Soft Skyrmions, Spontaneous Valence and Selection Rules in Nanoparticle Superlattices Alex Travesset* Department of Physics and Astronomy and Ames Laboratory, Iowa State University, Ames, Iowa 50011, United States S Supporting Information *
ABSTRACT: A number of bewildering paradoxes arise in the field of nanoparticle self-assembly: nominal low density superlattices, strong stability of low coordination sites, and a clear but imperfect correlation between lattice stability and the maximum of hard sphere packing, despite the fact that that nanocrystals themselves are, through their ligands, very much compressible. In this study, I show that by regarding nanocrystals as pseudotopological objects (“soft skyrmions”), it is possible to identify and classify the ligand textures that determine their bonding. These textures consist of interacting vortices, where the total vorticity defines a spontaneous valence (coordination). Furthermore, skyrmion interactions are governed by two simple assumptions, which lead to a set of selection rules for superlattice structure. Besides resolving all the above paradoxes, the predictions are completely supported by more than one hundred sixty experiments gathered from the literature, including a wide range of nanocrystal cores and ligands (saturated or unsaturated hydrocarbons, amines, polystyrene, etc.). How those results can be used for addressing more complex structures and guiding future experiments is also addressed. KEYWORDS: self-assembly, nanocrystal, superlattices, skyrmions, packing fraction, vortices other coarse-grained models21 remarkably expands the range of HS equilibrium lattices but does not conceptually resolve the flaws of the HS model.18 Most commonly observed BNSLs occur at high density and involve tetrahedra, icosahedra, and other polyhedral motifs packed very efficiently, thus raising fascinating connections between geometry, topology, defects, packing problems, or phase separation. Often, NCs display coordinations as low as three or four, pointing to an underlying patchiness or valence emerging from the isotropic ligands. Besides these obvious rivetting fundamental questions, a thorough understanding for all these problems is critical to our ability to advance these materials and develop their revolutionary applications. The goal for this paper is to provide rigorous selection rules for the structure of BNSLs and, along the process, resolve the many flaws and paradoxical results inherent to the HS approach. The starting point in this study is the optimal packing model(OPM),22,23 see Figure 1, which successfully described the structure of capping ligands in one-component superlattices. With the NC characterized by its core radius Ri and maximum ligand extension Li, the OPM predicts a NC radius ri given by
I
t is very challenging to control the interactions of raw NCs, so a general strategy for programming their self-assembly into relevant materials and structures is to functionalize their surface with capping ligands. The shell formed by the capping ligands thus determines the material structure and plays the role of the electronic orbitals and its ensuing valence in materials of simple elements. Typical ligands include DNA, polymers, or hydrocarbons,1 leading to periodic and quasiperiodic structures of NCs, i.e., superlattices, self-assembled by DNA hybridization,2,3 electrostatic phase separation,4 or solvent evaporation,5−16 just to name a few. Superlattices involving two NC species, i.e., binary nanoparticle superlattices (BNSLs), provide the most sophisticated structures assembled so far. Currently, more than 20 2D and 3D different structures of BNSLs have been reported.17 Rationalization of these lattices has been made by modeling NCs as hard spheres (HS) whose diameter is given by the NC nearest neighbor distance of a single-component, two-dimensional triangular lattice. Rather remarkably, a clear correlation between maximum of the HS packing fraction and the presumably equilibrium structures has been established. However, such correlation is somewhat rudimentary, as many BNSLs exist at low HS packing fraction, and furthermore, lattice distances are, more often than not, inconsistent with the assumed HS radius.18 Attempts to describe those disagreements in terms of more flexible inverse power law potentials19,20 or © 2017 American Chemical Society
Received: March 30, 2017 Accepted: May 17, 2017 Published: May 17, 2017 5375
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ACS Nano η(γ ) =
vol occupied by matter ⇒ η(γ ) = ηHS(γ ) within OPM volume (3)
Detailed free energy calculations18 show that the OPM model gives results that are basically equivalent to the predictions of HS or the more precise inverse power law models. It should be noted, however, that the closest possible approach between NCs would occur if η(γ) = 1. This condition defines the smallest NC radii (r̃i < ri). For a single-component lattice, it is 1/3 ⎛ A L⎞ 1/3 1/3 1/3 ri ̃ = ηHS R i⎜1 + 3 H i ⎟ ≡ ηHS R i(1 + 3ξiλ i)1/3 ≡ ηHS ri < ri Ai R i ⎠ ⎝
Figure 1. Representation of a NC as a qS = −1 skyrmion: Pure hedgehog (Nv = 0) and one with vorticity (Nv = 1). Representation of the OPM and NCs with vortices. The covered cones represent points where the packing density is η = 1. The quantities Li, Ri, ri, and Ai used extensively throughout the text are defined. AH (not shown) is the smallest possible molecular area for the corresponding ligand.
