DOI:
10.1039/C6SM00713A
(Paper)
Soft Matter, 2017,
13, 147157
Topological structure prediction in binary nanoparticle superlattices
Received
23rd March 2016
, Accepted 25th April 2016
First published on 27th April 2016
Systems of spherical nanoparticles with capping ligands have been shown to selfassemble into beautiful superlattices of fascinating structure and complexity. In this paper, I show that the spherical geometry of the nanoparticle imposes constraints on the nature of the topological defects associated with the capping ligand and that such topological defects control the structure and stability of the superlattices that can be assembled. All these considerations form the basis for the orbifold topological model (OTM) described in this paper. The model quantitatively predicts the structure of superlattices where capping ligands are hydrocarbon chains in excellent agreement with experimental results, explains the appearance of low packing fraction lattices as equilibrium, why certain similar structures are more stable (bccAB_{6}vs. CaB_{6}, AuCu vs. CsCl, etc.) and many other experimental observations.
1 Introduction
Strategies for programming the selfassembly of nanoparticles consist in grafting ligands that enable the control of their interactions. DNA has proven to be a very versatile ligand, and many superlattices have been successfully assembled.^{1–4} Using hydrocarbons as ligands has also been a successful route for nanoparticle crystallization, both in single component^{5} and binary^{6–9} systems. While much theoretical work has been devoted to DNA programmable crystallization, see ref. 10 and 11 for recent reviews, this paper will attempt to fill the existing gap in the theoretical study of the selfassembly of nanoparticles with hydrocarbon ligands, herein called nanocrystals (NCs) into binary superlattices (BNSL).
Early studies of Murray and Sanders^{12} recognized the strong correlation between crystal equilibrium phases of binary micronsize particles and packing fraction in hard sphere (HS) models. Such approach has also been applied to BNSLs, where a clear but imperfect correlation between the packing fraction and equilibrium phases has been observed.^{6–9,13} Yet, the phase diagram of BNSLs is far more complex than the one reported for HS,^{14,15} as exemplified by the recent study of Boles and Talapin^{13} where different binary systems with similar effective NC radius (with different hydrocarbon chain length, for example) have been shown to exhibit different equilibrium phases at the same temperature and stoichiometry.
In a recent study, my group has investigated binary systems interacting with inverse power law potentials.^{16–18} The rationale for such study is that a sufficiently shortranged potential should have a phase diagram whose equilibrium phases are strongly correlated to high packing fractions, and account for the flexibility of NC interactions and phonon modes. Our studies have shown that an inverse power law model provides a surprisingly successful description of the experimental results, although a few shortcomings have been noted, such as, for example, the inability to account for lattices with low packing fraction as equilibrium phases or the dominance of the CsCl over the experimentally favored AuCu lattice.
While “universal” approaches, like HS or inverse power law potentials are of great value, there is an obvious need for more microscopic models. After all, such universal models cannot predict how the ligands are organized within the lattice or what are the expected NC separations within the unit cell. Models for the interaction of two NCs with grafted polymers with tilt have been discussed before,^{19} but actual capping ligands in NCs contain few (less than 5) Kuhn lengths, so a description in terms of a polymer is not suitable. Still, simple but predictive microscopic models have been developed for single component systems: in ref. 20, the optimal packing model (OPM) was introduced (see Fig. 1), which provides a detailed formula for the closest separation between NCs, and correctly predicts a fcc to bcc transition for increasing ligand length. An alternative model, the overlap cone model (OCM)^{21–23} brings NCs much closer than predicted by OPM. Further below (see Sections 3.1 and 3.2) I provide a detailed description of these models. The beautiful results of Boles and Talapin^{13} have shown that two dimensional systems are well described by OPM when particles have a coordination number of six or greater while for lower coordination number results seem more consistent with the OCM. The motivation for this paper is to develop a model that can account for all these experimental results, and in this way, provide a rigorous framework for the prediction of crystal structure in general NCs systems.

 Fig. 1 Figure illustrating the assumptions of the OPM model: within the shaded cone, the ligand chain is maximally packed. If NCs come closer than predicted by OPM (_{A} < r_{A}), ligand chains must be pushed aside. This forms the basis for the OTM model discussed in this paper.  
The model described in this paper is general and is applicable to spherical nanoparticles with any ligands. For the purpose of validating the model, however, special emphasis will be made to capping ligand that consist of saturated hydrocarbon chains, as those are easier to analyze theoretically and there is very detailed experimental data that can be used to validate the model. Application to other type of ligands, including the important case of hydrocarbons with some degree of unsaturation, will be left for a subsequent publication.
2 The orbifold topological model (OTM)
2.1 NC as compressible spheres
The parameters defining a single NC are the core radius R and the two properties defining the ligand: its footprint A_{0} and maximal length L. If the ligand consists of n_{H} monomers (a hydrocarbon, for example), their positions are labeled as 
{^{H}_{i}}_{i=1…nH}.  (1) 
The volume within the NC available for a single ligand, see Fig. 1 
 (2) 
where it has been assumed a dense packing of ligands where the grafting density coincides with the footprint. Consideration of more sparse grafting densities may also be of interest,^{24} but this possibility will not be explored in this paper. The dimensionless parameters, defined in ref. 20, are

