagram is remarkable.

ConclusionsThe phase diagram for IPL potentials has been described andprovides a paradigm for any general short-ranged soft potential;introduction of soft repulsive potential allows a characterizationof a significant morphological diversity in self-assembled super-lattices. In the region where phase separation into A + B can beovercome, the phase diagram is much richer than the one reportedfor HS (15, 17, 18). It remains an open question as to whethermore general HS models (where, for example, the additivity of theradius is dropped) can account for these additional phases. Moreimportantly, the study provides a semiquantitative description ofthe equilibrium crystals observed in NPS experimental systemsby considering two parameters only: PF and the SA; Fig. 6. Howrobust predictions based on SA are is a subject for further exploration.This paper will be followed by further studies where more so-

phisticated potentials or free energies are considered. There havebeen many important developments relating the relation of po-tentials to given lattice structures (32–34) or characterization ofoptimal PFs (35–37) that will definitely synergize with this work.It is my contention that the presented calculations capture the

conceptual relevant aspects of the phase diagram with soft inter-actions, thus providing at least a qualitative description of mesoscaleNPS. Furthermore, the methods presented are completely gen-eral and can accommodate any continuous potential. In addition,this paper highlights the clearly unappreciated value of DLT:Calculation of these phase diagrams by standard thermodynamic

integrations would have required enormous computationalresources.

Materials and MethodsDLT (38) provides an exact low-temperature expansion to investigate thethermodynamic properties of any phase with long-range order (23, 24).Quantitative comparisons of excess free energies in soft potentials havedemonstrated agreement within less than 0.1%, extending in some cases tothe liquid–solid transition (25, 26). Within DLT, the potential energy is ex-panded to quadratic order

U=NU0 +12

Xn

Xn′

uα,sðnÞDα, sjβ,tðn−n′Þuβ,tðn′Þ, [11]

where U0 is the potential energy at zero temperature (lattice sums), D is theDM, and uα,sðnÞ are the displacement of the particle at the s-lattice basis atthe unit cell n along the α= 1,2,3 direction. All thermodynamic quantitiescan be calculated from the DM and U0.

Both the lattice sums and the DM were calculated using the HOODLTsoftware, which is described elsewhere (25). The calculations were run in aCPU cluster using Message Passing Interface (MPI) as implemented in thePython package PyPar. A calculation of a phase diagram at all γ for givenvalues of ð«ABÞ using 16 cores takes of the order of 6 h.

ACKNOWLEDGMENTS. A.T. acknowledges interesting discussions withM. Boles, C. Calero, N. Horst, C. Knorowski, O. Gang, G. Miller, D. Talapin,and D. Vaknin. I thank T. Kennedy for many important discussions andencouragement during the course of this work. Warm thanks for the use ofcomputer cluster Cystorm at Iowa State, which has been purchased frommultiple National Science Foundation (NSF) grants. This work is supported bythe US Department of Energy (DOE), Basic Energy Sciences, Materials Scienceand Engineering Division. The research was performed at the Ames Labora-tory, which is operated for the US DOE by Iowa State University underContract DE-AC02-07CH11358.

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