(4)
The equation above is a slight generalization of the optimal cone model(OCM) introduced in ref.24 for the case of two or three NCs. In a remarkable paper, Boles and Talapin16 showed conclusive evidence that NCs in two-dimensional lattices are approximately described by the OPM result for NCs with coordination 6, while for lower coordination they are more consistent with eq 4. In this paper, it will be assumed that the NC are spherical and ligands are flexible geometric cylinders. The technicalities of mapping this description from more realistic NCs that are not perfectly spherical, contain ligands with kinks, etc., is fully discussed in the Supporting Information (Experimental Parameters). It is useful to think of a given NC as a skyrmion. Here, each ligand chain grafted to the NC defines a vector whose origin is at the NC and whose end is at the center of the last group. Mapping this vector in a sphere defines a map S2 → S2; see Figure 1. Because of the way the ligands are attached, this skyrmion has a topological charge qS = −1. Still, by focusing on the projection of the ligand vector to the sphere surface, the order parameter may define vortices, with an associated total vorticity Nv. It cannot be stressed enough that the vortices are not topological, as they may contain singular points where the order parameter escapes to the third dimension (perpendicular to the sphere), and as a result, there are no topological
1/3 ⎛ A L⎞ ri = R i⎜1 + 3 H i ⎟ ≡ R i(1 + 3ξiλi)1/3 ≡ R iτOPM(λi) Ai R i ⎠ ⎝
(1)
where λ ≡ L/R is called the “softness” and ξi ≡ AH/Ai ≤ 1 is the ratio between the maximum possible molecular area (AH) to the actual molecular area (Ai). For reasons that will become more clear below, I identify the ri defined above by the OPM as the equivalent HS NC radii. In binary lattices, the ratio of the smallest (rB) to the largest (rA) HS radius is defined as the γparameter
γ=
rB ≤1 rA
(2)
The HS packing fraction is a function ηHS(γ) of this parameter only. The packing fraction of a generic superlattice η(γ) is defined as the actual volume occupied by matter, the NC core plus the ligands:
Figure 2. Representation of the MgZn2 lattice with hard spheres. Sphere A is red, and Sphere B is blue. The HS contacts form a diamond and B spheres a pyrochloric lattice. The Wigner−Seitz cell at γ = γc (calculated using the radical Voronoi Tesselation in Voro++25) shows the Frank−Kasper dodecahedra (particles are scaled down for proper visualization). The skyrmion texture is tetrahedral (n-tetra). Note that the OTM branch corrects the low HS packing fraction. The program ovito26 was used for visualization. 5376
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providing explicit values for ri̅ , η, and other quantities. Their actual form, however, differs from lattice to lattice, and there are additional constraints, explicitly discussed in the Supporting Information (Additional Constraints). The implementation of the OTM model is illustrated for the MgZn2 lattice as a paradigm, leaving the details for all other BNSLs in the Supporting Information (Lattices). MgZn2 Lattice As a Paradigm. The MgZn2 lattice is described by space group P63/mmc (194) with a unit cell with nA = 4 in Wyckoff positions 4f and nB = 8 in positions 2a and 6h; see Figure 2. The maximum of ηHS(γ) is reached for γ ≡ γc = 2/3 . Because this BNSL is a Frank−Kasper phase, the Wigner−Seitz cells are dodecahedrons (with additional four disclination lines for the A particles). Despite the fact that the respective coordinations are 16 and 12 for A and B particles, see Figure 2, at γ = γc with NC regarded as HS, each A-NC is in contact with four other A-NCs while B-NC with 6-B-NCs. According to assumption II, OTM solutions are only possible for the A-NC contacts, that is, for γ < γc, where Nv,A = 4 and Nv,B = 0. Within OTM, the Nv = 4 solution consists of four vortices arranged in a texture with tetragonal symmetry; see Figure 2. The maximum packing fraction occurs at γc, so by assumption I γ ̅ = γc. From assumption II, Nv,B = 0 so that rB̅ = rB. It is
constraints about the total number of vortices on that sphere. Within this representation, the OPM result represents local deformations of a simple hedgehog Nv = 0, while closest approach below 2 ri requires at least one NC with nonzero vorticity Nv ≠ 0, see Figure 1. Because ligands themselves are flexible, the concept of skyrmion topological charge qS is only approximate, so the more proper name of soft skyrmion is used through the paper.