 (3) 
The actual volume occupied by the ligand is V_{h} = A_{0}L. The OPM model^{20} assumes that for two nearest neighbor NCs, their contact ligands are occupying all the available volume, that is V_{n} = V_{h} orThe generalization of this equation to two nearest neighbor NCs (labeled k and l) is: 
s_{kl} = r_{k} + r_{l} = τ_{k}R_{k} + τ_{l}R_{l},  (5) 
where R_{i} is the radius of NC i = A, B and τ_{i} is the solution to eqn (4). With these definitions of particle radii, the ratio of two NC radii γ is defined aswhere r_{(A/B)} is the radius of a large (A) or small (B) NC.
The monomer packing fraction (η_{m}) is defined as the total packing fraction, including core and ligand monomers. For a BNSL whose unit cell contains n_{A} NC of type A and n_{B} of type B, it is given as

 (7) 
where
V_{uc}(
γ,
r_{A}) =
V_{uc}(
γ)
r_{A}^{3} is the volume of the unit cell. If the nearest neighbor distances are given by the OPM formula
eqn (4), then the monomer packing fraction
η_{m} is the same as the HS packing fraction
η_{HS}^{13} 
 (8) 
The free energy of the system results from a competition between the attractive van der Waals (vdW) forces and the repulsive entropic ligands. The vdW forces are approximated as two spheres of radius r_{k} and r_{l} in contact, whose free energy is given by^{25}

 (9) 
where
D = 0.2 (nm) (spheres in contact) and
A_{h} is the Hamaker constant. With these assumptions, the OPM free energy reads

 (10) 
where
(
i) runs over all NCs that are in physical contact with NC
i and
χ is a dimensionless coupling constant. As for the entropic hydrocarbon chain contribution, I assume that the approximation of modeling the NC as a deformable sphere already captures part of this contribution. More sophisticated free energies are available,
^{26} but they will not be necessary for the goals of this paper, as the limitations of the OPM model in describing BNSLs are of a different nature, as I extensively discuss below. The function
f(i,j) follows from
eqn (9) 
 (11) 
The definition of physical contact may include nearest neighbors, whose distance is
s_{ij} or further away neighbors at a distance
νs_{ij}, with
ν > 1. Two NCs are defined to be in contact if their mutual distance is less than their fully extended hydrocarbon chains. This condition can be stated as

ν_{ij}s_{ij} ≤ R_{i}(λ_{i} + 1) + R_{j}(λ_{j} + 1).  (12) 
The free energy
eqn (10) and (11) together with the constraint
eqn (12) define the free energy of the OPM model. By construction, the lattices are incompressible, and I will assume that the external pressure is zero. Thus, the chemical potential is numerically the same as the free energy.
Given the chemical potentials for all the pure phases, first the lowest chemical potential for each lattice with a given stoichiometry is determined. Then, those lattices are checked for possible phase separation against two phases X + Y, following the standard rules of thermodynamics (see for example, the formulas in ref. 17 and 18). Those lattices that survive all phase separation conditions define equilibrium.
2.2 NCs and topological defects
The closest approach within NCs would occur if the monomer packing fraction would become one η_{m} = 1. From eqn (7), this would correspond to a NC radius A r_{A}′ given by 
 (13) 
and a minimal separation 
s_{kl}′ = η_{HS}^{1/3}(γ′)s_{kl},  (14) 
with s_{kl} given by the OPM formula eqn (5), which corresponds to the OCM model.^{23} Such model would definitely optimize the vdW attraction, but would incur, as I show below, in a prohibitively large free energy cost.
This large free energy cost is a result of inherent topological constraints. First I define a vector joining the first monomer (the one attached to the core) to the last (see notations in eqn (1))

 (15) 
As clear from
Fig. 1, within the OPM model, the vector
is perpendicular to the core surface, thus defining a hedgehog structure. When interdigitation occurs the vector field
cannot be perpendicular to the surface, at least over a finite domain, as illustrated in
Fig. 2. I find it useful to consider the projection of
to the tangent plane of the sphere
thus defining a two dimensional field. When two NCs interdigitate, the
field in at least one NC will contain a topological defect located at their contact point, see
Fig. 2. If there is more than one topological defect, then there maybe neutral lines, see
Fig. 3 (left), where the ligands are perpendicular to the sphere, and the vector field
is identical to zero.

 Fig. 2 Illustration on how the projection of the vector field , given by as defined in eqn (16) becomes nonzero and defines a topological defect. NC A (left) has a topological index q_{T} = 1 while NC B (right) has q_{T} = 0.  

 Fig. 3 (left) Two defect separated by a neutral line. (right) Different possible defect configurations on the sphere: (A) vortices surrounded by neutral lines, where topology does not constrain the number of defects, and two types of configurations with four ½disclinations: tetrahedral (B) and along the equator (major circle) (C).  
I now consider the case where there are n_{S} defects and no antidefects, as those very quickly annihilate on the surface of a sphere.^{27} If the defects have charge Q = 2πp, topological constraints^{28} dictate that

 (17) 
If defects are vortices
p = 1 then
n_{S} = 2 and there must be two of them. The order parameter
is 3dimensional, however, so it is possible to allocate an arbitrary number of vortices
n_{S} > 2, circumventing
eqn (17) by separating vortices with neutral lines, as illustrated in
Fig. 3A.
It is also possible that each vortex will split into two ½ disclinations connected by a neutral line, see Fig. 3B and C. Such defect configuration has neutral lines that are much shorter than a configuration with four vortices, see Fig. 3A. It is also apparent that ½disclinations are joined by straight neutral lines, compared to the kinks and vertices for a configuration with vortices. I will therefore assume that the cost of neutral lines is such that only ½ disclinations joined by neutral lines are possible (which includes a vortex, as the limit of two disclinations located at the same point). In this case the vector can be identified with its − (as the direction can be changed along the neutral line), thus becoming an effective two dimensional nematic on a sphere, whose free energy is

 (18) 
Here,
K is the Frank constant, and
g^{ab} and ∇
_{a} define the differential geometry of the surface (in this case the sphere), namely the metric and the covariant derivative. In this way, nematic textures describe the tilt of the ligand chains from its normal. These nematic textures have been extensively investigated in the liquid crystal community and can be read from existing literature.
^{27–29}
2.3 The equations defining the OTM model: topological index
I define the topological index q_{T} of a NC as the total number of defects of its capping ligands. Considerations discussed above have served to motivate the hypothesis that the topological index of a single NC must be such that q_{T} ≤ 4. The OTM defines the NC radius _{A}, _{B} of a given NCs within a unit cell as the smallest distance that maximizes η_{m} with the q_{T} ≤ 4 constraint. I note that if the topological index is q_{T} = 0, the chains must be all perpendicular to the sphere and the closest separation between two NCs is given by the OPM result. For two isolated NCs, the topological index is not restricted and the OTM and OCM results coincide. The OTM monomer packing fraction of a given BNSL is given as 
 (19) 
and satisfies η_{HS}() ≤ η_{m} ≤ 1. The interparticle distances are also in between the OCM and OPM results, namely 
s_{kl}′ ≤ _{kl} ≤ s_{kl}.  (20) 
Thus, the OTM model predictions are always in between the OCM and OPM values and coincide only in some limits. Note that the parameter is different than γ as defined by eqn (6), and is determined by the OTM radius _{A} and _{B}. In the results section I show how these considerations completely determine all NCs distances, coordination numbers and the monomeric packing fraction(density). I emphasize that although for q_{T} = 0 the NC interacting potential is two body (as it is defined by eqn (9)), for q_{T} ≠ 0, the model includes many body effects.

 Fig. 4 Color code for the lattices used.  
3 Results
3.1 Single component phase diagram
I consider the simple lattices shown in Table 1. The chemical potential is given by eqn (10) and is computed for each value of λ. The results are shown in Fig. 5. The discrete jumps correspond to λ where the fully stretched capping ligand reaches across additional nearest neighbors as defined by eqn (5), thus increasing the coordination number q of the corresponding NC. A fcc/hcp transition to bcc is found at around λ = 0.66, as previously discussed in ref. 20. Neither fcc (q = 12) nor bcc (q = 8 + 6) can support solutions with nontrivial topological index.
Table 1 Simple lattices considered (ST: lattice type, SG: space group, S: strukturberich symbol, P: pearson symbol, A: alternative name)
N 
ST 
SG 
S 
P 
A 
fcc 
Simple 
Fmm 
A1 
cF4 

hcp 
Simple 
P6_{3}mmc 
A3 
hP2 
bcc 
Simple 
Im3m 
A2 
cI2 
sc 
Simple 
Pmm 
A_{h} 
cP1 
cubdiamond 
Simple 
Fdm 
A4 
cF8 
hexdiamond 
Simple 
P6_{3}mmc 
A3′ 
hP4 
A15 
Simple 
Pmm 
A15 
cP8 