RESULTS I consider 2D or 3D BNSLs with a unit cell containing nA and nB NCs. An example is MgZn2, where nA = 4, nB = 8; see Figure 2. The case of single-component systems corresponds to nB = 0. The interaction of soft skyrmions (the NCs) is governed by the Orbifold Topological Model (OTM), see below, whose early formulation was introduced in ref 18. The concept of soft skyrmion has been introduced at the end of the Introduction. OTM Model. OTM is defined by the following two assumptions: (1) For a given lattice, the minimum free energy occurs at the lattice constant with maximum HS packing fraction. (2) Skyrmion textures only support up to five vortices or Nv ≤ 5. The first assumption, which has an enthalpic origin, steams from the overall effective interaction among NCs being attractive, so that the maximum packing fraction consistent with existing constraints (for a given SL) is equivalent to a minimization of the free energy. The second assumption, which has an origin in the entropy of the chains, reflects that vortices in soft skyrmions are repulsive and extend over a long range. Application of the Euler theorem would restrict the number of vortices to 2,27 so Nv ≠ 2 requires the presence of singular points “neutral zones or lines”, where capping ligands are perpendicular to the sphere surface. Although a rigorous justification for this condition will certainly require detailed numerical simulations, very compelling evidence is provided, not only from existing preliminary results18 and the analysis of the hundreds of experiments presented further below but also from comparison to the experimental results of ref 16 of the separation distances between planar 6-coordinated and Q-coordinated NCs, where Q = 3,4,5,6, see the Supporting Information (Planar Systems). The possibility that Nv = 6, 7, ... become less favorable, as opposed to the strict prohibition of the hard statement Nv ≤ 5, has been analyzed in the Supporting Information (Planar Systems) and in the discussion below (see the case of NaCl, for example). There is no evidence that requires considering Nv = 6 or larger, so I will stick to Nv ≤ 5. A very important result to bear in mind is that the packing fraction and lattice separations of the OTM model in the trivial case Nv = 0 coincides with the HS packing fraction, as extensively discussed in ref 18; see eq 3. The implementation of the OTM proceeds by first identifying those NCs within the lattice unit cell that can sustain Nv ≤ 5 while still respecting the space group symmetries defining the lattice. This defines a set of NCs radius ri̅ such that rĩ < ri ̅ ≤ ri
rA̅ =
rB r Rτ γ = B = B B = rA ≤ rA γ̅ γc γc γc
(6)
As γ gets smaller, the condition γ ̅ = γc cannot hold once the A− B distance becomes less than rA + rB. This imposes a further constraint, defining a critical γc γc =
12 2 11 /γc −
12
=
2 11 −
2
≈ 0.7434 (7)
where for γ ≤ γ , consistently with assumption I, it is c
rA̅ =
⎛ 11 ⎞1/2 γ ⎛ 3 ⎞1/2 ⎜ ⎟ (1 + γ )rA orγ ̅ = ⎜ ⎟ forγ ≤ γ c ⎝ 11 ⎠ ⎝ 3⎠ 1+γ (8)