 Fig. 5 Chemical potential as a function of λ. The only equilibrium phases are fcc/hcp and bcc.  
Diamond q = 4 + 4 can support solutions with q_{T} ≤ 4, and at sufficiently large λ such OTM solutions can be competitive with bcc. However, this possibility occurs for large λ > 1.5 (very hairy), where other considerations may apply, as single component NCs are observed to develop noncubic unit cells.^{20} Investigation of large λ > 1.5 will be left for a future work.
3.2 Phase diagram of binary nanocrystals with topological index q_{T} = 0
I will consider NCs with R_{A} = 35 R_{B} = 20.5 (35, 20.5) and R_{A} = 50.1, R_{B} = 20.5 (50.1, 20.5), units in Angstrom. The phase diagram will be computed as a function of λ_{1} = λ_{A} and λ_{2} = λ_{B}. All the lattices considered are listed in Tables 1 and 2. The calculations used the software HOODLT as it has been described elsewhere.^{16}
For the (35, 20.5) case, the phase diagram, shown in Fig. 6, contains significant regions that include the following phases: AlB_{2}, CaB_{6}, NaCl, MgCu_{2}, AuCu, NaZn_{13}, CsCl, bccAB_{6} and CaCu_{5}. Three more equilibrium phases: cubAB_{13}, Cr_{3}Si and Li_{3}Bi are also found, but only in discrete points. The region λ_{1} ≈ λ_{2} corresponds to γ ∼ 0.55, which is where the HS packing fraction for AlB_{2} and NaZn_{13} becomes maximum, and indeed, those are the phases found. Similarly, λ_{1} ∼ 0.3, λ_{2} ∼ 1 corresponds to γ ∼ 0.75, where the equilibrium phases CsCl, AuCu and MgZn_{2} have large packing fraction. The bottom left corner of the figure corresponds to γ ∼ 0.35, where the packing fraction for bccAB_{6} and CaB_{6} is large, and those two phases compete as equilibrium phases, as it has also been pointed out experimentally.^{30} There is a central region (λ_{1} > 0.7, λ_{2} > 0.7) where no binary lattices exist and equilibrium corresponds to two pure A and B bcc lattices. This is a result that bcc has 14 neighbors, and when it becomes the equilibrium phase for both pure A and B, no other binary lattices within this range of λ exist with so many nearest neighbors.

 Fig. 6 Phase diagram for nanoparticles with radius R_{A} and R_{B} as a function of (λ_{1}, λ_{2}). The dots correspond to the experimental cases of ref. 13. Color codes are as shown in Fig. 4.  
For the (50.1, 20.5) case, seven phases exist over significant regions: AlB_{2}, CaB_{6}, NaCl, Cr_{3}Si, NaZn_{13}, bccAB_{6} and CaCu_{5}, and three more: AuCu, MgZn_{2} and cubAB_{13} are found in somewhat restricted points. The same correlation with HS packing fraction holds. Perhaps the most remarkable aspect is the appearance of the Cr_{3}Si phase at around γ ∼ 0.3, competing with bccAB_{6}. This phase is ubiquitous in DNA nanoparticle crystallization^{3,31} but, to my knowledge, has not been observed when the ligands are hydrocarbon chains. Compared with (35, 20.1), (50.1, 35) NCs exhibit the phases AuCu_{3} and Li_{3}Bi.
The stability of the different phases is investigated in Fig. 7 and 8 and will be compared against experimental results.^{13} For each of the phases considered, the quantity

 (21) 
is shown. Here,
μ_{binaryphase} is the chemical potential of the binary phase, and
μ_{equilibrium} is the value of the equilibrium chemical potential for the corresponding stoichiometry. Thus, the above quantity is 1 if the phase is equilibrium, and if it is not, it measures how far it is from actually becoming equilibrium. In the top figure (
λ = 0.578), AlB
_{2} and NaZn
_{13} are correctly predicted, but AuCu, which is reported experimentally, does not even exist at this packing fraction
γ as for
, the AuCu phase is, in reality, a CsCl.
^{17} The prediction for
γ = 0.77 shows only one equilibrium binary phase MgZn
_{2}/MgCu
_{2} (within the free energy
eqn (10) the chemical potential is the same for both), which is among the ones reported experimentally. Two more phases: Cr
_{3}Si and CaCu
_{5} are close to being equilibrium, but only the CaCu
_{5} phase is experimentally reported. If one where to assume, without any evidence, that the Cr
_{3}Si might be a difficult phase to nucleate, the results seem in excellent agreement with the experiments at
γ = 0.77. It should be noted, however, that experiments for the MgZn
_{2} phase report a NC radius that is 15% lower than the one predicted for
q_{T} = 0
eqn (5), clearly pointing to a more nuanced scenario than the
q_{T} = 0 predicts.

 Fig. 7 Relative value of chemical potentials as defined in eqn (21).  

 Fig. 8 Relative value of chemical potentials as defined in eqn (21).  
The relative chemical potentials follow similar trends for γ = 0.435 and γ = 0.558, see Fig. 8. Even though at γ = 0.435 the experiment reports bccAB_{6} (and two other quasicrystal structures that are not investigated in this paper), the theory predicts CaB_{6} with bccAB_{6} close. As previously noticed, those two phases compete for equilibrium, but the q_{T} = 0 clearly favors CaB_{6}. For γ = 0.558, the experimental phases reported are NaZn_{13}, which is very close to equilibrium, and Li_{3}Bi, which is clearly very far from equilibrium, pointing to a clear failure of the q_{T} = 0 assumption.
Overall, predictions for q_{T} = 0 turn out to be in fair agreement with experiments, but a number of discrepancies are obvious. Before discussing how allowing NCs to have a nontrivial topological index may modify the overall scenario, it is worth exploring whether reasonable variations of the model eqn (10) may solve any of these pending issues. For this purpose I consider the following six potentials, with f_{AA} = 1 (see eqn (11)):

 (22) 
where the relative strength of AA, AB and BB interactions are altered. As shown in
Fig. 9 the original predictions turn out to be very robust against these variations and do not fix any of the discrepancies noted. If anything, potential
f_{5}, which enhances the AB attractions over AA and BB does seem to fare slightly better than the other cases. At the risk of over interpreting the results, this might come from the fact that whenever two NC of different sizes come in contact, the smaller (or larger) NC may have more than one ligand in close contact, as depicted in
Fig. 1.

 Fig. 9 Relative value of chemical potentials for variations of the potential eqn (11), as defined in eqn (22).  
3.3 Binary nanoparticle supercrystals with nontrivial topological index
A given ANC can be nearest neighbor with q_{AA} ANCs and q_{AB} BNCs, giving a total coordination number q = q_{AA} + q_{AB}. Similarly, for a B NC, the total coordination number will be q = q_{BA} + q_{BB}. A necessary condition for BNSLs with q_{T} ≠ 0 to exist is that at least one of the coordination numbers q_{AA}, q_{AB}, q_{BA}, q_{BB} is less or equal than four. This condition, by itself, is less restrictive than it may appear: out of the 21 binary lattices in Table 2, 18 admit, in principle, nontrivial OTM solutions, at least for a range of γratios. Thus, a first prediction from OTM is that the following 3 lattices: NaCl, CsCl, CaB_{6} are the only ones rigorously described by the q_{T} = 0 results discussed above.
Table 2 Binary lattices considered (ST: lattice type, SG: space group, S: strukturberich symbol, P: pearson symbol, A: alternative name)
N 
ST 
SG 
S 
P 
A 
NaCl 
AB 
Fmm 
B1 
cF8 

AuCu 
AB 
P4/mmm 
L1_{0} 
cP4 

CsCl 
AB 
Pmm 
B2 
cP2 

ZnS 
AB 
F3m 
B3 
cF8 

ZnO 
AB 
P6_{3}/mc 
B4 
hP4 

AlB_{2} 
AB_{2} 
P6/mmm 
C32 
hP3 

CaF_{2} 
AB_{2} 
Fmm 
C1 
cF12 

MgCu_{2} 
AB_{2} 
Fdm 
C15 
cF24 
Laves 
MgZn_{2} 
AB_{2} 
P6_{3}/mmc 
C14 
hP12 
Laves 
AuCu_{3} 
AB_{3} 
Pmm 
L1_{2} 
cP4 