The packing fraction for the OTM branch, eq 3 is ⎧⎛ 3 ⎞ ⎪ ⎜ 1/γ + 2 ⎟η (γ ) if γ c ≤ γ ≤ γ ⎜ ⎟ HS c c 3 ⎪ ⎪ ⎝ 1/γc + 2 ⎠ η(γ ) = ηOTM (γ ) = ⎨ ⎪ 113/2 1 + 2γ 3 ⎪ if γ ≤ γ c ⎪ π 48 (1 + γ )3 ⎩ (9)
Among references included in this study, the MgZn2 phase has been reported 12 (one for polymer ligands) times. With the values of γ as provided in the literature, which are in agreement with the ones calculated by OTM for single-component lattices (more details on this below), the experimental results on a HS curve are shown in Figure 2. The two most obvious features are that the experiments appear to the left of γc, and that quite a few of those occur at a very low packing fraction. Both observations are easily explained within the OTM model: Experiments to the right of γc would require Nv,B = 6, which according to assumption II are forbidden, thus providing yet another evidence for this assumption. The paradoxical result of low packing fraction BNSLs is also resolved, as it is given by eq 9, which actually predicts a packing fraction that largely exceeds
(5)
and a corresponding packing fraction 1 > η(γ) ≥ ηHS(γ). The effective-γ, defined as γ ̅ = rB̅ /rA̅ , is also used. The two assumptions defining the OTM model give raise to a set of algebraic equations that can be solved analytically, thus 5377
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a
d̅ij (in nm)
BNSL
γ
contact
exp
OTM
HS
exp
OTM
HS
MgZn2 CaCu5 NaZn13 NaZn13 AlB2 Li3Bi bccAB6 AuCu
0.77 0.77 0.57 0.62 0.62 0.57 0.45 0.62
A−A B−B B−B B−B A−B A−B A−B A−A
0.79 0.76 0.73 0.79 0.77 0.88 0.83 0.83
0.78 0.67 0.75 0.82 0.74 0.84 0.80
0.65 0.67 0.74 0.68 0.74 0.55 0.74 0.65
1.30 2.40 2.48 1.75 2.29 0.90 1.96 1.67
1.30 2.24 2.20 1.87 2.15 0.98 2.18
1.80 2.24 2.70 1.80 2.15 2.24 2.40 2.50
The quantity dij (ligand shell length) is defined in eq 10.
Table 2. Characteristic Lattices and Their OTM Tranchesa BNSL NaCl CsCl AuCu
SG
WycA
WycB
Fm3m ̅ Pm3̅m P4/mmm
4a 1a 1a 1b
4b 1b 2e
texture
γc
(B)hexa**
√2−1
(B)n-equat
√3−1
γc
1 2
γc
[γc,γc)
γ 2 + 2γ 4 − γ 2 1 − γ2
γ≤γ AlB2 MgZn2
P6/mmm P63mmc
1a 4f
2d 2a, 6h
(A)n-tetra
1/2
( 23 )
√2−1
Cu3Au
Pm3̅m
1a
3c
(B)n-equat
Li3Bi
Pm3̅m
4a
4b, 8c
(8c-B)n-tetra
−1
Fe4C
P4̅3m
1a
4e
(B)n-fe4c
CaCu5
P6/mmm
1a
2c, 3g
(3g-B)n-equat
1 4/ 3 −1
(3g-B)n-equat
1 + 2 19 15
2
(2a-B)t-triang
CaB6 bccAB6
NaZn13 cAB13 a
6 2
Pm3̅m Im3̅m
Fm3c̅ Pm3m ̅
1a 2a
8a 1a
6f 12d
8b, 96i 1b, 12i
2 3
−1
(B)n-tetra
5 3
(B)n-tetra
1 10 − 1
(96i-B)t-nazn13 (B)n-collec**
1 2
√2−1
3 −1 3+ 2
√2−1
(B)p-cab6
2 11 − 2
−1
γc,2 1
8 + 19 10 3
1 + 19 3 (8 + 19 )
5−2 2 −1 1
1/√6 2 10 − 1
γ̅
branch γ > γc
c
1
[γc,γc) γ ≤ γc
γc 1/2
( 113 )
(γc,γc] γ ≥ γc (γc,γc] γ ≥ γc
γc 2γ−1 γc
γ > γc
γc
γ 1+γ
3 (1 + γ ) 2
(γc,2,γc,3]
4 3
(γc,2, γ ]
γc,2
c,2
(γ ,γc,2] (γc,1, γc,1]
γc,2
[γc , γ c)
γc
γ≥γ (γc,γc] (γc,1, γc,2]
γc,1
c,1
−1
γ−1
3γ 2 − 3γ
c
10 γ − 1
(γc,γc] γ > γc
γc
γ > γc,2 γ > γc
eq S71 (SI) γc
γ 2
5+2 2 −1
The hexa** is an extension of the OTM explored for NaCl and the n-collec** for the cub-AB13; see the discussion in the text.
CsCl and AlB2). The AlB2 phase is ubiquitous in the HS phase diagram, as it has very high packing fraction and entropy.28 Indeed, the bulk of the experimental points occur at packing fractions that exceed the fcc limit. It is less clear what stabilizes the CsCl phase, a phase absent in HS, but the experiments appear in a region where the packing fraction exceeds that of the simple bcc lattice, the stable single component phase. The bulk of the experimental results in MgZn2, AuCu3, Li3Bi, cub-AB13, and NaZn13 occur at values where ηHS is significantly below the fcc and bcc packing fraction. In all cases, without a single exception, the OTM branch brings the actual packing fraction to values that exceed the fcc result. This is also the case for Fe4C, although at this γ unphysical values η ≥ 1 result. Given that this BNSL has been reported in one paper only,6
the fcc. There are further quantitative comparisons that are discussed further below; see also Table 1. Comparison with Experiments. Data was collected from the following 11 references.5−14,16 Polymeric BNSLs15 are also discussed. In total, the analysis encompasses more than 160 independent experiments, from which the data Ri, ri is available, with softness parameter in the range λ ∈ [0.15, 7.5]. With these values, γ is computed from eq 2 and is in agreement with the experimentally reported values. Note that ri is given by eq 1, so the experimentally reported ri can be self-consistently checked if the grafting density σ is known. A critical analysis of this point is given in SI-OPM NC separation and grafting density. Analysis of Packing Fraction. There are three trivial latices (those that do not have any OTM branches), namely (NaCl, 5378
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Figure 3. Actual packing fraction η(γ) = ηOTM(γ). The OTM branches that differ from the HS are plotted in solid lines. The two dashed horizontal lines correspond to η(γ) for fcc and bcc, respectively, which are the equilibrium SLs for single-component systems.18 Points corresponding to η ≥ 1 are marked with light gray and are discussed in the test. A total of 126 experiments are shown. The value of γ is taken from the experimental data, which agrees with the OTM prediction for a triangular lattice (with a slight adjustment of the correct grafting density, when necessary), as discussed in SI-OPM NC separation and grafting density.
and the unusual “orbital” (see Figure 4) and also the Wigner cell (Supporting Information (Wigner Cell)), further studies will be necessary to clarify this phase. The experimental results in three additional lattices CaCu5, bccAB6, and CaB6 are also captured within the OTM, although in this case, the difference from the HS predictions is not as dramatic. The most interesting case is CaB6 as it is the only with Nv = 5; see Table 2. Such a structure may also have
implications for the DDQC/AT quasicrystal, which is a hybrid between CaB6 and the AlB2 BNSLs. Two BNSLs (NaCl and AuCu) show unusual features that are further analyzed below. AuCu lattice. The AuCu lattice stands as a clear outlier to the OTM model; see Figure 3 and Table 1. Except for two experiments (corresponding to 6.2 nm PbSe and 5.2 nm Au6 and 4.9 nm Pd9), the other nine correspond to η(γ) ≥ 1. The first aspect to note is that all 11 AuCu BNSLs occur when the 5379
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Results are summarized in Table 1. Within HS(OPM prediction), only the AlB2 and CaCu5 BNSL are well described, but for many others: MgZn2, Li3Bi, NaZn13 the HS predictions are grossly inaccurate. Within OTM, all results are correctly described, with separations predicted within an accuracy of 2 Å, and packing fraction within 0.03, which is most likely within experimental errors(nonspherical cores, resolution in EM images, etc.). BNSLs with Polymer Ligands. NCs with polymer ligands(atactic polystyrene) have been reported in ref 15, thus expanding the library of BNSLs to include softness values λ ∈ [1, 7.5]. The reported nearest-neighbor distances for triangular lattices are in agreement with the OPM prediction eq 1 (see the Supporting Information (OPM NC separation and grafting density) for a discussion). Among the six different BNSLs reported, several of them (MgZn2, AuCu3, etc.) occur at low nominal ηHS < ηbcc, but they fall into OTM branch that corrects this into a large packing fraction, thus demonstrating that spontaneous valence also occurs in long polymers; see also the figure in the SI. In summary, the polymeric BNSLs reported in ref 15 are fully described by the OTM model.