Li_{3}Bi 
AB_{3} 
Fmm 

cF16 

ReO_{3} 
AB_{3} 
Pmm 
D0_{9} 
cP4 

Cr_{3}Si 
AB_{3} 
Pmn 
A15 
cP8 

Fe_{4}C 
AB_{4} 
P3m 

cP5 

Pt_{3}O_{4} 
A_{3}B_{4} 
Pmn 

cP14 

CaCu_{5} 
AB_{5} 
P6/mmm 
D2_{d} 
hP6 

CaB_{6} 
AB_{6} 
Pmm 
D2_{1} 
cP7 

bccAB_{6} 
AB_{6} 
Imm 

cI14 
Cs_{6}C_{60} 
NaZn_{13} 
AB_{13} 
Fm3c 
D2_{3} 
cF112 

cubAB_{13} 
AB_{13} 
Pmm 

cP14 

cubfccAB_{13} 
AB_{13} 
Fmm 
D2_{f} 
cF56 

The detailed solution for each of the eighteen lattices considered and their free energy will be described elsewhere. Instead, I will discuss four inconsistencies that have been emphasized during the q_{T} = 0 analysis: smaller NC radius for MgZn_{2}, low packing fraction Li_{3}Bi absent as an equilibrium phase, the CaB_{6} being slightly favored over the bccAB_{6} and the AuCu lattice, which does not even exist at the corresponding γ ratio.
3.3.1 MgZn_{2}.
This lattice is described by space group P6_{3}/mmc (194) with a unitcell of four Aparticles in Wyckoff positions 4f and eight Bparticles in positions 2a and 6h. If the A and B particles are hard spheres, the maximum packing fraction is reached for . At this value there are A–A and B–B contacts, but no A–B contacts. For < γ_{c}, the shortest separation within Bparticles is given as 
 (23) 
For < γ_{c} A–A hard spheres are in contact, with q_{AA} = 4 and those are the NCs with q_{T} = 4. Thus, for this lattice, OTM solutions will exist for ≤ γ_{c} only.
The corresponding nematic texture (from the free energy eqn (18)) consists of four ½disclinations sitting at the corners of a perfect tetrahedra, as shown in Fig. 3C. Because the Bparticles are six coordinated their radii is given by r_{B} (eqn (5)). Therefore, from eqn (23), the NC radius for the Aparticles will satisfy

 (24) 
where γ is always defined by eqn (6), that is, with respect to the q_{T} = 0 case. Note that if γ ≥ γ_{c}, then _{A} becomes r_{A}, as topology prevents the appearance of defects for six coordinated particles.
3.3.3 bccAB_{6}.
This lattice is described by space group Imm (229) with a unit cell containing two A particles in Wyckoff positions 2a and twelve B particles in Wyckoff positions 12d. If the A and B particles are hard spheres, there are two maxima of the packing fraction. The first occurs at , and the second for . If γ_{c,1} < < γ_{c,2}, the minimum A–B distance is given by 
 (30) 
while for > γ_{c,2}, it is given by
OTM solutions with nontrivial topological index may exist for any > γ_{c,1}. In all situations, q_{T} = 0 for the Aparticles. In the first branch γ_{c,1} < < γ_{c,2} the solution corresponds to Bparticles with q_{T} = 4. The four disclinations follow a distorted tetrahedra and the shortest separation among A–B NCs reads

 (32) 
In this case, a further restriction exists among BNCs, which reads 
 (33) 
The monomeric packing fraction η_{u} can be computed by eqn (29), adapted to this case. There is a second branch for > γ_{c,2}. In that case, contacts are among Bparticles, and the analysis can be done along the same lines.
3.3.4 AuCu.
This lattice is described by space group P4/mmm (123) with a unit cell of two A particles in Wyckoff positions 1a and 1b and two Bparticles in Wyckoff positions 2e. If the A and B particles are hard spheres, the maximum packing fraction is reached for . At this value, each A particle is in contact with six other Aparticles and eight Bparticles, in a structure that is identical to CsCl. For > γ_{c} the shortest separation between A and B particles is given by 
 (34) 
The smallest distance between A–A particles along the caxis is 
 (35) 
Each ANC is in contact with four other ANC while still maintaining contact with eight B NCs. Thus, the ANCs are those with q_{T} = 4, and OTM solutions exist for only.
In this case, the nematic texture for each ANC corresponds to four disclinations contained within a single plane on a great circle in the sphere, as shown in Fig. 3C. Although such configuration is not the minimum of the free energy eqn (18), it has been found in numerical simulations of nematics on the sphere,^{32} and it is a minor variation of the tetrahedral configuration obtained by stretching the neutral lines all the way till the equators, which presumably may incur in a relatively low free energy cost.
A first relation follows from the observation that two A NCs separated along the caxis will touch each other along the north and south poles, where according to the nematic texture in Fig. 3C there is a neutral line, so ^{c}_{AA} = s_{AA} = 2r_{A}. Thus, from eqn (35), it follows

 (36) 
Another equation is obtained from noting that topology forbids disclinations for the BNCs so their smallest possible radius is r_{B}. Therefore 
_{A} ≥ r_{B} = τ_{B}R_{B}.  (37) 
If I assume that the equality in the above equation holds, the pair of equations eqn (36) and (37) admit a continuous solution for , which are shown in Fig. 10. At any other , = 1, (the cubic AuCu) and the smallest A–A separation is given by

 (38) 
independently of the actual value of γ. In this case, however, the BNCs ligands are not fully compressed and the corresponding vdW energy is not optimal.