Figure 4. Summary of all skyrmion textures (“atomic orbitals”) involved in bonding. The four structures below are possible by OTM but appear very rarely; see the discussion in the text. The BNSLs where they are found are shown in Table 2.
core A-NC is either PbS or PbSe, and in fact, it has been pointed out before18 that ligands are loosely coupled for this NC and may be lost before the self-assembly takes place. Clear evidence for this point is given in ref 17, where explicit images show face-to-face contact without ligands between PbS as well as PbSe NCs. Thus, the AuCu lattice represents another type of BNSL, where the assembly is not dictated by the ligands only, but also, by the atomic interactions between NC cores. The fact that the OTM model discriminates such features is a definite success. Does NaCl (or CaCu5) Show a Possible Violation of the OTM Assumptions? The NaCl lattice can be regarded as a limiting test to the OTM, as each NC has six HS contacts. The experiments that exhibit this BNSL include many diverse cores, such as Fe2O3, PbSe, PbS, Ag, or Co as A-NC or Au, Pd, CdSe, or Ag as B-NC. In contrast with AuCu, the appearance of the NaCl is not related to a particular chemistry of the core. The analysis of Figure 3 reveals three experimental points that are clearly below the bcc line, which at this low packing fractions would only be possible to reconcile with the OTM if Nv = 6. Those experimental points correspond to (A = 5.8 PbSe, B = 3 Pd) and (A = 7.2 PbSe, B = 3 Pd)6 and (A = PbSe, B= 3.4 Pd)9). Rather interestingly, those NC cores are the same that display ligand loss in the AuCu BNSLs. Given these considerations, it is very likely that those NaCl outliers are the result of ligand loss. Once these three experiments are excluded, the available data is entirely consistent with OTM. As for the CaCu5 lattice, those data that display a low packing fraction are all related to NCs with the PbSe A core again. A clear emerging conclusion is that NC with PbSe or PbS cores are special in that they show a tendency to ligand loss before self-assembly, and for that reason, their location in the packing fraction diagrams may be misplaced. Further experiments will be necessary to fully clarify this point, but presently, these BNSLs do not challenge the OTM assumptions. Quantitative Analysis and Lattice Constant. The OTM provides precise predictions for quantities such as packing fraction η and NC separations within the BNSL, among many others. Unfortunately, only the paper of Boles and Talapin16 provides this type of quantitative information. Their paper quotes values for the experimental packing fraction η and the size of the ligand shell dij̅ = sij̅ − R i − R j
CONCLUSIONS In this paper, I identified NCs as soft skyrmions interacting according to two simple OTM rules and provided detailed predictions on NC separations, density (or packing fraction), or ligand textures among other quantities. It also establishes the following selection rules based on the packing fraction: Any candidate equilibrium BNSL must have a packing fraction that exceeds the fcc/bcc limit. The more than one hundred 160 independent experiments analyzed provide quantitative, see Table 1 and the Supporting Information (Planar Systems), and qualitative, see Figure 3 verifications for these statements, with the only exceptions given by some NCs whose cores are PbS or PbSe where the evidence points that are subject to ligand loss before assembly, a point that has been documented in different experimental results.17 The paradoxes raised in BNSL self-assembly discussed in the introduction are resolved: The nominal low density (low packing fraction) superlattices simply do not exist, as all experimental points are located at OTM branches with high packing fraction, above the simple bcc lattices, see Figure 3. The stability of low coordination sites is a result of spontaneous valence, resulting in stronger bonding through an accumulation of ligands at a maximal local packing fraction; see Figure 1. The compressibility of NCs is dictated by the “quantization” of vortices Nv ≤ 5, which determine the situations in which they behave as HS. In the classical work of Murray and Sanders on μ-scale NCs,29,30 binary BNSLs were identified at the critical values of γ(see eq 2) where the HS packing fraction is maximum. In this work, I have shown that addition of soft ligands greatly expands the range and stability of superlattices phases as the presumably equilibrium structures extend to a wider range of γ, where the HS packing fraction maybe quite small. Pushing the analogy between ligand structure and atomic orbitals further, it is rather remarkable that only seven skyrmion textures are needed Figure 4, the analogous“atomic orbitals”, to explain all the BNSLs. Rather interestingly, out of this seven, only three of them: n-tetra, n-equat, and t-nazn13 describe a significant number of experiments. The orbital p-cab6 appears in CaB6 at an already large packing fraction, there is only one experimental point in support of n-triang and all points in support of n-fe4c
(10)
where si̅ j is the NC−NC separation for particles of type i = A, B. 5380
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ACS Nano
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occur at unphysical packing fractions. Finally, n-collec is a complex orbital that consists of a collective vortex structures that at this point, should be regarded as an speculation. In any case, the existence of these “orbitals” may reveal an underlying description in terms of disclination or dislocation networks that can be used as a starting point to look for new structures. It is my hope that this paper will stimulate further detailed quantitative comparisons along the lines of ref 16. These studies will provide detailed verifications to compare against the predictions presented in this paper. A fully predictive theory is fundamental to understand the factors that determine the equilibrium and self-assembly of NCs. Many practical and technological implications hinge on achieving such understanding.