 Fig. 10 Solution of the couple equations eqn (36) and (37) determining the structure of the AuCu lattice.  
4 Comparison with experimental results
To make contact with experiments, the maximum length L of the capping ligands needs to be determined as a function of their actual chemical composition. For a ligand consisting of n saturated hydrocarbons, such length is given as 
L = nr_{0}cos(α/2) + r_{1} = 0.128n + 0.2 (in nm)  (39) 
where α = 180 − 112 = 68 is the C–C angle, r_{0} = 0.154 is the C–C bond distance and r_{1} = 0.2 is the radius for an endcarbon.^{25} It has been argued^{33} that this formula tends to overestimate the maximum total hydrocarbon length. Actually, ref. 13 usedwhich predicts a slightly shorter maximum length than eqn (39). Results using both formulas will be quoted, serving the purpose of quantifying the sensitivity of the OTM in regards to maximum ligand length.
The quantities that will be compared to the experiments are the monomer packing fraction η_{u} and

_{ij} = _{ij} − R_{i} − R_{j},  (41) 
which are quoted by Boles and Talapin
^{13} in the supplementary information Tables S3, S4, S7 and S8. When using
eqn (39) the OTM predictions will be quoted as OTM, and as OTM* when using
eqn (40). The comparison between theory and experiment is summarized in
Table 3. I now discuss each of the cases analyzed in more detail.
Table 3 Comparison between theory and experiment η_{u} (dimensionless) is the monomer packing fraction eqn (7). _{ij} is the quantity eqn (41) (in nm). For MgZn_{2} (i = A, j = B), Li_{3}Bi (i = A, j = B), bccAB_{6} (i = A, j = B), AuCu (i = A, j = A). Experimental results marked with † were incorrect in the original reference and have been corrected
BNSL 
η
_{u}