METHODS The formulas needed to reproduce the calculations in this paper are provided in the SI. They all have been incorporated into the Python software HOODLT,31 which will be made publicly available soon. HOODLT as well as all the scripts necessary to reproduce the calculations and figures in this paper can be obtained from the author upon request. The molecular parameters (bonds, radius, etc.) are taken from ref 32.
ASSOCIATED CONTENT S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.7b02219. Detailed description of determination of the experimental parameters, application of OTM formulas for planar systems, OTM solutions for every lattice, and Wigner− Seitz cell for every lattice (PDF)
AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. ORCID
Alex Travesset: 0000-0001-7030-9570 Notes
The author declares no competing financial interest.
ACKNOWLEDGMENTS I acknowledge many discussions and clarifications of their work with M. Boles, D. Talapin, and D. Vaknin. I also thank my students N. Horst and C. Waltmann for many insights and discussions. This work is supported by the NSF, DMR-CMMT 1606336 “CDS&E: Design Principles for Ordering Nanoparticles into Super-crystals”. REFERENCES (1) Whetten, R. L.; Shafigullin, M. N.; Khoury, J. T.; Schaaff, T. G.; Vezmar, I.; Alvarez, M. M.; Wilkinson, A. Crystal Structures of Molecular Gold Nanocrystal Arrays. Acc. Chem. Res. 1999, 32, 397− 406. (2) Nykypanchuk, D.; Maye, M. M.; van der Lelie, D.; Gang, O. DNA-Guided Crystallization of Colloidal Nanoparticles. Nature 2008, 451, 549−552. (3) Park, S. Y.; Lytton-Jean, A. K. R.; Lee, B.; Weigand, S.; Schatz, G. C.; Mirkin, C. A. DNA-Programmable Nanoparticle Crystallization. Nature 2008, 451, 553−556. (4) Zhang, H.; Wang, W.; Mallapragada, S.; Travesset, A.; Vaknin, D. Macroscopic and Tunable Nanoparticle Superlattices. Nanoscale 2017, 9, 164−171. 5381
DOI: 10.1021/acsnano.7b02219 ACS Nano 2017, 11, 5375−5382
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ACS Nano (28) Eldridge, M. D.; Madden, P. A.; Frenkel, D. Entropy-Driven Formation of a Superlattice in a Hard-Sphere Binary Mixture. Nature 1993, 365, 35−37. (29) Sanders, J. V.; Murray, M. J. Ordered Arrangemens of Spheres of Two Different Sizes in Opal. Nature 1978, 275, 201. (30) Murray, M. J.; Sanders, J. V. Close-packed Structures of Spheres of Two Different Sizes II. The packing densities of likely arrangements. Philos. Mag. A 1980, 42, 721−740. (31) Travesset, A. Phase Diagram of Power Law and Lennard-Jones Systems: Crystal Phases. J. Chem. Phys. 2014, 141, 164501. (32) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: London, 2000.
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DOI: 10.1021/acsnano.7b02219 ACS Nano 2017, 11, 5375−5382