_{ij}

Exp. 
OTM 
OTM* 
Exp. 
OTM 
OTM* 
MgZn_{2} 
0.79 
0.77 
0.83 
1.30 
1.35 
1.13 
Li_{3}Bi 
0.88 
0.87 
0.86 
0.90 
0.98 
1.00 
bccAB_{6} 
0.83 
0.84 
0.86 
1.90 
2.20 
2.01 
AuCu 
(—)^{†} 
>1 
>1 
0.05^{†} 
0.02 
0.01 
The MgZn_{2} is already the equilibrium phase at q_{T} = 0, see Fig. 7, although in this case η_{u} = η_{HS}(γ = 0.77) = 0.64 and _{AA} = 1.85 (nm) in clear disagreement with the experimental results in Table 3. The solution for q_{T} = 4 brings both results to agreement.
The Li_{3}Bi at q_{T} = 0 has a very low monomer packing fraction η_{u} and it is very far from equilibrium, see Fig. 8. Even with reasonable variations of the potential, this BNSL cannot become equilibrium with a trivial topological index. The OTM solution makes use of eqn (28), which gives = 0.34, 0.35*, with results quoted in the table. As noted in experiment, the actual monomer packing fraction is very high (close to 0.9), in contrast with the nominal HS packing fraction η_{HS} = 0.54.
The bccAB_{6} BNSL is not the equilibrium for q_{T} = 0, but rather, the competing CaB_{6} phase is. However, the latter does not admit nontrivial OTM solutions while the bccAB_{6} phase does. Given that the nontrivial OTM solution has much lower vdW attraction, the bccAB_{6} phase should become equilibrium in agreement with experiments. The results quoted in the table fully support this statement, although, admittedly, a rigorous proof will require a free energy calculation, which is not developed in this paper.
The AuCu phase at γ = 0.58 imply = 1. For this value of γ = 0.58, the ligands for the B NCs are not fully compressed, which points to some inconsistency for this solution. Within the OTM, the AuCu consists of a cubic unit cell with _{AA} = 0.3 nm within the basal plane (perpendicular to the caxis) and a caxis length of 10.2 nm in reasonable agreement with the 10.6 nm value quoted by the experiment results^{13} (Fig. S10 in supplementary information). The resulting monomeric packing fraction is η_{u} > 1, which strongly suggests that ligands were lost during or before the measurement process. The PbS NCs have ligands bound to both the {111} and {100} planes, and different calculations^{24,34} show a much weaker binding to the latter, which may result in desorption during sample preparation after successive washing steps. Further evidence for ligand desorption is provided in ref. 7, where AuCu BNSLs are observed to display orientational core–core registry, as shown by lobes in the wideangle electron diffraction.
5 Conclusions
In this paper I have introduced the orbifold topological model (OTM) which recognizes the role played by the topological defects associated to the capping ligand. The model provides detailed predictions on the structure of superlattices and quantitatively describes the experimental data with excellent precision, as summarized in Table 3. In addition, the OTM provides clear explanations for a number of experimental observations:
• Lattices with nontrivial topological index q_{T} > 0 dominate over competing phases with q_{T} = 0 (such as bccAB_{6} over CaB_{6} or AuCu over CsCl)
• Lattices with low hard sphere packing may become equilibrium phases with maximal monomer packing fraction η_{u} (such as Li_{3}Bi).
• Ligand reorganization is a factor that may determine equilibrium in some cases (such as AuCu).
The first statement follows from the fact that CaB_{6} and CsCl do not admit OTM solutions with nontrivial topological index, while both AuCu and bccAB_{6} do, and those solutions with nontrivial topological index have a much lower van der Waals energy, which generally determines the equilibrium state.^{26} The second statement has been illustrated with the Li_{3}Bi case, see Table 3. It should be noted, however, that for such low nominal packing fraction phases to become equilibrium, those phases must have a large packing fraction maximum at some other value of = γ_{c}, see discussion in Section 4.
Dynamical ligands and their effect on the phase diagram has recently been investigated in ref. 35. It has been shown that for two different Ag NCs coated with the same ligand, resulting phases are consistent with the HS model predictions, while for Ag NCs each coated with a different ligand other phases: CaB_{6}, AuCu_{3} and DDQC were observed. It was argued that there is an exchange of ligands between NCs and that this process stabilizes those phases not present in the HS model phase diagram. Within this scenario, low packing fraction phases are interpreted as kinetically trapped during ligand exchange. The detailed predictions of the OTM model have shown excellent agreement with the experimentally observed NC distances, see Table 3. If ligand exchange would become a dominant process, those distances would differ very significantly from those predictions and I must conclude that for the Au, PbS and Fe_{2}O_{3} NCs ligand exchange is not significant, except for the AuCu (PbS and Au NC) phase. Furthermore the OTM shows that low nominal packing fraction phases (Li_{3}Bi) do not need to be metastable and can be equilibrium. Finally, as our previous results^{17,18} and this paper shows, the HS model does not provide a reliable description of BNSLs, as phase diagrams depend not only on γ, but on the characteristics of the ligand also. Of course, there is always the possibility that ligand exchange may be more relevant for specific Ag NCs (similarly as found for the {100} plane for PbS), but this will need to be substantiated with detailed atomistic studies, such as ref. 34.
The OTM also provides many other predictions that should be verifiable with current experimental techniques. For example, the actual ligand textures, with the ½disclinations and the location of neutral lines, see Fig. 3 should be visible with microscopy techniques. Other predictions, such as the detailed NC separations within the unit cell for ligands with unsaturated hydrocarbons and others^{36} will be discussed elsewhere.
Because of the role played by topological defects, the OTM includes many body effects whenever the topological index q_{T} is different from zero. It also generalizes two previous models, the optimal packing model (OPM)^{20} and the overlap cone model (OCM).^{23} The OTM reduces to the OCM for two isolated nanoparticles and to the OPM when nanoparticles have trivial topological index q_{T} = 0. It is interesting to note that the OTM for q_{T} = 0 basically predicts the same equilibrium phases, see Fig. 6, as found with inverse power law potentials^{17,18} and with similar inconsistencies: absence of low packing fraction phases and the predominance of some phases such as CsCl over AuCu or CaB_{6} over bccAB_{6}. This probably points out to the limitations of “universal” approaches, which cannot account for the nuances in the structure of the ligands.
Certainly, the role of topological defects, the interactions of neutral lines, see Fig. 3, and the precise structure of the corresponding ligand textures will require a more detailed and quantitative analysis. There are here other possibilities that I have not discussed in this paper. For example, how ½ disclinations might split into defects with lower charges or the role of antidefects. Also, the case of topological index q_{T} = 3, which may become relevant for some BNSLs, such as the NaZn_{13} or AlB_{2} and last, but not least, the necessary free energy for a rigorous quantitative analysis, will be discussed with full detail in the near future. This last part serves to emphasize the role of softness, as it has been discussed with simpler binary mixtures models of colloids.^{37,38}
In summary, the OTM is a significant progress towards a prediction of the structure in nanoparticle superlattices and reveals fascinating relations between packing of hard spheres, lattice structure, geometry and topology.
Acknowledgements
I am indebted to M. Boles for many clarifications of the experimental results and existing literature of BNSLs. I also acknowledge interest and discussions with C. Calero, N. Horst, O. Gang, S. Mallapragada, D. Talapin and D. Vaknin. I want to thank T. Kennedy for many discussions and encouragement during the course of this work. This work is supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials, Materials Science and Engineering Division. The research was performed at the Ames Laboratory, which is operated for the US DOE by Iowa State University under contract number DEAC0207CH11358.
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